A Multistage Decision-Dependent Stochastic Bi-level Programming

A Multistage Decision-Dependent Stochastic Bi-level Programming Approach for Power Generation Investment Expansion Planning Yiduo Zhan

∗

Qipeng P. Zheng

†

February 11, 2018

Abstract In this paper, we study the long-term power generation investment expansion planing problem under uncertainty. We propose a bilevel optimization model that includes an upper-level multistage stochastic expansion planning problem and a collection of lower-level economic dispatch problem. This model seeks for the optimal sizing and siting for both thermal and wind power units to be built to maximize the expected profit for a profit-oriented power generation investor. To address the future uncertainties in the decision-making process, this paper employs decision-dependent stochastic programming approach. In the scenario tree, we calculate the nonstationary transition probabilities based on discrete choice theory and the economies of scale theory in electricity systems. The model is further reformulated as a single-level optimization problem, and solved by decomposition algorithms. The investment decisions, computational times, and the optimality of the decision-dependent model are evaluated by case studies on IEEE reliability test systems. The results show that the proposed decision-dependent model provides effective investment plans for long-term power generation expansion planning.

Key words: Multistage Stochastic Programming, Bilevel Optimization, Generation Expansion, Endogenous Uncertainty, Non-Stationary Transition Probability, Decision-Dependent Probability, Model Reformulation, Decomposition Algorithms.

1

Introduction

In recent years, the number of newly installed renewable generators, especially for wind capacity, has increased rapidly. As a result, wind power has been participating in the electricity market by a large percentage. Hence, the investors will need to consider the possible investment decisions on both conventional and renewable power generators when making investment decisions, especially in long-term. Although there are plenty of researches studying the generation investment decisions, the majority of them focus on conventional energy sources. The rapid growth of renewable energy penetration in power systems has stimulated a large number of studies that address the investment problems related to wind power (Murphy et al. 1982, Kennedy 2005, Ivanova et al. 2005, Baringo and Conejo 2013, Valenzuela and Wang 2011). Many studies focus on long-term investment expansion planning of wind power. For example, Kennedy (2005) analyzes the long-term costs and benefits of wind power planning, and Zhan et al. (2017b) presents a multi-stage stochastic approach for expansion planning while considering wind power plants. ∗

Department of Industrial Engineering and Management Systems, University of Central Florida, Orlando, FL 32816; email: [email protected]; † Department of Industrial Engineering and Management Systems, University of Central Florida, Orlando, FL 32816; email: [email protected];

1

There two different approaches for investment problems in electricity generation expansion planning: a centralized framework (Zhan et al. 2017b) and a market framework (Baringo and Conejo 2011). The centralized approach solves for the system-optimal solutions while considering the electricity system as a whole. Whereas, the market approach represents the electricity market in a way that each participant (e.g., generation companies) optimize its own goal. The advantage of the market framework is that the electricity price is determined by the electrical power market. This is extremely important for investment decisions in the deregulated electrical power market because the electricity selling price is a major factor that affects the investor’s objective, i.e., the overall profit. Therefore, in this study, we use the market approach to represent the perspective of a profit-oriented investor. However, one of the major hurdles of using market framework is that the formulation contains the bilevel structure, making it very difficult to solve. Bilevel programming, which originally arises from the economic problem in the field of game theory (Colson et al. 2007), captures the hierarchical relationship between two decision levels. Because of the unconventional structure of bilevel programming, the solution approach are not straightforward. A series of studies have been conducted in this direction. These studies addressed multiple topics in investment planning of wind power, including investment within market environment (Baringo and Conejo 2011), investment on transmission and wind generators (Baringo and Conejo 2012), investment with risk consideration (Baringo and Conejo 2013), and strategic wind power investment with the aim of altering the market clearing prices (Baringo and Conejo 2014). A common feature is that the problems are all set up under the market framework, i.e., that they are formulated as a bilevel model where the upper level represents wind investment decisions and the lower level represents the market clearing problems. They also present solution approaches to handle the bilevel structure that transform the bilevel structure into a Mathematical Program with Equilibrium Constraints (MPEC). The MPEC is then converted to a Mixed-Integer Linear Program (MILP) by introducing binary variables. However, these studies also have limitations. For example, they are limited to only considering the expansions of wind power generation. They have not considered the options of building both conventional and wind power generators. Additionally, the stochastic model in these studies only contain two stages, i.e., a planning stage and a operational stage. However, in the real world long-term expansion planning, the investment decisions are made multiple times in the future planning horizon, which can be better represented by multistage stochastic programming models. Within this context, we consider a power investor who invests on expansions of both conventional and wind generators. It is aimed at deciding the optimal sizing, timing, siting and the type the electrical power generation units to be newly built or expanded within an electrical power network. The objective of the electricity power investor is to maximize the expected profit from selling the electrical power production in the long term. Because of the variable and uncertain behavior of wind, generation planning with wind power often needs to deal with tremendous future uncertainty (Krishnan et al. 2016). To address these uncertainties, stochastic programming is one of the most popular approaches for power system generation planning problems (Wallace and Fleten 2003). Most of the existing stochastic programming approaches were designed to deal with exogenous uncertainty which are described by random variables with some predetermined probability distributions. However, in real-world generation planning, both uncertain parameters and the decision variables will affect the future uncertainties, which is known as endogenous uncertainties. Ivanova et al. (2005) shows that the decisions on power plant expansion are affected by several variable criteria including capital costs, current costs, budget deduction, and electricity prices. For a power system with a large amount of wind power involved, the study of Valenzuela and Wang (2011) shows that different installed capacities of wind power will influence the entire power system. In the study of Jonsbr˚ aten et al. (1998), the decision-dependent approach was applied to a mixed-integer stochastic programming model where 2

