Gibbs sampling for game-theoretic modeling of private network upgrades with distributed generation

Gibbs sampling for game-theoretic modeling of pri v ate network upgrades with distrib uted generation Merlinda Andoni, V alentin Robu, Da vid Flynn and W olf-Gerrit Fr ¨ uh Heriot-W att Univ ersity , Edinb urgh, UK Email: { m.andoni,v .robu,d.flynn,w .g.fruh } @hw .ac.uk Abstract —Renewable energy is incr easingly being curtailed, due to oversupply or network constraints. Curtailment can be partially av oided by smart grid management, but the long term solution is network reinfor cement. Network upgrades, however , can be costly , so recent interest has focused on incentivising private in vestors to participate in network inv estments. In this paper , we study settings where a private r enewable in vestor constructs a power line, b ut also provides access to other generators that pay a transmission fee. The decisions on optimal (and interdependent) renewable capacities built by in vestors, affect the resulting curtailment and profitability of projects, and can be formulated as a Stackelberg game. Optimal capacities rely jointly on stochastic variables, such as the renewable resour ce at project location. In this paper , we show how Markov chain Monte Carlo (MCMC) and Gib bs sampling techniques, can be used to generate observ ations from historic resource data and simulate multiple future scenarios. Finally , we validate and apply our game-theoretic formulation of the in vestment decision, to a real network upgrade problem in the UK. I . I N T R O D U C T I O N Renew able ener gy sources (RES) play a key role in the climate change mitigation agenda. RES are v ariable, depend on weather patterns and are difficult to predict, hence raise techni- cal challenges regarding network management. Moreov er , grid infrastructure is often inadequate to support RES development, especially in the area of distribution networks. For instance, often RES projects are clustered in remote areas of the grid, where planning approv al may be f av ourable and renew able resources abundant. T ypically , in the UK, such areas are windy islands with constrained connections to the main grid. In these areas, technical limitations and imbalance of renewable supply to demand, often result in RES curtailment, i.e. the energy that could hav e been generated is wasted as the system cannot absorb it or transfer it where required. Curtailment can affect implementation of future RES projects, result in lost rev enues and instigate socio-economic challenges, especially when local community-owned projects are in volv ed. T ypically , RES generators are granted firm connections to the grid and recei ve compensation for incurred curtailment, the cost of which is eventually borne by all system users. Howe ver in se veral occasions, RES generators are offered interruptible, non-firm connections, as an alternativ e to expensiv e or time consuming reinforcements. The so-called flexible commercial arrangements are increasingly offered by network operators, as an alternativ e, and are creating a shift in network access rules. A detailed revie w on flexible commercial arrangements can be found in [1], [2]. As shown in [3], arrangements that are fair and equally share curtailment and access among generators, can maximise the generation capacity b uilt at a certain location and minimise discouragement for future inv estors. Solutions to reduce curtailment include demand side man- agement, energy storage or smart grid techniques such as Dynamic Line Rating (DLR) or real-time monitoring of the thermal state of the lines [4] and Acti ve Network Management (ANM), i.e., the automatic control of the power system by control devices and data that allo w real time operation and optimal power flows [5]. The long term solution, howe ver , is network upgrade. Anaya & Pollit (2015) compared smart interruptible connections to traditional grid reinforcements in [5]. It is estimated that, by 2030 in the US alone, up to $2 trillion will be required for network upgrades [6]. As grid expansion is expensi ve, it is desirable to provide incenti ves to priv ate parties for taking part in such in vestments. Priv ate network upgrades hav e been studied in sev eral works [7], [8]. Ho wev er , they raise the question for system operators of defining the frame work within which these pri vate lines are incentivised, built and accessed by competing generators. New pri vate lines constructed, often follow a ‘single access’ principle, i.e. lines for sole-use that suffice only to accommo- date the RES capacity of each project. Ho wever , as shown in [3], it is possible for system operators to encourage RES generators to install larger capacity lines under a ‘common ac- cess’ principle, i.e. a priv ate in vestor is licensed to build a line only if it grants access to smaller generators, which are subject to transmission charges. In these settings, curtailment and line access rules can play a significant role in the resulting grid expansion. Crucially , this leads to a leader-follower or a Stack- elber g game between the line investor and local generator s respectiv ely . Stackelberg games in network upgrades and RES settings hav e been presented in [9], [10]. Our previous work was one of the first to introduce Stackelberg game formulations in settings that combine network upgrades, curtailment and line access rules, howe ver the model presented in [3] did not take into account stochastic generation and demand required for equilibrium estimation. Subsequent work [11] improv ed by utilising real data, howe ver , followed a one-shot, single scenario approach. W e build on this work by dev eloping a principled framew ork, based on game-theoretic and state-of- the-art sampling techniques, i.e. Markov chain Monte Carlo (MCMC). Several authors used MCMC for modelling of wind speeds or wind power outputs [12], [13]. Our framework allows modelling multiple renewable in vestment scenarios that reduce the uncertainty of future generation and demand. In more detail, the main contributions of the work are: • W e develop a new methodology that generates observa- tions from renewable resource data. While historic data, such as wind speeds may be av ailable, they might have considerable gaps and joint distributions cannot be ex- pressed in simple, closed-form equations. For this reason we dev elop a MCMC methodology (Gibbs sampling) that can draw samples from av ailable data and run multiple scenarios of potential futures. • W e establish a methodology that can determine optimal generation capacity inv estments through use of real de- mand and wind speed data. This work is one of the first to combine Stackelberg equilibria to a large-scale realistic game with MCMC techniques. Our model designates players’ actions, depending on RES output correlation and expected curtailment, and studies the cost parameters effects on the equilibrium of the game. • Previous work has been extended to account for varying demand data instead of an average approach followed in both [3], [11]. The model is applied and validated in a real network upgrade problem in the UK. Section II of the paper presents the theoretical formulation of the game, Section III introduces Gibbs sampling, Section IV demonstrates the methodology based on a real-world case study , Section V shows the results and Section VI concludes. I I . S TAC K E L B E R G G A M E M O D E L D E S C R I P T I O N Consider a simplified two-node network: location A is a net consumer (area of high demand) and location B is a net producer (area of high wind resource). Moreover , consider two players: a line