Title Generation Capacity Expansion Planning under Demand Uncertainty Using Stochastic MixedInteger Programming Authors William Gandulfo and Esteban Gil Universidad T´ecnica Federico Santa Mar´ıa, Valpara´ıso, Chile Conference: 2014 IEEE Power & Energy Society General Meeting (IEEE-PES-GM 2014), National Harbour, USA, Jul. 27-31, 2014. URL: http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6939368 DOI: 10.1109/PESGM.2014.6939368 Abstract Generation Capacity Expansion Planning (GCEP) decides about generation capacity investments to adequately supply the future loads, while minimizing investment and operation costs satisfying a set of technical and security constraints. This paper presents a Stochastic Mixed-Integer Programming formulation (SMIP) for suggesting future generation investments considering demand uncertainty. The method was applied to the Chilean Northern Interconnected System (SING) with a planning horizon of 14 years considering uncertainty on the possible future connection of large industrial and mining loads. The computational challenges posed by GCEP under uncertainty required compromising between the detail of the stochastic demand representation and the detail of the transmission system. Thus, scenario-reduction was applied to keep the problem of a manageable size without losing too much transmission detail. Our results for the SING showed that use of SMIP can bring expected savings of about 1.1% on the total investment plus expected operational cost with respect to optimization using an average demand scenario. Furthermore, the stochastic plan showed less variability across scenarios and proved to be more resilient to changes in the modeling assumptions than the other plans. Keywords Generation planning, capacity expansion planning, stochastic optimization, stochastic mixedinteger programming. (c) 2014 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.

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2014 IEEE Power & Energy Society General Meeting (IEEE-PES-GM 2014), National Harbour, USA, Jul. 27-31, 2014. URL: http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6939368. DOI: 10.1109/PESGM.2014.6939368

Generation Capacity Expansion Planning under Demand Uncertainty Using Stochastic Mixed-Integer Programming William Gandulfo, Esteban Gil, and Ignacio Aravena Universidad T´ecnica Federico Santa Mar´ıa Valpara´ıso, Chile Abstract—Generation Capacity Expansion Planning (GCEP) decides about generation capacity investments to adequately supply the future loads, while minimizing investment and operation costs satisfying a set of technical and security constraints. This paper presents a Stochastic Mixed-Integer Programming formulation (SMIP) for suggesting future generation investments considering demand uncertainty. The method was applied to the Chilean Northern Interconnected System (SING) with a planning horizon of 14 years considering uncertainty on the possible future connection of large industrial and mining loads. The computational challenges posed by GCEP under uncertainty required compromising between the detail of the stochastic demand representation and the detail of the transmission system. Thus, scenario-reduction was applied to keep the problem of a manageable size without losing too much transmission detail. Our results for the SING showed that use of SMIP can bring expected savings of about 1.1% on the total investment plus expected operational cost with respect to optimization using an average demand scenario. Furthermore, the stochastic plan showed less variability across scenarios and proved to be more resilient to changes in the modeling assumptions than the other plans. Index Terms—generation planning, capacity expansion planning, stochastic optimization, stochastic mixed-integer programming.

I. I NTRODUCTION ENERATION Capacity Expansion Planning (GCEP) refers to deciding future investments in generation capacity to maintain an adequate margin between supply and demand, considering possible future changes in demand and the transmission network. Thus, GCEP is concerned with deciding the type, size, location, and timing of future generation plants. Uncertainty is always present in any modeling effort, especially when the decision horizon is so extended as in GCEP. Demand growth variability, fuel prices volatility, primary energy availability, and technology change are some of the aspects introducing uncertainty in GCEP. Traditionally, the problem is solved deterministically (usually employing a single demand scenario) and then uncertainty is addressed by running sensitivity studies to analyze the impacts of variations in the modeling assumptions and to evaluate the robustness and resilience of the solution. The deterministic GCEP problem (assuming perfect foresight) has been widely studied and several methods to solve it have been proposed, either by using traditional linear programming (LP), mixed integer linear

