A three-level MILP model for generation and transmission expansion planning

Transcription

1 A three-level MILP model for generation and transmission expansion planning David Pozo Cámara (UCLM) Enzo E. Sauma Santís (PUC) Javier Contreras Sanz (UCLM)

2 Contents 1. Introduction 2. Aims and contributions 3. The three-stage transmission planning model: 1. Third stage: Market clearing 2. Second stage: Generation investment equilibria 3. First stage: Transmission investment plan 4. Case studies 5. Conclusions 2

3 Introduction Several techniques have been applied to investigate power systems transmission planning: linear programming, MILP, Benders decomposition, dynamic programming. Other authors propose the use of heuristics: genetic algorithms, simulated annealing, agent-based systems and game theory. Other methods integrate transmission expansion planning within a pool-based market. One of this works by Sauma and Oren (26) introduces a methodology to assess the economic impact of transmission investment anticipating the strategic response of oligopolistic generation companies. 3

4 Introduction Sauma and Oren (26) formulate a three-period model for studying how the exercise local market power by generation firms affects the equilibrium between generation and transmission investments and the valuation of different transmission expansion projects. The methodology is based on an iterative process to find the equilibrium but does not solve the optimal transmission planning, only evaluates the social welfare impact of some predetermined transmission expansion projects. Other authors, Motamedi et al. (21), use an agent-based system where a generation company is a learning agent and uses a heuristic method to solve the same problem in four stages: bidding strategies, market clearing, generation investment and transmission expansion. 4

5 Introduction Additionally, other works propose multi-period models to characterize investments, like Murphy and Smeers (25). They use a two-stage model of investment in generation capacity. Generation investment decisions are made at the first stage (subject to equilibrium constraints) while spot market operations occurs at the second stage. Garcés et al. (29) propose a bilevel model where the transmission planner minimizes transmission investment costs in the upper level and the lower level represents the market clearing. The bilevel model is reformulated as a mixed-integer linear problem using duality theory. 5

6 Aims and contributions We present an approach that extends and transforms the three-level model by Sauma and Oren (26) into a one-level MILP optimization problem. Our model integrates transmission planning, generation investment and market operation decisions anticipating both the equilibria of generation investments in a decentralized market and the market clearing equilibria. We characterize the equilibria of generation investments made by decentralized firms (an EPEC: Equilibrium Problem subject to Equilibrium Constraints) as a set of linear inequalities. 6

7 Aims and contributions We consider the generation investment EPEC as a set of linear constraints that the network planner can impose in its transmission planning convex optimization problem. This makes possible to obtain an optimal transmission plan that anticipates both generation investments and market operation equilibria. We calculate all possible pure Nash equilibria of generation investment problem (EPEC). We linearize the entire model obtaining a MILP formulation. 7

8 Three-level model formulation We assume that the transmission planning model consists of three stages that are described in reversed order, since the first stage represents the final decision by the transmission planner. Our model is of the Stackelberg type, where the transmission planner (first level) anticipates generation expansions (second level), and the clearing of the market (third level). 8

9 Three-level model formulation In the third level we model the energy market operation equilibrium where the independent system operator (ISO) clears the perfectly competitive market and the generator companies (GENCOs) optimize profits from bidding at marginal costs. In the second level each GENCO anticipates to the result of the third level in order to plan his own capacity expansion. The problem is modeled as an MPEC (Mathematical Problem with Equilibrium Constraints) per GENCO. We obtain the Nash equilibrium when all firms optimize their investment strategies, each one running an MPEC model. The extension to consider all firms is an EPEC. In the first level the transmission planner invests on transmission lines anticipating the Nash equilibrium at the second level à la Stackelberg. 9

10 Three-level model formulation Level 1 Transmission Investment (Maximize social welfare minus investment cost on lines) Optimal decisions: transmission expansion plan: f l Level 2 Generation Investment (Maximize GENCOs profits minus investment cost on capacity) Optimal decisions: generation capacity expansion g i e Level 3 Pool-Based Market Operation (Equilibrium of ISO and GENCOs) Optimal decisions: market operation q ie, β i e 1

11 Three-level model formulation We assume there is a spot market in which the GENCOs are able to submit their energy bids. Our model produces locational marginal prices (LMPs) as a result of linear network constraints. Demand is inelastic and there are different demand profiles by selecting a set of equivalent scenarios for each demand profile. The model considers transmission network constraints through a lossless DC approximation assuming price-taker generators. LMPs are given by the Lagrange multipliers of the energy constraint at each node. All nodes can have both demand and generation and all generation capacity at a node is owned by a single GENCO. 11