the timing of information discovery can be influenced by decisions. A recent study from Zhan et al. (2017b) uses decision-dependent multistage stochastic model to formulate the long-term electricity expansion planning problem, where the probability distribution of future electricity prices is affected by the level of wind power. However, Zhan et al. (2017b) considers the electricity markets as a whole, which is more suitable for a centralized decision making system but not specifically for a deregulated electricity market . In this paper, we propose a long-term planning model through a bilevel, multistage, decisiondependent, stochastic nonlinear programming approach. The stochastic formulation considers both endogenous uncertainty (i.e., investment and maintenance costs) and exogenous uncertainty (i.e., demand and wind intensity). The endogenous uncertainty is represented by the decision-dependent process where the probability distributions of unit cost depends on the key decision variables: the generation amount of different types of generators. Our proposed model contains bilevel, multistage and nonlinear features that make the optimization process computationally very challenging. We have developed a hybrid transformation decomposition solution approach that transforms this model to a solvable format and reduce its computation complexity. We first presents a linear transformation that uses strong duality theorem to get rid of nonlinear terms in the revenue expression. Then, we employ the MPEC transformation that transforms the bilevel structure to a single level problem with the upper level being constrained by the KKTs of all lower-level problems. The MPEC is then converted to mixed-integer linear programming (MILP) by introducing binary variables. In order to improve the solution time of large scale problems, we employ Dantzig-wolfe decomposition (Dantzig and Wolfe 1960) to decompose the problem into a series of parallel subproblems. The major contributions of this paper are summarized as follows, 1. This paper presents a bilevel multistage decision-dependent stochastic programming optimization model for the long-term power generation expansion planning problem in an electricity market framework. 2. This paper considers both exogenous and endogenous uncertainties for nodal demands, wind intensity, and the investment and maintenance cost of wind generators. We have also proposed a new decision-dependent probability model to respect the scale of economy at each scenario. 3. This paper develops a hybrid solution approaches that addresses the computational difficulties of our proposed optimization model and solve the model within a reasonable time.

1.1

Organization of the Paper

The remainder of this paper is arranged as follows. Section 2 presents the consideration of model settings and assumptions. Section 3 provides the main features of the problem and the mathematical formulation in details. The solution approaches are discussed in Section 4. The results of an illustrative example and case studies are analyzed in Section 5. Section 6 concludes the paper with some relevant remarks.

2

Model Settings and Assumptions

In this study of generation asset planning, we assume an electricity network powered by two types of generation units: thermal and wind. This is typical in areas with abundant wind resources, such as the US midwest. In addition, we assume that the energy storage units are not considered in the network. This is because the energy storage units are mainly applied to deal with energy dispatch problems in the short-term market such as day-ahead unit commitment (Huang et al. 2014, Jiang 3

et al. 2012, Gil et al. 2010). The long-term planning horizon averages out the effect of storage units in the short-term. Several parameters in this research are considered as random variables, including electricity demand, wind intensity, investment and maintenance costs. In the following subsections, we discuss the uncertainty settings.

2.1

Uncertain Demands and Wind Energy

Although our model is for long-term generation asset planning, we also consider the short-term dispatch problems within the long-term expansion planning. In short-term, both demands and wind energy are very volatile, and do not affect and are not affected observably by the long-term investment decisions. This type of uncertainty is commonly referred to as “exogenous.” In this study, we use uncertain short-term data given in Baringo and Conejo (2011), who has modeled the demand and wind uncertainty based on the load- and wind-duration curves from history data. According to this study, the demand and wind intensity varies greatly between different seasons. Therefore, we consider to divide each planning stage into four demand blocks, which correspond to four different seasons in a year. The uncertainty within each demand block is represented via exogenous uncertain scenarios. For example, each demand block may have two uncertain exogenous uncertain scenarios, high and low. Similarly, the wind intensity is also treated as exogenous uncertainty associated with different wind intensity levels within each demand block. For the sake of simplicity, the length of each block are set to be equal to each other, and both demand and wind are assumed to be normally distributed within each block. The uncertain demand level and wind intensity level are generated by the C++’s default random normal number generator function in the “math.h” library. The mean and standard deviation for the uncertain demand and wind are acquired from IEEE test systems.

