Robust Transmission Network Expansion Planning Problem Considering Storage Units

Rob ust T ransmission Network Expansion Planning Problem Considering Storage Units ´ Alv aro Garc ´ ıa-Cerezo Department of Electrical Engineering Universidad de Castilla-La Mancha Ciudad Real, Spain Alvaro.Garcia29@alu.uclm.es Luis Baringo Department of Electrical Engineering Universidad de Castilla-La Mancha Ciudad Real, Spain Luis.Baringo@uclm.es Raquel Garc ´ ıa-Bertrand Department of Electrical Engineering Universidad de Castilla-La Mancha Ciudad Real, Spain Raquel.Garcia@uclm.es Abstract —This paper addresses the transmission network expansion planning problem considering storage units under uncertain demand and generation capacity . A two-stage adaptive rob ust optimization framework is adopted whereby short- and long-term uncertainties ar e accounted for . This work differs from pre viously reported solutions in an important aspect, namely , we include binary recourse variables to a void the simultaneous charging and discharging of storage units once uncertainty is re vealed. T wo-stage r obust optimization with discr ete recourse problems is a challenging task, so we propose using a nested column-and-constraint generation algorithm to solve the r esulting problem. This algorithm guarantees con vergence to the global optimum in a finite number of iterations. The performance of the proposed algorithm is illustrated using the Gar ver’s test system. Index T erms —Storage units, transmission netw ork expansion planning, two-stage rob ust optimization, uncertainty . I . I N T RO D U C T I O N The T ransmission Network Expansion Planning (TNEP) problem identifies the optimal reinforcements to be made in the transmission network of a power system. This problem is generally solved by a central entity , such as the system operator , that determines the transmission network in vestment decisions that are optimal for the power system as a whole, e.g., those inv estment decisions that minimize inv estment and operating costs. Since the pioneering work by Garver [1], the relev ance of the TNEP problem has motiv ated the de velopment of many models in the technical literature. Over the last decade, the integration of rene wable energy resources in power systems has given rise to the dev elopment of TNEP models that consider an uncertain en vironment in the decision-making process [2]. In this regard, tw o frame- works have been used to deal with this type of problem. Firstly , stochastic programming [3], which characterizes the uncertainty using a discrete set of realizations or scenarios. This uncertainty framew ork is adequate if the probabilistic distribution of uncertain parameters is kno wn; ho wev er, this task is not trivial. On the other hand, the number of scenarios This work has been partially funded by the Ministry of Science, In- nov ation, and Universities of Spain under Projects R TI2018-096108-A-I00 and R TI2018-098703-B-I00 (FEDER, UE), the Ministry of Education and Professional T raining of Spain under Grant 998142, and the Universidad de Castilla-La Mancha under Grant BI2018. needed to obtain an accurate representation of the uncer- tainty is generally very large, which may result in tractability issues. Secondly , robust optimization [4], which ov ercomes tractability issues associated with stochastic programming, but in contrast, solutions might be too conservati ve. Ne vertheless, computational tractability is more restrictiv e. Moreover , one of the aims of the TNEP problem is guaranteeing the supply of demands in all situations so that a conserv ativ e solution is not a drawback in this type of problems. Thus the rob ust framework is preferred by scientist and researchers to deal with this type of problems. W e also advocate and use a rob ust framework. In particular , we propose a two-stage adapti ve rob ust optimization approach. There are many references in the technical literature about the TNEP problem considering a robust optimization frame- work, such as, [5], [6]. Howe ver , none of them consider storage units, and we consider that this is a significant as- pect, especially since penetration of rene wable units in power systems is expected to significantly increase in the next years. Thus, it is relev ant to consider the impact of including storage units to store the renew able energy production during low- demand