Storage or No Storage: Duopoly Competition Between Renewable Energy Suppliers in a Local Energy Market

1 Storage or No Storage: Duopoly Competition Between Rene wable Ener gy Suppliers in a Local Ener gy Market Dongwei Zhao, Student Member , IEEE, Hao W ang, Member , IEEE, Jianwei Huang, F ellow , IEEE, and Xiaojun Lin, F ellow , IEEE Abstract Renew able energy generations and energy storage are playing increasingly important roles in serving consumers in power systems. This paper studies the market competition between renewable ener gy suppliers with or without energy storage in a local ener gy market. The storage in vestment brings the benefits of stabilizing rene wable energy suppliers’ outputs, b ut it also leads to substantial inv estment costs as well as some surprising changes in the market outcome. T o study the equilibrium decisions of storage in vestment in the renewable energy suppliers’ competition, we model the interactions between suppliers and consumers using a three-stage game-theoretic model. In Stage I, at the beginning of the in vestment horizon (containing many days), suppliers decide whether to in vest in storage. Once such decisions ha ve been made (once), in the day-ahead market of each day , suppliers decide on their bidding prices and quantities in Stage II, based on which consumers decide the electricity quantity purchased from each supplier in Stage III. In the real-time market, a supplier is penalized if his actual generation f alls short of This work is supported by the Presidential Fund from the Chinese Uni versity of Hong Kong, Shenzhen, China, the Shenzhen Institute of Artificial Intelligence and Robotics for Society (AIRS), and in part by the NSF award ECCS-1509536. Part of the results have appeared in IEEE ICC 2019 [1]. Dongwei Zhao is with the Department of Information Engineering, The Chinese University of Hong Kong, Hong Kong, China (e-mail: zd015@ie.cuhk.edu.hk). Hao W ang is with the Department of Civil and En vironmental Engineering and the Stanford Sustainable Systems Lab, Stanford Univ ersity , CA 94305 USA (e-mail: haowang6@stanford.edu). Jianwei Huang is with the School of Science and Engineering, The Chinese University of Hong K ong, Shenzhen, China, the Shenzhen Institute of Artificial Intelligence and Robotics for Society (AIRS), and the Department of Information Engineering, The Chinese University of Hong K ong, Hong Kong, China (e-mail: jianweihuang@cuhk.edu.cn). Xiaojun Lin is with the School of Electrical and Computer Engineering, Purdue Uni versity , W est Lafayette, IN 47907, USA (e-mail: linx@ecn.purdue.edu). 2 his commitment. W e characterize a price-quantity competition equilibrium of Stage II in the local energy market, and we further characterize a storage-in vestment equilibrium in Stage I incorporating electricity- selling revenue and storage cost. Counter-intuitiv ely , we show that the uncertainty of renewable energy without storage in vestment can lead to higher supplier profits compared with the stable generations with storage inv estment due to the reduced market competition under random energy generation. Simulations further illustrate results due to the market competition. For example, a higher penalty for not meeting the commitment, a higher storage cost, or a lo wer consumer demand can sometimes increase a supplier’ s profit. W e also show that although storage in vestment can increase a supplier ’ s profit, the first-mover supplier who in vests in storage may benefit less than the free-rider competitor who chooses not to inv est. Index T erms Local ener gy market, Renew able generation, Energy storage, Market competition, Market equilibrium N O M E N C L A T U R E Acr onyms S 1 S 1 the case where both suppliers in vest in storage S 0 S 0 the case where neither supplier in vests in storage S 1 S 0 the case where one supplier in vests in storage and the other does not V ariables ϕ i storage in vestment decision of supplier i p d,t i bidding price of supplier i at hour t of day d y d,t i bidding quantity of supplier i at hour t of day d x d,t i electricity quantity that consumers purchase from supplier i at hour t of day d Random variables X d,t i generation amount of supplier i at hour t of day d C D d,t i charge and discharge power of supplier i at hour t of day d P arameters/constants λ penalty price for the supply shortage ¯ p price cap for the bidding price 3 D d,t demand of consumers at hour t of day d c i unit storage in vestment cost of supplier i ov er the in vestment horizon κ i scaling factor of supplier i S i storage capacity of supplier i C i storage in vestment cost (scaled in one hour) of supplier i Symbols of payoffs π R,d,t i supplier i ’ s rev enue at hour t of day d π RE ,d,t i supplier i ’ s equilibrium revenue at hour t of day d π S 1 S 1 i supplier i ’ s expected equilibrium rev enue in S 1 S 1 case ov er the in vestment horizon π S 0 S 0 i supplier i ’ s expected equilibrium rev enue in S 0 S 0 case ov er the in vestment horizon π S 1 S 0 | Y i with-storage supplier i ’ s expected equilibrium re venue in S 1 S 0 case o ver the in vestment horizon π S 1 S 0 | N i without-storage supplier i ’ s expected equilibrium rev enue in S 1 S 0 case over the in- vestment horizon Π i supplier i ’ s profit over the in vestment horizon I . I N T RO D U C T I O N A. Backgr ound and motivation Rene wable energy , as a clean and sustainable energy source, is playing an increasingly impor- tant role in po wer systems [2]. For e xample, from the year 2007 to 2017, the global installed capacity of solar panels has increased from 8 Gigaw atts to 402 Gigawatts, and the wind power capacity has increased from 94 Gigaw atts to 539 Gigaw atts [2]. Compared with traditional lar ger- scale generators, rene wable ener gy sources can be more spatially distributed across the po wer system, e.g., at the distribution lev el near residential consumers [2]. Due to the distributed nature of renew able energy generations, there has been growing interest in forming local energy markets for rene wable ener gy suppliers and consumers to trade electricity at the distribution le vel [3]. Such local energy markets will allow consumers to purchase electricity from the least costly sources locally [4], and allo w suppliers to compete in selling electricity directly to consumers (instead of dealing with the utility companies). 4 Ho we ver , man y types of rene wable ener gy are inherently random, due to f actors such as weather conditions that are difficult to predict and control. Under current multi-settlement energy market structures with day-ahead and real-time bidding rules (which are mostly designed for controllable generations) [5], renew able energy suppliers face a sev ere disadvantage in the competition by making forward commitment (in the day-ahead market) that they may not be able to deliv er in real time. For e xample, suppliers are often subject to a penalty cost if their real-time deliv ery de viates from the commitment in the day-ahead market [6]. Energy storage has been considered as an important type of fle xible resources for renewable energy suppliers to stabilize their outputs [7]. In vesting in storage can potentially improve the rene wable ener gy suppliers’ position in these energy mark ets. Ho we ver , in vesting in storage incurs substantial in vestment costs. Furthermore, the return of storage in vestment depends on the outcome of the market, which in turn depends on how suppliers with or without storage compete for the demand. Therefore, it remains an open problem regarding whether competing r enewable ener gy supplier s should in vest in energy storage in the market competition and what economic benefits the stora ge can bring to the suppliers. B. Main r esults and contributions In this paper, we formulate a three-stage game-theoretic model to study the market equilibrium for both storage in vestment as well as price and quantity bidding of competing renew able energy suppliers. In Stage I, at the beginning of the in vestment horizon, each supplier decides whether to in vest in storage. W e formulate a storage-in vestment game between two suppliers in Stage I, which is based on a bimatrix game to model suppliers’ storage-in vestment decisions for maximizing profits [8]. Giv en the storage-in vestment decisions in Stage I, competing suppliers decide the bidding price and bidding quantity in the (daily) local energy market in Stage II. W e formulate a price-quantity competition game between suppliers using the Bertrand-Edgeworth model [9] (which models price competition with capacity constraints) in Stage II. Giv en suppliers’ bidding strategies, consumers decide the electricity quantity purchased from each supplier in Stage III. T o the best of our knowledge, our work is the first to study the storage-in vestment equilibrium between competing rene wable ener gy suppliers in the two-settlement energy mark et. This problem 5 is quite nontrivial due to the penalty cost on the random generations of a general probability distribution. By studying this three-stage model, we rev eal a number of ne w and surprising insights that are against the prev ailing wisdom in the literature on the rene wable energy suppliers’ re venues in such a two-settlement market [6], [10] and on the economic benefits of storage supplementing in rene wable energy sources [11], [12]. • First, the uncertainty of the r enewable generation can be favorable to suppliers . Note that the pre v ailing wisdom is that storage in vestment (especially when the storage cost is low) will improv e suppliers’ rev enue by stabilizing their outputs [11], [12]. In contrast, we find that the opposite may be true when considering market competition. Specifically , without storage, suppliers with random generations always hav e strictly positiv e re venues when facing any positi ve consumer demand. Ho wev er , if both suppliers in vest in storage and stabilize their rene wable outputs, their rev enues reduce to zero once the consumer demand is belo w a threshold, which is due to the increased market competition after storage inv estment. • Second, a higher penalty and a higher storage cost can also be favorable to the suppliers . Note that the common wisdom is that a higher penalty [10] and a higher storage cost [11] will decrease suppliers’ profit. Howe ver , when considering market competition, the opposite may be true. W ith a higher penalty for not meeting the commitment, rene wable energy suppliers become more conservati ve in their bidding quantities, which can decrease market competition and increase their profits. Furthermore, a higher storage cost may change one supplier’ s storage-in vestment decision, which can benefit the other supplier . • Third, the first-mover supplier who in vests in ener gy stora ge can be at the disadvantage in terms of pr ofit incr ease , which is contrary to the first-mov er advantage gained by early in vestment of resources or new technologies [13]. W e find that although in vesting in storage can increase one supplier’ s profit, it may benefit himself less than his competitor (who does not in vest in storage). This is because the later mover becomes a free rider , who may benefit from the changed price equilibrium in the energy market (due to the storage in vestment of the other supplier) but does not need to bear the in vestment cost. In addition to these surprising and ne w insights, a k ey technical contribution of our work is the solution to the game-theoretic model for the price-quantity competition, which in volv es 6 a general penalty cost due to random generations of a general probability distribution. Note that such a price-quantity competition with the Bertrand-Edge worth model has been studied in literature under quite different conditions from ours. The works in [14]–[16] studied a general competition between suppliers with strictly con ve x production costs. They focused on the analysis of pure strategy equilibrium without characterizing the mixed strategy equilibrium. The study in [17] characterized both pure and mixed strategy equilibrium between suppliers with deterministic supply . Ho we ver , this work considered zero cost related to the production (i.e., no production cost or possible penalty cost). In electricity markets, the works in [18] and [19] also used Bertrand- Edge worth model to analyze the competition among renewable energy suppliers with random generations. Howe ver , both [18] and [19] considered the suppliers’ electricity-selling competition in a single-settlement energy market, and suppliers deli ver random generations in real time. These studies did not consider day-ahead bidding strategies and an y de viation penalty cost. In particular , the two-settlement markets with de viation penalty hav e been essential for ensuring the reliable operation of power systems. Our work is the first to consider the two-settlement energy market, characterizing both pure and mixed strategy equilibrium based on the Bertrand-Edgew orth model. Such a setting is nontrivial due to the penalty cost caused by the suppliers’ random production of a general probability distribution. The remainder of the paper is organized as follows. First, we introduce the system model in Section II, as well as the three-stage game-theoretic formulation between suppliers and consumers in Section III. Then, we solve the three-stage problem through backward induction. W e first characterize the consumers’ optimal purchase decision of Stage III in Section IV. Then, we characterize the price-quantity equilibrium of Stage II and the storage-in vestment equilibrium of Stage I in Sections V and VI, respectiv ely . W e propose a probability-based method to compute the storage capacity in Section VII. Furthermore, in Section IX, we extend some of the theoretical results and insights from the duopoly case to the oligopoly case. Finally , we present the simulation results in Section VIII and conclude this paper in Section X. I I . S Y S T E M M O D E L W e consider a local energy market at the distribution lev el as shown in Figure 1. Consumers can purchase energy from both the main grid and local rene wable energy suppliers. T o achiev e a 7 positi ve rev enue, the renew able energy suppliers (simply called suppliers in the rest of the paper) need to set their prices no greater than the grid price, and they will compete for the market share. Furthermore, suppliers can choose to in vest in energy storage to stabilize their renewable outputs and reduce the uncertainty in their deliv ery . Next, we will introduce the detailed models of timescales, suppliers and consumers, and characterize their interactions in the two-settlement local energy market. Loca l e nergy ma r k e t Supp l i er 1 Supp l i er 2 Consumer s Ma i n grid Fig. 1: System structure. A. T imescale W e consider two timescales of decision-making. One is the in vestment horizon D = { 1 , 2 , ..., D s } of D s days (e.g., D s corresponding to the total number of days for the storage in vestment horizon). Suppliers can decide (once) whether to in vest in energy storage at the beginning of the in vestment horizon. The in vestment horizon is divided into many operational horizons (many days), and each d ∈ D corresponds to the daily operation of the ener gy market, consisting of many time slots T = { 1 , 2 , ..., T } (e.g., 24 hours of each day). In the day-ahead market on day d − 1 , suppliers decide the electricity price and quantity to consumers for each hour t ∈ T of the next day d ∈ D . W e will introduce the market structure in detail later in Section II.D. B. Suppliers In Sections IV-VI, we focus on the duopoly case of two suppliers in our analysis. Later in Section IX, we further generalize to the oligopoly case with more than two suppliers. The reason for focusing on the duopoly case is twofold. First, our work focuses on a local ener gy mark et that is much smaller than a traditional wholesale energy market. The number of suppliers serving 8 one local area is also expected to be limited [20], compared with thousands of suppliers in the wholesale energy mark et [21]. In such a small local energy market, a fe w large suppliers may dominate the market [22]. Second, we consider two suppliers for analytical tractability , which is without losing key insights and can effecti vely capture the impact of competition among suppliers considering the storage in vestment. For example, we show that in the duopoly case, the uncertainty of renewable generation can be beneficial to suppliers. Such an insight is still valid in the oligopoly case. W e denote I = { 1 , 2 } as the set of two suppliers. For hour t of day d , the renewable output of supplier i ∈ I is denoted as a random variable X d,t i , which is bounded in [0 , ¯ X d,t i ] . W e assume that the random generation X d,t i has a continuous cumulativ e distribution function (CDF) F d,t i with the probability density function (PDF) f d,t i . The distribution of wind or solar po wer can be characterized using the historical data, which is known to the rene wable energy suppliers. 1 As renew ables usually hav e extremely lo w marginal production costs compared with traditional generators, we assume zero marginal production costs for the suppliers [18] [19]. C. Consumers W e consider the aggregate consumer population, and we denote the total consumer demand at hour t of day d as D d,t > 0 . Note that consumers in one local area usually face the same electricity price from the same utility . Thus, if the local market’ s electricity price is lower than the grid price, all the consumers will first purchase electricity from local suppliers. From the perspecti ve of suppliers, they only care about the total demand of consumers and how much electricity they can sell to consumers. Furthermore, our work conforms to the current energy market practice that suppliers make decisions in the day-ahead market based on the predicted demand. Thus, for the demand D d,t , we consider it as a deterministic (predicted) demand in our model. 2 Since the electricity demand is usually inelastic [5], we also assume the following. 1 In Section VIII of simulations, we use historical data to model the empirical CDF of renewable generations, which is explained in detail in Appendix.XIV. 2 The day-ahead prediction of consumers’ aggregated demand can be fairly accurate [24]. W e assume that the demand and supply mismatch due to the demand forecast error will be regulated by the operator in the real-time market. 9 Assumption 1. Consumers’ demand is perfectly inelastic in the electricity price. Consumers must purchase their demand D d,t either from the main grid (at a fixed unit price P g ) or from the local rene wable suppliers (with prices to be discussed later). 3 D. T wo-settlement local ener gy market W e consider a two-settlement local energy market, which consists of a day-ahead market and a real-time market [5]. In such an ener gy market, suppliers ha ve market power and can strategically decide their selling prices. 4 Consumers ha ve the flexibility to choose suppliers by comparing prices [4]. W e explain the two-settlement energy market in detail as follows. • In the day-ahead market on day d − 1 (e.g., suppliers’ bids are cleared around 12:30pm of day d − 1 , one day ahead of the deli very day d [25]), supplier i ∈ I decides the bidding price p d,t i and the bidding quantity y d,t i for each future hour t ∈ T of the deli very day d . Based on suppliers’ bidding strategies, consumers decide the electricity quantity x d,t i ( ≤ y d,t i ) purchased from supplier i . Supplier i will get the re venue of p d,t i x d,t i in the day-ahead market by committing the deli very quantity x d,t i to consumers. Thus, the day-ahead market is cleared through matching supply and demand. Any excessi ve demand from the consumers will be satisfied through energy purchase from the main grid. • In the real-time market at each hour on the next day d , if supplier i ’ s actual generation falls short of the committed quantity x d,t i (i.e., x d,t i > X d,t i ), he needs to pay the penalty λ ( x d,t i − X d,t i ) in the real-time mark et, which is proportional to the shortfall with a unit penalty price λ . For the consumers, although suppliers may not deliv er the committed electricity to them, the shortage part can be still satisfied by the system operator using reserve resources. The cost of reserve resources can be covered by the penalty cost on the suppliers. Note that the suppliers and consumers make decisions only in the day-ahead market. No acti ve decisions are made in the real-time market, b ut there may be penalty cost on the deliv ery shortage. 3 W e do not consider demand response for the consumers. 4 This price model is dif ferent from the usual practice of the wholesale energy market, where the market usually sets a uniform clearing price for all the suppliers through market clearing [5]. 