12 Third level: Market clearing In the third level we obtain the equilibrium that occurs when the ISO clears the market for given generation and transmission capacities. Marginal generation costs are constant and inversely proportional to the new installed capacity: c i g i, g i = a i b i (g i g i ). c i a i b i g i g i 12

13 Third level: ISO problem The ISO problem is modeled as a cost minimization one (dual variables are presented to the right of each constraint): min q i,r i i c i g i, g i q i = min q i,r i i a i q i b i g i g i q i (1) subject to: q i g i : ξ i i N (2) r i i N f l = : α (3) φ l,i i N r i f l : λ + l, λ l l L (4) q i r i = d i : β i i N (5) q i : γ i i N (6) 13

14 Third level: ISO problem The objective function (1) minimizes the total cost of generation. Constraint (2) establishes the maximum power that the GENCOs can produce. Constraint (3) indicates that network losses are negligible. Constraint (4) expresses the maximum flow through lines as a function of the power transfer distribution factors (PTDFs). Constraint (5) meets demand at every node as a function of the node injection and the flow coming from the lines connected to this node. Constraint (6) forces power generation to be non negative at every node. 14

15 Third level: ISO problem The Karush-Kuhn-Tucker (KKT) conditions equivalent to (1)-(6) are given by: a i b i g i g i γ i β i + ξ i = : q i i N (7) α + l L λ l + λ l φ l,i β i = : r i i N (8) γ i q i i N (9) ξ i g i q i i N (1) λ l f l + φ l,i r i l L (11) i N λ + l f l φ l,i r i l L (12) i N i N r i = : α (13) q i r i = d i : β i i N (14) 15

16 Third level: GENCO problem Each individual GENCO maximizes its profit considering the income from sales at nodal market prices provided by the ISO cost minimization: max q i i N G β i q i a i q i b i g i g i q i (15) s.t. q i g i : ξ i i N G (16) q i i N G (17) 16

17 Third level: GENCO problem Let s call primal to the problem in (15)-(17). Thus, from the duality theorem (Luenberger and Ye, 28), we know that if either the primal or the associated dual problem has an optimal solution, then the other one has the same optimal solution. Since both primal and dual problems are linear in this case, the problem is convex and we can also apply the strong duality theorem (Luenberger and Ye, 28). Thus, we get (18) from applying the strong duality theorem: β i q i a i q i b i g i g i q i i N G = g i ξ i i N G G (18) 17

18 Third level: Market clearing Using the Fortuny-Amat linearization formula, we have that the set of constraints (19)-(3) fully represents level 3 of our model: a i b i g i g i γ i β i + ξ i = : q i i N (19) α + l L λ l + λ l φ l,i β i = : r i i N (2) i N r i = : α (21) q i r i = d i : β i i N (22) γ i BM γ i (η γ i ) i N (23) q i BM γ i(1 η γ i ) i N (24) λ l BM λ l λ (η l ) l L (25) 18

19 Third level: Market clearing f l + φ l,i r i BM λ l λ (1 η l ) l L (26) i N λ l + BM λ l + (η l λ + ) l L (27) f l φ l,i r i BM λ l + λ (1 η + l ) l L (28) i N ξ i BM ξ i(η ξ i ) i N (29) g i q i BM ξ i(1 η ξ i ) i N (3) 19

20 Second level: Generation investment In the second level, each GENCO determines the investments in generation capacity to increase its profits due to the linear decrease in the generation marginal costs: max g i U G = β i q i c i g i, g i q i CIG g i, g i i N G = β i q i a i q i b i g i g i q i K i g i g i (31) i N G s.t. (19) - (3) Using (18), we can rewrite the problem as: max g i U G = g i ξ i K i g i g i i N G (32) s.t. (19) - (3) 2

21 Second level: Generation investment The only non-linear term in (32) is g i ξ i. Since the g i variables are controlled by the generation firms and it is possible for the generation expansion to be done in discrete steps, then we use a binary expansion (Pereira et al., 25) to discretize g i : Λ i g i = g i + Δ gi 2 k y ki k= i N G (33) Accordingly, the non-linear product g i ξ i can be replaced by the expression: Λ i g i ξ i = g i ξ i + Δ gi 2 k y ki k= i N G (34) where we define y ki by constraints (35) and (36), using the Fortuny-Amat linearization formula: ξ i y ki BM(1 y ki ) i N G, k =,1,, Λ i (35) y ki BM(y ki ) i N G, k =,1,, Λ i (36) 21