2.2

Uncertain Investment and Maintenance Costs

In this study, we also incorporate uncertainties lying in the various costs of wind generators. This consideration is due to the fact that the wind energy is one of the most rapidly growing energy sources. The cost of wind generators is mainly contributed by the investment and the maintenance costs, both of which have a very big chance to vary in the future. The technology of wind turbines has experienced immense improvements during the last 30 years. Due to the recent undergoing important progress of power electronic device technology (Chen et al. 2009), both construction and maintenance costs would likely decrease in future. On the other hand, the new technologies are less mature and may likely have some defects which may cause high maintenance costs. Moreover, as the age of existing wind turbine increases, more maintenance is required for the components under intense and variable mechanical stress (de Azevedo et al. 2016). These potential issues may also increase the wind generator’s cost. These costs are mainly the long-term costs as expansions are usually done every 5 to 10 years in the next several decades. One of the best ways to capture this uncertain dynamics (i.e., stochastic processes) to assist decision making is through a scenario tree. The uncertain investment and maintenance costs are characterized by different discrete levels, each of which is represented by a node of the scenario tree. We denote a node in the scenario tree as n. For every node n, it has a unique ancestor node as a(n). In contrast, Sn denotes the set of successors of node n. At each ancestor node a(n), each node in the child node set Sa(n) is corresponding to an outcome/realization of the discrete random cost. We use B a(n) and B n to represent the unit investment cost in ancestor node a(n) and node n, respectively. Similarly, we use ma(n) and mn to represent the unit maintenance cost in ancestor node a(n) and node n, respectively. Then, δ n and

4

σ n , a pair of prefixed parameters, are used to generate the outcome/realization of prices at node n, through the following equations, B n = B a(n) · (1 + δ n ), n

a(n)

m =m

n

· (1 + σ ),

∀n ∈ Sa(n) .

(1a)

∀n ∈ Sa(n) .

(1b)

For different nodes, δ n and σ n are chosen differently. For example, in a binary tree, the two child nodes of a(n) can have opposite values, e.g., ±5%, to represent an increase and a decrease for both parameters. 2.2.1 The Intrinsic “Endogeneity ” As the wind power technologies are constantly improving and costs are changing, the uncertainties are strongly related to the investments and popularity of wind energy. To model these endogenous uncertainties, we assume the probabilities of the discrete levels of costs (in the scenario tree) are driven by the investment (translated to the capacities of different types of generation assets) and usage of wind generators (a measure of popularity) based on discrete choice theory (Luce 1977, McFadden 1973). We assume the probability associated with each outcome/realization varies according to the value of Economies of Scale(EoS) of the corresponding cost outcome. 2.2.2

The Economies of Scale (EoS) Model

The concept of economic scale in electricity market was first introduced in (Nerlove 1961), it is described as a phenomenon that when the scale of production increases, the cost per unit production would decrease. (Christensen et al. 1973) has derived that the combined cost is a transcendental logarithmic function of key parameters, and the measurement of the economic scale is the elasticity of the cost. The study in Christensen and Greene (1976) introduced the range (between 0 and 1) for Economies of Scale (EoS). The EoS measures the potential of cost reduction in the electricity system. When EoS is close to 0, it means the system has well “exploited” the potential of cost reduction. On the other hand, if the EoS is close to 1, then the system should have a large potential to reduce the cost. It is defined as follows, EoSib = 1 −

∂ ln Cib ∂ ln gib

= 1 − (αY + γY Y ln gib +

X

γY v ln cib,v )

(2)

v

where the translog cost function for the total cost is ln Cib =

X 1 αv ln cib,v α0 + αY ln gib + γY Y (ln gib )2 + 2 v X 1 XX + γuv ln cib,u ln cib,v + γY v ln gib ln cib,v 2 u v v

(3)

where Cib and gib represent the total cost and the total amount of generation of generator i at bus b, respectively. cib,v is the combined cost of unit capital cost, unit operation and maintenance (OM) cost, and unit fuel cost with corresponding index v. α0 ,αY , γY Y , γY v are coefficients from empirical data as in Christensen and Greene (1976). Therefore, this EoS evaluates a decreasing effect of the overall cost Cib of the electricity system. For wind power specifically, this EoS depends on not only the amount of investment (i.e., capital cost) but also the amount of generation (i.e., maintenance cost). 5

70

65

Unit Cost($/Mwh)

60

55

50

45

40

35

0

0.5

1

1.5 2 2.5 Total Production(Mwh)

3

3.5

4 4

x 10

Figure 1: Unit Cost vs. the amount of generation To further study the properties of cost and EoS, we plot a cost-production curve using the data from IEEE reliability test system (Force 1999), shown in Figure 1. When the production level is low the unit cost decreases as in production growth. After production level passes a certain amount, i.e. g0 , the cost is no longer decreasing but starts to increase. In addition, the steepness of the cost curve is decreasing all the time as the production increases. The decrease of steepness indicates that the EoS value is decreasing as the production increases. All of the above observations can be shown as following lemma and theorem. Lemma 1. The Economies of Scale EoS is a decreasing function in terms of generation g, regardless of the cost level cv , i.e. ∂EoS ∂g < 0. Proof. By plugging in the definition of EoS, we have ∂(1 − αY − γY Y ln g − ∂EoS = ∂g ∂g Since γY Y > 0 and g ≥ 0, so in terms of generation g.