10 T o facilitate the analysis, we further make sev eral assumptions of this local energy market as follows. First, for the excessiv e amount of generations (i.e., x d,t i < X d,t i ), we assume the follo wing. Assumption 2. Suppliers can curtail any e xcessive r enewable ener gy generation (be yond any specific given level). Assumption 2 implies that we do not need to consider the possible penalty or rew ard on the excessi ve renewable generations in real time. 5 Second, the local energy market is much smaller compared with the wholesale energy market. Thus, the suppliers are usually small and hence may focus on serving local consumers. It is less likely for them to trade in the wholesale energy market. This is summarized in the follo wing assumption. Assumption 3. Suppliers only participate in the local ener gy market and serve local consumers. The y do not participate in the wholesale ener gy market. Third, for the bidding price p i and penalty price λ , we impose the following bounds. Assumption 4. Each supplier i ’s bidding price p i has a cap ¯ p that satisfies p i ≤ ¯ p < P g . Assumption 5. The penalty price satisfies λ > ¯ p . Assumption 4 is without loss of generality , since no supplier will bid a price higher than P g ; otherwise, consumers will purchase from the main grid. 6 Assumption 5 ensures that the penalty is high enough to discourage suppliers from bidding higher quantities beyond their capability . Note that price cap ¯ p and the penalty λ are exogenous fixed parameters in our model. Next, we introduce ho w suppliers in vest in the energy storage to stabilize their outputs. 5 There are different policies to deal with the surplus feed-in energy of renewables. In some European countries, the energy markets giv e rewards to the surplus energy [26]. In the US, some markets deal with the surplus energy using the real-time imbalance price that can be either penalties or rewards [10]. 6 W e av oid the case ¯ p = P g as it may bring ambiguity to the local energy market if the bidding price is equal to the main grid price P g , in which case it is not clear whether consumers purchase energy from the local energy market or from the main grid. 11 E. Stora ge in vestment Each supplier decides whether to in vest in storage at the beginning of the in vestment horizon. W e denote supplier i ’ s storage-in vestment decision variable as ϕ i , where ϕ i = 1 means in vesting in storage and ϕ i = 0 means not in vesting. If supplier i in vests in storage, we assume the follo wing. Assumption 6. The with-stor ag e supplier will utilize the storag e to completely smooth out his power output at the mean value of r enewable generations. Thus, supplier i with the rene wable generation X d,t i will charge and discharge his storage 7 to stabilize the power output at the mean value E [ X d,t i ] . The charge and discharge power C D d,t i is as follo ws. C D d,t i = X d,t i − E [ X d,t i ] , (1) where C D d,t i > 0 means charging the storage and C D d,t i < 0 means discharging the storage. Note that E X d,t i [ C D d,t i ] = 0 , which implies the long-term av erage power that the suppliers need to charge or dischar ge his storage is zero. Next, we introduce ho w to characterize the storage capacity and the storage cost. First, based on the charge and discharge random variable C D d,t i , we propose a simple yet ef fecti ve probability-based method to characterize the storage capacity S i using historical data of renew able generation X d,t i . In particular , we set a probability threshold, and then aim to find a minimum storage capacity S i such that the energy le vel in the storage exceeds the capacity with a probability no greater than the probability threshold. W e will explain this methodology in Section VII. Second, we calculate the storage cost of suppliers ov er the in vestment horizon (scaled into one hour) as C i = c i κ i S i , where c i is the unit capacity cost over the in vestment horizon and κ i is the scaling factor that scales the in vestment cost ov er years to one hour . The factor κ i is calculated as follows. W e first calculate the present v alue of an annuity (a series of equal annual cash flows) 7 There can be different ways to deal with the randomness of renewable generations, including the curtailment of renewable energy and the use of additional fossil generators to provide additional energy . It is interesting to combine energy storage with other mechanisms (such as renewable energy curtailment), which we will explore in the future work. 12 with the annual interest rate r i (e.g., r i = 5% ), and then we divide the annuity equally to each hour . This leads to the formulation of the factor κ i as follo ws [27]. κ i = r i (1 + r i ) y i (1 + r i ) y i − 1 · 1 Y d , (2) where y i is the number of years ov er the in vestment horizon (e.g., y i = 15 for Li-ion battery that can last for 15 years), and Y d is the total hours in one year (e.g., Y d = 365 × 24 ). Therefore, gi ven the parameter c i and κ i as well as the probability distribution of random gen- eration, the storage capacity and storage cost can be regarded as the fixed values for the supplier who in vests in storage. Note that a higher storage capacity leads to a higher storage in vestment cost, which can further affect the storage-in vestment decisions in the suppliers’ competition. Next, in the Section III, we will introduce the three-stage model between suppliers and consumers in detail. I I I . T H R E E - S TAG E G A M E - T H E O R E T I C M O D E L W e build a three-stage model between suppliers and consumers. In Stage I, at the beginning of the in vestment horizon, each supplier decides whether to in vest in storage. In the day-ahead energy market, for each hour of the next day , suppliers decide the bidding prices and quantities in Stage II, and consumers make the purchase decision in Stage III. Next, we first introduce the types of rene wable-generation distributions for computing suppliers’ electricity-selling rev enues ov er the in vestment horizon, and then we explain the three stages respectiv ely in detail. A. T ype of r enewable-gener ation distributions W e cluster the distrib ution of renewable generation into sev eral types. Note that suppliers’ re venues depend on the distribution of renew able generations. W e use historical data of renew able energy to model the generation distribution. Specifically , for the renewable generations at hour t of all the days over the in vestment horizon, we cluster the empirical distribution into M types, e.g., M = 12 for 12 months considering the seasonal effect. In this case, each type m ∈ M = { 1 , 2 , . . . , M } occurs with a probability ρ m = 1 12 considering 12 months. 8 W e use the data of rene wable energy of all days in month m at hour t to approximate the distribution of renew able 8 There can be other types of clustering with unequal probabilities. 13 Fig. 2: Three-stage model. generation at hour t for all the days in this month m . Then, to study the interactions between consumers and suppliers in the local energy market, we will assume that the renewable generation of day d follows a random type (month) m , uniformly chosen from m ∈ M . For notation con venience, we replace all the superscripts d, t into m, t . B. The thr ee-stage model W e illustrate the three-stage model between suppliers and consumers in Figure 2. • Stage I: at the beginning of the in vestment horizon, each supplier i ∈ { 1 , 2 } decides the storage-in vestment decisions ϕ i ∈ { 0 , 1 } . • Stage II: in the day-ahead market, for each hour t of the next day , each supplier i decides his bidding price p m,t i and bidding quantity y m,t i based on suppliers’ storage-in vestment decisions, assuming that the rene wable-generation distribution is of month m . • Stage III: in the day-ahead market, for each hour t of the next day , consumers decide the electricity quantity x m,t i purchased from each supplier i based on each supplier’ s bidding price and quantity , assuming that the renew able-generation distribution is of month m . This three-stage problem is a dynamic game. The solution concept of a dynamic game is kno wn as Subgame Perfect Equilibrium, which can be derived through backward induction [28]. Therefore, in the following, we will explain the three stages in detail in the order of Stage III, Stage II, and Stage I, respectiv ely . 14 1) Stage III: At hour t of month m , giv en the bidding price ( p m,t 1 , p m,t 2 ) and bidding quantity ( y m,t 1 , y m,t 2 ) of both suppliers in Stage II, consumers decide the electricity quantity ( x m,t 1 , x m,t 2 ) purchased from supplier 1 and supplier 2, respectiv ely . The objecti ve of consumers is to maximize the cost saving of purchasing ener gy from local suppliers compared with purchasing from the main grid only . W e denote such cost saving as follows: π m,t c ( x m,t 1 , x m,t 2 ) = ( P g − p m,t 1 ) x m,t 1 + ( P g − p m,t 2 ) x m,t 2 . (3) Recall that we model the collectiv e purchase decision of the entire consumer population together . Consumers must satisfy their demand either from the local energy market or from the main grid (at the fixed price P g ). The total cost of satisfying the entire demand by the main grid is fixed. Therefore, minimizing the total energy cost is equiv alent to maximizing the cost sa vings in the local energy market. W e present consumers’ optimal purchase problem as follows. Stage III: Consumers’ Cost Saving Maximization Problem max x m,t 1 ,x m,t 2 ( P g − p m,t 1 ) x m,t 1 + ( P g − p m,t 2 ) x m,t 2 , (4a) s.t. x m,t 1 + x m,t 2 ≤ D m,t , (4b) 0 ≤ x m,t i ≤ y m,t i , i = 1 , 2 . (4c) Constraint (4b) states that the total purchased quantity x m,t 1 + x m,t 2 is no greater than the demand D m,t . Constraints (4c) states that the quantity purchased from supplier i is no greater than his bidding quantity y m,t i . This problem is a linear programming and can be easily solved, which we show in Section IV. W e denote the optimal solution to Problem (4) as a function of suppliers’ bidding prices and quantities ( p m,t , y m,t ) , i.e., x m,t ∗ i ( p m,t , y m,t ) , ∀ i = 1 , 2 , where p m,t = ( p m,t 1 , p m,t 2 ) and y m,t = ( y m,t 1 , y m,t 2 ) . 2) Stage II: Gi ven the storage-in vestment decision ϕ = ( ϕ 1 , ϕ 2 ) in Stage I, both suppliers decide the bidding price p m,t and bidding quantity y m,t to maximize their rev enues in Stage II. W e denote supplier i ’ s electricity-selling revenue as π R,m,t i , which consists of two parts: the commitment rev enue p m,t i x m,t ∗ i ( p m,t , y m,t ) from committing the deli very quantity in the day-ahead market, and the penalty cost in the real-time market. Supplier i who in vests in storage (i.e., ϕ i = 1 ) will be penalized if the committed quantity x m,t ∗ i ( p m,t , y m,t ) is larger than his stable generation E [ X m,t i ] . Supplier i who does not in vest in storage (i.e., ϕ i = 0 ) will be penalized if the commitment x m,t ∗ i ( p m,t , y m,t ) is larger than his actual random generation X m,t i . 15 Note that the decisions of two suppliers are coupled with each other . If one supplier bids a lo wer quantity or a higher price, it is highly possible that consumers will purchase more electricity from the other supplier . W e formulate a price-quantity competition game between suppliers gi ven storage-in vestment decisions ϕ as follows. Stage II: Price-quantity competition game • Players: supplier i ∈ { 1 , 2 } . • Strategies: bidding quantity y m,t i ≥ 0 and bidding price p m,t i ∈ [0 , ¯ p ] of each supplier i . • Payof fs: supplier i ’ s rev enue at hour t of month m is π R,m,t i  p m,t i , x m,t ∗ i ( p m,t , y m,t ) , ϕ  =                    p m,t i x m,t ∗ i ( p m,t , y m,t ) − λ ( x m,t ∗ i ( p m,t , y m,t ) − E [ X m,t i ]) + , if ϕ i = 1; p m,t i x m,t ∗ i ( p m,t , y m,t ) − λ E X m,t i  ( x m,t ∗ i ( p m,t , y m,t ) − X m,t i ) +  , if ϕ i = 0 , (5) where we define ( g ) + = max( g , 0) . If both suppliers in vest in storage (i.e., P i ϕ i = 2 ), the equilibrium has been characterized in [17]. Howe ver , if there is at least one supplier who does not in vest in storage (i.e., P i ϕ i ≤ 1 ), characterizing the equilibrium is quite non-trivial due to the penalty cost on the random generation of a general probability distribution. W e will discuss ho w to characterize the equilibrium in detail in Section V. W e denote the equilibrium revenue of supplier i as π RE ,m,t i ( ϕ ) . 3) Stage I: At the beginning of the in vestment horizon, each supplier decides whether to in vest in storage to maximize his expected profit. W e denote supplier i ’ s expected profit as Π i , which incorporates the expected re venue in the local energy market and the possible storage in vestment cost. As one supplier v aries his storage-in vestment decisions, it leads to a different price-quantity subgame, which will affect both suppliers’ profits. Thus, suppliers’ storage-in vestment decisions are coupled and we formulate a storage-in vestment game between suppliers as follows. Stage I: Storage-in vestment game • Players: supplier i ∈ { 1 , 2 } . • Strategies: whether in vesting in storage ϕ i ∈ { 0 , 1 } . • Payof fs: supplier i ’ s expected profit (scaled in one hour) is 16 Π i ( ϕ ) = E m,t [ π RE ,m,t i ( ϕ )] − ϕ i C i . (6) This storage-in vestment game is a 2 × 2 bimatrix game where each supplier has two strategies. Although the Nash equilibrium of 2 × 2 bimatrix game can be easily solved numerically , the close-form equilibrium does not exist in all subgames of Stage II. It is challenging to analyze the storage-in vestment equilibrium with respect to the parameters, e.g., demand and storage cost, and we discuss it in detail in Section VI. W e solve this three-stage problem through backward induction. W e first analyze the solution in Stage III giv en the bidding prices and bidding quantities in Stage II. Then, we incorporate the solution in Stage III to analyze the price and quantity equilibrium in Stage II, giv en (arbitrary) storage-in vestment decisions in Stage I. Finally , we incorporate the equilibrium of Stage II into Stage I to solve the storage-in vestment equilibrium. In the next three sections of Section IV, Section V, and Section VI, we will analyze the three stages in the order of Stage III, Stage II, and Stage I, respecti vely . I V . S O L U T I O N O F S TAG E I I I In this section, we characterize consumers’ optimal purchase solution to Problem (4) in Stage III. W e use subscript i ∈ { 1 , 2 } to denote supplier i and we use − i to denote the other supplier . Note that in Stage III, the decisions are made independently for each hour of each day . For notation simplicity , we omit the superscript m, t in the corresponding variables and parameters. Gi ven the bidding price p and bidding quantity y of suppliers, we characterize in Proposition 1 consumers’ optimal decision x ∗ ( p , y ) = ( x ∗ i ( p , y ) , i = 1 , 2) in Stage III. Recall that we assume that the bidding price in the local energy market is lower than the main grid price (i.e., ¯ p < P g ). Proposition 1 (optimal purchase x ∗ ( p , y ) in Stage III) . • If p i < p − i for some i ∈ { 1 , 2 } , then x ∗ i ( p , y ) = min ( D , y i ) and x ∗ − i ( p , y ) = min ( D − min ( D , y i ) , y − i ) . • If p 1 = p 2 , then the optimal pur chase solution can be any element in the following set. X opt = { x ∗ ( p , y ) : 2 X i =1 x ∗ i ( p , y ) = min( D , 2 X i =1 y i ) , 0 ≤ x i ≤ y i , i = 1 , 2 } . 17 W e assume that the demand will be allocated to the suppliers accor ding to the condition either p 1 < p 2 or p 2 < p 1 . The condition p 1 < p 2 or p 2 < p 1 is selected based on maximizing the two suppliers’ total r evenue . 9 Proposition 1 shows that the consumers will first purchase the electricity from the supplier who sets a lower price. If there is remaining demand, then they will purchase from the other supplier . Furthermore, if consumers’ demand cannot be fully satisfied by the local suppliers, they will purchase the remaining demand from the main grid. W e sho w the proof of Proposition 1 in Appendix.XV. Next we analyze the strategic bidding of suppliers in Stage II by incorporating consumers’ optimal purchase decisions x ∗ ( p , y ) . V . E Q U I L I B R I U M A N A L Y S I S O F S TAG E I I In this section, we will characterize the bidding strate gies of suppliers for the price-quantity competition subgame in Stage II, giv en the storage-in vestment decision in Stage I. Note that, depending on the storage-in vestment decisions in Stage I, there are three types of subgames: (i) the both-in vesting-storage (S 1 S 1 ) case, (ii) the mixed-in vesting-storage (S 1 S 0 ) case, where one in vests in storage and one does not, and (iii) the neither-in vesting-storage (S 0 S 0 ) case. The competition- equilibrium characterization between suppliers is highly non-tri vial, due to the general distrib ution of rene wable generations and the penalty cost. In particular , the pure price equilibrium may not exist, which requires the characterization of the mixed price equilibrium. Next, we first sho w that each supplier’ s equilibrium bidding quantity is actually a weakly dominant strategy that does not depend on the other supplier’ s decision, based on which we further deriv e the suppliers’ bidding prices at the equilibrium for each subgame. Note that in Stage II, the decisions are made independently for each hour of each day . For notation simplicity , we omit the superscript m, t in the corresponding v ariables and parameters. A. W eakly-dominant strate gy for bidding quantity W e show that giv en the bidding price p , each supplier has a weakly dominant strategy for the bidding quantity that does not depend on the other supplier’ s quantity or price choice. This is 9 If there is no difference between p 1 < p 2 and p 2 < p 1 , the demand will be allocated by either p 1 < p 2 or p 2 < p 1 with equal probabilities. 18 rather surprising, and it will help reduce the two-dimensional bidding process (in volving both quantity and price) into a one-dimensional bidding process (in v olving only price). Deri ving the weakly dominant strategy is nontri vial due to the penalty cost on the renewable generation of a general probability distribution faced by the without-storage supplier . W e first define the weakly dominant strategy for the bidding quantity y ∗ i in Definition 1, which enables a supplier to obtain a re venue at least as high as any other bidding quantity y i , no matter what is the other supplier’ s decision. Definition 1 (weakly dominant strategy) . Given price p and stora ge-in vestment decision ϕ , a bidding quantity y ∗ i is a weakly dominant strate gy for supplier i if π R i ( p i , x ∗ i ( p , ( y ∗ i , y − i )) , ϕ ) ≥ π R i ( p i , x ∗ i ( p , ( y i , y − i )) , ϕ ) , for any y − i and y i 6 = y ∗ i . W e then characterize suppliers’ weakly dominant strategy y ∗ ( p , ϕ ) for the bidding quantity in Theorem 1. Theorem 1 (weakly dominant strategy for the bidding quantity) . The weakly dominant strate gy y ∗ ( p , ϕ ) is given by y ∗ i ( p i , ϕ i ) =      E [ X i ] , if ϕ i = 1 , F − 1 i  p i λ  , if ϕ i = 0 , (7) wher e F − 1 i is the in verse function of the CDF F i of supplier i ’ s random generation. Theorem 1 shows that a with-storage supplier i (i.e., ϕ i = 1 ) should bid the quantity at the stable production le vel E [ X i ] (independent of price p ) so that he can attract the most demand but do not face any penalty risk in the real-time market. For a without-storage supplier i (i.e., ϕ i = 0 ), howe ver , he has to trade of f between his bidding quantity and the penalty cost incurred by the random generation. His weakly dominant strategy y ∗ i ( p i , ϕ i ) depends on his own bidding price p i , b ut does not depend on the other supplier − i ’ s bidding price p − i or bidding quantity y − i . Note that when price p i = 0 , the bidding quantity y ∗ i (0 , ϕ i ) = F − 1 i (0) = 0 . Furthermore, the bidding quantity y ∗ i ( p i , ϕ i ) increases in price p i , which shows that the without-storage supplier i should bid more quantities when he bids a higher price. When price p i = ¯ p , the bidding quantity satisfies y ∗ i ( ¯ p, ϕ i ) = F − 1 i  ¯ p λ  < ¯ X i (i.e., the maximum generation amount) since we assume ¯ p < λ . 19 B. Equilibrium price-bidding strate gy: pure equilibrium W e will further analyze the price equilibrium between suppliers based on the weakly dominant strategies for the bidding quantities in Theorem 1. W e characterize the price equilibrium with respect to the demand that can affect the competition lev el between suppliers. For the S 1 S 0 and S 0 S 0 cases, we show that a pure price equilibrium exists when the demand D is higher than a threshold (characterized in the later analysis). Howe ver , when the demand D is lower than the threshold, there exists no pure price equilibrium due to the competition for the limited demand. For the S 1 S 1 case, the equilibrium structure is characterized by two thresholds of the demand (characterized in the later analysis). A pure price equilibrium will exist when the demand D is higher than the larger threshold or lower than the other smaller threshold. Ho wev er , when the demand D is in the middle of the two thresholds, there exists no pure price equilibrium. W e first define the pure price equilibrium of suppliers in Definition 2, where no supplier can increase his re venue through unilateral price deviation. Definition 2 (pure price equilibrium) . Given the storag e-investment decision ϕ , a price vector p ∗ is a pur e price equilibrium if for both i = 1 , 2 , π R i ( p ∗ i , x ∗ i ( p ∗ , y ∗ ( p ∗ , ϕ )) , ϕ ) ≥ π R i  p i , x ∗ i  ( p i , p ∗ − i ) , y ∗ (( p i , p ∗ − i ) , ϕ i )  , ϕ  , (8) for all 0 ≤ p i ≤ ¯ p , wher e y ∗ is the weakly dominant strate gies derived in Theor em 1. Then, we sho w the existence of the pure price equilibrium in Proposition 2. Proposition 2 (existence of the pure price equilibrium) . • Subgames of type S 1 S 0 and type S 0 S 0 (i.e., when P i ϕ i < 2 ): – If D ≥ P i y ∗ i ( ¯ p, ϕ i ) , ther e exists a pur e price equilibrium p ∗ i = ¯ p , with equilibrium r evenue π RE i = λ R F − 1 i ( ¯ p/λ ) 0 xf i ( x ) dx , for any i = 1 , 2 . – If 0 < D < P i y ∗ i ( ¯ p, ϕ i ) , ther e is no pure price equilibrium. • Subgame of type S 1 S 1 ( P i ϕ i = 2 ): – If D ≥ P i y ∗ i ( ¯ p, ϕ i ) , ther e exists a pur e price equilibrium p ∗ i = ¯ p , with equilibrium r evenue π RE i = ¯ p E [ X i ] , for any i = 1 , 2 . 