G

programming (MILP), or heuristic/meta-heuristic techniques [1]–[3]. Of particular interest lately has been the incorporation of reliability constraints [4], renewable integration [5], [6], emissions control [7], and decentralized decision making with incomplete information [8]–[10]. Increase in computational power has lately allowed to extend deterministic GCEP formulations to use stochastic programming paradigms capable of providing from the start capacity expansion plans that are more robust the uncertainty of the input parameters [11]. This paper proposes the use of Stochastic Mixed-Integer Programming (SMIP) to solve the stochastic GCEP problem under demand uncertainty. The proposed method is illustrated for the Chilean Northern Interconnected System (Sistema Interconectado del Norte Grande, SING). As the SING supplies mostly industrial and mining loads, planners suffer demand uncertainty about what new large industrial and mining projects might materialize (or not), creating great difficulty to suggest investors indicative generation expansion plans. The use of stochastic SMIP allows to suggest indicative generation capacity expansion plans that are more robust to changes in the demand assumptions by optimizing for a greater set of demand scenarios. This paper is structured as follows: Section II formulates the GCEP problem using SMIP; Section III describes the test system and discusses the generation of the demand scenarios; Section IV presents and discusses the simulation results and Section V presents the main conclusions of this work. II. S TOCHASTIC MIXED - INTEGER PROGRAMMING APPLIED TO THE GCEP PROBLEM A. Stochastic programming Stochastic optimization is understood as a set of techniques for modeling mathematical optimization problems involving uncertainty. Particularly, stochastic programming takes the deterministic formulation of a mathematical optimization problem and expands it to multiple realizations of the stochastic parameter, where each realization corresponds to a scenario s from the set of scenarios S. Thus, stochastic programming requires representing the probability distribution of the random parameters by a finite set of discrete scenarios [12]. In this sense, stochastic programming is capable of finding a unique solution that is feasible for all scenarios and that is optimal in some sense, such as minimizing/maximizing the expected IEEE-PES-GM 2014

value of a certain function. In 2-stage stochastic programming, decision variables are separated in 2 stages, where first stage decisions are the same across all scenarios while second stage decisions can be different. Stochastic programming is not the same than solving S different deterministic scenarios, as in stochastic programming the first stage variables are forced to be the same for the full set of scenarios by a set of nonanticipativity constraints. The same concept can be generalized for more than 2-stages. For modeling GCEP under demand uncertainty we are using 2-stage Stochastic Mixed-Integer Programming (SMIP) where the first stage decisions are the generation investments (integer variables) while the second stage decisions are operational decisions such as generation dispatch and power flows in the lines. Therefore, first-stage decisions are made based on partial information from the second stage (probability distribution of the stochastic parameter) and, after realization of the stochastic parameters, the second-stage decisions (or recourse decisions) take actions to optimize the objective function given the first-stage decisions. Although application of SMIP is being intensely researched for other power system applications (such as stochastic unit commitment under demand and/or wind uncertainty [13]–[15]), except for [11] it has still not been widely used for capacity expansion planning. A generic mathematical formulation of SMIP to the problem is given in (1). The problem consists of minimizing the expected value of the objective function over the discrete set of scenarios S. Each scenario s has a different weight or probability ωs , given by the discrete representation of the stochastic input parameter θ. Minimize:

X

ωs · F (θ s , {xs , y s , z s })

s∈S

 x1 y  A(θ ) 0 ··· 0   1   b(θ 1 )  1 z1  A(θ 2 ) · · · 0   0  .   b(θ 2 )      ..  Subject to:  . . . ..  =  ..   . .. ..   .   . . x  0 0 · · · A(θ S )  S  b(θ S ) yS zS xs ≥ 0, y s ≥ 0, z s ≥ 0 ∀s ∈ S (1) 

xs , y s , and z s are, respectively, the first-stage decisions, second-stage decisions, and slack variables in scenario s. Depending on the nature of the uncertainty, the objective function F (θ s , {xs , y s , z s }), the matrix of coefficients A(θ s ), and/or the right-hand side vectors b(θ s ) can depend on θ. However, in the particular case of demand uncertainty, only b(θ s ) is dependent on the stochastic input parameter θ s . Next, it is necessary to force the first-stage decisions to be the same across scenarios. This is achieved by defining nonanticipativity constraints forcing all vectors xs to be the same across the different scenarios. B. Objective function In the deterministic formulation the objective function seeks to minimize the total investment and operational costs. The

SMIP formulation extends the deterministic formulation and the SMIP objective function (2) is the expected value of the deterministic objective function. As first-stage decision variables (investment decisions) are the same across scenarios, the problem is equivalent to minimizing investment plus expected operational costs. Minimize: " X X ωs ∀s∈S