∂EoS ∂g

P

v

γY v ln cv )

=−

γY Y . g

< 0, therefore Economies of Scale EoS is a decreasing function

Theorem 1. For a generator, the unit cost Cunit is a decreasing function of the generation g if it unit is less than a certain amount of production g0 , i.e., ∂C∂g ≤ 0 when g ≤ g0 . Proof. The unit cost Cunit is equal to the total cost C divided by generation g, we have ∂Cunit ∂g

= = =

∂( Cg )

∂C ∂g

C = − 2 ∂g g g P γY v ln cv ) g · C αY γY Y ln g C ·( + + v )− 2 2 g g g g g X C · (αY + γY Y ln g + γY v ln cv − 1) g2 v

= −

C · EoS g2

If g ≤ g0 , then EoS ≥ 0, and Cunit is a decreasing function of g. According to the given data (see appendix), we found that g0 = 2053M wh for wind generation and g0 = 30379M wh for thermal generation. 6

2.2.3

Calculating the Decision-Dependent Transition Probabilities

As aforementioned, in this non-stationary stochastic process the transition probability from a parent node to a child node in the scenario tree is dependent on the capacities and generations of different types of generation technologies. EoS is a very good utility function to measure the aggregated effects of an outcome. Hence, we consider a utility function (of a node in the scenario tree) as the total P sum of the all EoS’s over all buses (indexed by b) and generator types (indexed by i), i.e., ib EoSib . In order to distinguish the notations of different nodes in the scenario tree, we add the index n to the equation (2) to facilitate the respresentation, # " X X X n n n γY v ln cib,v ) (4) 1 − (αY + γY Y ln gib + EoSib = ib

v

ib

where cnib,v are the realizations of uncertain costs (investment and maintenance) at node n. The n , on the other hand, is the amount of generated electricity from generator type i at bus variable gib b of node n. This EoS value is a function of both uncertain parameters (cost) and the decision variables (generation/capacity). As we have acquired the EoS for each node n in the scenario tree, we then can take advantage of discrete choice theory (Luce 1977, McFadden 1973) to determine the transition probabilities to each node from its parent node. We use P robn to denote the probability at which node n will happen when predicting the next time period costs at its parent node. The decision-dependent probability is formulated as follows, P n n ib EoS P ib t . P rob = P (5) t∈Sa(n) ib EoSib where the probability P robn is set as the ratio between the EoS at node n and the sum of the EoS’s of all sibling nodes (e.g., Sa(n) ). Note that the probability is a function of unit costs cnib,v and the n . Because g n is a decision variable, the values of g n are unknown before amount of generation gib ib ib the optimization problem is solved. 0.506 Low level cost High level cost

0.504

Probability

0.502

0.5

0.498

0.496 200

400

600

800

1000 1200 1400 Generation(Mwh)

1600

1800

2000

2200

Figure 2: Probabilities vs. the Amount of Total Generation Figure 2 shows an example of two-outcome probability curves that changes in response to the total amount of generation. The vertical axis represents the probability value, and the horizontal 7

axis is the total generation amount. Each curve corresponds to the probability value in each cost outcome. From the plot, one can observe that the probability is affected by both the cost level and the total generation amount. The two discrete outcome’s probabilities are reflected by two separate curves on the plot. When the generation amount is small, the low-cost outcome has a higher probability. When the total generation amount is increasing, the high-cost outcome’s probability starts to increase and the low-cost outcome’s probability starts to decrease. All of the above observations can be shown as the following theorems and corollary. Theorem 2. In the two-outcome case with a high cost outcome (cH v ) and a low cost outcome L H (cv ). The probability of high cost outcome P rob is an increasing function of generation g, i.e., ∂P robH > 0, where g is the total generation. ∂g Proof. By plugging in the definition of P robH , we have ∂P robH ∂g

=

∂EoS H ∂g EoS H + EoS L

γY Y g γY Y = − g = −

−

EoS H · ( ∂EoS ∂g

(EoS H + EoS L − EoS H · (EoS H + EoS L )2 P L γY v (ln cH v − ln cv ) · v (EoS H + EoS L )2

H

∂EoS L ∂g ) L EoS )2

+

H

∂P rob L Since − γYgY < 0 ,cH > 0. The probability of high cost outcome v > cv , and γY v < 0, therefore ∂g H P rob is an increasing function of the total generation g. L Corollary 1. In the two-outcome case with a high cost outcome (cH v ) and a low cost outcome (cv ), ∂P robL L the probability of low cost outcome P rob is a decreasing function of generation g, i.e., ∂g < 0, where g is the total generation.

3

Model Formulation

In this research, we are taking the market-based approach. We formulate our problem in a bilevel framework to identify the optimal investment plan for generation expansion.

3.1

Model Description

For this analysis, we formulate a bilevel multistage decision-dependent stochastic programming model considering thermal and wind power generator in long-term (10 to 20 years) planning horizon. This bilevel structure incorporates both long-term expansion planning and short-term generation and dispatch. It includes an upper-level stochastic expansion planning problem and a collection of lower-level problems that solves for Economic Dispatch. The upper-level long-term expansion planning problem is formulated as a multi-stage stochastic problem with decision-dependent uncertainties (i.e., non-stationary). This problem seeks to determine the optimal investment plan for generators with the aim of maximizing the expected total profit. This investment plan considers multiple aspects including optimal siting, optimal sizing, and the optimal timing (investing stage) for both thermal and wind generators. The newly built generators will participate in the electricity market, offering its production at Locational Marginal Price (LMP). Hence, the profit is achieved by selling the electricity to the market, and it depends on the market clearing prices which are determined by lower-level problems. 8

The lower-level problem represents the Economic Dispatch modeled by the DC optimal power flow (DCOPF) problem that seeks to find the minimum fuel cost by specifying the power generation and power flow within an electrical network (Frank et al. 2012). The optimal dual solutions of the lower-level problems provide the market clearing price, which are used in the upper-level problem to compute the expected profit. The market clearing prices are considered as LMPs (Baringo Morales 2013). Upper-level Maximize expected profit

Investment decisions

LMPs

Lower-level Minimize generation cost

OPF 1

. . .