periods to be of use when is needed. The TNEP problem considering storage units under a robust framew ork is analyzed in [7]–[9]. Prevention of the simulta- neous charge and discharge of storage units is modeled in [7] through first-stage binary variables. Howe ver , this assumption is unrealistic because the proper way to deal with these vari- ables must be once uncertainty is re vealed during the recourse problem, i.e., considering them as second-stage variables. In contrast, [8] and [9] do not include binary variables to deal with the issue of simultaneous charging and dischar ging of storage units. They basically assume that this is not relev ant or significant. As shown in [10], if the storage charging price at any bus with storage units is higher than or equal to the locational marginal price, simultaneous charging and discharging is likely to occur . For this reason, in systems with high penetration of renewable generating units where there are periods with very low or null marginal prices, it is necessary to prev ent this undesirable situation. The consideration of binary variables in the lower -level problem prev ents the use of the traditional column-and- constraint generation algorithm to solve the two-stage adaptiv e 978-1-7281-1156-8/19/$31.00 ©2019 IEEE robust optimization problem, because the two lowermost opti- mization levels cannot be recast as an equi valent single-lev el optimization problem. W e propose solving the resulting model using the nested column-and-constraint generation algorithm described in [11], which to our knowledge is the only exact solution procedure to date for this kind of problems. In summary , the main contribution of this paper is to propose an adaptive robust optimization approach with integer recourse v ariables for the TNEP problem, which avoids the simultaneous charging and discharging of storage units and makes the problem tractable ev en for medium-large size sys- tems. In addition, we also consider the impact of both short- and long-term uncertainties in the decision-making process. Short-term uncertainties in demand and renewable production are considered through a set of representativ e days, while long-term uncertainties in demand growth and capacity of generating units are represented using confidence bounds. The rest of the paper is organized as follo ws. Section II provides the formulation of the problem. Section III explains the solution procedure. Results from a case study are reported in Section IV. Finally , Section V concludes the paper with some relev ant remarks. I I . C O M PAC T P R O B L E M F O R M U L A T I O N The TNEP problem is formulated using the following three- lev el adaptive robust optimization formulation in compact form: min x ,c wc e T x + c wc (1a) subject to: Kx = f (1b) Lx ≤ g (1c) x ∈ Z n (1d) c wc = ( max u ,c c (2a) subject to: u ∈ U (2b) c = " min y , z b T y (3a) subject to: Ax + B ( x ) y + Du = a : λ (3b) Fx + Gy + Hu + I ( x ) z ≥ d : µ (3c) z ∈ { 0 , 1 } m #) , (3d) where x is a vector representing first-stage expansion deci- sion variables; c and c wc are variables representing operating and worst-case operating cost, respecti vely; u is the vector of worst-case uncertainty realizations; y and z are second- stage variables (continuous and binary) representing operating decisions; and λ and µ are the lower-le vel dual variables. In our particular problem, the aim of the vector of binary variables z is to avoid the simultaneous charging and dis- charging of storage units. Additionally , A , D , F , G , H , K , and L are coefficient matrices; B ( x ) and I ( x ) are matrices whose elements depend on expansion v ariables; and a , b , d , e , f , and g are coef ficient vectors. Finally , U represents the uncertainty set that contains all possible materializations of the uncertain parameters, n is the dimension of vector x , and m is the dimension of vector z . Problem (1)-(3) is a three-le vel model. The upper-le vel (1) determines the expansion decisions (first-stage variables) minimizing in vestment and operating costs; the middle-le vel (2) detects the worst-case realizations of uncertainty sources for the inv estment plan identified by the upper -lev el, i.e., in this le vel uncertainty is rev ealed; and the lo wer-le vel (3) allo ws system operators to tak e a recourse action optimizing operating costs once first-stage variables and