22 Second level: Generation investment The generation expansion problem of each GENCO can be set as a linear Mathematical Program subject to Equilibrium Constraints (MPEC), as shown in (37) (51): max U G = g i ξ i + Δ gi 2 k y ki g i s.t. i N G Λ i k= Λ i K i Δ gi 2 k y ki k= (37) Λ i a i b i Δ gi 2 k y ki k= γ i β i + ξ i = : q i i N (38) α + l L λ l + λ l φ l,i β i = : r i i N (39) i N r i = : α (4) q i r i = d i : β i i N (41) 22

23 Second level: Generation investment γ i BM γ i (η i γ ) i N (42) q i BM γ i(1 η i γ ) i N (43) λ l BM λ l (η l λ ) l L (44) f l + φ l,i r i BM λ l λ (1 η l ) l L (45) i N λ l + BM λ l + (η l λ + ) l L (46) f l φ l,i r i BM λ l + λ (1 η + l ) l L (47) i N ξ i BM ξ i(η i ξ ) i N (48) Λ i g i + Δ gi 2 k y ki k= q i BM ξ i(1 η i ξ ) i N (49) ξ i y ki BM(1 y ki ) i N G, k =,1,, Λ i y ki BM(y ki ) i N G, k =,1,, Λ i (5) (51) 23

24 Second level: Generation investment Level 2 problem can be formulated as an Equilibrium Problem with Equilibrium Constraints (EPEC) in which each firm a mixed integer linear programming (MILP) MPEC problem faces given the other firms commitments and the system operator s import/export decisions). This EPEC represents the equilibrium when all the GENCOs expand their capacity simultaneously subject to the market equilibrium of level 3. We enumerate the GENCOs investment strategies and express the Nash equilibria conditions as a finite set of inequalities. 24

25 Second level: Generation investment To characterize the EPEC equilibria as a set of linear inequalities, we discretize all generation investment strategic variables of the problem. The general expression for the Nash equilibrium is given by: U G e (g i e, i N) max g i U G g i, g e i, i N G, i N G G Ψ (52) where, for all GENCOs, U G e (g i e ) is the utility function of each GENCO G given its strategic decision variable g i e in the Nash equilibrium, which is always better than any other utility resulting from a different strategy, assuming that the other GENCOs use their Nash equilibrium strategies, g e i. Hence, the Nash equilibrium in (52) is solved by approximating its solution using discrete strategies. In doing that, we replace expression (52) by a set of inequalities, where the strategic variables are discretized for each GENCO. 25

26 Second level: Generation investment The Nash equilibria of the GENCOs capacity investment decisions are given by the following set of inequalities: U G e g i e, i N (53) s U G s G G gi, g e i, i G Ψ, s G S G N G, i N G where we have to distinguish between the left hand side (LHS) and the right hand side (RHS) of (53). 26

27 Second level: Generation investment The LHS in (53) is the utility function of each GENCO given its strategic decision variable in the Nash equilibrium. That is, the definition of the utility function for GENCO G in the equilibrium is given by: Λ i Λ i e U G = g i ξ e i + Δ gi 2 k e y ki K i Δ gi 2 k e y ki G Ψ (54) i N G k= k= subject to the linearized constraints of stage 3 in the equilibrium, which correspond to constraints (38)-(51), but replacing y ki, y ki, q i, r i, γ i, β i, ξ i, α, λ l +, λ l, η i γ, η l λ +, η l λ, and η i ξ by y e ki, y e ki, q e i, r e i, γ e i, β e i, ξ e i, α e +, λ e l, λ e γ l, η e λ i, η + e λ l, η e ξ l, and η e i, respectively, and considering (5) and (51) for all i N. 27

28 Second level: Generation investment The RHS in (53) is the utility function of each GENCO given a particular value of the strategic decision variable. That is, considering firm G chooses strategy s G (which involves investing in s generation capacity at node i up to the capacity g G i, with i N G, G Ψ), the definition of the utility function for GENCO G is given by: U G s G = g i s G ξ i s G Ki g i s G g i i N G G Ψ, s G S G (55) subject to the corresponding constraints of stage 3, which correspond to constraints (38)- (51), but considering them G Ψ, s G S G, replacing y ki, y ki, q i, r i, γ i, β i, ξ i, α, λ l +, λ l, η i γ, η l λ +, η l λ, and η i ξ s y G s ki, y G s ki, q G s i, r G s i, γ G s i, β G s i, ξ G i, α s G +, λ sg s l, λ G γ s l, η G λ i, η + sg λ l, η sg ξ s l, and η G i, by respectively, and replacing (38) by (56) and (57), (49) by (58) and (59), (5) by (6), and (51) by (61): 28