OPFΞ

Figure 3: Bilevel structure The interaction between upper-level and lower-lever problem is illustrated in Figure 3. The electricity power investor makes expansion plans in the upper-level problem. The information of investment decisions are sent to the lower-level problems, i.e. the electricity market. The lower-level problems calculate the optimal power generation and flows according to the investment decisions. The LMPs are determined by the optimal dual solution of lower-level problems under different demand and wind intensity scenarios. The upper-level problem calculates the expected profit using the LMPs. Note that the upper-level and the lower-level problems are interconnected and must be solved jointly.

3.2

Upper-level Problem

In the upper-level formulation, we address the the long-term generation expansion problem by maximizing the expected total profit. The decision variables are the investment decision of electricity generator units (both thermal and wind generators). The parameters and variables related to the upper-level model are summarized in Table 1 and 2. The total profit to be maximized is defined as the difference between the operational profit and the total investment cost. The formulation of the investment cost (denoted as IC) in time period

9

Table 1: Upper-level Parameters and Indices t = 1, . . . , T ω t = 1 . . . , St k = 1, . . . , Kt ξ = 1, . . . , Ξk i = 1, 2 b = 1, . . . , BNum t Bib,ω cib,ωk x0ib xmax ib B max

The time periods (or stages) in the upper-level model Possible realizations of endogenous uncertainties (i.e., scenarios) at stage t Demand block (usually refer as season) at stage t Possible lower-level realizations of exogenous uncertainties at season k. Generator type. 1 for conventional, 2 for wind. Bus number. Unit investment cost for generator i at bus b in scenario ω of stage t, in $/MW. Unit production cost for generator i at bus b in scenario ω of stage t, in $/MWh. Initial capacity of generator i at bus b, in MW. Maximum allowed capacity for generator i at bus b, in MW. Investment Budget, in $.

Table 2: Upper-level Variables t αib,ω t xib,ω t∗ gib,kωξ

λt∗ b,kωξ ICωt t REVωtt kξ

Expansion decision for generator i at bus b in scenario ω of stage t, in MW. Capacity for generator i at bus b in scenario ω of stage t, in MW. Optimal generation amount for generator i at bus b in scenario ω and ξ of stage t in demand block k, in MWh. It is the optimal solution of the lower-level problems. Locational Marginal Price (LMP) at bus b in scenario ω and ξ of stage t in demand block k, in $/MWh. It is the optimal dual solution of the lower-level problems. Investment cost at stage t of scenario ω, in $. The operational profit in scenario ω and ξ of stage t in demand block k, in $.

10

t is given below, ICωt t (αt ) =

X

t t Bib,ω αib,ω , ∀, t, ω

(6)

ib t t where αib,ω is the investment decisions and Bib,ω represents unit investment cost. In our model, the investment costs are calculated on all nodes except the nodes associated in the last stage T . This is because the investment decisions are made to accommodate the future power system operations, and we assume an invested infrastructure is only available in the next stage. t The operational profit is calculated as the product between the generation (denoted as gib,kωξ ) t t and unit price cost difference (denoted as (λb,kωξ − cib,ωk )). The λb,kωξ is the Locational Marginal Price (LMP) that is solved in the lower-level model. cib,ωk is the unit production cost that includes both fuel cost and maintenance cost. As we assumed in Section 1 that we focus on the effect of expanded generators, therefore the profit only includes the generation from newly built generators since stage 0. . The formulation of total operation profit (denoted as REV ) in the time period t is given below, X
t∗ REVωtt kξ = λt∗ (7) b,kωξ − cib,ωk · gib,kωξ , ∀ω, t, ξ, k ib

The upper-level model varies in the time scale t, which represents each planning stage. In our model, we set the length of each stage to be 5 years. This model can also be applied to compute under other scales of planning horizon by changing the values of parameters without loss of generality. Each planning horizon is divided into 4 demand block (denoted as k) to represents different demand and wind uncertain levels which is discussed in Section 2. The upper-level model is formulated as follows, max F(α; ω) := −IC 0 (α0 )+ α     X 1 1  + ··· Eω1 max −ICω1 1 (α1 ) + Eξ max REVω1,k 1 (λk , gk ) 1 1  α1 λk ,gk k∈K1        X T  T +EωT max −ICωTT (αT ) + ) Eξ max REVωT,k , g · · · (λ T k k T  αT   λT k ,gk k∈KT

(8a) s.t.

T X

Eωt IC t (αt ) ≤ B max

(8b)

t=0 t−1 t−1 xtib,ω = xib,ω + αib,ω , ∀i, b, t, ω 0 t max xib ≤ xib,ω ≤ xib , ∀i, b, t, ω t∗ λt∗ b,kω,ξ , gib,kωξ ∈ arg min{Lower-level

(8c) (8d) Problem}, ∀t, ∀k, ∀ω,

(8e)

In the objective function (8a), the notation Eωt (·) represents the expected value of different outcomes for endogenous uncertainty. Our stochastic model uses discrete probability distributions for each corresponded uncertain scenario. Thus, this notation of expected value Eωt (·) is the same P 1 1 as n∈Sa(n) P robn · (·). The notation Eξ maxλ1 ,g1 REVω1,k 1 (λk , gk ) represents the expected value is k k taken upon the realization of exogenous uncertainty ξ, which is reflected on the uncertain demand and wind intensity, as discussed in Section 2. The constraints (8b) set the limit of total investment cost should not exceed the investment budget. Constraint (8c) states that the current available capacity (xtib,ω ) is equal to the sum of 11

t−1 capacity from last planning stage (xt−1 ib,ω ) and the invested generators (αib,ω ). The capacity of each generator is restricted in constraint (8d). Finally, constraints (8e) states that the values of variables t λtb,kω,ξ and gib,kωξ are determined by lower-level problems defined in (9).