X

BuildCostg,y · GenBuildg,y,s

∀y∈Y ∀g∈GB

+

X X X

V oLLy · U SEn,t,y,s (2)

∀y∈Y ∀t∈T ∀n∈N

+

X X X

GenCostg,t,y · GenLoadg,t,y,s

X X X

P enaltyc,t,y · Slackc,t,y

∀y∈Y ∀t∈T ∀g∈G

+

∀y∈Y ∀t∈T ∀c∈C

#

where G and GB are, respectively, the set of all generators and the set of generators with building decisions; S is the set of electric demand scenarios; N is the set of buses; T and Y are the set of time blocks and the set of years in the planning horizon, respectively; C is the set of constraints. GenBuildg,y,s and BuildCostg,y are generator g investment decisions and its associated costs; V oLLy is the Value of Lost Load; U SEn,t,y,s is the unserved energy in bus n; GenLoadg,t,y,s and GenCostg,t,y,s are the generation and the operating cost of generator g; Slackc,t,y and P enaltyc,t,y are the slack and its associated penalty for violating constraint c. C. Constraints 1) Energy balance at each bus: Incumbent and new generation capacity must be able to supply future demand and transmission losses ∀i ∈ N, ∀t ∈ T, ∀y ∈ Y, ∀s ∈ S. X

GenLoadg,t,y,s + U SEi,t,y,s

∀g∈Gi

X

F lowlij ,t,y,s + 0.5

∀j∈Ωj

= Demandi,t,y,s

X

Losslij ,t,y,s

(3)

∀j∈Ωj

where Ωj and Gi are the set of buses adjacent to bus i and the set of generators connected to bus i; F lowlij ,t,y,s and Losslij ,t,y,s are the flow and losses in transmission line from i to j; Demandi,t,y,s is the demand in bus i. 2) Transmission constraints: Relationships between voltage angles and flows and transmission limits are defined ∀lij ∈ L, ∀t ∈ T, ∀y ∈ Y, ∀s ∈ S by equations (4) to (6). F lowlij ,t,y,s = Yij · Angi,t,y,s − Yij · Angj,t,y,s

(4)

M inF lowlij ,t,y ≤ F lowlij ,t,y,s ≤ M axF lowlij ,t,y

(5)

M inAngi ≤ Angi,t,y,s ≤ M axAngi

(6)

where L is the set of transmission lines; Yij is the admittance from node i to j; Angi,t,y,s is the voltage angle at node i and M inAngi and M axAngi are its limits; M inF lowlij and M axF lowlij are the min and max flow at line lij .

3) Generation limits: Constraints associated to generation limits depend on the type and size of the generating units ∀g ∈ G, ∀t ∈ T, ∀y ∈ Y, ∀s ∈ S. M inGeng · GenU nitg,t,y,s ≤ GenLoadg,t,y,s GenLoadg,t,y,s ≤ M axGeng,t,y,s · GenU nitg,t,y,s

(7)

where M inGeng and M axGeng are the min stable level and max capacity of generator g, and GenU nitg,y,s is an integer variable with the number of units of the generator. 4) Non-anticipativity constraints: Forcing first-stage decision variables (investments) to be the same for all scenarios. GenBuildg,y,1 = GenBuildg,y,s ∀s ∈ S

(8)

D. Solution method and problem size The mathematical program in equations (2)-(8) corresponds to a Mixed-Integer Linear Program (MILP). Once formulated, a MILP can be solved by a number of available optimization solvers through a combination of Branch-and-Bound and cutting planes techniques. However, as the number of variables grows these problems can easily become computationally intractable, requiring compromises on the modeling detail to reduce the number of variables. This can be achieved in a number of ways, such as aggregating transmission buses, simplifying the modeling of transmission losses, and/or reducing the number of stochastic scenarios. In this case study, we aggregated the transmission system down to a level that ensured proper representation of the most important transmission constraints, but as the mathematical problem remained too large we also had to reduce the number of demand scenarios, as section III will discuss. III. T EST SYSTEM AND DEMAND SCENARIOS The SING is a system supplying mainly industrial load (about 90%) with 4.6 GW installed generation capacity in 2012. Electric energy supply comes mainly from coalfired units (83.0% of total generation in 2012), some newer combined-cycle and open-cycle gas-fired units (13.6%), some fuel-oil and diesel peaking plants (2.8%), and a small amount of hydro generation (0.5%). The CDEC-SING, the SING independent system operator (ISO), provides on their website databases of their system containing detailed production and network data. These databases were adapted for the purposes of this work and the outputs of the simulations were benchmarked against actual system outputs to check for correctness and consistency. Our SING model was composed of 100 buses (from 13.8 kV to 345 kV), 124 transmission lines, 23 transformers, 53 incumbent generators, and 67 candidate generation projects for expansion. The planning horizon is 14 years, from 2013 to 2026. The transmission system was expanded exogenously based on the transmission expansion plan provided by the system operator. Industrial and domestic demands were grown independently based on growth rates based on historical data. Besides the existing demand, there exist many large industrial and mining projects that may or may not materialize during the planning