in vestor , who can be merchant-type or a utility company and is building i) the AB interconnection and ii) P N 1 renew able capacity at B, and a local player representing all local RES generators located at B, and who builds rene wable capacity equal to P N 2 . This second player can be thought of as in vestors from the local community , who do not hav e the technical/financial capacity to build a line, b ut may hav e access to cheaper land, find it easier to get social approval to build turbines etc., hence may have lower costs for RES deployment. Building the line will elicit a reaction from local in vestors. Crucially , the line in vestor has a first mover advantage in building the grid infrastructure, which is expensi ve, technically challenging, and only few in vestors (e.g. DNO-approved) hav e the expertise and regulatory approv al to carry it out. From a game-theoretic perspecti ve, this is a bi-le vel Stackelber g game, between the leader (line in vestor or first player) and follower (local generators or second player). W e assume that players’ actions are driv en by profit maximisation criteria. The profit functions of players can be expressed as in [11]: Π 1 = ( E G 1 − E C 1 ) p G − E G 1 c G 1 + ( E G 2 − E C 2 ) p T − C T (1) Π 2 = ( E G 2 − E C 2 )( p G − p T ) − E G 2 c G 2 (2) In these equations, E G i represents player’ s i expected ener gy produced over the project lifetime, if no curtailment occurred, while E C i is the energy lost through curtailment. Line cost is estimated as C T = I T + M T , where I T is the cost of building the line (or initial in vestment) and M T the cost of operation and maintenance. The monetary value of the power line is proportional to the ener gy flowing from B to A, charged for local generators and ‘common access’ rules with p T transmission fee per energy unit transported through the line. Moreover , the cost of expected generation per unit is defined as c G i = ( I G,i + M G i ) /E G i , where I G i is the cost of building the plant and M G i the operation and maintenance costs. Finally , the energy generated by a RES unit is sold at a constant generation tariff price, equal to p G . As seen in Eq. (1), the line inv estor has two streams of rev enue, the self-produced ener gy and the energy produced by local generators and transported through the line. The costs of the line in vestor relate to installing and operating the RES capacity ( c G 1 ) and to building the po wer line ( C T ). Similarly from Eq. (2), the profits of the local generators depend only on the energy produced (generation cost c G 2 ) and transmitted through the line at a charge of p T . The research question our model tries to answer is ‘How to determine the optimal generation capacities P N i built by the two players, so that pr ofits ar e maximised?’ T o answer this question, the profit equations, Eq. (1) and Eq. (2), need to be expressed in terms of generation capacities. Follo wing the analysis presented in [11], expected genera- tion E G i and curtailment E C i can be expressed as functions of the players’ generation capacities. If x i is the per unit power generated by i player, then this represents a stochastic variable that depends on the wind speed distribution, and is equal to x i = P G i /P N i , where P G i is the actual po wer output of i generator and P N i its rated capacity . If E ( P G i,t ) is the expected po wer generated at time interval t , then the total energy generated for duration equal to the project’ s lifetime is E G i = X t x i P N i = X t E ( P G i,t ) , ∀ t . Similarly , the total energy curtailed is E C = X t E ( P C t ) , ∀ t , where E ( P C t ) is the expected power curtailed at time step t . The resulting curtailment depends on wind resources at location B and demand at A, denoted as P D,t . Curtailment ev ents happen when