3.3

Lower-level Problem

The lower-level problems are DC optimal power flow (DCOPF) that seeks to find the minimum fuel cost under different outcomes of demand and wind intensity. The parameters and variables related to the lower-level model are summarized in Table 3 and 4. The model is formulated in Equation (9).

i = 1, 2 b = 1, . . . , Nb l = 1, . . . , L o(l) r(l) ref b \ b : ref. cib,ωk dtb,kξ Hk Sl βi,kξ flmax π

Table 3: Lower-level Parameters and Indices Generator type. 1 for conventional, 2 for wind. Bus number. Transmission line number. sending-end bus of line l. receiving-end bus of line l. Reference bus. bus except for the reference bus. Unit production cost for generator i at bus b in scenario ω of stage t, in $/MWh. Demand level at bus b in scenario ξ of stage t in season k, in MWh. Number of hours in demand block k. Susceptance of line l, in MW. Utilization rate of generator i in demand block k of uncertain scenario ξ. The transmission limit of line l, in MW. Mathematical constant, the ratio of a circle’s circumference to its diameter.

Table 4: Lower-level Variables Generation amount for newly expanded generators i at bus b in scenario ω and ξ of stage t in demand block k, in MWh. Generation amount for existing generators i at bus b in scenario ω and ξ of stage t in demand block k, in MWh. Power flow via transmission line l in scenario ω and ξ of stage t in demand block k, in MWh. Phase angle at bus b in scenario ω and ξ of stage t in demand block k.

t gib,kωξ

ptib,kωξ t fl,kωξ t δb,kωξ

min

∀t,k,ω,ξ

s.t.

X

t cib,ω (gib,kωξ + ptib,kωξ )

(9a)

i,b

X

t (gib,kωξ + ptib,kωξ ) −

i

X

t fl,kωξ +

l|o(l)=b

X

−

≤

t fl,kωξ

≤

Hk flmax

∀b

(9b)

l|r(l)=b

t t t fl,kωξ = Hk Sl (δo(l),kωξ − δr(l),kωξ ), : φtl,kωξ

Hk flmax

t fl,kωξ = dtb,kξ : λtb,kωξ

: 12

φmin,t l,kωξ ,

∀l

φmax,t l,kωξ

(9c) ∀l

(9d)

min,t max,t t 0 ≤ gib,kωξ ≤ Hk βi,kξ xtib,ω : θib,kωξ , θib,kωξ max,t 0 ≤ ptib,kωξ ≤ Hk βi,kξ x0ib : ϕmin,t ib,kωξ , ϕib,kωξ

∀i, b ∀i, b

min,t max,t t − π ≤ δb,kωξ ≤ π, : ηb,kω , ηb,kω ∀b \ b : ref. t δb,kωξ

= 0, :

χtb,kω

∀b : ref.

(9e) (9f) (9g) (9h)

The objective function (9a) represents the generation cost of the existing and newly invested generators. Constraints (9b) enforce the supply-load balance at each node. The transmission flows are defined and limited in (9c) and (9d), respectively. The power productions of generation units are bounded in constraints (9e) and (9f). The parameters βi,kξ represents the capacity factor of generators, which is a ratio of the capacity that are effective and utilized for generation. The wind intensity level is related to the capacity factor. Finally, constraints (9g) and (9h) enforce voltage angles be bounded at every node. Because the electricity network is considered as a DC network, the LMPs can be considered as market clearing price (Baringo Morales 2013). The LMPs are the optimal solution of the dual variables λtb,kωξ corresponding to the constraints (9b).

4

Proposed Solution Approach

The model proposed in Section 3 is a multi-stage stochastic bilevel nonlinear program. Its bilevel, stochastic, nonlinear and nonconvex features bring tremendous solution challenges. To our best knowledge, there is no existing method that can directly solve our model. Therefore, we have developed a hybrid transformation and decomposition solution approach that transforms this model to a solvable format and reduce its computational complexity. This approach includes linear transformation of the profit term, linearization heuristics for decision-dependent probability, and the implementation of the Dantzig-Wolfe decomposition. Generally, the approaches in Section 4.1-4.3 eliminates the bilevel, nonlinear and nonconvex features of our model and transform our model to a solvable format. The approach in Section 4.4 decomposes the stochastic structure thus reduces computational complexity.

4.1

The Mathematical Program with Equilibrium Constraints (MPEC)

The bilevel formulation requires the upper-level problem (8) and the lower-level problem (9) to be jointly solved. In this section, we transform the bilevel problem to a single-level optimization problem. Because of the lower-level problems (9) are continuous and linear, so they can be replaced by their Karush-Kuhn-Tucker (KKT) optimality conditions. Thus, the bilevel structure is transformed to a single-level problem via embedding the KKT conditions in the upper-level problem. Therefore, our bilevel problem is recast as a single-level optimization problem. This problem is a Mathematical Program with Equilibrium Constraints (MPEC). Its formulation is provided below in (10). max (8a)

(10a)

α

s.t.