horizon, depending on exogenous factors (such as international metal prices and capital costs). This fact introduces plenty of uncertainty in the generation capacity expansion planning process and is a cause of concern for system operators and for both generation and industrial project investors. Table I shows a list of 10 large industrial and mining projects being considered, and the probabilities of materialization assigned to each project by a panel of experts based on their state of execution and market conditions and projections. Generating all possible combinations for the projects would create a unmanageable number of demand scenarios. As some projects are mutually exclusive while some others are interdependent, we could significantly reduce the size of the probability tree. At the end, we were capable of reduce the number of demand scenarios to 12, with each scenario being a combination of industrial/mining projects (see Table II). The reduced set of 12 scenarios was then further reviewed by a set of experts and found to be consistent and plausible. Thus, each demand scenario will determine the right hand side of equation (3) for the bus where the demand associated to the project would connect, and the probabilities in Table II will determine the value of ωs in the objective function (2) for the respective scenario. TABLE I M INING DEMAND PROJECTS AND THEIR PROBABILITIES Project name Collahuasi Phase II expansion Patache port expansion Spencer Hip´ogeno Project Desalination Sierra Gorda Sierra Gorda Escondida Services Escondida desalination project Zald´ıvar primary sulfur Mine Ministro Hales Mine Antucoya

Stage of the project In construction Potential To start soon In construction Potential In construction In construction Potential In construction In construction

Probability 0.9 0.6 0.8 0.95 0.6 0.85 0.85 0.6 0.95 0.8

TABLE II P ROBABILITY OF EACH DEMAND SCENARIO Scenario 1 2 3 4 5 6

Probability 0.096 0.081 0.089 0.094 0.091 0.091

Scenario 7 8 9 10 11 12

Probability 0.079 0.081 0.08 0.073 0.081 0.063

IV. S IMULATION RESULTS The results for the proposed SMIP method were compared against results using a deterministic MILP formulation and against the generation investment plan suggested by the system operator. Thus, in this section we present results for 3 different generation capacity expansion plans: CDEC-SING plan: Generation investment plan informed by the system operator. The plan did not consider some of the industrial and mining demands considered by the other models, so it will have some unserved energy when detailed

Stochastic Plan Deterministic Plan CDEC-SING− Plan

3000

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Investment Cost [MMUS$]

5000 4500 4000 3500 3000 2500 2000 1500 1000 500

Stochastic Plan Deterministic Plan CDEC-SING Plan

2500

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25

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Fig. 1. Annual generation investments for the different plans

operations are evaluated for some of the demand scenarios. Deterministic plan: Generation investment plan optimized from a deterministic formulation using the weighted average of the 12 demand scenarios. Stochastic plan: Generation investment plan obtained by SMIP, optimized for the full set of demand scenarios. The deterministic and stochastic generation investment plans were obtained using the mathematical formulation described in Section II. The databases were implemented in the software PLEXOS [16] and the mathematical programs were solved using the optimization solver XPress [17] in an Intel i7960 processor with 4 cores and 24 GB of RAM. Both GCEP models were implemented using monthly load duration curves discretized in 5 blocks with a planning horizon from 2013 to 2026 and linearized transmission losses. They also used the same solver parameters (MIP gap of 0.1%). Simulation times and RAM requirements are reported in Table III. TABLE III S IMULATION TIMES AND COMPUTATIONAL RESOURCES

Model Simulated SMIP Deterministic CDEC-SING

Expansion Time Max. RAM [hh:mm:ss] [GB] 26:23:28 13.95 0:15:07 3.76 -

Operations Time Max. RAM [hh:mm:ss] [GB] 00:36:27 2.22 00:35:39 2.19 00:43:42 4.35