x 1 P N 1 + x 2 P N 2 − P D,t > 0 . Players’ generating plants are located at neighbouring locations at B, therefore experience correlated wind speeds. The stochastic variables x 1 and x 2 , follow a joint probability distribution function f ( x 1 , x 2 ) and expected curtailment at time interval t can be expressed as (detailed analysis shown in [11]): E ( P C,t ) = Z 1 0 Z 1 P D,t − x 1 P N 1 P N 2 ( x 1 P N 1 + x 2 P N 2 ) f ( x 1 , x 2 ) dx 2 dx 1 − P D,t Z 1 0 Z 1 P D,t − x 1 P N 1 P N 2 f ( x 1 , x 2 ) dx 2 dx 1 (3) Crucially , Eq. (3) sho ws that curtailment depends on both players’ strategies i.e. the generation capacities built. Curtail- ment expressions for each player under a ‘common access’ regime can be reasonably can be approximated by E C i = E G i E G i + E G − i E C . This concludes the expression of profit equations as functions of players’ rated capacities. Optimal capacities installed are determined in the equilib- rium of the game, which is found by bac kwar d induction . The line inv estor or leader assesses and ev aluates the second player’ s reaction, in order to determine his strategy (i.e. P N 1 ) and influence the equilibrium price. The leader estimates the follower’ s best response , giv en his own capacity P N 1 : P ∗ N 2 = arg max P N 2 Π 2 ( P N 1 , P N 2 ) (4) Next, the leader estimates which solution from the set of the local generators’ best response P ∗ N 2 maximises his own profit: P ∗ N 1 = arg max P N 1 Π 1 ( P N 1 , P ∗ N 2 ) (5) In other words, the leader moves first by installing their own capacity . In the second le vel, followers respond to the capacity built, as anticipated by the leader . The equilibrium of the game ( P ∗ N 1 , P ∗ N 2 ) satisfies both Eq. (4) and Eq. (5) and is giv en by the notion of the subgame perfect equilibrium. In practical settings the joint distribution of stochastic renew able resources is often unknown, but historic data may be av ailable. In addition, due to the interdependency in re- sulting curtailment and multiple parameters a nice closed- form solution of the game cannot be found or expressed analytically . In [11] we presented an empirical algorithm that utilises directly real data and approximates the solution of the game following a one-shot approach. In addition, data may experience important gaps. In this paper we show how we can utilise real data to simulate scenarios that approximate the real distribution with a state-of-the-art MCMC technique. I I I . G I B B S S A M P L I N G Markov chain Monte Carlo (MCMC) is a class of methods for simulation of stochastic processes. Gibbs sampling can be thought of as a particular case of the Metropolis-Hastings algorithm used for MCMC [14]. The Gibbs sampler uses the conditional distributions as proposal distributions with acceptance probability equal to 1 [15] and can be easily imple- mented in various applications. Using this technique we can generate, from historic data, observ ations that are dependent. W ind data samples from the project locations form a Markov chain (MC). W e can experiment with the length of the chain or sampling size n and we can repeat the process for multiple MCs or number of r ealisations N . In practice, n and N need to be determined in such way that the resulting MC conv erges to the real distribution, is ergodic and computationally efficient. Ergodicity means that all possible states of the MC can be visited and are independent of the starting state [15]. The methodology applied is described in detail in the next sections with the help of the Kintyre-Hunterston case study . Fig. 1. Kintyre-Hunterston project map 1 I V . C A S E S T U DY A N A L Y S I S A. Kintyre-Hunter ston link The case study analysis is based on a real £ 230m grid rein- forcement project in the UK that links the Kintyre peninsula to the Hunterston