uncertainty are known. Upper-le vel problem (1) minimizes the total cost (1a), which includes both the in vestment and worst-case operating costs. Constraints (1b) define which transmission lines are initially built in the network. Constraints (1c) impose an inv estment budget and limit the number of storage facilities to be built. Constraints (1d) set out the integer nature of the vector of expansion variables x . The middle-le vel problem (2) determines the worst-case uncertainty realization, i.e., the uncertainty realization that leads to the largest operating cost (2a) once the expansion decisions are made by the upper-le vel. Constraints (2b) impose that uncertain variables (in our case demands and generation capacities) are characterized by uncertainty set U . It should be noted that we use a cardinality-constrained uncertainty set as described in [12]. In this reference, the uncertainty sets are modeled through the uncertainty budget Γ , representing the maximum number of random parameters that may reach their lower or upper limits. In our model, we consider uncertainty in demands, con ventional, and wind-po wer generating units, therefore we deal with three uncertainty budgets Γ D , Γ G , and Γ W , respectiv ely . The lower-le vel problem (3) identifies the minimum operat- ing cost given upper- and middle-level decisions; i.e., problem (3) identifies the system operating decisions that minimize the operating cost (3a). Constraints (3b) and (3c) model the operating feasibility set, comprising the power balance at each bus, capacity of con ventional and renewable generating units, power flo w limits through each transmission line, limits on load shedding, operation of each energy storage, limits on the energy storage le vels of each storage unit, no simultaneous charging and discharging of energy storage units, limits on the charging and discharging po wer of storage units, and definition of reference b us. Finally , constraints (3d) set out the binary nature of the vector of variables z . I I I . S O L U T I O N A P P ROA C H Problem (1)-(3) is an instance of mixed-integer three-lev el programming with lower -lev el binary variables. The proposed solution approach is the nested column-and-constraint gener- ation algorithm described in [11]. This method inv olves two loops. The outer loop comprises the iterative solution of a master problem and a max-min subproblem with lower-le vel binary variables. The solution of the max-min subproblem in volv es the iterative solution of two optimization problems, namely the inner-loop master problem and the inner-loop subproblem. Next, the master problem and the subproblem are presented. A. Master Pr oblem The master problem constitutes a relaxation for the original problem (1)-(3) where the second-stage problem is iterativ ely approximated by a set of valid operating constraints. The master problem at iteration j of the outer loop is formulated as the following mixed-integer problem: min x , y i , z i ,η e T x + η (4a) subject to: Kx = f (4b) Lx ≤ g (4c) x ∈ Z n (4d) η ≥ b T y i ; i = 1 , ... , j − 1 (4e) Ax + B ( x ) y i + Du ( i ) = a ; i = 1 , ... , j − 1 (4f) Fx + Gy i + Hu ( i ) + I ( x ) z i ≥ d ; i = 1 , ... , j − 1 (4g) z i ∈ { 0 , 1 } m ; i = 1 , ... , j − 1 , (4h) where the additional vectors of decision variables y i and z i , respectiv ely corresponding to y and z , are associated with the uncertain realizations identified by the subproblem at outer- loop iteration i through u ( i ) . The objective function (4a) is equiv alent to (1a) except for the term η , which is an approximation of the worst- case operating cost. Constraints (4b)-(4d) are equiv alent to (1b)-(1d), respectively . Constraints (4e) represents a lower bound for η . Constraints (4f)-(4h) correspond to lower -lev el constraints (3b)-(3d), respectiv ely . Note that non-linear terms B ( x ) y i in constraints (4e) in- clude the product of binary expansion-decision variables of matrix B ( x ) and continuous operating variables y , while non- linear terms I ( x ) z i in constraints (4f) include the product of integer expansion-decision variables of matrix I ( x ) and binary variables z . Both terms can be linearized as sho wn in [13] so that problem (4) is finally recast as a mixed-integer linear programming problem. B. Subpr oblem At each iteration j of the outer loop, the subproblem determines the worst-case uncertainty realizations yielding the maximum v alue of the minimum operating cost for a giv en upper-le vel decision pro vided by the previous master problem (4). The subproblem is a max-min problem comprising the two lowermost levels (2)-(3) parameterized in terms of giv en upper-le vel decision variables x ( j ) . Due to the presence of binary variables in the lower level, we propose solving such a max-min problem through an inner loop comprising the iterativ e solution of two optimization problems as described in [11], namely the inner-loop master problem and the inner-loop subproblem. 