20 – If D ≤ min i y ∗ i ( ¯ p, ϕ i ) , ther e exists a pur e price equilibrium p ∗ i = 0 , with equilibrium r evenue π RE i = 0 , for any i = 1 , 2 . – If min i y ∗ i ( ¯ p, ϕ i ) < D < P i y ∗ i ( p i , ϕ i ) , ther e is no pure price equilibrium. W e summarize the existence of pure price equilibrium and the weakly dominant strategy of bidding quantity in T able I. Subgame W eakly dominant strategy of bidding quantity Existence of pure price equilibrium Non-existence of pure price equilibrium S 1 S 1 y ∗ i ( p i , ϕ i ) , ∀ i = 1 , 2 (a) D ≥ P i y ∗ i ( ¯ p, ϕ i ) : p ∗ i = ¯ p , ∀ i = 1 , 2 (b) D ≤ min i y ∗ i ( ¯ p, ϕ i ) : p ∗ i = 0 , ∀ i = 1 , 2 min i y ∗ i ( ¯ p, ϕ i ) < D < P i y ∗ i ( ¯ p i , ϕ i ) : no pure price equilibrium S 1 S 0 , S 0 S 0 y ∗ i ( p i , ϕ i ) , ∀ i = 1 , 2 D ≥ P i y ∗ i ( ¯ p, ϕ i ) : p ∗ i = ¯ p , ∀ i = 1 , 2 . 0 < D < P i y ∗ i ( ¯ p, ϕ i ) : no pure price equilibrium T ABLE I: W eakly dominant strategy of bidding quantity as well as the conditions for the existence of pure price equilibrium. According to Proposition 2, for all the types of subgames, when the demand D is higher than the summation of the suppliers’ maximum bidding quantities (i.e., D ≥ P i y ∗ i ( ¯ p, ϕ i ) ), both suppliers will bid the highest price ¯ p . The reason is that both suppliers’ bidding quantities will be fully sold out in this case, and the highest price will giv e the highest re venue to each supplier . Basically there is no impact of market competition in this case. Howe ver , for the S 1 S 0 and S 0 S 0 subgames, if the demand D is lower than the threshold P i y ∗ i ( ¯ p, ϕ i ) , there exists no pure price equilibrium. In contrast, for the S 1 S 1 subgame, it is also possible that when the demand D is smaller than a threshold (i.e., D < min i y ∗ i ( ¯ p, ϕ i ) ), both suppliers hav e to bid zero price and get zero rev enue. The intuition is that the competition lev el of the S 1 S 1 subgame is higher than that of the S 1 S 0 and S 0 S 0 subgames due to both suppliers’ stable outputs, which leads to zero 21 bidding prices if the demand is low . The result of the subagame S 1 S 1 has been proved in [17]. W e present the proofs of subgames of type S 1 S 0 and type S 0 S 0 in Appendix.XVI. C. Equilibrium price-bidding strate gy: mixed equilibrium When the demand is such a le vel that there is no pure price equilibrium as shown in Proposition 2, we characterize the mixed price equilibrium between suppliers. First, we define the mixed price equilibrium under the weakly dominant strategy y ∗ ( p , ϕ ) in Definition 3, where µ denotes a probability measure 10 of the price ov er [0 , ¯ p ] [17]. Definition 3 (mixed price equilibrium) . A vector of pr obability measures ( µ ∗ 1 , µ ∗ 2 ) is a mixed price equilibrium if, for both i = 1 , 2 , Z [0 , ¯ p ] 2 π R i  p i , x ∗ i (( p i , p − i ) , y ∗ ( p i , p − i )) , ϕ ) d ( µ ∗ i ( p i ) × µ ∗ − i ( p − i )  ≥ Z [0 , ¯ p ] 2 π R i  p i , x ∗ i (( p i , p − i ) , y ∗ ( p i , p − i )) , ϕ ) d ( µ i ( p i ) × µ ∗ − i ( p − i )  , for any measur e µ i . Definition 3 states that the expected rev enue of supplier i cannot be increased if he unilaterally de viates from the mix ed equilibrium price strate gy µ ∗ i . Let F e i denote the CDF of µ ∗ i , i.e., F e i ( p i ) = µ ∗ i ( { p ≤ p i } ) . Let u i and l i denote the upper support and lo wer support of the mixed price equilibrium µ ∗ i , respectiv ely , i.e., u i = inf { p i : F e i ( p i ) = 1 } and l i = sup { p i : F e i ( p i ) = 0 } . T o characterize the mixed price equilibrium, we need to fully characterize the CDF function F e i (including u i and l i ) for each i ∈ { 1 , 2 } . Then, we sho w that the mixed price equilibrium exists for each type of subgames and char - acterize some properties of the mixed price equilibrium in Lemma 1. Lemma 1 can be deriv ed follo wing the same method for the S 1 S 1 case in [17]. Later , we discuss how to compute the mixed price equilibrium of the S 1 S 1 , S 1 S 0 , and S 0 S 0 cases, respecti vely . Lemma 1 (characterization of the mixed price equilibrium) . F or any pair ( ϕ i , ϕ − i ) , when the demand D falls in the range where no pur e price equilibrium exists as shown in Pr oposition 2, the mixed price equilibrium exists and has pr operties as follows. 10 A probability measure is a real-valued function that assigns a probability to each event in a probability space. 22 (i) Both suppliers have the same lower support and the same upper support: l 1 = l 2 = l > 0 , u 1 = u 2 = ¯ p. (9) (ii) The equilibrium electricity-selling r evenues π RE i satisfy: π RE i ( ϕ ) = π R i ( l , min( D , y ∗ 2 ( l , ϕ i )) , ϕ ) . (10) (iii) F or any i = 1 , 2 , F e i is strictly incr easing over [ l , ¯ p ] , and has no atoms 11 over [ l , ¯ p ) . Also, F e i cannot have atoms at ¯ p for both i = 1 , 2 . Lemma 1 sho ws that both suppliers’ mixed-price-equilibrium strategies ha ve the same support and have continuous CDFs ov er [ l , ¯ p ) . Based on Lemma 1, we next characterize the mixed price equilibrium for the subgames of each type S 1 S 1 , S 1 S 0 , and S 0 S 0 . 1) S 1 S 1 subgame (i.e., P ϕ i = 2 ): As sho wn in Proposition 2, when the demand satisfies min i y ∗ i ( ¯ p, ϕ i ) < D < P i y ∗ i ( ¯ p, ϕ i ) , there is no pure price equilibrium. W e can characterize a close-form equilibrium re venue for each supplier at the mixed price equilibrium, which has been prov ed in [17]. Furthermore, under the mixed price equilibrium, both suppliers get strictly positi ve revenues, while they may get zero re venues under the pure price equilibrium as shown in Proposition 2. W e show the close-form equilibrium rev enue in Appendix.XI. 2) S 1 S 0 subgame (i.e ., P i ϕ i = 1 ): In the S 1 S 0 subgame, a mixed price equilibrium arises when 0 < D < P i y ∗ i ( ¯ p, ϕ i ) . Ho we ver , we cannot characterize a close-form equilibrium rev enue, as in the S 1 S 1 case due to the penalty cost on the general rene wable generations for the without-storage supplier . Instead, we can first characterize the CDF of the mixed price equilibrium assuming the lo wer support l in Theorem 2, and then show ho w to compute the lo wer support l in Proposition 3. W e present the proofs in Appendix.XVI. Theorem 2 (S 1 S 0 : CDF of the mixed price equilibrium) . In the S 1 S 0 subgame (i.e., P i ϕ 1 = 1 ), when 0 < D < P i y ∗ i ( ¯ p, ϕ i ) , suppose that the common lower support l 1 = l 2 = l of the mixed price equilibrium is known. Then, the suppliers’ mixed equilibrium price strate gies ar e char acterized by the following CDF F e i : • If ϕ i = 1 , we have F e i ( p ) = π R − i  p, min  y ∗ − i ( p, ϕ − i ) , D  , ϕ  − π RE − i ( ϕ ) π R − i ( p, min  y ∗ − i ( p, ϕ − i ) , D  , ϕ ) − π R − i ( p, ( D − E [ X i ]) + , ϕ ) . (11) 11 The atom at p means that the left-limit of CDF at p satisfies F e i ( p − ) , lim p 0 ↑ p F e i ( p 0 ) < F e i ( p ) . 23 • If ϕ i = 0 , we have F e i ( p ) = Z ¯ p l π RE − i ( ϕ ) p 2 · min ( y ∗ i ( p, ϕ i ) , D ) − p 2 · ( D − E [ X − i ]) + dp. (12) for any l ≤ p < ¯ p . As shown in Theorem 2, supplier i ’ s mixed strategy F e i is coupled with the other supplier’ s equilibrium rev enue π RE − i . Next, we will explain how to compute the lo wer support l . T ow ard this end, in (11) and (12), we replace the equilibrium lower support l by a v ariable l † i , and replace F e i ( p ) by F e i ( p | l † i ) to emphasize that F e i ( p | l † i ) is a function of l † i . Lemma 1 (iii) implies that there exists a solution l † i to the equation F e i ( ¯ p − | l † i ) = 1 for at least one of the suppliers. Furthermore, we can prove that F e i ( ¯ p − | l † i ) decreases in l † i , and hence the solution (in l † i ) to F e i ( ¯ p − | l † i ) = 1 is unique. Then, we can compute the lower support l in Proposition 3. Proposition 3 (S 1 S 0 : computing the lower support l ) . Based on the solution l † i such that F e i ( ¯ p − | l † i ) = 1 , ∀ i = 1 , 2 , we consider two cases and compute the lower support l as follows. 1) If F e i ( ¯ p − | l † i ) = 1 has a solution l † i for both suppliers, then the equilibrium lower support is l = max i ( l † i ) . 2) If F e i ( ¯ p − | l † i ) = 1 has a solution l † i for only one supplier i , we have this unique solution l † i as the equilibrium lower support l . Through Theorem 2 and Proposition 3, we can compute the lo wer support and suppliers’ equilibrium rev enues. Although we cannot obtain a close-form equilibrium re venue, in Theorem 3, we can show that in the S 1 S 0 subgame, if two suppliers’ random generations have the same mean v alue, then the with-storage supplier’ s equilibrium re venue is always strictly higher than that of the without-storage supplier . Theorem 3 (S 1 S 0 : rev enue comparison) . If ϕ i = 1 , ϕ − i = 0 and E [ X i ] = E [ X − i ] , then π RE i ( ϕ ) > π RE − i ( ϕ ) for both pur e and mixed price equilibrium. P articularly , if X − i follows a uniform distribution over [0 , ¯ X − i ] , we have π RE i ( ϕ ) π RE − i ( ϕ ) ≥              2 , if 0 < D ≤ E [ X i ] , 4 , if D = E [ X i ] , λ ¯ p , if D > E [ X i ] . (13) 24 Theorem 3 shows the dominance of the with-storage supplier in the S 1 S 0 subgame, whose electricity-selling rev enue can be much higher than that of the without-storage supplier . The intuition is that the random generation makes the without-storage supplier at the disadv antage in the market (due to the penalty cost). This suggests potential economic benefits of storage in vestment for the supplier . 12 Ho we ver , in vesting in storage does not always bring benefits. If both suppliers in vest in storage, it may reduce both suppliers’ rev enues compared with the case that at least one supplier does not in vest in storage. W e will discuss it later in Proposition 5. 3) S 0 S 0 subgame (i.e., P i ϕ i = 0 ): In the S 0 S 0 case, both suppliers do not in vest in storage and face the penalty cost. When 0 < D < P i y ∗ i ( ¯ p, ϕ i ) , for the mixed price equilibrium, we can neither obtain the close-form equilibrium re venue as in the S 1 S 1 case nor obtain the equilibrium strategy CDF as in Theorem 2 of the S 1 S 0 case. Note that in the S 1 S 1 and S 1 S 0 subgames, at least one supplier is not subject to the penalty cost, which makes it possible to characterize the equilibrium strategy CDF or e ven close-form equilibrium re venue. In this S 0 S 0 subgame, we will characterize a range of the lower support l in Proposition 4. Proposition 4 (S 0 S 0 : lower support) . In the S 0 S 0 subgame (i.e., P i ϕ 1 = 0 ), when 0 < D < P i y ∗ i ( ¯ p, ϕ i ) , the lower support l of the mixed price equilibrium satisfies min i y ∗ i ( l , ϕ i ) < D ≤ X i y ∗ i ( l , ϕ i ) and l < ¯ p. (14) The bidding quantity y ∗ i ( l , ϕ i ) is the minimal bidding quantity of supplier i when he uses the mixed price strategy . Proposition 4 shows that this minimal bidding quantity cannot be too lower or too higher for both suppliers. Note that the mix ed price equilibrium has a continuous CDF ov er [ l , ¯ p ) shown in Lemma 1, but we cannot deriv e it in close form. T o hav e a better understanding of the CDF , we discretize the price to approximate the original continuous price set, and compute the mix ed equilibrium for the discrete price set. The details are shown in Appendix.XII. 12 Note that Theorem 3 only compares the revenue of the two suppliers. When considering the storage in vestment cost in Stage I and comparing the suppliers’ profit, we will have some surprising results shown in Section VI and Section VIII. 25 D. Strictly positive r evenue in the S 1 S 0 and S 0 S 0 subgames Analyzing the equilibrium rev enues of the three types of subgames, we show in Proposition 5 that in the S 1 S 0 and S 0 S 0 subgames, both suppliers always get strictly positi ve rev enues. Proposition 5 (strictly positiv e rev enue with randomness) . In the S 1 S 0 and S 0 S 0 subgames, each supplier i always gets strictly positive r evenue at (both pur e and mixed) equilibrium, i.e., π RE i > 0 . This result is counter-intuiti ve for the following reason. Recall that in the S 1 S 1 subgame, both suppliers can get zero rev enue if the demand is belo w a threshold as shown in Proposition 2. The common wisdom is that when the generation is random, the rev enues of suppliers tend to be low due to the penalty cost. In contrast, Proposition 5 sho ws that the suppliers’ rev enues are always strictly positiv e when the generation is random. Thus, the randomness can in fact be beneficial. The underlying reason should be understood from the point of vie w of market competition. The randomness makes suppliers bid more conserv ati vely in their bidding quantities, which leads to less-fierce market competition and thus increases their rev enues. V I . E Q U I L I B R I U M A N A LY S I S O F S TAG E I In Stage I, each supplier i has two strategies: (i) in vesting in storage, i.e., ϕ i = 1 , and (ii) not in vesting storage, i.e., ϕ i = 0 , which leads to a bimatrix game. For this bimatrix game, we can analyze the equilibrium strategy by simply comparing the profits for each strategy pair of the two suppliers. Note that while the electricity-selling re venue is gi ven in the results of Section V, the profit also depends on the storage cost. T o calculate the storage in vestment cost, we also propose a probability-based method using real data to characterize the storage capacity for the with-storage supplier in Section VII. Each supplier’ s profit can be calculated by taking the expectation of the equilibrium re venue in the local energy market at each hour , and subtracting storage in vestment cost over the in vestment horizon (scaled into one hour). Note that suppliers’ storage-in vestment strate gy pairs ϕ = ( ϕ 1 , ϕ 2 ) lead to four possible subgames: S 1 S 1 subgame (i.e. P i ϕ i = 2 ), S 1 S 0 subgame (i.e., P i ϕ i = 1 , including two cases: ( ϕ 1 , ϕ 2 ) = (1 , 0) and ( ϕ 1 , ϕ 2 ) = (0 , 1) ), and S 0 S 0 subgame (i.e. P i ϕ i = 0 ). T aking the expectation of equilibrium rev enue ov er all the hours in the in vestment horizon, we denote supplier i ’ s equilibrium rev enue in the S 1 S 1 and S 0 S 0 subgames as π S 1 S 1 i and π S 0 S 0 i , 26 respecti vely . For the S 1 S 0 subgame, we denote the with-storage and without-storage supplier i ’ s equilibrium rev enue as π S 1 S 0 | Y i and π S 1 S 0 | N i , respectiv ely . For illustration, we list the profit table with all four strategy pairs in T able II. Supplier 2: in vest Supplier 2: not in vest Supplier 1: in vest ( π S 1 S 1 1 − C 1 , π S 1 S 1 2 − C 2 ) ( π S 1 S 0 | Y 1 − C 1 , π S 1 S 0 | N 2 ) Supplier 1: not in vest ( π S 1 S 0 | N 1 , π S 1 S 0 | Y 2 − C 2 ) ( π S 0 S 0 1 , π S 0 S 0 2 ) T ABLE II: Supplier’ s profits under dif ferent ϕ . Next, we will first deri ve the conditions for each storage-in vestment strategy pair to be an equilibrium, respectiv ely . Then, we analyze the equilibrium with respect to the parameters of storage cost and demand. Finally , we show that both suppliers can get strictly positiv e profits in this storage-in vestment game. A. Conditions of pur e storage-in vestment equilibrium W e will characterize the conditions on the storage cost and the subgame equilibrium rev enue for each strategy pair to become an equilibrium, respectiv ely . First, we define the pure storage-in vestment equilibrium in Definition 4, which states that neither supplier has an incenti ve to deviate from his storage-in vestment decision at the equilibrium. Definition 4 (pure storage-in vestment equilibrium) . A storage-in vestment vector ϕ ∗ is a pur e stora ge-in vestment equilibrium if the pr ofit satisfies Π i  ϕ ∗ i , ϕ ∗ − i  ≥ Π i  ϕ i , ϕ ∗ − i  , for any ϕ i 6 = ϕ ∗ i , and any i = 1 , 2 . Based on Definition 4, we characterize the conditions on the storage cost and the subgame equilibrium rev enue for the storage-in vestment pure equilibrium in Theorem 4, the proof of which is presented in Appendix.XVII. Theorem 4 (conditions of pure storage-in vestment equilibrium) . • S 0 S 0 case is an equilibrium if C i ∈ [ π S 1 S 0 | Y i − π S 0 S 0 i , + ∞ ) , for both i = 1 , 2 . • S 1 S 0 case is an equilibrium (wher e ϕ i = 1 and ϕ − i = 0 ) if C i ∈ [0 , π S 1 S 0 | Y i − π S 0 S 0 i ] and C − i ∈ [ π S 1 S 1 − i − π S 1 S 0 | N − i , + ∞ ) . 27 • S 1 S 1 case is an equilibrium if C i ∈ [0 , π S 1 S 1 i − π S 1 S 0 | N i ] , for both i = 1 , 2 . If C i satisfies none of the conditions above , ther e exists no pur e stora ge-in vestment equilib- rium. 13 Theorem 4 shows that the storage-in vestment equilibrium depends on the comparison between the storage cost and the re venue difference between the cases S 1 S 0 and S 1 S 0 , or the cases S 1 S 0 and S 1 S 1 . Also, Theorem 4 implies that a lower storage cost will incentivize the supplier to in vest in storage. According to Theorem 4, giv en the storage cost and the expected equilibrium rev enue of each subgame, we can characterize the pure equilibrium for nearly all values of C i . Howe ver , if storage cost C i satisfies none of the conditions in Theorem 4, there will be no pure price equilibrium. Note that when there is no pure storage-in vestment equilibrium, we can always characterize the mixed equilibrium as the game in Stage I is a finite game [28]. W e show how to compute the mixed equilibrium in Appendix.XVII. Since we cannot characterize close-form equilibrium re venues for the S 1 S 0 and S 0 S 0 subgames, it remains challenging to characterize the storage-in vestment equilibrium with respect to the system parameters, e.g., the storage cost and demand. In the ne xt subsection, we will focus on deri ving insights of the storage-inv estment equilibrium in some special and practically interesting cases. B. Impact of stora ge cost and demand on storage-in vestment equilibrium W e analyze the impact of storage cost and demand on the storage-in vestment equilibrium and hav e the analytical results for the cases when: (i) the storage cost C i is sufficiently large; (ii) the demand D m,t is suf ficiently large or small. W e present all the proofs in Appendix.XVII. T o better illustrate the storage-in vestment equilibrium, we sho w one simulation result of the equilibrium split (i.e., the storage-in vestment equilibrium with respect to parameters such as the demand and the storage cost) in Figure 3, and the details of the simulation setup are presented in Section VIII. In this simulation, for the illustration purpose, we consider the same demand D for an y hour t of any month m . W e also consider two homogeneous suppliers (with the same 13 Note that if π S 1 S 0 | Y i − π S 0 S 0 i < 0 or π S 1 S 1 i − π S 1 S 0 | N i < 0 , then the set [0 , π S 1 S 0 | Y i − π S 0 S 0 i ] = ∅ or [0 , π S 1 S 1 i − π S 1 S 0 | N i ] = ∅ . This means that the condition C i ∈ [0 , π S 1 S 0 | Y i − π S 0 S 0 i ] or C i ∈ [0 , π S 1 S 1 i − π S 1 S 0 | N i ] cannot be satisfied. 28         0 0.2 0.4 0.6 0.8 1    0 5 10 15                  Fig. 3: Equilibrium split with storage cost and demand at λ = 1 . 5 HKD/kWh. storage cost, the same rene wable ener gy capacity and the same renew able energy distribution) to rev eal the impact on storage-in vestment decision. 14 In Figure 3 (where the penalty price is λ = 1 . 5 Hong K ong dollars (HKD) per kWh), with respect to the demand and storage cost, the storage-in vestment equilibrium is di vided into three regions: Re gion I of S 1 S 1 (the left side of the red curve), Region II of S 1 S 0 (between the red curve and the blue curve), and Region III of S 0 S 0 (the right side of the blue curve). First, for the impact of the storage cost, a higher storage cost will discourage suppliers from in vesting in storage as implied in Theorem 4. W e will further sho w that when the storage cost is higher than a threshold, no suppliers will in vest in storage no matter what the demand or penalty . Ho we ver , counter-intuiti vely , we also find that in the case of a zero storage cost, not both suppliers will in vest in storage once the demand is lo wer than a certain threshold. As sho wn in Figure 3, when the storage cost is larger than a threshold, i.e., C > 0 . 86 × 10 3 HKD, the S 0 S 0 case will be the only equilibrium (independent of the demand D ) and no suppliers in vest in storage. W e show this property in Proposition 6. The reason is that the benefit from in vesting in storage is bounded. When the storage cost is greater than a threshold corresponding to the bounded benefit, no suppliers will choose to in vest in storage. 14 W e can prov e that a pure Nash equilibrium of storage in vestment always exists in this homogeneous case. Howe ver , for the heterogeneous case, we cannot theoretically prove that the pure Nash equilibrium always exists. In Appendix, we simulate an example with two heterogeneous suppliers (with different capacities of rene wables) and sho w the storage-inv estment equilibrium in such a heterogeneous case. 29 Proposition 6. There exists a thr eshold C S 0 S 0 i such that if the storage cost satisfies C i > C S 0 S 0 i for both i = 1 , 2 , the S 0 S 0 case will be the unique pur e storage-in vestment equilibrium. Ho we ver , as sho wn in Figure 3, when the demand is smaller than a certain threshold, i.e., D < 2 . 