substation on the Scottish mainland (see Fig. 1). Kintyre is a region that has attracted vast renew able in vestment, predominantly wind generation, resulting in the necessity of a newly built po wer line, which provided space for 150 MW of additional renewable capacity . Follo wing the analysis described in Section II, we assume that the demand region or Location A is Hunterston and location B is the geographical re gion covering the Kintyre peninsula. The line inv estor and local generators install gen- eration capacities at different sub-regions of B. T wo weather stations were selected by the UK Met Of fice database 2 , the station with ID 908 located in the Kintyre peninsula (wind farm of line in vestor) and with ID 23417 located in Islay (wind farm of local generators), with a distance between them of 44 km . These weather stations provide data ov er a common period of 17 years (1999–2015). Demand data used in simulations are based on real UK National Demand data 3 in the time period of 2006–2015. UK demand data are normalised to represent a lower local demand. More details on the case study and data processing can be found in [11]. Literature in wind forecasting commonly uses W eibull dis- tributions for the representation of actual wind distributions. Howe ver , the joint probability distribution of the wind speed (and of the players’ power outputs), exhibits correlation and in practice is not kno wn. If there are suf ficient wind speed mea- surements for both players locations, then the joint probability distribution can be approximated directly from the av ailable historic data. The method described in the following section can be used to draw observations from av ailable data and simulate dif ferent scenarios. The technique can generate large datasets as required for the intended analysis. B. Gibbs sampling applied W e apply Gibbs sampling to the joint biv ariate distribution of wind speeds at the players’ locations. Players’ wind speeds 2 https://badc.nerc.ac.uk/search/midas stations/ 3 http://www2.nationalgrid.com/UK/Industry- information/ Electricity- transmission- operational- data/Data- Explorer/ T ABLE I R E SU LT S F O R N = 100 R E AL I S A T I O N S A N D A N IN C R E AS I N G N U M BE R O F O B S ERV AT IO N S n Sample size ¯ w 1 σ ¯ w 1 WCI ¯ w 1 ME ¯ w 1 ¯ w 2 σ ¯ w 2 WCI ¯ w 2 ME ¯ w 2 ¯ P D σ ¯ P D WCI ¯ P D ME ¯ P D n = 1 , 000 12 . 1321 0 . 6569 0 . 2607 18 . 91% 12 . 2297 0 . 6226 0 . 2470 17 . 61% 108 . 5722 0 . 7859 0 . 3119 2 . 63% n = 5 , 000 12 . 0853 0 . 2903 0 . 1152 6 . 65% 12 . 1762 0 . 2797 0 . 1110 6 . 47% 108 . 6271 0 . 3395 0 . 1348 1 . 49% n = 10 , 000 12 . 0929 0 . 2262 0 . 0897 4 . 63% 12 . 1842 0 . 2187 0 . 0869 4 . 36% 108 . 6000 0 . 2403 0 . 0953 1 . 03% n = 50 , 000 12 . 1125 0 . 0874 0 . 0347 2 . 02% 12 . 2028 0 . 0857 0 . 0340 1 . 97% 108 . 5979 0 . 1033 0 . 0410 0 . 73% n = 100 , 000 12 . 1155 0 . 0631 0 . 0251 1 . 28% 12 . 2075 0 . 0602 0 . 0239 1 . 16% 108 . 5954 0 . 0663 0 . 0263 0 . 70% n = 200 , 000 12 . 1065 0 . 0441 0 . 0175 0 . 80% 12 . 1986 0 . 0427 0 . 0169 0 . 76% 108 . 5915 0 . 0453 0 . 0180 0 . 62% n = 500 , 000 12 . 1049 0 . 0272 0 . 0108 0 . 68% 12 . 1968 0 . 0262 0 . 0103 0 . 62% 108 . 5930 0 . 0288 0 . 0114 0 . 57% T ABLE II R E SU LT S F O R n = 5 , 000 SA M P LI N G SI Z E AN D A N I N CR E A S IN G N UM B E R O F RE A L I SAT IO N S N Realisations ¯ w 1 σ ¯ w 1 WCI ¯ w 1 ME ¯ w 1 ¯ w 2 σ ¯ w 2 WCI ¯ w 2 ME ¯ w 2 ¯ P D σ ¯ P D WCI ¯ P D ME ¯ P D N = 100 12 . 0850 0 . 2774 0 . 0840 6 . 64% 12 . 2017 0 . 2671 0 . 0809 6 . 01% 108 . 5703 0 . 3037 0 . 0919 1 . 34% N = 170 12 . 1120 0 . 2217 0 . 0879 5 . 35% 12 . 1812 0 . 2145 0 . 0851 5 . 15% 108 . 6275 0 . 3279 0 . 1301 1 . 26% N = 500 12 . 0867 0 . 2709 0 . 0476 7 . 22% 12 . 1764 0 . 2618 0 . 0460 7 . 16% 108 . 5932 0 . 3052 0 . 0536 1 . 47% N = 1 , 000 12 . 