1) Inner-loop master pr oblem: The inner-loop master prob- lem at iteration j of the outer loop and iteration ` of the inner loop is formulated as follow: c wc = max ξ, u , λ k , µ k ξ (5a) subject to: u ∈ U (5b) B T ( x ( j ) ) λ k + G T µ k = b ; k = 1 , ... , ` (5c) µ k ≥ 0 ; k = 1 , ... , ` (5d) ξ ≤ ( λ k ) T  a − Ax ( j ) − Du  + ( µ k ) T  d − Fx ( j ) − Hu − I ( x ( j ) ) z ( k )  ; k = 1 , ... , `, (5e) where µ k and λ k are additional vectors of decision v ariables associated with the fixed values of z ( k ) , which are obtained from the solution of the inner-loop subproblems pre viously solved in the current inner loop iteration indexed by k . The objective function (5a) is the approximation of the worst-case operating cost at iteration j of the outer -loop, which is maximized subject to: 1) Constraints (5b) comprising the characterization of the uncertainty set. 2) Constraints (5c) and (5d), which correspond to lo wer- lev el dual feasibility constraints for fixed v alues of z . 3) Constraints (5e), where the v alue of the lower -level dual objectiv e function for fixed values of z is considered as an upper bound for ξ . Note that non-linear terms ( µ k ) T Hu and ( λ k ) T Du included in constraints (5e) are the product of binary uncertainty vari- ables u and continuous dual variables λ k and µ k , respectively . Both terms can be linearized as shown in [13]. 2) Inner-loop subpr oblem: The inner -loop subproblem at iteration j of the outer loop and iteration ` of the inner loop is formulated as follows: c ( ` ) = min y , z b T y (6a) subject to: Ax ( j ) + B ( x ( j ) ) y + Du ( ` ) = a (6b) Fx ( j ) + Gy + Hu ( ` ) + I ( x ( j ) ) z ≥ d (6c) z ∈ { 0 , 1 } m . (6d) The objectiv e function (6a) is identical to (3a), where c ( ` ) corresponds to the operating cost of the inner -loop iteration ` . Constraints (6b)-(6d) are identical to constraints (3b)-(3d), where the values of the expansion-decision v ariables x ( j ) are obtained by the master problem (4) at outer-loop iteration j and the values of uncertainty realizations vector u ( ` ) are identified by the inner-loop master problem (5) at inner-loop iteration ` . C. Algorithm The proposed nested column-and-constraint generation al- gorithm works as follow: 1) Initialization of the outer loop. • Initialize the outer-loop lower and upper bounds: LB o = −∞ and U B o = + ∞ . • Set the outer-loop iteration counter j to 1. • Set the vector of uncertain variables u ( j ) to their nominal values. 2) Master problem solution. • Solve problem (4). Obtain the optimal solution of the vector of expansion variables x ( j ) . • Update LB o to the objectiv e function of the master problem (4): LB o ← e T ( x ( j ) ) + η ( j ) . 3) Subproblem solution. 3.1) Initialization of the inner loop. • Initialize the inner-loop lower and upper bounds: LB i = −∞ and U B i = + ∞ . • Set the inner-loop iteration counter ` to 1. • Set the vector of uncertain variables u ( ` ) to their nominal values. 3.2) Solution of the inner-loop subproblem. • Solve problem (6) for given u ( ` ) and x ( j ) . Ob- tain the optimal solution of the v ector of binary variables z ( ` ) . • Update LB i to the maximum between the cur- rent value of LB i and the objecti ve function of problem (6): LB i ← max { LB i , c ( ` ) } . 3.3) Solution of the inner-loop master problem. • Solve problem (5) for given z ( k ) , k = 1 , ... , ` and x ( j ) . Obtain the optimal solution of the uncertain variables vector u ` . • Set u ( ` +1) ← u . • Update U B i to the objective function of problem (5): U B i ← c wc . 3.4) Inner-loop conv ergence checking. If U B i − LB i is lower than a predefined tolerance ε i , stop and set u ( j +1) ← u ( ` +1) . Otherwise, increase the inner-loop iteration counter: ` ← ` + 1 and go to step 3.2. 