8 MW , the S 1 S 1 case cannot be a pure equilibrium ev en when the storage cost C = 0 . W e sho w this property in Proposition 7. The reason is that when the demand is smaller than a certain threshold, in the S 1 S 1 case, both suppliers can only get zero rev enues (as shown in Proposition 2) due to the competition. Thus, if the S 1 S 1 case is the storage-in vestment state where both suppliers in vest in storage, one supplier can always de viate to not in vesting in storage, which can bring him a strictly positi ve profit as implied in Proposition 5. Proposition 7. If the demand satisfies 0 < D m,t ≤ min i E [ X m,t i ] for any t and m , the S 1 S 1 case cannot be the equilibrium. Second, for the impact of demand, we already show that at a suf ficiently lo w demand, the S 1 S 1 case cannot be the equilibrium in Proposition 7. W e will further show that if the demand is higher than a certain threshold, each supplier has a dominant strategy of whether to in vest in storage based on his storage cost, which does not depend on the other supplier’ s decision. For example, at D > 11 MW in Figure 3, for these two homogeneous suppliers, if the storage cost is higher than a threshold, i.e., C > 0 . 63 × 10 3 HKD, each supplier will not in vest in storage (i.e., S 0 S 0 ); otherwise, each supplier will in vest (i.e., S 1 S 1 ). W e show this property in Proposition 8. The reason is that if the demand is large enough, both suppliers can bid the highest price and sell out the maximum bidding quantity . Thus, there is no competition between suppliers, and they will make storage-in vestment decisions based on their own storage costs. Proposition 8. Ther e exists D m,t,th > 0 and C th i > 0 , such that when the demand satisfies D m,t ≥ D m,t,th for any t and m , supplier i has the dominant strate gy ϕ ∗ i as follows. 15 ϕ ∗ i =      1 , if the storage cost C i ≤ C th i , 0 , if the storage cost C i > C th i . (15) 15 W e characterize the close-form threshold D m,t,th > 0 and C th i > 0 in Appendix.XVII. 30 C. Strictly positive pr ofits of suppliers W e sho w that in suppliers’ competition f acing the cost of storage in vestment, both suppliers can get strictly positi ve profits. Proposition 9 (strictly positi ve profit) . Both suppliers will get strictly positive pr ofits at the stora ge-in vestment equilibrium. This proposition also shows the benefit of the uncertainty of rene wable generation, which is similar to Proposition 5. Recall that if both suppliers hav e stable outputs, they may get zero re venue (sho wn in Proposition 2) and thus get negati ve profit considering the storage cost. Ho we ver , with the random generation, both suppliers will get strictly positi ve profits at the storage-in vestment equilibrium even facing the storage cost. W e will explain it as follows. Note that in the S 0 S 0 case or the S 1 S 0 case, the without-storage supplier always gets a strictly positiv e re venue (shown in Proposition 5) with a zero storage cost. In the S 1 S 0 case or the S 1 S 1 case, if the with-storage supplier gets a non-positiv e profit, he can always deviate to not in vesting in storage. This deviation provides him a strictly positi ve profit, which implies that the supplier will always get strictly positiv e profit. V I I . C H A R A C T E R I Z A T I O N O F S T O R A G E C A PAC I T Y W e propose a probability-based method using historical data of renew able generations to compute the storage capacity . Note that suppliers charge and dischar ge the storage to maintain his output at the mean value of the random renewable generations as sho wn in (1). 16 Therefore, the charge and discharge amounts are also random variables, and we characterize the storage capacity such that its energy lev el will not exceed the storage capacity with a targeted probability . In this part, we focus on the storage with 100% charge and discharge efficienc y and no degradation cost. In Appendix.XIII, we show that a lower charge/dischar ge efficienc y and the consideration of degradation cost will increase the total storage cost of a supplier , which further affects the storage-in vestment equilibrium. 16 It is interesting to size the variable storage capacity considering the possibility of not completely smoothing out the renewable output. Ho wever , it is quite challenging to characterize such an equilibrium storage capacity in closed-form, which we will study as future work. 31 T o begin with, we set a probability target α , and we aim to find a storage capacity S i such that the energy lev el in the storage exceeds the capacity with a probability no greater than α . Specifically , the with-storage supplier i will char ge and dischar ge storage with value C D m,t i at hour t of month m as shown in (1). W e assume that the initial energy lev el of storage is fixed for all the months and denote it as S l i . Note that the energy le vel of storage is the sum of the charge and discharge over the time, and is constrained by the storage capacity . Starting from the initial energy le vel S l i , the probability that ener gy lev el exceeds the minimum capacity (i.e., zero) and the maximum capacity (i.e., S i ) of the storage in a day of month m is max t 0 ∈T Pr ( P t 0 t =1 C D m,t i + S l i < 0) and max t 0 ∈T Pr ( P t 0 t =1 C D m,t i + S l i > S i ) , respecti vely . Considering all months m , we aim to choose the storage capacity S i so that the follo wing hold: E m  max t 0 ∈T Pr ( t 0 X t =1 C D m,t i + S l i < 0)  ≤ α, (16) E m  max t 0 ∈T Pr ( t 0 X t =1 C D m,t i + S l i > S i )  ≤ α. (17) Then, we describe ho w to use historical data [29] to compute the storage capacity that satisfies the probability threshold as in (16) and (17). we will first characterize an upper bound for the probability that energy le vel exceeds the giv en storage capacity in terms of the random variable C D m,t i , and then we propose Algorithm 1 to compute the required storage capacity to satisfy (16) and (17). First, gi ven the underflo w capacity S l i > 0 and overflo w capacity S u i , S i − S l i > 0 , we characterize an upper bound P r l,m ( S l i ) for max t 0 Pr ( P t 0 t =1 C D m,t i + S l i < 0) and an upper bound P r u,m ( S u i ) for max t 0 Pr ( P t 0 t =1 C D m,t i + S l i > S i ) , respecti vely . W e characterize these upper bounds based on Marko v inequality [30], which is shown in Proposition 10. Proposition 10 (Markov-inequality-based upper bound) . Given S l i > 0 and S u i > 0 , the Markov- inequality-based upper bounds ar e shown as follows. • F or the upper bound P r l,m ( S l i ) : P r l,m ( S l i ) = max t 0 min s> 0 B l ( s ) , (18) wher e B l ( s ) , e − sS l i · E h e s P t 0 t =1 − C D m,t i i . 32 • F or the upper bound P r u,m ( S u i ) : P r u,m ( S u i ) , max t 0 min s> 0 B u ( s ) , (19) wher e B u ( s ) , e − sS u i · E h e s P t 0 t =1 C D m,t i i . Note that P r l,m ( S l i ) and P r u,m ( S u i ) are decreasing in S l i and S u i , respecti vely . Also, P r l,m ( S l i ) → 0 as S l i → + ∞ , and P r u,m ( S u i ) → 0 as S u i → + ∞ . These show that a larger capacity will decrease the probability that the charge/discharge exceeds the capacity . Also, for any probability threshold α > 0 , we can always find a capacity , such that the probability that ener gy lev el exceeds the capacity is belo w α . Second, we propose Algorithm 1 to characterize the storage capacity S i based on the historical data of C D m,t i (deri ved from the rene wable generation data of X m,t i ). W e use the underflow capacity S l i for supplier i as an e xample for illustration, and the o verflo w capacity S u i follo ws the same procedure. Specifically , for the underflow capacity S l i , we search it in an increasing order from zero as in Step 4. Gi ven S l i , for each month m , we calculate the exceeding probability P r l,m ( S l i ) according to (18) as in Steps 5-7. Note that based on the data samples of P t 0 t =1 − C D m,t i , B l ( s ) is strictly con ve x in s . Thus, for any S l i > 0 , the value of min s> 0 B l ( s ) can be efficiently computed using Ne wton’ s method [31]. Further , we conduct an exhausti ve search for t 0 ∈ T to obtain P r l,m ( S l i ) . W e calculate the expected e xceeding probability E m [ P r l,m ( S l i )] ov er months as in Step 8. W e obtain the minimal underflow capacity S l i if the exceeding probability satisfies E m [ P r l,m ( S l i )] ≤ α as in Step 9. Similarly , we can get the minimal o verflo w capacity S u i . The required storage capacity is calculated as in Step 11. As an illustration, we calculate and show the underflow probability E m [ P r l,m ( S l i )] and o verflo w probability E m [ P r u,m ( S u i )] in the blue solid curve and red dashed curve respectiv ely in Figure 4. The probability of E m [ P r l,m ( S l i )] ( E m [ P r u,m ( S u i )] , respecti vely) decreases with respect to the capacity S l i ( S u i , respecti vely). If the capacity S l i ( S u i , respecti vely) is small and close to zero, the exceeding probability E m [ P r l,m ( S l i )] ( E m [ P r u,m ( S u i )] , respecti vely) will approach one. Howe ver , when the capacity is large and close to a certain value (e.g., 6 in Figure 4), the corresponding exceeding probability will be close to zero. W e choose the probability threshold α = 5% and obtain the corresponding minimal capacity S l ∗ i and S u ∗ i as marked in Figure 4. 33 Algorithm 1 Storage capacity S i 1: initialization : set iteration index S l i = S u i = 0 , step size ∆ S ; 2: f or each k ∈ { l , u } do 3: repeat 4: S k i := S k i + ∆ S ; 5: f or each m ∈ M do 6: Supplier i calculates P r k,m ( S k i ) according to (18) or (19); 7: end f or 8: Supplier i calculates E m [ P r k,m ( S k i )] ; 9: until E m [ P r k,m ( S k i )] ≤ α ; 10: end f or 11: Each supplier i computes S i = S l i + S u i ; 12: output : S i . V I I I . S I M U L AT I O N In simulations, in addition to some analytical properties of storage-in vestment equilibrium sho wn in Section VI, we will further in vestig ate the impact of the penalty , storage cost, and demand on suppliers’ profits. W e will sho w some counter-intuiti ve results due to the competition between suppliers. For example, a higher penalty , a higher storage cost, and a lo wer demand can ev en increase a supplier’ s profit at the storage-in vestment equilibrium. Furthermore, the first supplier who in vests in storage may benefit less than the competitor who does not in vest in storage. W e will illustrate the detailed results in the follo wing. A. Simulation setup In simulations, we consider two homogeneous suppliers (with the same renew able capacity , generation distribution, and storage cost) to show the storage-in vestment equilibrium. W e also 34         0246       0 0.2 0.4 0.6 0.8 1               Fig. 4: Characterization of storage capacity . consider a fixed demand D for all the hours and months for illustration. Next, we explain the empirical distribution of rene wable generation as well as parameter configurations of the penalty price λ , demand D , and storage cost C . 1) Empirical distribution of r enewable generation: W e use the historical data of solar energy generation in Hong K ong from the year 1993 to year 2012 [29] to approximate the continuous CDF of suppliers’ renewable generations. Specifically , we cluster the rene wable generations at hour t of all days into M = 12 types (months) considering the seasonal effect. W e use daily data (from the year 1993 to year 2012) of renewable energy in month m at hour t to approximate the distribution of rene wable generation at hour t of month m . Based on the discrete data, we characterize a continuous empirical CDF to model the distribution of renewable power . W e present the details of the characterization of empirical CDF in Appendix.XIV. Furthermore, to check the reliability of the empirical distribution, we consider two sample data sets: one set consists of all the data samples from the year 1993 to 2012, and the other consists of the data samples from another specific year (e.g., 2013). W e conduct K olmogorov-Smirnov test [32] using the Matlab function kstest2 to test whether these two data sets are from the same continuous distribution [33]. The result shows that most of the hours of a month can pass the test. Also, our model is general for any continuous distrib ution of rene wable generations. Interested readers can also use other data or other distributions of renew able energy to test the results. 35 May: Hour 0 5 10 15 20 25 Power (MW) 0 1 2 3 4 Fig. 5: A verage solar energy of different hours in May . 2) P arameters configuration : W e explain the configuration of the parameters of the penalty price λ , demand D , and storage cost C , respectively . W e set the parameters to reflect the real- world practice, and study the impact of the parameters on the market equilibrium. • The penalty λ : W e choose the price cap ¯ p = 1 HKD/kWh, since the electricity price for residential users in Hong K ong is around 1 HKD/kWh [34]. Note that a penalty price satisfies λ > ¯ p . In Figure 6(a), we will consider a wide range of the ratio λ ¯ p ∈ [1 . 2 , 20] to demonstrate the impact of the penalty . In Figures 6(b)(c)(d), we fix the penalty price λ = 1 . 5 HKD/kWh and focus on illustrating the impact of other parameters. • The demand D : In Figure 6(d), we will discuss a wide range of demand from 0 MW to 15MW to show the impact of the demand. As a comparison, in Figure 5, we show the av erage renew able power across hours in May . In Figure 6(a) and (b), we fix the demand at D = 1 MW to show the impact of other parameters ( λ and C ). In Figure 6(c), we choose a lar ger demand D = 12 MW and a smaller demand D = 6 MW to show the impact of demand on the equilibrium profit. • The Storage cost C i : Recall that the storage in vestment cost is C i = c i κ i S i . There are dif ferent types of storage technologies with diverse capital costs and lifespans. For example, the pumped hydroelectric storage is usually cheap, and can last for 30 years with the capital cost c i = 40 ∼ 800 HKD/kWh, while the Li-ion battery can last 15 years with the capital cost about c i = 1600 ∼ 9000 HKD/kWh [35]. W e choose the annual interest rate r i = 5% , and 36 the storage capacity for the with-storage supplier is characterized as 43 MWh by Algorithm 1. W e capture the impact of parameters c i and κ i through the storage cost C i . According to the calculation of storage in vestment cost C i = c i κ i S i , we can calculate that the (hourly) in vestment cost C i of the pumped hydroelectric storage is 0 . 012 × 10 3 − 0 . 255 × 10 3 HKD and the cost of the Li-ion battery is 0 . 76 × 10 3 − 4 . 36 × 10 3 HKD. This shows that the storage cost can hav e a wide range. 17 Then, in Figures 6(c), we will consider a wide range of storage costs from 0 to 2 × 10 3 HKD. Although zero storage cost is not very practical, we use it to show a low storage cost and capture the entire range of the impact of the storage costs. In Figure 6(a)(b)(d), we choose lower storage costs ( 0 . 1 × 10 3 and 0 . 15 × 10 3 HKD) and higher storage costs ( 1 × 10 3 and 1 . 5 × 10 3 HKD) to show the dif ferent results under dif ferent storage costs. B. Simulation r esults W e will discuss the impact of penalty , storage cost, and demand on suppliers’ profits, and show some counter-intuiti ve results due to the competition between suppliers. 1) The impact of penalty on suppliers’ pr ofits: Although a higher penalty λ can incr ease the penalty cost on the without-stora ge supplier , surprisingly , we find that a higher penalty can also incr ease this supplier’ s pr ofit, due to the r educed market competition in the ener gy market. W e show how suppliers’ profits and e xpected bidding prices at the storage-in vestment equi- librium change with the penalty (at demand D = 1 MW) in Figure 6(a) and 6(b), respectiv ely . Dif ferent colors represent different storage costs. The diamond marker shows that S 0 S 0 is the storage-in vestment equilibrium, and the circle marker shows that S 1 S 0 is the equilibrium. Also, when S 1 S 0 is the equilibrium, the solid lines and dashed lines distinguish the with-storage supplier and without-storage supplier , respectiv ely . First, we show that at the equilibrium where both suppliers do not in vest in storage (i.e., S 0 S 0 ), a higher penalty λ can increase both suppliers’ profits. As shown in Figure 6(a), when the storage cost is high at C = 1 . 5 × 10 3 HKD, both suppliers will not in vest in storage for any value of the penalty λ from 1 . 2 HKD/kWh to 20 HKD/kWh (in blue curve with diamond marker). In 17 Note that we only consider the in vestment cost in the storage cost. In practice, there are also other costs that need to be included, such as maintenance cost. 37         0 5 10 15 20        0 0.1 0.2 0.3 0.4 0.5 0.6                                         (a)              0 5 10 15 20         0.3 0.4 0.5 0.6 0.7 0.8 0.9                                         (b)                00 . 511 . 52        0 0.5 1 1.5 2                                         (c)            0 5 10 15        0 0.5 1 1.5 2 2.5                                               (d) Fig. 6: (a) Profit of suppliers with penalty ( D = 1 MW); (b) Expected bidding price of suppliers with penalty ( D = 1 MW); (c) Profit of suppliers with storage cost ( λ = 1 . 5 HKD/kWh); (d) Profit of suppliers with demand ( λ = 1 . 5 HKD/kWh). this case, both suppliers’ profits can first increase (at λ < 11 HKD/kWh) and then decrease (at λ > 11 HKD/kWh) with λ (in blue curve). The intuition for the increase of profit at λ < 11 HKD/kWh is that a higher penalty decreases both suppliers’ bidding quantity if the bidding price remains the same. This reduces the market competition and enables both suppliers to bid a higher price in the local energy market as shown in Figure 6(b) (in blue curve). Howe ver , the increased penalty also increases the penalty cost on suppliers, so the suppliers’ profits will also decrease if the penalty is too high (at λ > 11 HKD/kWh). Second, we show that at the equilibrium where one supplier in vests in storage and one does 38 not (i.e., S 1 S 0 ), a higher penalty λ can also increase both suppliers’ profits. W e consider a low storage cost C = 0 . 15 × 10 3 HKD as in red curves in Figure 6(a) and Figure 6(b). W e see that if λ is low (at λ < λ a ), both suppliers will not in vest in storage (i.e., S 0 S 0 ), and their profits increase with penalty sho wn in Figure 6(a) (at λ < λ a in red curve with diamond marker that o verlaps with blue curve). As the penalty increases (at λ > λ a ), the equilibrium will change from S 0 S 0 to S 1 S 0 , since a higher penalty and a lower storage cost can enable a supplier to enjoy more benefits by in vesting in storage. W e discuss the profit of the with-storage supplier and without-storage supplier respecti vely as follows. • For the with-storage supplier , as shown in Figure 6(a), when λ > λ a , his profit increases as penalty increases (in red solid curv e), which can be much higher than the without-storage supplier (in red dashed curve). The reason is that in the S 1 S 0 case, the penalty cost makes the with-storage supplier dominate ov er the without-storage one. The with-storage supplier can bid higher prices than the without-storage supplier as shown in Figure 6(b) (in red solid curve and red dashed curve), and he also does not need to pay the penalty cost. • Ho we ver , for the without-storage supplier , as sho wn in Figure 6(a), his profit also slightly increases as the penalty increases around λ a < λ < 10 HKD/kWh (in red dashed curve). The intuition is that a higher penalty giv es the advantage to the with-storage supplier , which reduces the market competition and increases both suppliers’ bidding price as shown in Figure 6(b) (in red curves). Thus, it can also benefit the without-storage supplier . Howe ver , as shown in Figure 6(a), if the penalty further increases to λ > 10 HKD/kWh (in red dashed curve), the without-storage supplier’ s profit will also decrease due to the increased penalty cost. 2) The impact of storag e cost on suppliers’ pr ofits: Intuitively , a higher storage cost will discoura ge a supplier fr om in vesting in stora ge, whic h generally decr eases a supplier’ s pr ofit. However , we find that it may also incr ease a supplier’ s pr ofit if the other supplier changes his strate gy due to the increased storag e cost. W e show how suppliers’ profits at the storage-in vestment equilibrium change with the storage cost in Figure 6(c). Different colors represent different demands. The diamond marker , circle marker , and star marker correspond to different storage-in vestment equilibria of S 0 S 0 , S 1 S 0 , and S 1 S 1 , respectiv ely . For the S 1 S 0 case, the solid lines and dashed lines distinguish the with-storage 39 supplier and without-storage supplier , respectiv ely . As shown in Figure 6(c) (in both red curve and blue curve), generally the higher storage cost decreases suppliers’ profits. Howe ver , we show that the opposite may be true using the example of D = 6 MW (in red curve). When the demand is at D = 6 MW (in red curve), as the storage cost increases, the equilibrium changes from S 1 S 1 (when C < C a ), to S 1 S 0 (when C a < C < C b ), and finally to S 0 S 0 (when C > C b ). When the equilibrium changes from S 1 S 1 to S 1 S 0 at the threshold C = C a , one with-storage supplier in the original S 1 S 1 case has a higher (upward jumping) profit, after the other supplier chooses not to in vest in storage due to the high storage cost. This changes the equilibrium from S 1 S 1 to S 1 S 0 , which reduces the competition and giv es more adv antages to the with-storage supplier . 