29 Second level: Generation investment a i b i g i s G g i γ i s G β i s G + ξ i s G = i N G, G Ψ, s G S G (56) Λ i a i b i Δ gi k= 2 k e y ki γ i s G β i s G + ξ i s G = i N G, G Ψ, s G S G (57) g i s G q i s G BM ξ i(1 η i ξ s G ) i N G, G Ψ, s G S G (58) Λ i g i + Δ gi 2 k e y ki k= q i s G BM ξ i (1 η i ξ s G ) i N G, G Ψ, s G S G (59) s ξ G s i y G ki BM(1 y e ki ) s y G ki BM(y e ki ) k =,1,, Λ i, i N G, G Ψ, s G S G k =,1,, Λ i, i N G, G Ψ, s G S G (6) (61) 29

30 First level: Transmission investment In level 1, the network planner which we model as a Stackelberg leader in our 3-level game maximizes the social welfare subject to the transmission constraints while anticipating the solutions from levels 2 and 3. Since we have considered inelastic demands, this problem is equivalent to minimize the total cost: sum of generation dispatch costs and transmission investment costs. Thus, the objective function of the transmission planner in level 1 is: min f l,l L inv i c i g i, g i q i + CIL f l, f l l L inv (62) = min f l,l L inv i a i q i b i g i g i q i + K l f l f l l L inv 3

31 First level: Transmission investment The problem of minimizing (62) subject to the transmission constraints and the constraints representing the solutions from levels 2 and 3 is non linear. Moreover, the variables that e represent the solution to the EPEC are equilibrium results, thus, having q i instead of q i, and e g i instead of g i. The non-linear term in the objective function, g e i q e i, can be decomposed e using the binary expansion applied to g i and linearized using the Fortuny-Amat formulation. This yields: g i e q i e = g i q i e + Δ gi Λ i k= 2 k e y ki i N G (63) q e i y e ki BM q (1 y e ki ) i N G, k =,1,, Λ i (64) y e ki BM q (y e ki ) i N G, k =,1,, Λ i (65) where y e ki is a continuous variable taking the values of either or q e i. 31

32 First level: Transmission investment In (62), we have considered that there is a set of transmission lines that are candidates for investment L inv. That means that the previously-constant maximum active flows f l are now variables of the problem in level 1. Note that, contrary to the assumptions in (Sauma and Oren, 26), the network planner now solves level 1 for the optimal transmission expansion capacities in both new and existing lines within the set of candidate locations. Therefore, we can formulate level 1 problem as a mixed integer linear programming optimization program subject to EPEC and other equilibrium constraints. 32

33 Finding all Nash equilibria in level 2 The EPEC for the level 2 problem may have multiple equilibria. The model described finds only one EPEC equilibrium, but we could be interested in detecting more than one equilibrium, or even all of them. We modify the previous section model of the level 2 problem in order to find all pure strategy EPEC equilibria. To do that, we generate holes in the feasible region for each solution found within the set of discrete strategies: y kie. Given a solution vector of the EPEC problem of level 2, we include a new constraint to generate a hole in the solution already found. 33

34 Finding all Nash equilibria in level 2 i,k (y ki (n) y e ki ) 2 ε n (66) Each one of the quadratic terms in (66) is expanded as: y ki e y 2 ki = y 2 e ki + y 2 e ki 2 y ki n y ki (67) and, using the fact that y ki and y e ki are binary numbers, (67) is equivalent to: y ki + y e ki e 2 y ki n y ki (68) which is a linear expression. Thus, (67) becomes: y ki + y e ki e 2 y ki n y ki ε 2 n (69) i,k 34

35 Variation of line impedance in level 1 Impedance x l x l 2 f l 2f l Line capacity Link impedance as a function of transmission capacity. 35

36 PTDFs calculation in level 1 In order to apply this modification to the model proposed for stage 1, and keep the level-1 formulation as a mixed integer linear programming optimization program, continuously changing the power transfer distribution factors (PTDFs) seems unviable due to the nonlinearities involved. Instead, we consider a discretization of the equivalent impedance in the potentially-expanded lines and calculate the associated PTDFs. 36

37 Variation of line impedance in level 1 If the initial transmission capacity of link l is f l and investments can be done up to a capacity whose value is f l max, we can approximate the equivalent impedance by performing a discrete approximation, between f l and f l max Equivalent impedance x l 1 x l 2 x l 3 x l f l f l 1 f l 2 f l 3 = f l max Final capacity Discretization of the equivalent impedance as a function of installed transmission capacity. 37