4) Adjustment of the outer-loop upper bound. Update U B o to the minimum between the current value of U B o and the sum of U B i and annualized inv estment costs: U B o ← min { U B o , e T ( x ( j ) ) + U B i } . 5) Outer-loop con ver gence checking. If U B o − LB o is lower than a predefined tolerance ε o , stop. Otherwise, increase the outer-loop iteration counter: j ← j + 1 and go to step 2. I V . C A S E S T U DY This section provides the results of a case study . A. Data The proposed approach is analyzed using the Garver’ s six- node test system [1] depicted in Fig. 1. This netw ork comprises Fig. 1. Garver’ s six-node test system. six buses, three con ventional generating units, one wind-power generating unit, one existing storage unit, fiv e demands, and six existing transmission lines. Note that bus 6, which includes a con ventional generating unit, is initially isolated. Howe ver , new transmission lines can be built connecting this bus with the rest of the network. In particular , we consider that the number of existing and prospective transmission lines per corridor can be equal to three at most. Additionally , we consider the possibility of building new storage units at b us 6. Generating units, storage units, demand, and transmission line data are giv en in T ables I, II, III, and IV, respectiv ely . W e consider that conv entional and wind-power generating units can experience a maximum deviation of 50% from their nominal capacities, while demand lev els can increase up to a maximum of 20%. W e consider an inv estment return period of 25 years and a discount rate of 10%, which results in an annual amortization rate of 11%. W e assume a total in vestment b udget of e 60 million. Furthermore, we consider that the con ver gence tolerances for the outer and the inner loops are 10 − 6 . Although in this paper the formulation of the problem has focused on modeling the long-term uncertainty , special attention should be also given to short-term uncertainty , which plays a relev ant role in the lower -level problem. W e consider that the short-term uncertainty has influence on the demand and wind-po wer production. In order to model the short-term uncertainty , we apply a modified version of the K-means clustering technique described in [14] to the demand and wind historical data during 2016 in T exas [15]. W e obtain 10 representativ e days of demand and wind-power production conditions, each one of them composed by 24 hourly data. Simulations were run using CPLEX 12.7.0.0 [16] under GAMS 24.8.3 [17] on a Gigabyte R280-A3C with 2 Intel Xeon E5-2698 at 2.3 GHz and 256 GB of RAM. T ABLE I G E NE R A T I N G U N IT D A TA Nominal Operation T echnology Bus capacity cost (MW) ( e /MWh) Con ventional 1 150 60 Con ventional 3 350 65 Con ventional 6 500 70 W ind-power 3 400 0 T ABLE II S TO R AG E U N I T D AT A Bus 3 Bus 6 Maximum energy (MWh) 100 200 Initial energy (MWh) 1 2 Charging po wer capacity (MW) 20 40 Discharging po wer capacity (MW) 20 40 Charging ef ficienty (%) 82 82 Discharging ef ficienty (%) 100 100 Maximum number of - 3 units that can be b uilt In vestment cost ( 10 3 e ) - 10,000 T ABLE III D E MA N D D A T A Bus Nominal Load-shedding lev el (MW) cost ( e /MWh) 1 100 11,250 2 300 11,500 3 50 12,000 4 200 11,000 5 300 11,200 T ABLE IV T R AN S M I SS I O N L I NE D A TA From T o Reactance Capacity In vestment Bus Bus (p.u.) (MW) cost ( 10 3 e ) 1 2 0.40 100 7,723.20 1 3 0.38 100 7,337.04 1 4 0.60 80 11,584.80 1 5 0.20 100 3,861.60 1 6 0.68 70 13,129.44 2 3 0.20 80 3,861.60 2 4 0.40 100 7,723.20 2 5 0.31 100 5,985.48 2 6 0.30 100 5,792.40 3 4 0.59 82 11,391.72 3 5 0.20 70 3,861.60 3 6 0.48 100 9,267.84 4 5 0.63 75 12,164.04 4 6 0.30 100 5,792.40 5 6 0.61 78 11,777.88 T ABLE V C A SE S T U DY R E S U L T S U N D ER D IFF E R E NT U N C E RT A I N T Y L EV E L S Uncertainty lev el Γ D = 0 Γ D = 2 Γ D = 3 Γ D = 5 Γ G = 0 Γ G = 1 Γ G = 2 Γ G = 3 Γ W = 0 Γ W = 0 Γ W = 1 Γ W = 1 2-3 (x2) 1-5 (x1) 2-3 (x2) 2-5 (x1) 2-3 (x1) 2-3 (x1) New lines built 3-5 (x2) 3-5 (x2) 2-6 (x3) 2-6 (x3) 4-6 (x2) 3-6 (x1) 3-5 (x2) 3-5 (x2) 4-6 (x3) 4-6 (x1) 5-6 (x1) New storage units 2 0 2 3 built at node 6 In vestment costs (M e ) 47 60 59 59 T otal annual costs (M e ) 94 162 1,497 2,624 CPU (s) 28 49,255 16,549 636 # outer-loop iterations 1 5 4 2 # inner-loop iterations 1 2, 2, 2, 2, 3 2, 3, 2, 2 11, 2 B. Results W e have solved the