3) The impact of demand on suppliers’ pr ofits: Intuitively , a higher demand will incr ease a supplier’ s pr ofit. However , we show that a higher demand may also decrease a supplier’ s pr ofit if the other supplier chang es his strate gy due to the incr eased demand. W e show how suppliers’ profits at the storage-in vestment equilibrium change with the demand in Figure 6(d). Different colors represent different storage costs. The diamond marker , circle marker , and star marker correspond to different storage-in vestment equilibria of S 0 S 0 , S 1 S 0 , and S 1 S 1 , respectiv ely . For the S 1 S 0 case, the solid lines and dashed lines distinguish the with-storage supplier and without-storage supplier respecti vely . As shown in Figure 6(d) (in both red curve and blue curve), generally a higher demand increases a supplier’ s profit. Howe ver , we sho w that the opposite may be true using the example of C = 0 . 1 × 10 3 HKD (in red curve). When the storage cost is low at C = 0 . 1 × 10 3 HKD (in red curve), as the demand increases, the equilibrium changes from S 0 S 0 (when D < D a ), to S 1 S 0 (when D a < D < D b ), and finally to S 1 S 1 (when D > D b ). When the equilibrium changes from S 1 S 0 to S 1 S 1 at the threshold D = D b , the with-storage supplier in the original S 1 S 0 case has a smaller (do wnward jumping) profit, after the other supplier also chooses to in vest in storage due to the high demand. This changes the equilibrium from S 1 S 0 to S 1 S 1 , which increases the market competition and weakens the advantage of the with-storage supplier in the original S 1 S 0 case. Furthermore, when the storage cost is high at C = 1 . 5 × 10 3 HKD (in blue curve with diamond marker), both suppliers will not in vest in storage independent of the demand. 40 4) F irst-mover disadvantage and advantage: Intuiti vely , the first supplier who in vests in stor- age can benefit more than the without-storage competitor . However , we find that if the stor age cost is high, the first-mover supplier in in vesting storage can also benefit less than the fr ee-rider competitor who does not in vest in storage . As shown in Figure 6(c) at D = 6 MW (in red curve), the S 1 S 0 case is the equilibrium when the storage cost is in the range C a < C < C b . If the storage cost is low at C a < C < 0 . 7 × 10 3 HKD, the with-storage supplier’ s profit is higher than the without-storage supplier’ s profit. Howe ver , if the storage cost is high at 0 . 7 × 10 3 HKD < C < C b , the with-storage supplier’ s profit is lower than the without-storage supplier . This shows both adv antage and disadvantage of the first-mover . Although in some situations in vesting storage will increase the supplier’ s profit, he can get more profits if he waits for the other to in vest first when the storage cost is high. Howe ver , if the storage cost is lo w , he should be the first to in vest storage in order to get a higher profit. I X . E X T E N S I O N S : A M O R E G E N E R A L O L I G O P O L Y M O D E L W e build a more general oligopoly model and extend some of the theoretical results and insights from the duopoly case to the oligopoly case. Compared with the duopoly model, the only dif ference of the oligopoly model is that the number of suppliers can be more than two, i.e., |I | ≥ 2 . F ollo wing the analysis of the duopoly model, we also analyze the equilibrium in Stage II and Stage I in the oligopoly case and deriv e some insights. Specifically , in Stage II, we extend the theoretical results of the price-quantity competition equilibrium. In Stage I, we generalize analytical results of the impact of storage cost and demand on the storage-inv estment equilibrium. Furthermore, we sho w that some of the ke y insights from the duopoly case, e.g., the uncertainty of renew able generation can be beneficial to suppliers, still hold in the oligopoly case. Next, we will discuss the extensions of Stage II and Stage I in detail, respecti vely . W e include all the proofs of the propositions in Appendix.XVIII. A. Stage II Analysis For Stage II, the weakly dominant strategy of bidding quantities still hold for the case of more than two suppliers. W e generalize the conditions on the existence of the pure price equilibrium and show that the mixed price equilibrium also exists in the oligopoly case. Furthermore, we 41 sho w that suppliers get positiv e rev enues at the mixed price equilibrium. W e show the extended analysis in detail as follo ws. 1) W eakly dominant strate gy for bidding quantities: The weakly dominant strategies for bid- ding quantities still hold as in Theorem 1. 2) Existence of the pur e price equilibrium: W e deriv e the conditions for the existence of the pure price equilibrium among suppliers, and generalize Proposition 2. Specifically , we consider a general subgame in Stage II denoted as S U |V , where suppliers in the set U in vest in storage and suppliers in the set V do not in vest. Recall we denote the set of all the suppliers as I , and we have U S V = I . The case U = I means that all the suppliers in vest in storage, and the case V = I means that no supplier in vests in storage. W e show the existence of the pure price equilibrium in Proposition 11. Proposition 11 (existence of the pure price equilibrium in the oligopoly case) . Considering a subgame S U |V of storage in vestment among suppliers in Stage II, the existence of the pur e price equilibrium depends on the demand D as follows: • If D ≥ P i ∈I y ∗ i ( ¯ p, ϕ i ) , ther e exists a pur e price equilibrium p ∗ i = ¯ p , with an equilibrium r evenue π RE i = λ R F − 1 i ( ¯ p/λ ) 0 xf i ( x ) dx for any i ∈ V and π RE i = ¯ p E [ X i ] for any i ∈ U . • If D ≤ P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j ) for any j ∈ U , ther e exists a pure price equilibrium p ∗ i = 0 , with an equilibrium r evenue π RE i = 0 , for any i ∈ I . • If there exists j ∈ U such that P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j ) < D < P i ∈I y ∗ i ( ¯ p, ϕ i ) , there is no pur e price equilibrium. Similar to the duopoly case, the result of this proposition can be interpreted as follo ws. If the demand is higher than the threshold P i ∈I y ∗ i ( ¯ p, ϕ i ) , all the suppliers can bid the price cap to sell the maximum quantities. If the demand is very lo w such that D ≤ P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j ) for any j ∈ U , the competition is fierce and all the suppliers bid zero price. Ho we ver , if the demand is in the middle, there will be no pure price equilibrium. Note that if the number of with-storage suppliers is no greater than one, i.e., | U |≤ 1 , the condition that there exists j ∈ U such that D ≤ P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j ) cannot be satisfied. It means that there will be no pure equilibrium of p ∗ i = 0 for any demand D > 0 . 42 3) Existence of the mixed price equilibrium: For the case in Proposition 11 that there exists no pure price equilibrium, we sho w that there exists a mixed price equilibrium. Ho we ver , the characterization of mix ed strategy is highly non-trivial for the oligopoly case and it is dif ficult to completely generalize Lemma 1. W e generalize it partially as Proposition 12 to show the existence of the mixed price equilibrium and show that all the suppliers get positiv e rev enues at the mixed price equilibrium. Proposition 12 (mixed price equilibrium in the oligopoly case) . F or any ϕ , when there is no pur e price equilibrium, a mixed price equilibrium exists and the equilibrium electricity-selling r evenues π RE i satisfies π RE i ( ϕ ) > 0 , for any i ∈ I . The equilibrium rev enue for the case where all the suppliers in vest storage (i.e., U = I ) has been characterized in [17]. When there are two suppliers, we can also characterize the cumulativ e distribution function (CDF) of the mixed price strategy for the case of one in vesting storage and one not in vesting in storage as in Theorem 2. Howe ver , when I > 2 , for any case where | U | < I , it is highly non-tri vial to characterize the corresponding CDF analytically . B. Stage I Analysis For Stage I, for the general oligopoly case, we sho w that a mix ed storage-in vestment equilibrium always exists. W e can also generalize the analytical results of the impact of storage cost and demand on the storage-inv estment equilibrium for those settings where (i) the storage cost is suf ficiently large; and (ii) the demand is sufficiently lar ge or small. Furthermore, some of the ke y insights, e.g., the uncertainty of rene wable generation can be beneficial to suppliers, will still hold for the oligopoly case. W e discuss the extensions in details in the following. 1) Existence of the storage-in vestment equilibrium: A mixed equilibrium of storage in vestment always exists. Note that each supplier has two strategies: in vesting in storage and not inv esting in storage. Numerically , we can check the pure storage-in vestment equilibrium by the Nash equilibrium definition. Also, a mixed equilibrium of storage in vestment always exists due to the finite numbers of storage-in vestment strategies [28]. 2) Impacts of the storage cost and demand on storag e-in vestment equilibrium: Some analysis of the impact of the storage cost and demand on storage-in vestment equilibrium in the duopoly 43 case can also be extended. Specifically , we can extend Propositions 6, 7 and 8 and to the oligopoly case, which generalizes the analytical results for the settings where (i) the storage cost is suf ficiently large; and (ii) the demand is sufficiently large or small. First, since the benefit from in vesting in storage is bounded, we can show that when the storage cost is greater than a threshold, no suppliers will choose to in vest in storage. Proposition 13. There exists a thr eshold C no i such that if the storage cost satisfies C i > C no i for any i ∈ I , the S ∅|I case (i.e., no suppliers in vesting in storage) will be the unique pur e stora ge-in vestment equilibrium. Second, in the subgame S U |V where | U |≥ 2 , if the demand is too low , all the suppliers may get zero re venue in the energy market as implied in Proposition 11. This will make the with-storage suppliers de viate to not in vesting in storage. Thus, we hav e the proposition as follows. Proposition 14. In the subgame S U |V , if the demand satisfies 0 < D m,t ≤ min j ∈U ( P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j )) for any t and m , the case S U |V (i.e., suppliers in set U in vest in storag e and suppliers in set V do not in vest in storag e) cannot be a pur e storage-in vestment equilibrium. Third, as in Proposition 11, when the demand is higher than certain threshold, all the suppliers can bid the price cap to sell all his bidding quantity . In this case, there is no competition between suppliers, and they will make storage-in vestment decisions independently based on their o wn storage costs. W e show this proposition as follows. Proposition 15. Ther e exist D m,t,th > 0 and C th i > 0 , such that when the demand satisfies D m,t ≥ D m,t th for any t and m , supplier i has the dominant strate gy ϕ ∗ i as follows. ϕ ∗ i =      1 , if the storage cost C i ≤ C th i , 0 , if the storage cost C i > C th i . (20) 3) P ositive pr ofits at the storage-in vestment equilibrium: W e can further extend Proposition 9 to show the benefit of the uncertainty to the equilibrium profit. W e sho w that in suppliers’ competition (e ven with the potential cost of the storage in vestment), all the suppliers can get strictly positi ve profits at the equilibrium. 44 Proposition 16 (strictly positiv e profit) . All the suppliers will get strictly positive pr ofits at the stora ge-in vestment equilibrium. This proposition sho ws the benefit of the rene wable generation randomness. If all the suppliers hav e stable outputs, they may get zero revenue as implied in Proposition 11 and thus get neg ati ve profit under possible storage cost. Howe ver , with the random generation, all the suppliers will get strictly positi ve profit at the storage-in vestment equilibrium ev en considering the storage cost. The intuition is that if one supplier in vests in storage and gets non-positiv e profit, he can always choose not to in vest in storage. This at least sa ves him the cost of storage inv estment, which increases his profit. Also, note that when no supplier in vests in storage, all the suppliers can get positi ve profits. Therefore, only the state where all the suppliers get positi ve profits can be an equilibrium. In summary , we can extend some of our major theoretical results and insights to the oligopoly case of more than two suppliers. Some of the ke y insights from the duopoly case, e.g., the uncertainty of renew able generation can be beneficial to suppliers, still hold in the oligopoly case. Ho wev er , we are not able to analytically extend all insights to the oligopoly case due to the complexity of analysis. W e would like to explore it in our future work. X . C O N C L U S I O N W e study a duopoly two-settlement local energy market where rene wable energy suppliers com- pete to sell electricity to consumers with or without energy storage. W e formulate the interactions between suppliers and consumers as a three-stage game-theoretic model. W e characterize a price- quantity competition equilibrium in the local energy market, and further characterize a storage- in vestment equilibrium at the beginning of the in vestment horizon between suppliers. Surprisingly , we find the uncertainty of renew able generation can increase suppliers’ profits compared with the case where both suppliers in vest in storage and stabilize the outputs. In simulations, we sho w more counterintuiti ve results due to the market competition. For example, a higher penalty , a higher storage cost, and a lower demand may increase a supplier’ s profit. W e also sho w that the first-mov er in in vesting in storage may benefit less than the free-rider competitor who does not in vest in storage. In the future w ork, we will size the variable storage capacity considering the possibility of not completely smoothing out the renew able output. 45 A P P E N D I X This appendix is organized as follo ws: • Section XI: W e show the equilibrium rev enue of suppliers in the S 1 S 1 case, when the demand satisfies min i y ∗ i ( ¯ p, ϕ i ) < D < P i y ∗ i ( ¯ p, ϕ i ) and there is no pure price equilibrium b ut the mixed price equilibrium. • Section XII: W e sho w ho w we discretize the continuous price set to approximate the mixed price equilibrium in the S 0 S 0 case. • Section XIII: For the storage capacity characterization, we first show the proof of the propositions in Section VII, and then we present the model of the imperfect storage. • Section XIV: For the simulations, we first show the characterization of the continuous CDF for the renew able-generation distribution using historical data, and then we simulate an example of two heterogeneous suppliers. • Section XV: W e prove the theorems and propositions of Stage III. • Section XVI: W e prove the theorems and propositions of Stage II. • Section XVII: W e prove the theorems and propositions of Stage I. • Section XVIII: W e prove the propositions in the oligopoly model. X I . A P P E N D I X : M I X E D P R I C E E Q U I L I B R I U M O F S 1 S 1 S U B G A M E As sho wn in Proposition 2, when the demand satisfies min i y ∗ i ( ¯ p, ϕ i ) < D < P i y ∗ i ( ¯ p, ϕ i ) , there is no pure price equilibrium. W e can characterize a close-form equilibrium revenue for each supplier at the mixed price equilibrium in Proposition 17, which has been proved in [17]. Proposition 17 (S 1 S 1 : mixed-equilibrium re venue) . In the S 1 S 1 case (i.e., P i =1 ϕ i = 2 ), if min i y ∗ i < D < P i y ∗ i , ther e exists no pur e price equilibrium but exists the mixed price equilib- rium, with the equilibrium r evenue as follows. π RE i ( ϕ ) =        ¯ p ( D − y ∗ − i ) , if y ∗ i > y ∗ − i , ¯ p ( D − y ∗ i ) y ∗ i min( y ∗ − i , D ) , other w ise, wher e y ∗ i = E [ X i ] and y ∗ − i = E [ X − i ] as char acterized in Theorem 1. 46 According to Proposition 17, one supplier’ s equilibrium re venue is related to the other supplier’ s bidding quantity (i.e., mean value of generations). Specifically , one supplier’ s equilibrium rev enue decreases if the other supplier’ s bidding quantity increases. Furthermore, under the mixed price equilibrium, both suppliers get strictly positiv e rev enues while they may get zero rev enues when the demand is below the threshold min i y ∗ i as shown in Proposition 2 under the pure price equilibrium. X I I . A P P E N D I X : M I X E D P R I C E E Q U I L I B R I U M O F S 0 S 0 S U B G A M E In the S 0 S 0 case, both suppliers do not in vest in storage and face the general penalty cost. When 0 < D < P i y ∗ i ( ¯ p, ϕ i ) , the mixed price equilibrium has a continuous CDF ov er [ l , ¯ p ) sho wn in Lemma 1, but we cannot deriv e it in close form. T o have a better understanding of the CDF , we discretize the price to approximate the original continuous price set, and compute the mixed equilibrium for the discrete price set. Specifically , we discretize the price between (0 , ¯ p ] into { ∆ p, 2∆ p, 3∆ p, ..., ¯ p − ∆ p, ¯ p } with a small ∆ p > 0 . W e search for the lower support in the range gi ven in (14) in the following way . Gi ven a lower support l 0 , the mix ed strate gy of each supplier has the support { l 0 , l 0 + ∆ p, l 0 + ∆ p, ..., ¯ p } that approximates the original continuous support [ l , ¯ p ] . For each supplier , each of price strategies in the support yields the same expected rev enue, which can be used to construct a set of linear equations and calculate the mixed equilibrium. If the probability of each price for each supplier is between (0 , 1) , then the lower support l 0 is feasible; otherwise, there exists the price that should be excluded from the support { l 0 , l 0 + ∆ p, l 0 + ∆ p, ..., ¯ p } and the lower support l 0 is not feasible. W e calculate the equilibrium rev enue according to Lemma 1 (ii). X I I I . A P P E N D I X : C H A R A C T E R I Z A T I O N O F S T O R A G E C A P AC I T Y W e will first prov e Proposition 10 and show some properties of the upper bound P r l,m ( S l i ) and P r u,m ( S u i ) . Then, we discuss the imperfect storage model and sho w ho w it af fects the storage cost. 47 A. Pr oof of Pr oposition 10 Proof : Belo w , we illustrate the upper bound P r l,m ( S l i ) . The upper bound P r u,m ( S u i ) can be deri ved analogously . Gi ven t 0 ∈ T , we hav e Pr ( t 0 X t =1 − C D m,t i > S l i ) = Pr  e s P t 0 t =1 − C D m,t i ≥ e sS l i  ≤ e − sS l i · E h e s P t 0 t =1 − C D m,t i i , B l ( s ) , (21) for any s > 0 . The inequality in (21) is due to the Markov inequality . 18 Gi ven S l i > 0 , we can find a tight upper bound for the probability Pr ( P t 0 t =1 − C D m,t i > S l i ) by minimizing the RHS in (21) ov er s . Therefore, P r l,m ( S l i ) = max t 0 min s> 0 B l ( s ) . B. Pr operties of some pr operties of the upper bound P r l,m ( S l i ) and P r u,m ( S u i ) . W e have properties for P r l,m ( S l i ) and P r u,m ( S u i ) as follo ws. Proposition 18 (properties of the upper bounds) . Given S l i > 0 and S u > 0 , the Markov- inequality-based upper bounds have pr operties as follows. 1) P r l,m ( S l i ) ≤ 1 and P r u,m ( S u i ) ≤ 1 . 2) P r l,m ( S l i ) and P r u,m ( S u i ) ar e decr easing in S l i and S u i , r espectively . 3) P r l,m ( S l i ) → 0 as S l i → + ∞ , and P r u,m ( S u i ) → 0 as S u i → + ∞ . Proof : The first property is because min s> 0 B l ( s ) ≤ B l (0 − ) = 1 and min s> 0 B u ( s ) ≤ B u (0 − ) = 1 . The second property is straightforward from the function B l ( s ) and B u ( s ) . The third property is because C D m,t i is bounded. Thus, B l ( s ) → 0 as S l i → + ∞ , and B u ( s ) → 0 as S u i → + ∞ . Proposition 18 shows that a larger capacity decreases the charge/discharge exceeding probabil- ity . Also, for any positiv e probability threshold α , we can always find a suf ficiently large capacity to let the exceeding probability below α . This lays the foundation for Algorithm 1. C. Generalization of imperfect storag e model W e consider the imperfect energy storage in two aspects: (i) less-than-100% charge and discharge ef ficiency and (ii) the degradation cost incurred by the charge and discharge. Next, we 18 This inequality is also known as Chernof f bound, which can achieve a tight probability bound [36]. 