38 Case studies: 3-node example 2 l 1 l 3 1 l 2 3-node case study data 3 Node Demand Generation units: Production costs parameters Unit cost of investment i d i [MW] g i [MW] a i [$/MWh] b i [$/(MW MWh)] K i [$/MW]

39 Case studies: 3-node example PTDFs for the four considered states in the 3-node network, when investing in line 1 only Case of no link investment Case of investment in interval [7 8.4] MW Case of investment in interval [ ] MW Case of investment in interval [1.5 14] MW

40 Case studies: 3-node example The thermal capacity for each line is 7 MW and the unit transmission investment cost (K l ) is $25/MW for each line. We consider 4 possibilities for transmission investment: investment in line 1, investment in line 2, investment in line 3, and investment affecting lines 1, 2, and 3 simultaneously. The flow limit according to our discretization process is 14 MW and there are four states for the expansion line capacity: no investment, line flow bound between 7 and 8.4 MW, line flow bound between 8.4 and 1.5 MW, and line flow bound between 1.5 and 14 MW. For the second level of our model, we assume the three GENCOs can invest in generation capacity from 3 MW up to 54 MW at intervals of 1.6 MW. 4

41 Link impedance (p.u.) Case studies: 3-node example Capacity investment factor Link impedance as a function of the capacity in line 1. 41

42 Case studies: 3-node example Solving the problem of level 1, for the case of investing in line 1 only, we obtain that the optimal value is to invest up to 7.4 MW of capacity for line 1. The GENCO in node 1 invests 14.4 MW in generation capacity, meaning that its total production becomes 44.4 MW in level 3. The GENCO in node 1 becomes the most economic unit, whose marginal cost is $2.68/MWh, and the production of this GENCO is partially consumed at node 1 (3 MWh) and partially sent through lines 1 and 2. This yields the same LMPs for all the nodes and the minimum cost of dispatch. 42

43 Case studies: 3-node example Optimal market clearing values given the solutions of level 1 and 2 in the 3-node network Available capacity for Node Profits [$] each GENCO [MW] LMP [$/MWh] Production [MWh] Optimal values of the problem for level 1 of the 3-node network Case Cost for the transmission planner [$] Line capacity [MW] Available capacity [MW] g 1 g 2 g 3 l (l 1 ) l (l 2 ) l (l 3 ) l 1, l 2 & l (l 1 ) (l 2 )

44 Case studies: 3-node example Note that if we fix the investment in line 1 and we solve the EPEC problem by applying the methodology to find all pure Nash equilibria, we obtain two more equilibria. In the optimization process of level 1, the transmission planner attempts to anticipate the EPEC equilibrium by choosing the best possible solution for level 2. However, this cannot be guaranteed. Hence, we solve the level 1 problem using what we call an optimistic solution for the transmission planner, which considers that the transmission planner anticipates the best (from the social welfare viewpoint) EPEC equilibria. There is also a pessimistic solution for the transmission planner, which considers the worst EPEC equilibria. 44

45 Dispatch plus line investment cost Case studies: 3-node example Capacity of line 1 Optimistic and pessimistic level-1 solutions for the case of investing only in line 1. 45

46 Case studies: 4-node example 2 l 1 l 3 4 l 4 1 l node example data Node Demand Generation units: Production cost parameters Unit cost of investment i d i [MW] g i [MW] a i [$/MWh] b i [$/(MW MWh)] K i [$/MW]

47 Case studies: 4-node example 4-node example line data Line Initial thermal limit capacity Unit transmission investment cost Maximum thermal limit capacity l f l [MW] K l [$/MW] max f l [MW]

48 Case studies: 4-node example Optimal values of the problem for level 1 of the 4-node network Case Cost for the transmission planner [$] Line capacity [MW] Available capacity [MW] g 1 g 2 g 3 g 4 l 1, l 2, l 3 & l (l 1 ) (l 2 ) (l 4 )

49 Conclusions A MILP three-level model for transmission investment is proposed: Level 1: Transmission planner Level 2: Equilibrium in generation expansion Level 3: Market clearing The transmission planner anticipates the generation expansion decisions and market clearing à a la Stackelberg. The Nash equilibrium of generation expansion uses an EPEC framework. All possible pure Nash equilibrium are obtained. Line impedance approximation as a function of installed capacity. 49

50 A three-level MILP model for generation and transmission expansion planning David Pozo Cámara (UCLM) Enzo E. Sauma Santís (PUC) Javier Contreras Sanz (UCLM)