TNEP problem under different uncer- tainty levels, obtaining the expansion decisions, inv estment costs, total annual costs, computation times, and number of outer- and inner-loop iterations provided in T able V. From this table we reach the conclusion that expansion decisions are clearly influenced by uncertainty lev els. In particular, building storage units is more attracti ve than building transmission lines for the case of maximum uncertainty le vel according to the results obtained. Moreo ver , in all cases e valuated new lines are built connecting bus 6 with the rest of the network. Regarding computational times, which correspond to the running time used until the outer-loop conv ergence is attained, we observe that the time needed is lower when the uncer- tainty b udgets are null or equal to the number of demands, con ventional, and wind-power generating units, respecti vely . This is an expected result because in those cases the w orst- case scenario is known in adv ance. In addition, the number of outer-loop iterations needed to solve the problems follows the same behavior as the computational time. Regarding the inner- loop problems, the number of iterations required to solve the subproblem is generally between one and three except for the case considering that all random parameters may reach their lower or upper limits, which requires 11 iterations during the first outer-loop iteration. In order to demonstrate the importance of including binary variables in the model to av oid simultaneous charging and discharging of storage units, we hav e solved the problem without considering them. This problem has been solved using the traditional column-and-constraint generation algorithm, obtaining simultaneous charging and discharging in some cases, which highlights the importance of including these binary variables due to their potential impact on expansion decisions. Finally , note that the solution of the problem would be different depending on the number of representati ve days con- 3 4 5 6 7 8 9 10 11 12 2 , 200 2 , 300 2 , 400 2 , 500 2 , 600 2 , 700 Number of representativ e days T otal annual cost (M e ) Fig. 2. T otal annual cost considering dif ferent number of representative days. sidered. This f act has been analyzed using dif ferent numbers of representativ e days in order to find when the total annual cost of the TNEP problem becomes stable. W e have considered the uncertainty budgets Γ D = 5 , Γ G = 3 , and Γ W = 1 , obtaining the results depicted in Fig. 2. It can be seen that the total annual cost presents a stable value selecting a number of representative days greater than 8. For that reason, we hav e chosen 10 representativ e days to model the short-term uncertainty . V . C O N C L U S I O N S This paper proposes a new model to solve the TNEP prob- lem considering storage units and avoiding the simultaneous charging and discharging of those units. T o that end, we adopt a two-stage adaptive robust optimization frame work and include binary variables in the second stage v ariable set. The resulting three-level formulation forces us to use a nested column-and-constraint generation algorithm, whose descrip- tion is also giv en. Finally , we hav e analyzed the influence of the uncertainty le vel in expansion decisions, in vestment and operating costs, computational times, and number of outer- and inner-loop iterations on a case study using the Garver’ s six-node test system. Giv en the theoretical framew ork and the results of the case study , the conclusions below are in order: 1) It is necessary to use binary variables in the third- lev el to avoid the simultaneous charging and dischar ging of storage units, which prev ents the use of traditional column-and-constraint generation algorithms. 2) Disregarding the pre vention of simultaneous char ging and discharging of storage units may have a great impact on expansion decisions, especially in systems with high renew able penetration. 3) There is no clear pattern among uncertainty le vels, ex- pansion decisions, and the computational b urden of the problem. Future work comprises a more detailed description of the effect of simultaneous charging and discharging of storage units in larger systems, as well as the inclusion of transmission line contingencies in the formulation. R E F E R E N C E S [1] L. L. 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