1097 0 . 2764 0 . 0343 8 . 20% 12 . 2025 0 . 2666 0 . 0331 7 . 57% 108 . 5949 0 . 3198 0 . 0397 1 . 47% N = 5 , 000 12 . 1044 0 . 2793 0 . 0155 9 . 06% 12 . 1966 0 . 2717 0 . 0151 8 . 81% 108 . 5951 0 . 3244 0 . 0180 1 . 58% N = 10 , 000 12 . 1032 0 . 2754 0 . 0108 9 . 41% 12 . 1956 0 . 2672 0 . 0105 8 . 60% 108 . 5917 0 . 3268 0 . 0128 1 . 70% N = 50 , 000 12 . 1022 0 . 2787 0 . 0049 9 . 87% 12 . 1943 0 . 2707 0 . 0048 9 . 26% 108 . 5901 0 . 3241 0 . 0057 1 . 70% Algorithm 1 G I B B S S A M P L I N G 1: w 1 , w 2 , P D  wind speed 1,2, power demand 2: T  number of samples 3: h w ( k ) 1 , w ( k ) 2 , P ( k ) D i , k ∈ { 1 , 2 , ..., k max }  historic data 4: F ( w 1 , w 2 )  wind distribution from data 5: G  P D , w 1 + w 2 2   demand cond. distrib . on mean wind 6: t ← 1 7: h w ( t ) 1 , w ( t ) 2 , P ( t ) D i ← sample ( w 1 , w 2 , P D )  initialise 8: repeat 9: w ( t +1) 1 ← sampl e F ( w 1 | w ( t ) 2 ) 10: w ( t +1) 2 ← sampl e F ( w 2 | w ( t +1) 1 ) 11: P ( t +1) D ← sampl e G P D | w ( t +1) 1 + w ( t +1) 2 2 ! 12: t ← t + 1 13: until t > T 14: return h w ( t ) 1 , w ( t ) 2 , P ( t ) D i , t ∈ { t burn , t burn + 1 , ..., T } at each time t form a MC. The methodology is described in Alg. 1. From av ailable historic data, we create a joint distribution table of wind speeds at the players’ locations. For ev ery possible wind speed w 1 of first player , we record the subset of w 2 wind speed, and vice versa (Line 4 in Alg. 1). This represents the conditional distrib utions. In practice,the probabilities for certain combinations of wind speeds can be low (e.g. it is unlikely to have extremely high wind at one location and lo w wind speed to a proximal location), therefore some subsets can be sparse, either because some of observations represent rare e vents or due to correlation. W e overcome this difficulty by merging sparse bins of rare ev ents and outliers, and ensure ergodicity of the MC. The MC is initialised by randomly selecting a sample from the joint distribution table (Line 7 in Alg. 1). Each iteration step in volves replacing the v alue of one v ariable by a v alue se- lected randomly by the conditional F ( w i | w − i ( t ) ) . In addition, demand is randomly selected by the conditional distribution of demand over the average wind speed (Line 11 in Alg. 1). The procedure is c ycled through the v ariables forming n samples of h w ( t ) 1 , w ( t ) 2 , P ( t ) D i , t ∈ { 1 , 2 , ..., T = n } . T o ensure that the MC con verges, we run Alg. 1 for se veral sampling sizes n (small, moderate, large) and repeat the procedure for N realisations. Results are shown in T ables I and II. Columns represent the sample mean, standard deviation of the sample mean, width of the 95% confidence interval (WCI) and maximum error (ME) from mean of historic data, i.e. µ w 1 = 12 . 1029 , µ w 1 = 12 . 1950 and µ P D = 108 . 1830 . As the sampling size increases the sample mean follows a normal distribution and the standard deviation decreases, as expected by the central limit theorem (T able I). W e perform the same analysis for an increasing number of realisations that use a different starting point (T able II). W e also adopt a burn-in or warm-up period of 20% of samples to make sure that our results are independent off the starting state [15]. The results sho w that MC con verges to the distribution from data and that a large n is required but N can be chosen to be relatively small. For this reason and driv en by computational limitations, for the estimation of the Stackelber g equilibrium, we selected n = 50 , 000 , N = 170 and burn-in of 10 , 000 samples. C. Stackelber g equilibrium estimation W ind speed data generated by the Gibbs sampler are used to estimate the per unit po wer output of wind generators. Estima- tion is based on a generic power curve 4 and a sigmoid function approximation (see Alg. 2 Line 4). Players’ strategies are the capacities they can install. The maximum feasible solution for a single player was set equal to P N max = 500 . 