48 will explain how the storage charge and discharge are determined in our work, and then further discuss ho w the imperfect storage impacts the total storage cost and in vestment equilibrium. T o begin with, we explain the model of the storage charge and discharge as well as the energy le vel of the storage in our work. Specifically , the with-storage supplier charges and dischar ges the energy storage to stabilize his renewable output at the mean value. Thus, the charge and discharge po wer is only dependent on the random variable of rene wable generations. At hour t of renew able-generation-type (month) m , we denote the charge amount as C D m,t + i ≥ 0 and the discharge amount as C D m,t − i ≥ 0 . These v alues are characterized based on the random generation X m,t i as follo ws: C D m,t + i = ( X m,t i − E [ X m,t i ]) + , (22) C D m,t − i = ( X m,t i − E [ X m,t i ]) − , (23) where g + , max( g , 0) and g − , max( − g , 0) . Furthermore, we denote the char ge ef ficiency as η c i and the discharge efficienc y as η d i . The energy lev el in the storage can be calculated by adding the charge and discharge over time at month m as follows. e m,t i = e m,t − 1 i + η c i C D m,t + i − C D m,t − i /η d i . (24) Next, we discuss ho w the de gradation cost and the less-than-100% charge and discharge ef ficiency impact the total storage cost. 1) De gradation cost: W e sho w that the de gradation cost will increase the total cost of deploying the storage for the with-storage supplier . The degradation cost is caused by the charge and discharge of the storage. In the ideal case, we do not include the degradation cost as part of the storage cost. W ith the degradation, the total cost of deploying the storage will be higher . One widely used model in the literature for the degradation cost is a linear model [37] [38]. W e denote the unit cost of char ge and dischar ge as c o i . Thus, the expected degradation cost C o i (in each hour) is C o i = E m,t [ c o i C D m,t + i + c o i C D m,t − i ] , (25) which can be calculated based on the historical data of X m,t i . Therefore, W e can simply add (25) to the original storage cost. W e calculate the total storage cost as C 0 i = C i + C o i , which includes both in vestment cost and the degradation cost. 49 2) Char ge and dischar ge efficiency: The lo wer charge and discharge efficienc y will increase the storage capacity and thus increase the total storage cost. Our goal is to characterize a minimum storage capacity such that the energy lev el e m,t i will exceed the storage capacity with a probability no greater than α . As shown in (24), the charge and discharge efficienc y ( η c i , η d i ) will affect the energy lev el e m,t i . Compared with the perfect storage model with η c i = η d i = 100% , the difference in the imperfect storage model is that η c i < 100% and η d i < 100% . W ith the charge and discharge ef ficiency , we modify (16) and (17) in Section XIII.A into the follo wing. E m  max t 0 ∈T Pr ( t 0 X t =1 η c i C D m,t + i − C D m,t − i /η d i + S l i < 0)  ≤ α, (26) E m  max t 0 ∈T Pr ( t 0 X t =1 η c i C D m,t + i − C D m,t − i /η d i + S l i > S i )  ≤ α. (27) Similarly , we can follow Algorithm 1 in Section XIII.A to compute S i gi ven the probability threshold α . According to Algorithm 1 that computes the storage capacity , we sho w how charge/dischar ge ef ficiency impacts the storage capacity in Figure 7. The blue curve shows the case where the probability that the energy le vel exceeds the capacity is smaller than 5% and the red curve sho ws the case where the probability that the energy le vel e xceeds the capacity is smaller than 10%. W e see that as the efficienc y decreases, the required storage capacity increases (which further increases the storage in vestment cost). efficiency (  c =  d ) 0.6 0.7 0.8 0.9 1 Storage capacity (MW) 30 35 40 45 50  =5%  =10% Fig. 7: Storage capacity with charge/discharge ef ficiency . 50 In summary , compared with the case of perfect storage, a lower charge/discharge ef ficiency with the degradation cost will increase the total storage cost of a supplier . In Section VI, we present some analytical results of the storage cost’ s impact on the storage-in vestment equilibrium. In Section VIII, we also show the simulation results of the impact of the storage cost on the suppliers’ profits. These discussions can capture the impact of the imperfect storage. X I V . A P P E N D I X : S I M U L A T I O N S W e will first sho w the details of how we approximate the continuous CDF for the renew able- generation distribution using historical data. Then, we show a simulation result for two hetero- geneous suppliers. A. Empirical distribution of r enewable gener ations W e use the historical data of solar ener gy in Hong K ong from the year 1993 to year 2012 [29] to approximate the continuous CDF of suppliers’ renew able generations. Specifically , we cluster the rene wable generations at hour t of all days into M = 12 types (months) considering the seasonal ef fect. W e use daily data (from the year 1993 to year 2012) of renew able energy in month m at hour t to approximate the distribution of rene wable generation at hour t of month m . Based on the discrete data, we first use an empirical cumulative distribution function (ECDF) to model the rene wable power distribution. 19 Note that our model is b uilt on the continuous CDF of suppliers’ rene wable generations. Thus, we further use linear interpolation to set up the continuous ECDF from the ECDF [40]. W e illustrate the ECDF and linearly-interpolated ECDF in Figure 8(a), where the stepwise blue solid curve represents the ECDF and the red dotted curve represents the linearly-interpolated ECDF . For the illustration of rene wable generation distribution, we show the ECDF and linearly-interpolated ECDF of hour t = 9 of month m = 5 (May) in Figure 8(b). Through the linearly-interpolated ECDF F i , we can also compute the value F − 1 i ( · ) ef ficiently . 19 Giv en a sample of real-world data X 1 , X 2 , . . . , X m , the standard ECDF b F ( x ) : R → [0 , 1] is defined as b F ( x ) = 1 m P m i =1 I ( X i ≤ x ) , where I ( · ) is the indicator function [39]. 51 X 0 0.2 0.4 0.6 F(x) 0 0.2 0.4 0.6 0.8 1 1.2 ecdf interploted ecdf (a) X 0 2 4 6 F(x) 0 0.2 0.4 0.6 0.8 1 1.2 ecdf interploted ecdf (b) Fig. 8: (a) Illustration of ECDF and linearly-interpolated ECDF; (b) ECDF and linearly- interpolated ECDF at hour 9 of May . B. Simulations of two heter ogeneous suppliers W e simulate an example with two heterogeneous suppliers. Note that we can prov e that a pure Nash equilibrium of storage in vestment will always exist in the homogeneous case (with the same storage cost, the same renewable energy capacity and the same renew able energy distribution). Ho we ver , for the general heterogeneous case, we cannot theoretically pro ve that the pure Nash equilibrium always exists. In our follo wing example of heterogeneous suppliers, the pure Nash equilibrium of storage in vestment still exists. Specifically , we consider that supplier 2’ s rene wable generation capacity is twice as much as the capacity of supplier 1, where both suppliers have the same distribution of renew able energy . For comparison, we consider the homogeneous case as in the simulation of the main text where each supplier’ s renewable generation capacity is equal to supplier 1’ s capacity of the heterogeneous case. In the follo wing, we first assume that the storage in vestment cost is the same across the two suppliers, and study the storage-in vestment equilibrium with respect to the storage cost and demand in the homogeneous (capacity) case and heterogeneous (capacity) case, respecti vely . Then, we allo w the storage in vestment cost to also dif fer across the two suppliers in the heterogeneous case, and study the storage-in vestment equilibrium with respect to the two suppliers’ dif ferent storage costs. W e first consider the case that two suppliers’ bear the same in vestment cost of storage, so as 52         0 0.2 0.4 0.6 0.8 1     0 5 10 15                  (a)          00 . 511 . 5    0 5 10 15 (b) Fig. 9: (a) Equilibrium split in the homogeneous case; (b) Equilibrium split in the heterogeneous case. to focus on sho wing the impact of different capacities of renew ables. 20 Figure 9(a) shows the equilibrium split in terms of demand and storage cost under the homogeneous case. Note that this figure has been sho wn as Figure 3 of the main text. Figure 9(b) shows the equilibrium split in terms of demand and storage cost under the heterogeneous case. • In Figure 9(a), in Region I, both-in vesting-storage is one equilibrium; in Region III, neither- in vesting-storage is one equilibrium; in Region II, one in vesting in storage and one not in vesting in storage will be one equilibrium. • In Figure 9(b), in the solid-grid region, both-in vesting-storage is one equilibrium; in the dash-grid region, neither-in vesting-storage is one equilibrium; in the region bounded by the red curve, supplier 1 does not in vest in storage while supplier 2 should in vest in storage; and in the region bounded by the blue curve, supplier 1 in vests in storage while supplier 2 does not in vest in storage. Generally , in the heterogeneous case, we see that the region where supplier 2 should in vest in storage is larger than the region of supplier 1. The intuition is that supplier 2 has a larger capacity of renew ables, which gi ves her advantage in the competition. When both suppliers face 20 Note that the two suppliers have different storage capacities due to the different capacities of renew ables. W e choose dif ferent unit costs of storage capacity and let two suppliers have the same storage in vestment cost. 53           0 0.2 0.4 0.6 0.8 1           0 0.2 0.4 0.6 0.8 1 A B C D E Fig. 10: Equilibrium split with storage cost. the same high storage cost greater than 1000 HKD as in Figure 9(b), supplier 1 will not in vest in storage at the equilibrium for any demand D but supplier 2 may still in vest in storage when the demand is high. Also, the region of only supplier 1 in vesting in storage and only supplier 2 in vesting in storage can overlap under the heterogeneous case, which means that only supplier 1 in vesting in storage and only supplier 2 in vesting in storage are both equilibria. Next we consider the case that two heterogeneous suppliers bear different storage in vestment costs. W e choose a certain demand ( D = 4 MW) and show the equilibrium split with respect to the storage cost of the two suppliers in Figure 10. In Figure 10, if the storage costs of supplier 1 and supplier 2 lie in Region A, neither supplier will in vest in storage. In Region B, both suppliers will in vest in storage. In Region D, only supplier 2 in vests in storage and supplier 1 will not in vest in storage. In Region C, only supplier 1 in vests in storage and supplier 2 will not in vest in storage. Howe ver , in Region E, only supplier 1 in vesting in storage and only supplier 2 in vesting in storage, are both equilibria. X V . A P P E N D I X : P R O O F S O F S T AG E I I I T o prove Proposition 1, we will discuss the following two cases and analyze the objectiv e function of Problem (4) based on linear functions. For notation simplicity , we omit the superscript m, t in the corresponding variables and parameters. 54 • If p 1 = p 2 = p , we rewrite the objectiv e function (4a) as ( P g − p )( x 1 + x 2 ) . (28) Since P g − p > 0 , the optimal v alue is achiev ed at the maximum value of x 1 + x 2 , i.e., min( D , y 1 + y 2 ) according to the constraints (4b) and (4c). • If p 1 6 = p 2 , we assume p 1 > p 2 without loss of generality . W e rewrite the objectiv e function (4a) as ( P g − p 2 )( x 1 + x 2 ) + ( p 1 − p 2 ) x 1 . (29) Since P g − p 2 > 0 and p 1 − p 2 > 0 , the optimal v alue is achiev ed at the maximum value of x 1 + x 2 and the maximum v alue of x 1 as follo ws: x ∗ 1 + x ∗ 2 = min( D , y 1 + y 2 ) , (30) x ∗ 1 = min( y 1 , D ) . (31) Then, we obtain the optimal solution x ∗ 2 = min( D , y 1 + y 2 ) − min( y 1 , D ) , which is equiv alent to x ∗ 2 = min( D − min( y 1 , D ) , y 2 ) . Combining the abov e two cases, we hav e Proposition 1 prov ed. Remark 1 : Proposition 1 can be easily e xtended to the oligopoly case with more than 2 suppliers. Remark 2 : Gi ven the other supplier − i ’ s bidding price p − i and bidding quantity y − i , the supplier i ’ s payof f function generally is not continuous in price p i at p i = p − i due to the discontinuous change of the optimal capacity x ∗ i . This shows that giv en the other supplier − i ’ s decisions, supplier i ’ s payoff function generally is discontinuous. X V I . A P P E N D I X : P R O O F S O F S TAG E I I A. Pr oof of Theor em 1 T o pro ve Theorem 1, the k ey step is to sho w that gi ven price p i , the re venue function π R i ( p i , x ∗ i ( p , y ) , ϕ ) of supplier i with respect to x ∗ i ( p , y ) is increasing on the interval (0 , y ∗ i ) and decreasing on the interv al ( y ∗ i , + ∞ ) . Then, combined with Proposition 1, we can prove that y ∗ i will be the weakly dominant strategy for the bidding quantity . W e discuss the weakly dominant strategy for supplier i with ϕ i = 1 and ϕ i = 0 , respectiv ely . 55 1) Case of ϕ i = 1 : W e will prove that the weakly dominant strategy of bidding quantity for the with-storage supplier i (i.e., ϕ i = 1 ) is y ∗ i ( p i , ϕ i ) = E [ X i ] . Giv en any price p i ≤ ¯ p < λ , the function π R i ( p i , x ∗ i ( p , y ) , ϕ ) with respect to x ∗ i ( p , y ) is linearly increasing on the interval (0 , E [ X i ]) and linearly decreasing on the interv al ( E [ X i ] , + ∞ ) . Thus, giv en any price p i , we always hav e π R i ( p i , x ∗ i ( p , y ) , ϕ ) ≤ π R i ( p i , E [ X i ] , ϕ ) (32) Then, we discuss a total of three cases to sho w that with-storage supplier’ s re venue cannot be better off if he chooses strategy y i other than y ∗ i ( p i , ϕ i ) = E [ X i ] . For notation simplicity , we use y ∗ i to represent y ∗ i ( p i , ϕ i ) in the later discussion. (a) If y i < y ∗ i = E [ X i ] , according to Proposition 1, we hav e x ∗ i ( p , ( y i , y − i )) ≤ x ∗ i ( p , ( y ∗ i , y − i )) ≤ E [ X i ] , for any y − i , (33) which (according to (32)) implies π R i ( p i , x i ( p , ( y i , y − i )) , ϕ ) ≤ π i ( p i , x i ( p , ( y ∗ i , y − i )) , ϕ ) . (34) (b) If y i > y ∗ i = E [ X i ] and x ∗ i ( p , ( y i , y − i )) > E [ X i ] , according to Proposition 1, we hav e x ∗ i ( p , ( y ∗ i , y − i )) = E [ X i ] , which (according to (32)) implies π R i ( p i , x ∗ i ( p , ( y i , y − i )) , ϕ ) ≤ π R i ( p i , E [ X i ] , ϕ ) = π R i ( p i , x ∗ i ( p , ( y ∗ i , y − i )) , ϕ ) . (c) If y i > y ∗ i = E [ X i ] and x ∗ i ( p , ( y i , y − i )) ≤ E [ X i ] , according to Proposition 1, we hav e x ∗ i ( p , ( y i , y − i )) = x ∗ i ( p , ( y ∗ i , y − i )) , which implies π R i ( p i , x ∗ i ( p , ( y i , y − i )) , ϕ ) = π R i ( p i , x ∗ i ( p , ( y ∗ i , y − i )) , ϕ ) . Combining the above three conditions (a)-(c), we complete the proof that y ∗ i = E [ X i ] if ϕ i = 1 . 56 2) Case of ϕ i = 0 : W e prove the weakly dominant strategy of bidding quantity for the without- storage supplier i (i.e., ϕ i = 0 ) is y ∗ i = F − 1 i ( p i λ ) . W e take the deri v ati ve of π R i ( p i , x ∗ i ( p , y ) , ϕ ) with respect to x ∗ i ( p , y ) and gi ve any p i > 0 , it is easy to show that the function π R i ( p i , x ∗ i ( p , y ) , ϕ ) is increasing on the interval interval (0 , F − 1 i ( p i λ )) and decreasing on the interval ( F − 1 i ( p i λ ) , + ∞ ) . Thus, gi ven any price p i , we always hav e π R i ( p i , x ∗ i ( p , y ) , ϕ ) ≤ π R i  p i , F − 1 i ( p i λ ) , ϕ  . (35) Then, we can follow the proof step for y ∗ i in the case of ϕ i = 1 and prov e that y ∗ i = F − 1 i ( p i λ ) for supplier i with ϕ i = 0 . B. Pr oof of Pr oposition 2 W e verify the pure price equilibrium according to Definition 2 that the supplier cannot be better of f if he deviates unilaterally . T o wards this end, note that for supplier i with or without storage, the revenue function π R i ( p i , x ∗ i ( p , y ) , ϕ ) is strictly increasing with respect to both the price p i and the selling quantity x ∗ i ( p , y ) that is in the range [0 , y ∗ i ( p i , ϕ i )] (without considering the other supplier’ s coupled decisions). W e will discuss the three types of subgames respectiv ely . 1) The type S 0 S 0 (i.e., P i ϕ i = 0 ): W e first prove that when D ≥ P i y ∗ i ( ¯ p, ϕ i ) , p 1 = p 2 = ¯ p is a pure price equilibrium and sho w that this pure price equilibrium is unique. Then, we show that when D < P i y ∗ i ( ¯ p, ϕ i ) , there exists no pure price equilibrium. (a) The case of D ≥ P i y ∗ i ( ¯ p, ϕ i ) . W e first prove that when D ≥ P i y ∗ i ( ¯ p, ϕ i ) , p 1 = p 2 = ¯ p is a pure price equilibrium. When p 1 = p 2 = ¯ p , according to Proposition 1, the total selling energy quantities of supplier 1 and supplier 2 satisfy X i x ∗ i (( ¯ p, ¯ p ) , ( y ∗ 1 ( ¯ p, ϕ 1 ) , y ∗ 2 ( ¯ p, ϕ 2 ))) = min( D , y ∗ 1 ( ¯ p, ϕ 1 ) + y ∗ 2 ( ¯ p, ϕ 2 )) (36) = y ∗ 1 ( ¯ p, ϕ 1 ) + y ∗ 2 ( ¯ p, ϕ 2 ) . (37) Since x ∗ i ( p , y ) ≤ y ∗ i ( p i , ϕ i ) always holds for any i = 1 , 2 , based on (37), we hav e x i , x ∗ i (( ¯ p, ¯ p ) , ( y ∗ 1 ( ¯ p, ϕ 1 ) , y ∗ 2 ( ¯ p, ϕ 2 ))) = y ∗ i ( ¯ p, ϕ i ) . (38) 57 W e will show that both suppliers cannot be better off if they deviate from such a bidding strategy . W ithout loss of generality , if supplier 1 bids a price p 0 1 < ¯ p unilaterally , according to Proposition 1, we hav e x 0 1 , x ∗ 1 (( p 0 1 , ¯ p ) , ( y ∗ 1 ( p 0 1 , ϕ 1 ) , y ∗ 2 ( ¯ p, ϕ 2 )) = min { D, y ∗ 1 ( p 0 1 , ϕ 1 ) } (39) = y ∗ 1 ( p 0 1 , ϕ 1 ) (40) < x 1 . (41) Since the rev enue function π R i ( p i , x ∗ i ( p , y ) , ϕ ) is strictly increasing with respect to the price p i and the selling quantity x ∗ i ( p , y ) ≤ y ∗ , we hav e π R 1 ( p 0 1 , x 0 1 , ϕ ) < π R 1 ( ¯ p, x 1 , ϕ ) , (42) which shows that supplier 1’ s re venue decreases if he deviates from the price ¯ p . This proves that p 1 = p 2 = ¯ p is a pure price equilibrium. Next, we sho w that this equilibrium is unique. Without loss of generality , suppose that supplier 1 bids a price p 0 1 < ¯ p while the other supplier bids a price p 0 2 ≤ ¯ p . Since D ≥ P i y ∗ i ( ¯ p, ϕ i ) , according to Proposition 1, each supplier’ s maximum bidding quantity will be sold out and we hav e x ∗ 1 ( p 0 , y ∗ ( p 0 , ϕ )) = y ∗ 1 ( p 0 1 , ϕ 1 ) ≤ y ∗ 1 ( ¯ p, ϕ 1 ) . (43) Therefore, supplier 1 can always increase his price p 0 i to ¯ p , which will increase his rev enue due to the increased price and non-decreasing selling quantity . Thus, any price pair ( p 1 , p 2 ) 6 = ( ¯ p, ¯ p ) can’t be an equilibrium. (b) Case of 0 < D < P i y ∗ i ( ¯ p, ϕ i ) . W e will assume that both suppliers bid the pure prices and will discuss a total of three cases in the follo wing to sho w that no pure price strategy can be an equilibrium. First, suppliers’ bidding prices are not equal, and we assume p i < p − i without loss of generality . the lo wer-price supplier can al ways increase the price by a small ε > 0 such that p 0 i = p i + ε < p − i . Then, the bidding price p 0 i > p i and the selling quantity at p 0 i denoted as x 0 i satisfies x 0 i = min { D , y ∗ i ( p i + ε, ϕ i ) } ≥ x i = min ( D , y ∗ i ( p i , ϕ i )) . In this case, we denote the rev enue at the original price p i as π i , and the rev enue at the price p 0 i as π 0 i . W e have π 0 i > π i since p 0 i > p i and x 0 i ≥ x i . Thus, the unequal bidding price cannot be an equilibrium. 58 Second, two suppliers bid the same positiv e price, i.e., p 1 = p 1 = p > 0 . Based on Proposition 1, the selling quantities of two suppliers satisfy the following condition X i x ∗ i ( p , y ∗ ( p , ϕ )) = min( D , y ∗ 1 ( p, ϕ 1 ) + y ∗ 2 ( p, ϕ 2 )) . (44) For simplicity , we denote the original selling quantity of supplier 1 and supplier 2 as x 1 and x 2 , respecti vely when p 1 = p 1 = p > 0 . Then we discuss two cases (i) and (ii). • (i) When D < y ∗ 1 ( p, ϕ 1 ) + y ∗ 2 ( p, ϕ 2 ) , we hav e x 1 + x 2 = D . (45) In this case, if supplier 1 reduces the price by a small ε 1 > 0 to a price p 0 1 = p − ε 1 unilaterally , we hav e x 0 1 , x ∗ 1 (( p − ε 1 , p ) , ( y ∗ 1 ( p − ε 1 , ϕ 1 ) , y ∗ 2 ( p, ϕ 2 )) = min { D, y ∗ 1 ( p − ε 1 , ϕ 1 ) } . (46) If supplier 2 reduces the price by a small ε 2 > 0 to a price p 0 2 = p − ε 2 unilaterally , we hav e x 0 2 , x ∗ 2 (( p, p − ε 2 ) , ( y ∗ 1 ( p, ϕ 1 ) , y ∗ 2 ( p − ε 2 , ϕ 2 )) = min { D, y ∗ 2 ( p − ε 2 , ϕ 2 ) } . (47) W e choose small ε 1 and ε 2 such that D < y ∗ 1 ( p − ε 1 , ϕ 1 ) + y ∗ 2 ( p − ε 2 , ϕ 2 ) holds. Then, we hav e x 0 1 + x 0 2 = min { D , y ∗ 1 ( p − ε 1 , ϕ 1 ) } + min { D , y ∗ 2 ( p − ε 2 , ϕ 2 ) } . (48) Combining (45) and (48), we see that at least one supplier i can always reduce the price by a small ε i > 0 unilaterally such that the selling quantity increases by x 0 i − x i > 1 2 min( D , y ∗ 1 ( p − ε 1 , ϕ 1 ) , y ∗ 2 ( p − ε 2 , ϕ 2 )) . Since we can choose a suf ficiently small ε i , ∀ i = 1 , 2 , the rev enue π i will increase due to the increased selling quantity x 0 i − x i (with an upward jumping). • (ii) When y ∗ 1 ( p, ϕ 1 ) + y ∗ 2 ( p, ϕ 2 ) ≤ D < y ∗ 1 ( ¯ p, ϕ 1 ) + y ∗ 2 ( ¯ p, ϕ 2 ) , we hav e x 1 + x 2 = y ∗ 1 ( p, ϕ 1 ) + y ∗ 2 ( p, ϕ 2 ) . (49) Both suppliers can sell out the bidding quantities completely as follows. x 1 = y ∗ 1 ( p, ϕ 1 ) , x 2 = y ∗ 2 ( p, ϕ 2 ) . (50) 59 Note that D − y ∗ 2 ( p, ϕ 2 ) ≥ y ∗ 1 ( p, ϕ 1 ) = x 1 . Supplier 1 can always increase his price p to p 0 = ¯ p > p unilaterally , and x 0 1 = min( y ∗ 1 ( p 0 , ϕ 1 ) , D − y ∗ 2 ( p, ϕ 2 )) . Since we also hav e y ∗ 1 ( p 0 , ϕ 1 ) ≥ y ∗ 1 ( p, ϕ 1 ) = x 1 , supplier 1 ’ s obtained demand x 0 1 at p 0 will not decrease, i.e., x 0 i ≥ x i . Thus, the re venue of supplier i after increasing the price will also increase. In summary , when two suppliers bid the same positiv e price, one supplier can always deviate so as to obtain a higher re venue, which sho ws that the equal positiv e bidding prices cannot be pure price equilibrium. Third, both suppliers bid the price at zero: p 1 = p 2 = 0 . In this case, both suppliers hav e zero re venues: π R 1 = π R 2 = 0 . Note that both without-storage suppliers will also bid the zero quantity y ∗ i ( p i , ϕ i ) = 0 as shown in Theorem 1. Thus, any supplier i can always set a positi ve price p 0 i > 0 to obtain the positi ve demand since the other supplier bid zero quantity . This makes his rev enue π R 0 1 > 0 after increasing the price. There, the pure price strategy p 1 = p 2 = 0 cannot be the equilibrium So far , for the case of 0 < D < P i y ∗ i ( ¯ p, ϕ i ) , we hav e discussed all the three cases of the pure price strategies but none of them is an equilibrium. Thus, there exists no pure price equilibrium when 0 < D < P i y ∗ i ( ¯ p, ϕ i ) . 