Figure 1 shows the investment costs for each expansion plan. While the CDEC-SING plan investments are steadier, both the deterministic and the SMIP plans concentrate their investments between 2016 and 2018. While the deterministic and the stochastic plans show some similarities, the second one tends to invest more and to move some of the investments to earlier years to avoid unserved energy in some of the most extreme demand scenarios. By the end of the planning horizon (2026), the stochastic plan built 4028 MW of new capacity, against 3724 MW and 4126 MW of the deterministic and the CDEC-SING plan. The simulation results allowed obtaining some metrics to evaluate the performance of the SMIP method. One of

Fig. 2. Annual expected operational costs for the different plans

these metrics is the Value of the Stochastic Solution (V SS), corresponding to the difference between the Expected Result of Using the Expected Value (EEV ) and the value of the objective function of the SMIP problem (Recourse Problem, RP ). The V SS corresponds to the increase in the expected costs incurred for not taking into account the uncertainties during the optimization. The V SS for this case study is 354 MMU$, or 1.13% of the EEV . Another metric is the Expected Value of Perfect Information (EV P I), representing the value of having perfect foresight when solving the optimization problem, and corresponds to RP minus the Wait-and-See (W S) solution. The EV P I is 1,035 MMU$. After obtaining the generation capacity expansion plans, each one of them was tested for the 12 demand scenarios in a more detailed operational model in order to obtain their operational costs had each of the demand scenarios materialized. Then, the expected operational costs were obtained by calculating the weighted average cost, where the weights are given by the probabilities in Table II. As Figure 2 shows, the expected operational costs (fuel costs plus unserved energy cost) are quite different for the CDEC-SING plan and the optimized plans (deterministic and stochastic), especially after 2017. Although the stochastic plan has lower operational costs for all demand scenarios (as it builds more capacity and builds it sooner), for the moderate demand growth scenarios it has relatively similar costs with the deterministic plan. However, figure 3 shows that their operational costs can vary significantly for the scenarios where most or all the industrial/mining projects materialize (e.g. scenario 1 and 4). Figure 3 also shows that the stochastic plan is the one showing less variability across scenarios, suggesting that it is more resilient to different demand assumptions. Figure 4 shows total net present value (NPV) of the investment and expected operation costs for each of the capacity expansion plans. The stochastic plan is the one with the lowest NPV, while the CDEC-SING plan cost is the highest. When simulating operations a posteriori in more detail for the set of demand scenarios, the stochastic plan has significantly less expected energy not served, indicating a more robust and resilient capacity expansion plan.

Stochastic Plan Deterministic Plan CDEC-SING Plan

1 2

12

Scenario

3 36000 [MMUS$] 32000 [MMUS$] 28000 [MMUS$] 24000 [MMUS$] 20000 [MMUS$] 4

11

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Boxplot of annualized operational costs of each plan Annualized Operation Costs [MMUS$]

34000

more, the stochastic plan was more robust for the full set of scenarios and showed less variability across scenarios. Although simulation time for the SMIP method was almost 100 times the time of the deterministic one, the 26 hours it took to run was still reasonable considering the length of the planning horizon, the transmission detail, the number of integer decisions involved, and the expected savings. Furthermore, parallelization of SMIP problems for stochastic unit commitment problems has shown to be quite promising [14], [15], so the parallelization of the stochastic GCEP could potentially reduce significantly simulation times and allow the solution for larger systems.

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R EFERENCES

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Fig. 3. Annualized expected operational cost variability across demand scenarios

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Fig. 4. Comparison of NPV of total expected costs for each capacity expansion plan

V. C ONCLUSIONS This paper presented a SMIP formulation for deciding future generation investments considering uncertainty on the possible future connection of large industrial and mining loads in the Chilean SING. The methodology required to represent the demand uncertainty as a finite set of scenarios, and to extend the deterministic formulation of an investment problem to a 2stage stochastic mixed-integer program. In the future we intend to improve our discrete representation of the demand and the GCEP modeling by using multi-stage stochastic programming, and exploring the sensitivity of the stochastic GCEP problem to the number of demand scenarios. The stochastic plan showed expected savings of over 350 million dollars with respect to the deterministic plan (1.1% of the total investment plus expected operational cost). Further-

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