5 MW and the incremental capacity to step = 0 . 5 MW. For every possible 4 Parameters deriv ed by a 2 . 05 MW Enercon E82 wind turbine: http://www . enercon.de/en/products/ep- 2/e- 82/ Algorithm 2 G E N E R A T I O N & C U RTA I L M E N T E S T I M AT I O N 1: P N max  max rated capacity in search space 2: P N i ← [0 : step : P N max ]  strategy space i=1,2 player 3: α , β  power curve sigmoid parameters 4: P ( t ) G i ← 1 1 + e − α ( w ( t ) 1 − β ) · P N i  generation i player 5: for all P N 1 ∈ { 0 , ..., P N max } do 6: for all P N 2 ∈ { 0 , ..., P N max } do 7: RD ← P D − ( P G 1 + P G 2 )  residual demand 8: if RD > 0 then  no curtailment 9: RD ← 0 10: end 11: P C 1 ← P G 1 · RD P G 1 + P G 2  curtailment gen 1 12: P C 2 ← P G 2 · RD P G 1 + P G 2  curtailment gen 2 13: end 14: end 15: E G i ( P N i ) ← X P G i  total gen i 16: E C 1 ( P N 1 , P N 2 ) ← X P C 1  total curt 1 17: E C 2 ( P N 1 , P N 2 ) ← X P C 2  total curt 2 18: return ( E G 1 , E G 2 , E C 1 , E C 2 ) combination of the rated capacities installed ( P N 1 , P N 2 ) , we estimate the po wer generated and curtailed for each player on an hourly basis. Next, we estimate the aggregate power generated and curtailed by each player as the summation of 40 , 000 data points. The procedure is described in Alg. 2. For several cost parameters ( c G 1 , c G 2 , p T ) and feed-in tarif f price p G , we estimate the profits as defined in Eq. (1) and Eq. (2). For e very possible P N 1 , we find the capacity P ∗ N 2 that maximises the follower’ s profits Π ∗ 2 (follower’ s best response). From this set of solutions, the leader selects the one that maximises its own profit i.e. P ∗ N 1 (leader’ s best response). The equilibrium of the game is gi ven by the pair ( P ∗ N 1 , P ∗ N 2 ) , which satisfies best response functions as described in Alg. 3. V . R E S U LT S A N D D I S C U S S I O N The methodology described in previous sections was fol- lowed to run sev eral experiments. Fig. 2 shows the optimal rated capacities built by the players at the equilibrium of the game for N = 170 realisations. Recall here that e very realisation represents a completely different MC generated. The results are satisfactory and show a 10 M W range in the estimated solutions for optimal rated capacities. Similar results were observed for the optimal profits deriv ed. Moreov er , we study how the equilibrium results depend on v arying cost parameters. Fig. 3 shows the dependence on line in vestor’ s cost (first column), local generators’ cost (second column) and the transmission fee (third column). W e assume that both players can sell the energy generated for p G = £ 74 . 3 / MWh. For each scenario, the ke y parameter varies, while other parameters remain fixed ( Scenario 1 : c G 1 = 0 . 16 . . . 0 . 68 p G , c G 2 = 0 . 30 p G and p T = 0 . 26 p G , Algorithm 3 P R O FIT & S TAC K E L B E R G E Q U I L I B R I A 1: p G , p T  feed-in tariff, transmission fee 2: C T  cost of line 3: c G i  i player’ s generation cost 4: for all P N 1 ∈ { 0 , ..., P N max } do 5: for all P N 2 ∈ { 0 , ..., P N max } do 6: Π 1 ← ( E G 1 − E C 1 ) p G − E G 1 c G 1 + 7: ( E G 2 − E C 2 ) p T − C T 8: Π 2 ← ( E G 2 − E C 2 )( p G − p T ) − E G 2 c G 2 9: end 10: end 11: for all P N 1 ∈ { 0 , ..., P N max } do  best response gen 2 12: Π ∗ 2 ← max P N 2 Π 2 ( P N 1 , P N 2 ) 13: P ∗ N 2 ← arg max P N 2 Π 2 ( P N 1 , P N 2 ) 14: end 15: Π ∗ 1 ← max P N 1 Π 1 ( P N 1 , P ∗ N 2 )  best response gen 1 16: P ∗ N 1 ← arg max P N 1 Π 1 ( P N 1 , P ∗ N 2 ) 17: return Π ∗ 1 , Π ∗ 2 , P ∗ N 1 , P ∗ N 2 Number of iterations n 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 Rated capacity (MW) 100 105 110 115 120 125 130 135 140 145 150 Line investor Local generators Fig. 2. Optimal generation capacities of each player for several realisations ( c G 1 = 0 . 