Constraints (8b)-(8e)

(10b)

{Constraints (9b)-(9h),

(10c)

λtb,kωξ − λtb,kωξ −

min,t ϕmax,t ib,kωξ + ϕib,kωξ = cib,ω , ∀i, b max,t min,t θib,kωξ + θib,kωξ = cib,ω , ∀i, b

13

(10d) (10e)

min,t λto(l),kωξ − λtr(l),kωξ − φtl,kωξ + φmax,t l,kωξ − φl,kωξ = 0, ∀l X X max,t min,t Hk Sl φtl,kωξ + ηb,kω − ηb,kω = 0, ∀b \ b : ref. Hk Sl φtl,kωξ + −

−

Hk Sl φtl,kωξ

0≤ 0≤ 0≤ 0≤ 0≤ 0≤ 0≤

φmin,t l,kωξ φmax,t l,kωξ min,t θib,kωξ max,t θib,kωξ ϕmin,t ib,kωξ max,t ϕib,kωξ min,t ηb,kω max,t ηb,kω

X

+

Hk Sl φtl,kωξ + χtb,kω = 0,

∀b : ref.

(10h)

k|r(k)=b

l|o(l)=b

0≤

(10g)

k|r(k)=b

l|o(l)=b

X

(10f)

t + fl,kωξ ≥ 0,

∀l

(10i)

t ⊥ Hk flmax − fl,kωξ ≥ 0,

∀l

(10j)

⊥

Hk flmax

t ⊥ gib,kωξ ≥ 0,

∀i, b

(10k)

t ⊥ Hk βi,kξ xtib − gib,kωξ ≥ 0,

⊥ ptib,kωξ ≥ 0, ⊥

Hk βi,kξ x0ib

⊥ π+

t δb,kωξ

−

∀i, b

∀i, b ptib,kωξ

(10l) (10m)

≥ 0,

∀i, b

(10n)

≥ 0,

∀b \ b : ref.

(10o)

t ≥ 0, ⊥ π − δb,kωξ

∀b \ b : ref.

(10p)

}, ∀t, k, ω, ξ The KKT conditions are formulated as constraints (10c)-(10p). Constraints (10d)-(10h) correspond to the constraints the dual problem. The complementarity constraints (10i)-(10p) have the form of 0 ≤ α ⊥ γ ≥ 0 (i.e., at least one of them is zero).

4.2

Linear Transformation of the Operational Profit Terms and MILP Formulation

In the upper-level problem’s objective function (8a), the operational profit term is defined in et . An optiquation (7). It constitutes multiplication terms between two variables λtb,kωξ and gib,kωξ mization problem that contains the product of two decision variables are called a bilinear program (BLP). Bilinear programming belongs to a class of nonconvex and nonlinear optimization model (Zhan et al. 2017a). This nonlinear formulation will bring in great computational challenges to the solution process. Note that the dual constraints in (10e) can be rearranged as follows, max,t min,t λtb,kωξ − cib,ω = θib,kωξ − θib,kωξ .

(11)

t Then, we multiply the term gib,kωξ to the both sides of the above equation and then summing up both sides over all i and b will lead to the following, X X max,t min,t t t t (λtb,kωξ − cib,ω ) · gib,kωξ = (θib,kωξ gib,kωξ − θib,kωξ gib,kωξ ). (12) ib

ib

We then use the complementary slackness equations of (10k) and (10l) to replace the right hand side of equation (12). We have the following, X X max,t t (λtb,kωξ − cib,ω ) · gib,kωξ = Hk βi,kξ xtib · θib,kωξ . (13) ib

ib

By strong duality, we can obtain a linear expression for the profit term. This theorem states that if a problem is convex, the objective functions of the primal and the dual problems have the 14

same value at the optimum. Thus, X X X min,t t cib,ω (gib,kωξ + ptib,kωξ ) = dtb,kξ λtb,kωξ − Hk flmax (φmax,t l,kωξ + φl,kωξ ) i,b

b

−

l

X

max,t Hk βi,kξ · (xtib θib,kωξ + x0ib ϕmax,t ib,kωξ ) −

ib

X

min,t max,t π(ηb,kω + ηb,kω )

b\b:ref.

(14) P P max,t t In the following, we replace the term ib Hk βi,kξ xtib · θib,kωξ in (14) by ib (λtb,kωξ − cib,ω ) · gib,kωξ , which is provide in (13). Therefore, the nonlinear profit term is finally replaced by a series of linear terms, X X X min,t t (λtb,kωξ − cib,ω ) · gib,kωξ = dtb,kξ λtb,kωξ − Hk flmax (φmax,t l,kωξ + φl,kωξ ) ib

b

−

l

X

t cib,ω (gib,kωξ

+ ptib,kωξ ) −

i,b

−

X

Hk βi,kξ x0ib ϕmax,t ib,kωξ

ib min,t π(ηb,kω

X

+

max,t ). ηb,kω

(15)

b\b:ref.

