Jan 5, 2012 - Case studies: SIC. 26. 1 Diego de Almagro 220. 2 Carrera Pinto 220. 3 Cardones 220. 4 Maitencillo 220. 5 Pan de Azucar 220. 6 Los Vilos 220.
A three‐level MILP model for generation and transmission expansion planning David Pozo (UCLM) Enzo Sauma (PUC) Javier Contreras (UCLM) January 05th, 2012
Outline 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
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 (2006) introduces a methodology to assess the economic impact of transmission investment anticipating the strategic response of oligopolistic generation companies. 3
Introduction Sauma and Oren (2006) 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. (2010), 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
Introduction Additionally, other works propose multi‐period models to characterize investments, like Murphy and Smeers (2005). 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. (2009) 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
Aims and contributions We present an approach that extends and transforms the three‐level model by Sauma and Oren (2006) 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
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
Three‐level model formulation
Level 1
Transmission Investment (Maximize social welfare minus investment cost on lines)
Optimal decisions: transmission expansion plan: fl
Level 2
Generation Investment (Maximize GENCOs profits minus investment cost on capacity)
Optimal decisions: generation capacity expansion gie
Level 3
Pool-Based Market Operation (Equilibrium of ISO and GENCOs)
Optimal decisions: market operation qie , βie
8
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
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.
10
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:
,
0
0
.
ci
ai
bi
0 gi
gi 11
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
,
,
0
min
0
(1)
,
subject to: ∀ ∈
: :
0
(2) (3)
∈ ,
:
,
∀ ∈
(4)
:
∀ ∈
(5)
:
∀ ∈
(6)
∈
0
12
Third level: ISO problem The Karush‐Kuhn‐Tucker (KKT) conditions equivalent to (1)‐(6) are given by: 0
0 0
,
:
∀ ∈
(7)
:
∀ ∈
(8)
∀ ∈
(9)
∀ ∈
(10)
∈
0
0 0
0
0
,
0
∀ ∈
(11)
,
0
∀ ∈
(12)
∈
0 ∈
0
(13)
:
∈
:
∀ ∈
(14)
13
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: 0
max
(15)
∈
s.t. : 0
∀ ∈
(16)
∀ ∈
(17)
14
Third level: GENCO problem Let’s call primal to the problem in (15)‐(17). Thus, from the duality theorem (Luenberger and Ye, 2008), 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, 2008). Thus, we get (18) from applying the strong duality theorem: 0 ∈
∀
(18)
∈
15
Third level: Market clearing Using the Fortuny‐Amat linearization formula, we have that the set of constraints (19)‐(30) fully represents level 3 of our model: 0
0 0
,
:
∀ ∈
(19)
:
∀ ∈
(20)
∈
0
(21)
:
∈
: 0 0 0
1
∀ ∈
(22)
∀ ∈
(23)
∀ ∈
(24)
∀ ∈
(25)
16
Third level: Market clearing 0
1
,
∀ ∈
(26)
∀ ∈
(27)
∀ ∈
(28)
∀ ∈
(29)
∀ ∈
(30)
∈
0
0
1
, ∈
0 0
1
17
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
,
0
,
0
∈
(31) 0
–
0
∈
s.t. (19) - (30)
Using (18), we can rewrite the problem as: max
–
0
(32)
∈
s.t. (19) - (30)
18
Second level: Generation investment The only non‐linear term in (32) is
. Since the
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., 2005) to discretize : Λ 0
Δ
∈
2
(33)
0
Accordingly, the non‐linear product
can be replaced by the expression:
Λ 0
Δ
∈
2
(34)
0
where we define
by constraints (35) and (36), using the Fortuny‐Amat linearization formula:
0
1
0
∀ ∈
,
0,1, , … Λi
(35)
∀ ∈
,
0,1, , … Λi
(36) 19
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.
20
Second level: Generation investment The Nash equilibria of the GENCOs’ capacity investment decisions are given by the following set of inequalities:
,∀ ∈
(53) , ∈
,∀
,∀
∀
∈
,∀
∈
∉
where we have to distinguish between the left hand side (LHS) and the right hand side (RHS) of (53).
21
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 in the equilibrium is given by:
Λ 0
Λ
Δ
2
∈
–
Δ
2
0
∀ ∈Ψ
(54)
0
subject to the linearized constraints of stage 3 in the equilibrium, which correspond to ,
constraints (38)‐(51), but replacing ,
,
,
,
,
,
(50) and (51) for all ∈
,
,
,
,
,
,
,
, , ,
, and
, , ,
,
,
,
,
, and
by
, respectively, and considering
.
22
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 chooses strategy , with ∈
generation capacity at node i up to the capacity
(which involves investing in ,∀
∈ Ψ), the definition of
the utility function for GENCO is given by:
0
–
∀ ∈ Ψ, ∀sG ∈
∈
(55)
subject to the corresponding constraints of stage 3, which correspond to constraints (38)‐ (51), ,
but ,
,
, , ,
,
considering
,
, , , ,
,
, ,
, ,
,
, ,
∀ ∈ Ψ, ∀
them , and ,
,
∈
,
replacing
,
by
,
, and
,
respectively,
and replacing (38) by (56) and (57), (49) by (58) and (59), (50) by (60), and (51) by (61): 23
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
, ∈
,
0
,
0
∈
min
, ∈
(62) 0
0 ∈
24
First level: Transmission investment In (62), we have considered that there is a set of transmission lines that are candidates for investment maximum active flows
. That means that the previously‐constant are now variables of the problem in level 1.
Note that, contrary to the assumptions in (Sauma and Oren, 2006), 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.
25
Case studies: SIC 1 Diego de Almagro 220 2 Carrera Pinto 220 3 Cardones 220 10 A. Santa 220
9 San Luis 220
4 Maitencillo 220 5 Pan de Azucar 220 6 Los Vilos 220 7 Quillota 220
8 Polpaico 500/220 34 Loaguirre 220
12 Rapel 220 11 Cerro Navia 220 13 Chena 220 16 Colbún 220
14 Alto Jahuel 500/220 15 A. Jahuel 154 17 Paine154 20 Ancoa 500/220
18 Rencagua 154 19 San Fernando 154 21 Itahue 220/154 22 Parral 154 23 Chillán 154 25 Charrúa 154
26 Concepción 220/154 27 San Vicente 154 28 Hualpén 220/154 29 Coronel 154
24 Charrúa 500/220 30 Temuco 220 31 Valdivia 220 32 Barrio Blanco 220 33 Puerto Montt 220
26
Case studies: SIC
27
Case studies: SIC The model has been solved for different case studies: Case 1 is the benchmark case, without considering the expansion on capacity lines and generation; case 2 assumes the new hydro power plant in node 34 (see Table III) is not able to be built and only candidate lines 1 to 4 are possible (see Table II); case 3 considers the line expansion is limited to lines 1 to 4; case 4 considers all candidates for expansions in Tables II and III are allowed. The comparison of the results for these case studies is summarized in Table IV. 28
Case studies: SIC
29
Case studies: SIC Tables V and VI show the results of case 4 in detail. The capacity investment in each node is shown in the second column of Table V. The final marginal cost (third column) is compared with the average LMP in each node (fourth column). The average LMP is calculated by averaging the LMPs of the four demand patterns of the year. Table VI shows the solutions for the line capacity investment and the annualized investment cost per line in the second and third columns, respectively.
30
Case studies: SIC
31
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. The Chilen SIC can be solved with the proposed 3‐level model.
32
A three‐level MILP model for generation and transmission expansion planning David Pozo (UCLM) Enzo Sauma (PUC) Javier Contreras (UCLM) January 05th, 2012