2) The type S 1 S 0 (i.e., P i ϕ i = 1 ): Follo wing the same ar guments as in the type S 0 S 0 , we can first prove that when D ≥ P i y ∗ i ( ¯ p, ϕ i ) , p 1 = p 2 = ¯ p is a pure price equilibrium and sho w that this pure price equilibrium is unique. Furthermore, we can sho w that when D < P i y ∗ i ( ¯ p, ϕ i ) , there exists no pure price equilibrium. 3) The type S 1 S 1 (i.e., P i ϕ i = 2 ): The results hav e been proved in the paper [17]. In conclusion, we hav e Proposition 2 proved. C. Pr oof of Theor em 2 W e prove Theorem 2 based on Lemma 1 that has been shown in [17]. Howe ver , based on Lemma 1, deriving the mixed price equilibrium in our model is still not straightforward compared with [17]. That is because, in [17], supplier’ s bidding quantity is upper-bounded by his deterministic production quantity , while in our model, without-storage supplier’ s bidding quantity is upper-bounded by a function of price. The difference significantly increases the complexity of the analysis in our work. 60 T o prove Theorem 2, we will utilize a basic property of mix strategy equilibrium as sho wn in Lemma 2 [41]. In Lemma 2, we use π RM i ( µ i , µ − i , ϕ ) to denote the expected revenue of supplier i at any arbitrary mixed price strategy ( µ 1 , µ 2 ) , which is defined as follows. π RM i ( µ i , µ − i , ϕ ) = Z [0 , ¯ p ] 2 π R i ( p i , x ∗ i (( p i , p − i ) , y ∗ ( p i , p − i )) , ϕ ) d ( µ i ( p i ) × µ − i ( p − i )) Lemma 2. π RM i ( p i , µ ∗ − i , ϕ ) = π RE i ( ϕ ) , for all p i ∈ [ l , ¯ p ] , wher e π RE i is the equilibrium re venue [41]. Lemma 2 shows that the equilibrium re venue π RE i of supplier i is equal to the expected rev enue when he plays any pure strategy p i in the support, i.e., p i ∈ [ l, ¯ p ] , against the mixed strategy µ ∗ − i of the other supplier at the equilibrium. Based on Lemma 1 and Lemma 2, we will characterize the equilibrium re venue π RE i as well as the CDF of the mixed price equilibrium F e i ( p ) using the lower support l over p ∈ [ l , ¯ p ) . 21 W e make the analysis of the with-storage supplier (i.e., ϕ i = 1 ) and without-storage supplier (i.e., ϕ i = 0 ) as follows. 1) W ith-storage supplier i (i.e., ϕ i = 1 ): F or supplier i , based on Lemma 2, the equilibrium re venue π RE i can be characterized by the expected rev enue when he plays any pure strategy p i ∈ [ l , ¯ p ) against the mixed strategy of supplier − i (with CDF F e − i and PDF f e − i ) at the equilibrium as follo ws π RE i ( ϕ ) = π RM i ( p i , µ ∗ − i , ϕ ) = p i min( D , E [ X i ]) · (1 − F e − i ( p i )) | {z } p i ≤ p − i + p i Z p i l min  D − min( y ∗ − i ( p − i , ϕ − i ) , D ) , E [ X i ]  · f e − i ( p − i ) dp − i | {z } p i >p − i . (51) Note that in (51), D − min( y ∗ − i ( p − i , ϕ − i ) , D ) ≤ E [ X i ] will always hold for any p − i ∈ [ l , ¯ p ] , i.e., D − min( y ∗ − i ( l , ϕ − i ) , D ) ≤ E [ X i ] , or D ≤ E [ X i ] + y ∗ − i ( l , ϕ − i ) . (52) 21 Note that F e 2 ( p ) may not be continuous at p = ¯ p as indicated in Lemma 1. 61 This helps us simplify the second part “ p i > p − i " in (51). W e can prove this by contradiction as follo ws. If D − min( y ∗ − i ( l , ϕ − i ) , D ) > E [ X i ] , there exists a small ε > 0 such that D − min( y ∗ − i ( l + ε, ϕ − i ) , D ) > E [ X i ] still holds. Based on (51), we hav e π RE i = π RM i ( l , µ ∗ − i , ϕ ) = l · min( D , E [ X i ]) , (53) and for any ε > 0 , π RE i ( ϕ ) = π RM i ( l + ε, µ ∗ − i , ϕ ) = ( l + ε ) · min( D, E [ X i ])(1 − F e − i ( l + ε )) + ( l + ε ) Z l + ε l E [ X i ] · f e − i ( p − i ) dp − i = ( l + ε ) · min( D, E [ X i ])(1 − F e − i ( l + ε )) + ( l + ε ) · E [ X i ] · F e − i ( l + ε ) ≥ ( l + ε ) · min( D, E [ X i ]) . (54) Then, we can see that (53) and (54) contradict with each other , and thus D − min( y ∗ − i ( p − i , ϕ − i ) , D ) ≤ E [ X i ] will always hold for p 2 ∈ [ l, ¯ p ] , which enables us to simplify (51). Since π RE i ( ϕ ) = π RM i ( p i , µ ∗ − i , ϕ ) is constant ov er p i ∈ [ l, ¯ p ) , the deriv ativ e of π RM i ( p i , µ ∗ − i , ϕ ) with respect to p i is zero ov er p i ∈ [ l, ¯ p ) , i.e., ∂ π RM i ( p i , µ ∗ − i , ϕ ) ∂ p i =min( D , E [ X i ])(1 − F e − i ( p i )) + p i min( D , E [ X i ])( − f e − i ( p i )) + Z p i l  D − min( y ∗ − i ( p − i ) , D )  f e − i ( p − i ) dp − i + p i  D − min( y ∗ − i ( p i , ϕ − i ) , D )  f e − i ( p i ) =0 . (55) Combining (55) with (51), we ha ve the PDF of mixed price strategy at the equilibrium for without-storage supplier − i ’ s as follows. f e − i ( p ) = π RE i ( ϕ ) p 2 · min { y ∗ − i ( p, ϕ − i ) , D } − p 2 · [ D − E [ X i ]] + , (56) which is characterized by the equilibrium rev enue π RE i of supplier i . 2) W ithout-storage supplier i (i.e., ϕ i = 0 ): For supplier i without storage, similarly , based on Lemma 2, the equilibrium revenue π RE i ( ϕ ) can be characterized by the expected rev enue when 62 he plays any pure strategy p i ∈ [ l , ¯ p ) against the mixed strategy of supplier − i (with CDF F e − i ) at the equilibrium as follo ws π RE i ( ϕ ) = π RM i ( p i , µ ∗ − i , ϕ ) (57) = π R i ( p i , min ( D , y ∗ i ( p i , ϕ i ) , ϕ )) · (1 − F e − i ( p i )) | {z } p i ≤ p − i + π R i ( p i , min( D − min( E [ X − i ] , D ) , y ∗ i ( p i , ϕ i )) , ϕ ) · F e − i ( p i ) | {z } p i >p − i . (58) Similarly , we have that D − min( E [ X − i ] , D ) ≤ y ∗ i ( p i , ϕ i ) always holds for any p i ∈ [ l, ¯ p ] . Then, according to (58), we hav e the PDF of the mixed price strategy at the equilibrium for the with- storage supplier − i as follo ws. F e − i ( p ) = π R i ( p, min { y ∗ i ( p, ϕ i ) , D } , ϕ ) − π RE i ( ϕ ) π R i ( p, min { y ∗ i ( p, ϕ i ) , D } , ϕ ) − π R i ( p, [ D − E [ X − i ]] + , ϕ ) , (59) which is characterized by the equilibrium rev enue π RE i of supplier i . In conclusion, if ϕ i = 1 , we hav e F e i ( p ) = π R − i  p, min { y ∗ − i ( p, ϕ − i ) , D } , ϕ  − π RE − i ( ϕ ) π R − i ( p, min { y ∗ − i ( p, ϕ − i ) , D } , ϕ ) − π R − i ( p, ( D − E [ X i ]) + , ϕ ) . (60) If ϕ i = 0 , we hav e F e i ( p ) = Z ¯ p l π RE − i ( ϕ ) p 2 · min { y ∗ i ( p, ϕ i ) , D } − p 2 · ( D − E [ X − i ]) + dp. (61) for any l ≤ p < ¯ p . D. Pr oof of Pr oposition 3 T o prove Proposition 3, we first show that F e i ( ¯ p − | l † i ) is always decreasing in l † i , ∀ i , based on which we can prove Proposition 3 (1) by contradiction. Then, we can have Proposition 3 (2) prov ed directly from Lemma 1 (iii). W e now prove that F e i ( ¯ p − | l † i ) is always decreasing with l † i , for both ϕ i = 1 and ϕ i = 0 . 1) W ith-storage supplier i (i.e, ϕ i = 1 ): For the without-storage supplier i , according to (11), we hav e F e i ( ¯ p − | l † i ) = π R − i  ¯ p, min { y ∗ − i ( ¯ p, ϕ − i ) , D } , ϕ  − π RE − i ( ϕ ) π R − i ( ¯ p, min { y ∗ − i ( p, ϕ − i ) , D } , ϕ ) − π R − i ( ¯ p, ( D − E [ X i ]) + , ϕ ) . (62) Note that the equilibrium rev enue function π RE − i ( ϕ ) (shown in Lemma 1 (iii)) is increasing in the lo wer support l † i , and thus F e i ( ¯ p − | l † 1 ) is decreasing in l † i . 63 2) W ithout-storage supplier i (i.e, ϕ i = 0 ): For the without-storage supplier i , according to (12), we hav e F e i ( ¯ p − | l † i ) = Z ¯ p l † i l † i · min( D , E [ X i ]) p 2 · min { y ∗ i ( p, ϕ i ) , D } − p 2 · [ D − E [ X i ]] + dp. (63) W e take the first-order deriv ativ e of F e i ( ¯ p − | l † i ) with respect to l † i and obtain ∂ F e i ( ¯ p − | l † i ) ∂ l † i = Z ¯ p l † i min( D , E [ X i ]) p 2 · min { y ∗ i ( p, ϕ i ) , D } − p 2 · ( D − E [ X i ]) + dp − min( D , E [ X i ]) l † i · min { y ∗ i ( l † i , ϕ i ) , D } − l † i · ( D − E [ X i ]) + . (64) Further , we take the deri v ati ve of (64) with respect to l † i again and have ∂ 2 F e i ( ¯ p − | l † i ) ∂ l † i 2 = − 1 l † i · ∂ min( D, E [ X i ]) min { y ∗ i ( l † i ,ϕ i ) ,D }− ( D − E [ X i ]) + ∂ l † i . (65) Note that min( D, E [ X i ]) min { y ∗ i ( l † i ,ϕ i ) ,D }− ( D − E [ X i ]) + decreases in l † i because y ∗ i ( l † i , ϕ i ) increases in l † i . Thus, we always hav e ∂ 2 F e i ( ¯ p − | l † i ) ∂ l † i 2 ≥ 0 , (66) which sho ws that ∂ F e i ( ¯ p − | l † i ) ∂ l † i is non-decreasing with l † i . Then, we choose l † i = ¯ p and hav e ∂ F e i ( ¯ p − | l † i ) ∂ l † i = − min( D , E [ X i ]) ¯ p · min { y ∗ i ( ¯ p, ϕ i ) , D } − ¯ p · ( D − E [ X i ]) + < 0 , which holds for all l † i ≤ ¯ p . The reason is that D < E [ X i ] + y ∗ i ( ¯ p, ϕ i ) in the subgame S 1 S 0 without the pure price equilibrium. Therefore, we hav e that F e i ( ¯ p − | l † i ) decreases with l † i . T ill no w , we have shown that F e i ( ¯ p − | l † i ) is always decreasing in l † i for both ϕ i = 1 and ϕ i = 0 . Then, we can prov e Proposition 3 (1) by contradiction. According to Lemma 1 (iii), if F e i ( ¯ p − | l † i ) = 1 has a solution l †∗ i for both suppliers i = 1 , 2 , either l = max( l †∗ 1 , l †∗ 2 ) or l = min( l †∗ 1 , l †∗ 2 ) will hold. If l = min( l †∗ 1 , l †∗ 2 ) , without loss of generality , we assume l †∗ 1 < l †∗ 2 and l = l †∗ 1 . Note that F e 1 ( ¯ p − | l †∗ 1 ) = 1 and hence F e 2 ( ¯ p − | l †∗ 2 ) = 1 . Since F e 2 ( ¯ p − | l † 2 ) is decreasing with l † 2 , then F e 2 ( ¯ p − | l †∗ 1 ) > 1 , which is a contradiction of the CDF . Therefore, we can only choose l = max( l †∗ 1 , l †∗ 2 ) and we hav e Proposition 3 (1) prov ed. Furthermore, according to Lemma 1 (iii), we have that F e i ( ¯ p − ) = 1 is true for at least one of the suppliers. Thus, if we 64 hav e only one solution of l † i among i = 1 and i = 2 , it must be the equilibrium lower support, which has Proposition 3 (2) prov ed. E. Pr oof of Theor em 3 W e first prove that π RE i > π RE − i always holds for a general distribution for the renew able generation X i if ϕ i = 1 , ϕ − i = 0 and E [ X i ] = E [ X − i ] . Then, we consider the case that X − i follo ws a uniform distribution. 1) A general distribution for X i : W e consider the cases of pure price equilibrium and mixed price equilibrium respecti vely , and characterize suppliers’ rev enue as follows. (a) The case with pure price equilibrium: According to Proposition 2 and Lemma 1 (ii), we hav e π RE i ( ϕ ) =      ¯ p min( E [ X i ] , D ) , if ϕ i = 1 , π R i ( ¯ p, min( D , y ∗ i ( ¯ p, ϕ i )) , ϕ ) , if ϕ i = 0 . (67) Note that D ≥ E [ X i ] + y ∗ i ( ¯ p, ϕ i ) when there is the pure price equilibrium. Therefore, if ϕ i = 1 and ϕ − i = 0 , we hav e π RE i ( ϕ ) = ¯ p E [ X i ] . (68) π RE − i ( ϕ ) = π R − i ( ¯ p, y ∗ − i ( ¯ p, ϕ − i ) , ϕ ) (69) = λ Z F − 1 − i ( ¯ p λ ) 0 xf − i ( x ) dx. (70) = λ Z F − 1 − i ( ¯ p λ ) 0 xdF − i ( x ) (71) = ¯ pF − 1 − i ( ¯ p λ ) − λ Z F − 1 − i ( ¯ p λ ) 0 F − i ( x ) dx (72) < ¯ pF − 1 − i ( ¯ p λ ) − ¯ p Z F − 1 − i ( ¯ p λ ) 0 F − i ( x ) dx. (73) Based on (73), we consider the following function h ( x ) for any p > 0 and 0 ≤ x < ¯ X − i . Note that F − 1 − i ( ¯ p λ ) < ¯ X − i since ¯ p < λ . h ( x ) = px − p Z x 0 F − i ( x ) dx. (74) The, we hav e h 0 ( x ) = p − pF − i ( x ) > 0 , (75) 65 which sho ws that h ( x ) increases in x . Since F − 1 − i ( ¯ p λ ) < ¯ X − i , according to (73), we hav e π RE − i ( ϕ ) < ¯ p ¯ X − i − ¯ p Z ¯ X − i 0 F − i ( x ) dx (76) = ¯ p E [ X − i ] ≤ π RE i ( ϕ ) . (77) Based on (68) and (77), if E [ X − i ] ≤ E [ X i ] , then we always hav e π RE − i ( ϕ ) < π RE i ( ϕ ) . (78) (b) The case without pure price equilibrium: The proof procedure is the similar to the case (a) with pure price equilibrium. The difference is to replace ¯ p into the lo wer support l , i.e., π RE i ( ϕ ) =      l · min( E [ X i ] , D ) , if ϕ i = 1 , π R i ( l , min( D , y ∗ i ( l , ϕ i )) , ϕ ) , if ϕ i = 0 . (79) W e will discuss the following two cases. • E [ X i ] ≤ D : If ϕ i = 1 and ϕ − i = 0 , we hav e π RE i ( ϕ ) = l · E [ X i ] . (80) π RE − i ( ϕ ) = π R − i ( l , min( D , y ∗ − i ( l , ϕ − i )) , ϕ ) (81) ≤ π R − i ( l , y ∗ − i ( l , ϕ − i ) , ϕ ) . (82) W e can follo w the same argument as in (a) with the pure price equilibrium to sho w that π RE i > π RE − i if E [ X i ] ≥ E [ X − i ] . The only dif ference is to replace ¯ p by l . • E [ X i ] > D : If ϕ i = 1 and ϕ − i = 0 , we hav e π RE i ( ϕ ) = l · D . (83) π RE − i ( ϕ ) = π R − i ( l , min( D , y ∗ − i ( l , ϕ − i )) , ϕ ) . (84) – If y ∗ − i ( l , ϕ − i ) ≤ D , we hav e π RE − i ( ϕ ) = π R − i ( l , y ∗ − i ( l , ϕ − i ) , ϕ ) (85) ≤ ly ∗ − i ( l , ϕ − i ) − l Z y ∗ − i ( l,ϕ − i ) 0 F − i ( x ) dx ( as in (73) ) (86) < lD (87) = π RE i ( ϕ ) . (88) 66 – If y ∗ − i ( l , ϕ − i ) > D , we hav e π RE − i ( ϕ ) = π R − i ( l , D , ϕ ) (89) = l D − λ Z D 0 ( D − x ) f − i ( x ) dx (90) < lD (91) = π RE i ( ϕ ) . (92) Therefore, for the case (b) without pure price equilibrium, we also hav e that π RE i > π RE − i if E [ X i ] ≥ E [ X − i ] . Combining case (a) with the pure price equilibrium, for a general distribution of X i , we prov e that π RE i > π RE − i if E [ X i ] ≥ E [ X − i ] . 2) Uniform distrib ution of X − i : W e will deriv e the rev enues (at both pure and mixed price equilibrium) of suppliers under the uniform rene wable-generation distrib ution. For the pure price equilibrium, it is straightforward to calculate the equilibrium re venue when there is p 1 = p 2 = ¯ p when D ≥ P i y i ( ¯ p, ϕ i ) . For the case without pure price equilibrium, i.e., D < P i y i ( ¯ p, ϕ i ) , we will characterize the lower support for the mixed price equilibrium and characterize the equilibrium re venue based on Theorem 2 and Proposition 3. W e consider ϕ i = 1 and ϕ − i = 0 . W e have the PDF and CDF of the uniform distribution X − i as follo ws: f − i = 1 ¯ X − i , F − i ( x ) = x ¯ X − i . (93) According to Theorem 1, the weakly dominant bidding quantity strategy is y ∗ i = E [ X i ] , (94) y ∗ − i ( p − i , ϕ − i ) = F − 1 − i  p − i λ  = p − i λ ¯ X − i . (95) Next we discuss the case (a) with pure price equilibrium and the case (b) without pure price equilibrium respecti vely . (a) The case with pure price equilibrium: When D ≥ P i y i ( ¯ p, ϕ i ) , both suppliers’ bid price ¯ p and we hav e π RE i ( ϕ ) = ¯ p E [ X i ] , (96) π RE − i ( ϕ ) = π R − i ( ¯ p, y ∗ − i ( ¯ p, ϕ − i ) , ϕ ) = ¯ X − i 2 λ ¯ p 2 , (97) 67 which leads to the re venue ratio: π RE i ( ϕ ) π RE − i ( ϕ ) = λ E [ X i ] E [ X − i ] ¯ p . (98) If E [ X i ] ≥ E [ X − i ] , then π RE i ( ϕ ) π RE − i ( ϕ ) ≥ λ ¯ p . (99) (b) The case without pure price equilibrium: When D < P i y i ( ¯ p, ϕ i ) , based on the character- ization of CDF in Theorem 2, we discuss the following cases respectiv ely . • Case of 0 < D ≤ E [ X i ] : According to Theorem 2, we ha ve the CDF of the mix ed equilibrium price ov er p ∈ [ l, ¯ p ) as follows: F e i ( p ) = π R i  p, min { y ∗ − i ( p, ϕ − i ) , D } , ϕ  − π RE − i ( ϕ ) π R i  p, min { y ∗ − i ( p, ϕ − i ) , D } , ϕ  , (100) F e − i ( p ) = Z ¯ p l π RE i ( ϕ ) p 2 · min { y ∗ − i ( p, ϕ − i ) , D } dp. (101) W e can see that F e i ( p ) < 1 over p ∈ [ l , ¯ p ) since π RE − i ( ϕ ) > 0 . 22 According to Proposition 3, we solve the following equation to deriv e the equilibrium lower support l . F e − i ( ¯ p ) = Z ¯ p l π RE i ( ϕ ) p 2 · min { y ∗ − i ( p, ϕ − i ) , D } dp = 1 . (102) W e discuss the following two cases – 1) If D ≥ y ∗ − i ( ¯ p, ϕ − i ) , we hav e l = ¯ p 2 ¯ X − i D λ ( − 1 + s 1 + D 2 λ 2 ¯ p 2 ¯ X 2 − i ) . (103) – 2) If D < y ∗ − i ( ¯ p, ϕ − i ) , we hav e l = D λ ¯ X − i (1 + q 2 Dλ ¯ p ¯ X − i ) . (104) W e verify that in both cases (1) and (2), min( D , y ∗ − i ( l , ϕ − i )) = y ∗ − i ( l , ϕ − i ) . According to Lemma 1, the equilibrium re venue of both suppliers will be π RE i ( ϕ ) = l · ( D , E [ X i ]) = l · D , (105) π RE − i ( ϕ ) = π R − i ( l , min( D , y ∗ − i ( l , ϕ − i )) , ϕ ) = π R − i ( l , y ∗ − i ( l , ϕ − i ) , ϕ ) = ¯ X − i 2 λ l 2 , (106) 22 Note that π RE − i ( ϕ ) > 0 since the lower support l > 0 . 68 which leads to the re venue ratio: π RE i ( ϕ ) π RE − i ( ϕ ) = 2 λD l ¯ X − i . (107) In summary , we hav e π RE i ( ϕ ) π RE − i ( ϕ ) =              2 s 2 D λ ¯ p ¯ X − i + 2 , if D λ ¯ p ¯ X − i < 1 , 2 s 1 + D 2 λ 2 ¯ p 2 ¯ X 2 − i + 2 , if D λ ¯ p ¯ X − i ≥ 1 . (108) Therefore, when 0 < D ≤ E [ X i ] , we hav e – when 0 < D ≤ E [ X i ] , π RE i ( ϕ ) π RE − i ( ϕ ) ≥ 2 ; – when D = E [ X i ] and E [ X − i ] = ¯ X − i 2 ≤ E [ X i ] , π RE i ( ϕ ) π RE − i ( ϕ ) ≥ 4 (due to λ/ ¯ p > 1 ). • Case of E [ X i ] < D < P i y i ( ¯ p, ϕ i ) : W e characterize the re venue ratio between the two suppliers according to Lemma 1 as follows. π RE i ( ϕ ) = l · min( D , E [ X i ]) = l · E [ X i ] , (109) π RE − i ( ϕ ) = π R − i ( l , min( D , y ∗ − i ( l , ϕ − i )) , ϕ ) ≤ π R − i ( l , y ∗ − i ( l , ϕ − i ) , ϕ ) = ¯ X − i 2 λ l 2 . (110) Then, we hav e π RE i ( ϕ ) π RE − i ( ϕ ) ≥ l · E [ X i ] π R − i ( l , y ∗ − i ( l , ϕ − i ) , ϕ ) = 2 λ E [ X i ] l ¯ X − i . (111) If E [ X − i ] ≤ E [ X i ] , then π RE i ( ϕ ) π RE − i ( ϕ ) ≥ λ l > λ ¯ p . (112) Therefore, combining case (a), when D > E [ X i ] and E [ X − i ] ≥ E [ X i ] , we hav e π RE i ( ϕ ) π RE − i ( ϕ ) ≥ λ ¯ p . Finally , combining (a) and Subsection (b), we hav e Theorem 3 prov ed. F . Pr oof of Pr oposition 5 W e will discuss the equilibrium re venue with pure price equilibrium and without pure price equilibrium, respecti vely . 69 1) W ith the pure price equilibrium (i.e., D ≥ P i y ∗ i ( ¯ p, ϕ i ) : : If ϕ i = 1 , we hav e π RE i ( ϕ ) = ¯ p E [ X i ] > 0 . (113) If ϕ i = 0 , we hav e π RE i ( ϕ ) = π R i ( ¯ p, y ∗ i ( ¯ p, ϕ i ) , ϕ ) (114) = λ Z F − 1 − i ( ¯ p λ ) 0 xf i ( x ) dx (115) > 0 . (116) 2) W ithout the pur e price equilibrium (i.e ., D < P i y ∗ i ( ¯ p, ϕ i ) : :If ϕ i = 1 , due to the lo wer support l > 0 , we have π RE i ( ϕ ) = l min( D , E [ X i ]) > 0 . (117) If ϕ i = 0 , due to the lower support l > 0 , we hav e π RE i ( ϕ ) = π R i ( l , min( D , y ∗ i ( l , ϕ i )) , ϕ ) (118) > π R i (0 , min( D , y ∗ i (0 , ϕ i )) , ϕ ) (119) = 0 . (120) In conclusion, we hav e Proposition 5 proved. G. Pr oof of Pr oposition 4 W e prove Proposition 4 by contradiction. First, we prov e min i y ∗ i ( l , ϕ i ) < D by contradiction. Suppose that y ∗ i ( l , ϕ i ) ≥ D for both i = 1 , 2 and supplier − i ’ s mixed strategy F e − i has no atom at ¯ p based on Lemma 1 (iii). Then, against supplier − i ’ s bidding price p ∈ [ l, ¯ p ) , according to Proposition 1, supplier i ’ s selling out electricity quantity at the price ¯ p is x ∗ i ( p , y ) = min  D − min  D , y ∗ − i ( p, ϕ − i )  , y ∗ i ( ¯ p, ϕ i )  (121) =0 . (122) 70 Thus, the equilibrium re venue of supplier i can be characterized as follows π RE i ( ϕ ) = π RM i ( ¯ p, µ ∗ − i , ϕ ) = ¯ p Z ¯ p l x ∗ i ( p , y ) · f e − i ( p − i ) dp − i (123) = 0 . (124) Ho we ver , at the case of mixed price equilibrium, both suppliers’ equilibrium revenue is strictly positi ve as sho wn in Proposition 5, i.e., π RE i ( ϕ ) > 0 , which is contradiction to (124). Therefore, we hav e min i y ∗ i ( l , ϕ i ) < D . Second, we prove D ≤ P i y ∗ i ( l , ϕ i ) by contradiction. Suppose that D > P i y ∗ i ( l , ϕ i ) . Thus, there exists a small ε > 0 such that D > P i y ∗ i ( l + ε, ϕ i ) still holds. Note that min i y ∗ i ( l , ϕ i ) < D and we assume that y ∗ − i ( l , ϕ − i ) < D without loss of generality . W e also let this small ε satisfy y ∗ − i ( l + ε, ϕ − i ) < D . W e can characterize supplier i ’ s equilibrium rev enue using l and l + ε , respecti vely as follows. (a) W ith l : π RE i ( ϕ ) = π RM i ( l , µ ∗ − i , ϕ ) = l Z ¯ p l min( D , y ∗ i ( l , ϕ i )) f e − i ( p − i ) dp − i . (125) (b) W ith l + ε : π RE i ( ϕ ) = π RM i ( l + ε, µ ∗ − i , ϕ ) = ( l + ε ) · Z ¯ p l + ε min( D , y ∗ i ( l + ε, ϕ i )) f e − i ( p − i ) dp − i + ( l + ε ) Z l + ε l min  D − min  D , y ∗ − i ( p, ϕ − i )  , y ∗ i ( l + ε, ϕ i )  · f e − i ( p − i ) dp − i = ( l + ε ) Z ¯ p l + ε min( D , y ∗ i ( l + ε, ϕ i )) f e − i ( p − i ) dp − i + ( l + ε ) Z l + ε l y ∗ i ( l + ε, ϕ i ) · f e − i ( p − i ) dp − i > l Z ¯ p l + ε min( D , y ∗ i ( l , ϕ i )) f e − i ( p − i ) dp − i + l Z l + ε l min( D , y ∗ i ( l , ϕ i )) · f e − i ( p − i ) dp − i = l Z ¯ p l min( D , y ∗ i ( l , ϕ i )) f e − i ( p − i ) dp − i . (126) W e see that (125) and (126) contradict with each other . Therefore, we have D ≤ P i y ∗ i ( l , ϕ i ) . In conclusion, we hav e Proposition 4 proved. 71 X V I I . A P P E N D I X : P R O O F S O F S TAG E I A. Pr oof of Theor em 4 W e prov e Theorem 4 based on Definition 4 for the storage-in vestment equilibrium. W e first discuss the pure storage-in vestment equilibrium and then discuss the mix ed storage-in vestment equilibrium. First, for the pure price equilibrium, we use the example of the S 0 S 0 case. If the S 0 S 0 case is an equilibrium, each supplier will not be better off if he deviates to in vesting in storage, i.e., π S 1 S 0 | Y i − C i ≤ π S 0 S 0 i , ∀ i = 1 , 2 (127) Therefore, C i ∈ [ π S 1 S 0 | Y i − π S 0 S 0 i , + ∞ ) , for both i = 1 , 2 . Similarly , we can deri ve the conditions for the S 1 S 0 case and the S 1 S 1 case to be the equilibrium, respectiv ely . Second, if there is no pure storage-in vestment equilibrium, we can always compute the mixed storage-in vestment equilibrium [28]. Supplier i in vests in the storage with probability pr s i and does not in vest in storage with with probability pr n i , where pr s i + pr n i = 1 . W e construct the follo wing set of linear equations as follows to compute pr s i and pr n i [28].      