3 p G , c G 2 = 0 . 28 p G , p T = 0 . 26 p G and p G = £ 74 . 3 / MWh) Scenario 2 : c G 1 = 0 . 30 p G , c G 2 = 0 . 08 . . . 0 . 54 p G and p T = 0 . 26 p G and Scenario 3 : c G 1 = 0 . 26 p G , c G 2 = 0 . 20 p G and p T = 0 . . . 0 . 80 p G ). The results in Fig. 3 show the av erage equilibrium solution and min-max solutions, found for N = 170 realisations of the simulation procedure. In all sets of scenarios, the total capacity installed by all players decreases as the tested parameter value increases. Each player installs less capacity as their generation cost increases, while the other player benefits by increasing their capacity . The cost of local generators has a larger impact on the capacities installed for both players, as shown by comparing the first to the second column. Profits hav e similar behaviour to the op- timal rated capacities, but local generators face the additional cost of transmission char ges. If the follo wers’ generation cost is much lower than the line in vestor’ s (assuming for example that local generators might have access to cheaper land), the Cost of line investor c G1 (£/MWh) 10 15 20 25 30 35 40 45 50 Rated capacity (MW) 0 100 200 300 400 500 Line investor Local generators Cost of local generators c G2 (£/MWh) 5 10 15 20 25 30 35 40 Rated capacity (MW) 0 100 200 300 400 500 Line investor Local generator Transmission fee p T (£/MWh) 0 10 20 30 40 50 60 Rated capacity (MW) 0 100 200 300 400 500 Line investor Local generators Cost of line investor c G1 (£/MWh) 10 15 20 25 30 35 40 45 50 Profits (million £) -40 -20 0 20 40 60 80 100 120 Line investor Local generators Total profits Profit=0 Cost of local generators c G2 (£/MWh) 5 10 15 20 25 30 35 40 45 Profits (million £) -40 -20 0 20 40 60 80 100 120 Line investor Local generators Total profits Profit=0 Transmission fee p T (£/MWh) 0 10 20 30 40 50 Profits (million £) -40 -20 0 20 40 60 80 100 120 Line investor Local generators Total profits Profit=0 Fig. 3. Rows (1) and (2) sho w generation capacity b uilt and profits at Stackelberg equilibrium, respectively , column (1) sho ws dependency on generation cost of line inv estor, (2) on on generation cost of local generators and (3) on transmission fee line in vestor needs to charge a high transmission fee to have positiv e earnings. On the contrary , if the leader’ s cost is much lower , the generation capacity will mostly be installed by the line in vestor , as there is no room for profitable in vestment from local rene wable producers. As shown in Scenario 3, p T ' 0 . 16 p G or ' £ 12 MWh is the minimum v alue of transmission charges that allows profit for the line in vestor . Similarly , if the transmission fee is set too high, then it is not profitable for local in vestors to in vest in rene wable genera- tion. As p T is set by the system regulator , the methodology can be useful to determine a feasible range of charges that allo ws both transmission and generation in vestments to be profitable. V I . C O N C L U S I O N S & F U T U R E W O R K In this work we show how priv ately de veloped network upgrade for DGs can lead to a leader-follo wer game between the line and local inv estors. Curtailment and line access rules play a key role in the strategic game, the equilibrium of which determines optimal generation capacities and their profits. Settings where this model can be applied include numerous locations where demand and generation are not co-located. When real historic data is available, we can use MCMC and Gibbs sampling to simulate multiple future scenarios and reduce the uncertainty of the in vestment decisions. 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