pr s i + pr n i = 1 , ∀ i = 1 , 2 , pr s − i · ( π S 1 S 1 i − C i ) + pr n − i · ( π S 1 S 0 | Y i − C i ) = pr s − i · π S 1 S 0 | N i + pr n − i · π S 0 S 0 i , ∀ i = 1 , 2 . (128) By solving (128), we can obtain pr s i and pr n i for both i = 1 , 2 , which is the mixed storage- in vestment equilibrium. B. Pr oof of Pr oposition 6 W e prove Proposition 6 based on Theorem 4. Note that π S 1 S 0 | Y i − π S 0 S 0 i is bounded for both i = 1 , 2 . Thus, there alw ays exists C S 0 S 0 i such that C S 0 S 0 i > π S 1 S 0 | Y i − π S 0 S 0 i for each i = 1 , 2 . According to Theorem 4, the S 0 S 0 case will be the storage-in vestment equilibrium, which is also unique. C. Pr oof of Pr oposition 7 W e prov e Proposition 7 based on the storage-in vestment-equilibrium shown in Theorem 4 and suppliers’ equilibrium re venue in the case S 0 S 0 sho wn in Proposition 2. W e will show that if the 72 demand D m,t ≤ min i E [ X m,t i ] , the condition C i ∈ [0 , π S 1 S 1 i − π S 1 S 0 | N i ] , for both i = 1 , 2 cannot be satisfied. According to Proposition 2, in the S 0 S 0 case, if the demand D m,t ≤ min i E [ X m,t i ] , then both suppliers’ rev enue is zero. Therefore, if the demand 0 < D m,t ≤ min i E [ X m,t i ] for any m and t , we hav e π S 1 S 1 i = 0 , ∀ i = 1 , 2 . (129) Ho we ver , according to Proposition 5, we hav e that π S 1 S 0 | N i > 0 alw ays holds. Therefore, if the demand 0 < D m,t ≤ E [ X m,t i ] for any m and t , we hav e π S 1 S 1 i − π S 1 S 0 | N i < 0 , ∀ i = 1 , 2 . (130) Based on the condition of S 1 S 1 being the equilibrium in Theorem 4, the S 1 S 1 case cannot be a pure equilibrium if π S 1 S 1 i − π S 1 S 0 | N i < 0 , ∀ i = 1 , 2 . D. Pr oof of Pr oposition 8 W e will prove Proposition 8 based on Theorem 4. The key is to show π S 1 S 0 | Y i − π S 0 S 0 i = π S 1 S 1 i − π S 1 S 0 | N i > 0 for both i = 1 , 2 . When D m,t ≥ D m,t,th = max( P i y m,t ∗ i ( ¯ p, 1) , P i y m,t ∗ i ( ¯ p, 0)) , there exists the pure price equilibrium p 1 = p 2 = ¯ p for each type of subgame in Stage II according to Proposition 2. Therefore, for both i = 1 , 2 , π S 0 S 0 i = π S 1 S 0 | N i = E m,t [ π R,m,t i ( ¯ p, y ∗ i ( ¯ p, ϕ i ) , ϕ )] , where X i ϕ i = 0 (131) = E m,t [ λ Z y m,t ∗ i ( ¯ p, 0) 0 xf m,t i ( x ) dx ] (132) = E m,t [ ¯ py m,t ∗ i ( ¯ p, 0) − λ Z y m,t ∗ i ( ¯ p, 0) 0 F m,t i ( x ) dx ] , (133) which has been sho wn in (72). Furthermore, we also hav e π S 1 S 1 i = π S 1 S 0 | Y i = E m,t [ π R,m,t i ( ¯ p, y ∗ i ( ¯ p, ϕ i ) , ϕ )] , where X i ϕ i = 2 (134) = E m,t  ¯ py m,t ∗ i ( ¯ p, 1)  (135) = E m,t [ ¯ p ¯ X i − ¯ p Z ¯ X i 0 F m,t i ( x ) dx ] . (136) 73 Thus, we hav e π S 1 S 0 | Y i − π S 0 S 0 i = π S 1 S 1 i − π S 1 S 0 | N i = E m,t [ ¯ py m,t ∗ i ( ¯ p, 1) − λ Z y m,t ∗ i ( ¯ p, 0) 0 xf m,t i ( x ) dx ] (137) , C th i . (138) which is based on (132) and (135). Note that C th i > 0 always holds as implied in (78). According to Theorem 4, if C i ≤ C th i , then supplier i will in vest in storage (i.e., ϕ ∗ i = 1 ) while if C i > C th i , then supplier i will not in vest in storage (i.e., ϕ ∗ i = 0 ). E. Pr oof of Pr oposition 9 Suppliers always have strictly positi ve profit at the storage-in vestment equilibrium because the without-storage supplier can always hav e positi vely rev enue in the cases of S 1 S 0 and S 0 S 0 according to Proposition 5. W e show it as follows. • If the S 0 S 0 case is the equilibrium, both suppliers get strictly positiv e profit (with zero storage in vestment cost) according to Proposition 5. • If the S 1 S 0 case is the equilibrium, the without-storage suppliers get strictly positi ve profit (with zero storage in vestment cost) according to Proposition 5. If the with-storage supplier gets non-positi ve profit, he can always de viate to not in vesting in storage, which leads to the case S 0 S 0 and brings him strictly positi ve profit. • If the S 1 S 1 case is the equilibrium and one supplier gets non-positi ve profit, he can always de viate to not in vesting in storage, which leads to the case S 1 S 0 and brings him strictly positi ve profit. In summary , suppliers alw ays hav e strictly positiv e profits at the storage-in vestment equilibrium. X V I I I . A P P E N D I X : P R O O F S O F O L I G O P O L Y M O D E L A. Pr oof of Pr oposition 11 This proof can follow the same procedure in the proof of Proposition 2 by verifying the pure price equilibrium according to the definition of the Nash equilibrium. T ow ards this end, note that for supplier i with or without storage, the re venue function π R i ( p i , x ∗ i ( p , y ) , ϕ ) is strictly 74 increasing with respect to both the price p i and the selling quantity x ∗ i ( p , y ) that is in the range [0 , y ∗ i ( p i , ϕ i )] (without considering the other supplier’ s coupled decisions). W e will discuss the three cases, respecti vely . 1) The case of D ≥ P i ∈I y ∗ i ( ¯ p, ϕ i ) : W e can pro ve that when D ≥ P i y ∗ i ( ¯ p, ϕ i ) , p i = ¯ p is a pure price equilibrium . Also, this pure price equilibrium is unique. This proof can follow the same procedure in the Section XVI.B.1.a. of the proof of Proposition 2. The intuition is that when the demand is larger than the maximum bidding quantity , if any supplier deviates to a lo wer price, his selling quantity cannot be increased, which leads to a lower rev enue. 2) The case of D ≤ P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ i ) for any j ∈ U : W e first prov e by the definition of the Nash equilibrium that when D ≤ P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j ) for any j ∈ U , there exists a pure price equilibrium p ∗ i = 0 with an equilibrium rev enue π RE i = 0 , for any i ∈ I . Then, note that this equilibrium is not unique, b ut we show that suppliers always get zero re venue at any equilibrium. First, we prov e the pure price equilibrium p ∗ i = 0 . W e assume that p ∗ i = 0 , ∀ i ∈ I . W e will discuss two cases of with-storage supplier and without-storage supplier , respectiv ely . (a) For a supplier j ∈ U who in vests in storage, if he de viates to a higher price p 0 j > 0 , the demand that he gets is the following. min   D − min( D , X i ∈I \ j y ∗ i (0 , ϕ i )) , y ∗ j ( p 0 j , ϕ j )   , j ∈ U . (139) Note that according to Theorem 1, we hav e y ∗ k (0 , ϕ k ) = 0 , ∀ k ∈ V . Also, we have y ∗ k ( p k , ϕ k ) = E [ X k ] , ∀ k ∈ U . Therefore, (139) = min D − min( D , X i ∈U y ∗ i ( ¯ p, ϕ i ) − y j ( ¯ p, ϕ j )) , y ∗ j ( p 0 j , ϕ j ) ! , j ∈ U , (140) which is zero since D ≤ P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j ) , ∀ j ∈ U . Therefore, if this supplier deviates to a higher price, his rev enue will be still zero. (b) For a supplier j ∈ V who does not inv est in storage, if he deviates to a higher price p 0 j > 0 , 75 the demand that he gets is min   D − min( D , X i ∈I \ j y ∗ i (0 , ϕ i )) , y ∗ j ( p 0 j , ϕ j )   , j ∈ V , (141) = min D − min( D , X i ∈U y ∗ i ( ¯ p, ϕ i )) , y ∗ j ( p 0 j , ϕ j ) ! , j ∈ V (142) which is still zero since D ≤ P i ∈U y ∗ i ( ¯ p, ϕ i ) . Therefore, if this supplier deviates to a higher price, his re venue will be still zero. In conclusion, the bidding price p ∗ i = 0 , ∀ i ∈ I is an equilibrium where no supplier will de viate. Second, note that the equilibrium here is not unique, howe ver , each supplier always gets zero re venue at any equilibrium. W e show this by contradiction as follo ws. If supplier k gets positiv e re venue, it means that his bidding price and his obtained demand are both positi ve. W e assume that a set of suppliers P bid the price p > 0 the same as this supplier k . W e denote the set of suppliers whose prices are lower than p as P L and the set of suppliers whose prices are higher than p as P H . 23 Since this supplier gets positi ve demand, it means X i ∈P y ∗ i ( p i , ϕ i ) ≤ D − X i ∈P L y ∗ i ( p i , ϕ i ) , (143) or 0 < D − X i ∈P L y ∗ i ( p i , ϕ i ) < X i ∈P y ∗ i ( p i , ϕ i ) . (144) • Case (144) and |P | ≥ 2 : At least one of suppliers in P can decrease his price by a suf ficiently positi ve value, which can increase his obtained demand and increase his re venue. This shows that this case cannot be one equilibrium. • Case (144); |P | = 1 and p < ¯ p : This supplier can increase his price by a small positiv e v alue (which makes the bidding price smaller than the lo west bidding price in set P H S ¯ p ), which will not decrease his obtained demand. Thus, this deviation increases his rev enue and this case cannot be one equilibrium. • Case (144); |P | = 1 and p = ¯ p : Due to (144), we hav e P i ∈P L y ∗ i ( p i , ϕ i ) < D . Note that the set P L contains all the suppliers except the single supplier k . Thus, there alyways exists 23 Note that P L and P H can be both empty sets 76 j ∈ U such that P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j ) < P i ∈P L y ∗ i ( p i , ϕ i ) < D , which contradicts the condition D ≤ P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j ) , ∀ j ∈ U . This case is impossible. • Case (143) and p < ¯ p : any supplier in P can always increase his price by a small positi ve v alue (which makes the bidding price smaller than price cap ¯ p ) without decreasing his obtained demand, which increases his re venue. This shows that this case cannot be one equilibrium. • Case (143) and p = ¯ p : Due to (143), we ha ve P i ∈I y ∗ i ( p i , ϕ i ) ≤ D , which contradicts the condition D ≤ P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j ) , ∀ j ∈ U . Thus, this case is impossible. Therefore, we can draw the conclusion that at any equilibrium, suppliers get zero re venue. 3) The case that ther e exists j ∈ U such that P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j ) < D < P i ∈I y ∗ i ( ¯ p, ϕ i ) : In this case, there is no pure price equilibrium. This proof can follo w the similar procedure in the Section XVI.B.1.b of the proof of Proposition 2. W e can discuss three cases: (i) all the suppliers bid zero prices; (ii) suppliers’ bidding prices are all equal and positiv e. (iii) suppliers’ bidding prices are not equal for all the suppliers. W e sho w that all theses cases cannot be the pure price equilibrium. First, for case (i) , at least one supplier j (i.e., the j satisfying P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j ) < D ) who in vests in storage can increase his price, and he will get positiv e demand. This increases his re venue and shows that case (i) cannot be an equilibrium. Second, for case (ii), we can discuss two conditions P i ∈I y ∗ i ( p i , ϕ i ) ≤ D and P i ∈I y ∗ i ( p i , ϕ i ) > D , which is the same Section XVI.B.1.b . For P i ∈I y ∗ i ( p i , ϕ i ) ≤ D , any supplier can always increase his price without decreasing his obtained demand, which increases his rev enue. For P i ∈I y ∗ i ( p i , ϕ i ) > D , at least one supplier can alw ays reduce his price by a sufficiently small positi ve value, which can increase his demand and increase his re venue. Thus, case (ii) can not be an equilibrium. Third, for case (iii), we denote the set of suppliers with the lo west bidding prices p among all the suppliers as L . Similarly , we discuss two conditions P i ∈L y ∗ i ( p i , ϕ i ) ≤ D and P i ∈L y ∗ i ( p i , ϕ i ) > D . For P i ∈L y ∗ i ( p i , ϕ i ) ≤ D , any supplier can always increase his price by a small positiv e v alue (which makes the bidding price smaller than the second lo west price) without decreasing his obtained demand, which increases his rev enue. Thus, this case cannot be an equilibrium. For P i ∈L y ∗ i ( p i , ϕ i ) > D , there are three possibilities. 77 • The lowest price p > 0 and |L | = 1 : This supplier can increase his price by a small positiv e v alue (which makes the bidding price smaller than the second lowest bidding price), which will not decrease his obtained demand. Thus, it increases his re venue and this case cannot be one equilibrium. • The lo west price p > 0 and |L |≥ 2 : At least one of suppliers in L can decrease his price by a suf ficiently small positiv e value, which can increase his obtained demand and increase his re venue. This shows that this case cannot be one equilibrium. • The lo west price p = 0 : In this case, all the suppliers ha ve zero re venue, and P i ∈L y ∗ i (0 , ϕ i ) > D . Note that demand D also satisfies P i ∈U y ∗ i ( ¯ p, ϕ i ) − max j y ∗ j ( ¯ p, ϕ i ) < D , j ∈ U . W e denote arg max j ∈U y ∗ j ( ¯ p, ϕ j ) = j ∗ . Thus, there are two possibilities that lead to make this: – j ∗ ∈ L : The supplier j ∗ can increase his zero price to a positi ve price (which is smaller than the second lowest price) and get positiv e demand since P i ∈L\ j ∗ y ∗ i (0 , ϕ i ) ≤ P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ∗ ( ¯ p, ϕ i ) < D . This increases supplier j ∗ ’ s rev enue. – j ∗ / ∈ L : Any supplier k in L can increase his zero price to a positi ve price (which is smaller than the second lo west price) and get positi ve demand since P i ∈L\ k y ∗ i (0 , ϕ i ) < P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ∗ ( ¯ p, ϕ i ) < D . This increases supplier k ’ s rev enue. Therefore, the case that the lo west price p = 0 cannot be one equilibrium. Combining the case p = 0 and p > 0 , the condition P i ∈L y ∗ i ( p i , ϕ i ) > D is not an equilibrium. Combining cases (i)-(iii), we show that all theses cases cannot the equilibrium. Thus, there is no pure price equilibrium if there exists j ∈ U such that P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j ) < D < P i ∈I y ∗ i ( ¯ p, ϕ i ) . Finally , we hav e Proposition 11 prov ed. B. Pr oof of Pr oposition 12 W e first sho w the existence of mixed price equilibrium and then pro ve the positiv e rev enues for all the suppliers in the mixed price equilibrium. 1) Existence of mixed price equilibrium: This result can be deri ved from Theorem 5 [42]. 2) P ositive r evenue: Note that the case that there exists j ∈ U such that P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j ) < D < P i ∈I y ∗ i ( ¯ p, ϕ i ) is equi valent to the case P i ∈U y ∗ i ( ¯ p, ϕ i ) − max j ∈U y ∗ j ( ¯ p, ϕ j ) < D < P i ∈I y ∗ i ( ¯ p, ϕ i ) . W e will first prove by contradiction that for supplier n with n = arg max i ∈U y ∗ i ( ¯ p, ϕ i ) , 78 his equilibrium rev enue is positiv e. Then, we prove that other suppliers except supplier n also hav e the positiv e re venues. W e denote the support of supplier i ’ s mixed price strategy as S P i . First, we will prov e that for supplier n , his rev enue equilibrium π RE n > 0 . W e pro ve this by contradiction. W e assume that supplier n ’ s equilibrium rev enue π RE n = 0 , and discuss two cases. • For each supplier j 6 = n , the support S P j only contains 0, which means each supplier j 6 = i has the pure price strategy p j = 0 : Then, for supplier n , he can always set a pure price p n > 0 to achiev e positiv e demand and get positiv e rev enue since P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ n ( ¯ p, ϕ n ) < D , which contradicts the assumption that π RE n = 0 . • In all the suppliers except n , there exists at least one supplier k such that S P k contains positi ve price p k > 0 : For all the suppliers whose supports contain positiv e prices (except n ), we denote the set of those suppliers as P S . For any supplier k ∈ P S , we choose one positi ve price p k ∈ S P k . Thus, supplier n can always choose a pure price strategy 0 < p n < min k ∈P S p k , such that he can get positi ve demand and positi ve re venue with a positi ve probability . This contradicts the assumption that π RE n = 0 . Thus, we can ha ve the conclusion that at the equilibrium, supplier n ’ s re venue π RE n > 0 . This also implies that for supplier n , his support S P n does not contain zero. Second, we will prove that for any supplier j 6 = n , his equilibrium rev enue is positiv e. W e assume that supplier j ’ s equilibrium re venue π RE j = 0 . Note that among the suppliers except j , there exists at least one supplier n such that S P n contains positiv e price p n > 0 . For all the suppliers (except j ) whose supports contain positiv e prices, we denote the set of those suppliers as P S 0 . For an y supplier k ∈ P S 0 , we choose one positiv e price p k ∈ S P k . Thus, supplier j can always choose a pure price strategy 0 < p j < min k ∈P S 0 p k , such that he can get positiv e demand and positiv e rev enue with a positi ve probability . Therefore, at the equilibrium, supplier j ’ s re venue cannot be zero. Therefore, based on abov e discussions, we hav e that all the suppliers hav e the positive re venues in the case of P i ∈U y ∗ i ( ¯ p, ϕ i ) − max j ∈U y ∗ j ( ¯ p, ϕ j ) < D < P i ∈I y ∗ i ( ¯ p, ϕ i ) . C. Pr oof of Pr oposition 13 The proof follo ws the definition of Nash equilibrium. 79 It is straightforward that the benefit brought by in vesting storage for supplier i is bounded. At the case S U |V , we denote the equilibrium profit of supplier i as Π ∗ i ( S U |V ) and the expected equilibrium re venue (scaled in one hour) ov er the in vestment horizon as π RE E i ( S U |V ) . F or any case S U |V , one without-storage supplier i has the profit Π ∗ i ( S U |V ) = π RE E i ( S U |V ) , i ∈ V at the equilibrium. Ho we ver , if he deviates to in vesting in storage, he has the profit Π ∗ i ( S U S i |V \ i ) = π RE E i ( S U S i |V \ i ) − C i . Thus, for i ∈ V , we hav e Π ∗ i ( S U S i |V \ i ) − Π ∗ i ( S U |V ) (145) = π RE E i ( S U S i |V \ i ) − π RE E i ( S U |V ) − C i . (146) Note that π RE E i ( S U S i |V \ i ) − π RE E i ( S U |V ) is bounded for any S U |V . If the storage cost C i > C no i , where C no i is the maximum value of π RE E i ( S U S i |V \ i ) − π RE E i ( S U |V ) over all the cases S U |V , then this supplier i ∈ V will not deviate to in vesting in storage in any case of S U |V . Thus, no supplier in vesting in storage is the unique equilibrium. D. Pr oof of Pr oposition 14 The proof follo ws the definition of Nash equilibrium. Note that in the subgame S U |V , when 0 < D m,t ≤ min j ∈U ( P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j )) for any t and m , each supplier has zero re venue for any t and m as sho wn in Proposition 11. Thus, for each supplier i ∈ I , his expected equilibrium re venue π RE E i ( S U |V ) = 0 . Then, for supplier j ∈ U who in vests in storage, his profit is π RE E i ( S U |V ) − C i < 0 since C i > 0 . Therefore, this supplier i can always de viate to not in vesting storage which leads to a nonnegati ve profit. This sho ws that when 0 < D m,t ≤ min j ∈U ( P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j )) , the case S U |V (i.e., suppliers of set U inv esting in storage and suppliers of set V not in vesting in storage) cannot be a pure storage-in vestment equilibrium. E. Pr oof of Pr oposition 15 The intuition of this proposition is that when the demand D is sufficiently large, there is not competition between suppliers and they make decisions of storage in vestment independently . 80 As implied in Proposition 11, when demand D m,t ≥ P i ∈I y m,t ∗ i ( ¯ p, ϕ i ) in subgame S U |V , each supplier i can bid the price cap ¯ p to get his biding quantity y m,t ∗ i ( ¯ p, ϕ i ) . For con venience, at hour t of month m , we denote the bidding quantity of supplier i at price cap ¯ p in subgame S U |V as y m,t ∗ i ( ¯ p, ϕ i |S U |V ) . W e also denote the set of all the subgames as S Ω . Thus, if the demand D m,t ≥ max S U |V ∈S Ω P i ∈I y m,t ∗ i ( ¯ p, ϕ i |S U |V ) , D m,t,th 0 for any t and m , then each supplier i can bid the price cap ¯ p to get his biding quantity y m,t ∗ i ( ¯ p, ϕ i ) in any subgame for any t and m . This leads to the re venue π R,m,t i ( ¯ p, y m,t ∗ i ( ¯ p, ϕ i ) , ϕ ) that can be directly calculated based on supplier i ’ s parameter . In this case, we have the following. • If supplier in vests in storage, i.e., ϕ i = 1 , his equilibrium rev enue is E m,t [ π R,m,t i ( ¯ p, y ∗ i ( ¯ p, 1) , ϕ )] = = E m,t  ¯ py m,t ∗ i ( ¯ p, 1)  , (147) which has been sho wn in (135). • If supplier does not in vest in storage i.e., ϕ i = 0 , his equilibrium rev enue is E m,t [ π R,m,t i ( ¯ p, y ∗ i ( ¯ p, 0) , ϕ )] = E m,t [ λ Z y m,t ∗ i ( ¯ p, 1) 0 xf m,t i ( x ) dx ] , (148) which has been sho wn in (132). W e compared (147) and (148), and we characterize C th 0 i the same as (137) as follows. E m,t [ ¯ py m,t ∗ i ( ¯ p, 1) − λ Z y m,t ∗ i ( ¯ p, 0) 0 xf m,t i ( x ) dx ] , C th 0 i . (149) F . Pr oof of Pr oposition 16 W e prove this by contradiction and discuss a total of three cases. • If one supplier does not in vest in storage and gets zero profit (note that a without-storage supplier always has nonnegati ve profits), it only means the demand lies in the condition D ≤ P i ∈U y ∗ i ( ¯ p, ϕ i ) − y ∗ j ( ¯ p, ϕ j ) , ∀ j ∈ U as shown in Proposition 11 and Proposition 12, where all the suppliers get zero rev enues in the local energy market. This state is not stable because the with-storage supplier gets negati ve profit and he can always choose not to in vest in storage, which increases his profit. • If one supplier inv ests in storage and gets negati ve profit, he can always choose not to in vest in storage, which increases his profit. 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