The complementarity constraints in the MPEC (10) can be reformulated through exact equivalent mixed-integer linear equations using Fortuny-Amat transformation (Fortuny-Amat and McCarl 1981). All of the complementarity constraints have the equivalent form: α · γ = 0; α ≥ 0; γ ≥ 0. It can be linearized to two equivalent constraints: 0 ≤ α ≤ M · u and 0 ≤ γ ≤ M (1 − u), where M is a sufficiently large constant and u is a binary variable. In this way, the constraints related to KKT conditions are converted to mixed integer linear constraints. Finally, the our proposed model is linearized to be a Mixed Integer Nonlinear Program (MINLP) with only linear constraints as follows, max

(8a)

α

(16a)

subject to Constraints (8b)-(8e)

(16b)

Constraints (9b)-(9h)

(16c)

Constraints (10d)-(10h)

(16d)

{

φmin,t

φmin,t l,kωξ ≤ M · ul,kωξ , max,t

φ φmax,t l,kωξ ≤ M · ul,kωξ , min,t θib,kωξ max,t θib,kωξ

∀l

(16f)

∀i, b

(16g)

θmax,t

∀i, b

(16h)

∀i, b

(16i)

∀i, b

(16j)

≤ M · uib,kωξ , ≤ M · uib,kωξ ,

ϕmin,t ib,kωξ ≤ M · uib,kωξ , ϕmax,t

ϕmax,t ib,kωξ ≤ M · uib,kωξ , min,t

min,t ηb,kω ≤ M · uηb,kω ,

≤M·

(16e)

θmin,t

ϕmin,t

max,t ηb,kω

∀l

max,t uηb,kω ,

∀b \ b : ref.

(16k)

∀b \ b : ref.

(16l)

15

min,t

t Hk flmax + fl,kωξ ≤ M · (1 − uφl,kωξ ),

∀l

(16m)

max,t uφl,kωξ ),

∀l

(16n)

t Hk flmax − fl,kωξ ≤ M · (1 − θmin,t

t gib,kωξ ≤ M · (1 − uib,kωξ ),

∀i, b θmax,t

t Hk βi,kξ xtib − gib,kωξ ≤ M · (1 − uib,kωξ ),

ptib,kωξ ≤ M · (1 −

min,t uϕ ib,kωξ ),

∀i, b

(16p)

∀i, b ϕmin,t

Hk βi,kξ x0ib − ptib,kωξ ≤ M · (1 − uib,kωξ ), η min,t

t π + δb,kωξ ≤ M · (1 − ub,kω ), η max,t

(16o)

t π − δb,kωξ ≤ M · (1 − ub,kω ),

(16q) ∀i, b

(16r)

∀b \ b : ref.

(16s)

∀b \ b : ref.

(16t)

min,t max,t min,t min,t max,t ϕmax,t η min,t η max,t uφl,kωξ , uφl,kωξ , uθib,kωξ , uθib,kωξ , uϕ ib,kωξ , uib,kωξ , ub,kω , ub,kω , ∈

{0, 1},

}, ∀t, k, ω, ξ

(16u) (16v)

where M is a sufficiently large constant.

4.3

Linearization Heuristics for Decision-dependent Probability

From Section 2, we know that the transition probability P rob is a function of decision variables g. This makes the objective function both nonlinear and nonconvex. To the best of our knowledge, it is very difficulty to solve this nonlinear and nonconvex model to real optimality with an exact global optimal (Tseng et al. 2015). Given the rest of the model formulation already very computationally challenging with a large number of MILP constraints, any linear attempt will increase the computational complexity dramatically. Thus, we determine the best strategy is to employ an iterative heuristic method to tackle this nonlinear formulation. We first acquire an initial solution gini to compute the value of probability P rob(gini ). Then, we replace the decision-dependent probability P rob(g) by P rob(gini ), which is a fixed value to get rid of nonlinear term. After this step, the linear model is solved by MILP solver with the ˆ1 and objective value Zˆ1 . In the next step, the decision-dependent probability is optimal solution g ˆ2 and objective value Zˆ2 . replaced by P rob(ˆ g). The model is then solved with optimal solution g The heuristic process is then solved iteratively until the stopping criteria is reached at iteration i ˆ ˆ as |Zi −ˆZi−1 | < . This process is summarised in Algorithm 1 at the end of Section 4.4. Zi

4.4

Dantzig-Wolfe Decomposition Approach

The formulation in (16) with fixed (not variable) upper-level transition probabilities is a multistage stochastic Mixed Integer Linear Program (MILP) with a large number of constraints and variables. Although we have taken advantage of state-of-art MILP solver to solve the problem, but the computational time is still very long, and sometimes simply impossible. To identify the underlying computation complexity, we compare the computation time between deterministic model and the stochastic model, shown in Table 5. The computation time of deterministic model is much shorter than the time of stochastic model. This implies that the computational complexity is embedded in the stochastic structure.

16

Table 5: Deterministic vs. Stochastic Computation Times Bus

Computation time (sec) Deterministic Stochastic

3 30 57 118 ∗: Exceeded time

2.84 22.78 49.52 79116.53 18.52 (4.34%)* 73.16 (4.15%)* limit of 80000 seconds.

Non-anticipativity Constraints

Figure 4: Scenario splitting We first transform the nodal based formulation of the stochastic problem into scenario based formulation, i.e., the formulation is based on unique paths from the root node to the leave nodes. The scenarios are connected via non-anticipativity constraints shown in Figure 4, where, on the right, each parallel chain is a scenario of multiple stages. The structure of the scenario-based upper-level constraints is illustrated in Figure 5.

Nonanticipativity constraints purely on investment decisions. Deterministic problem under random outcome 1 Deterministic problem under random outcome Ξ

Figure 5: Problem Structure

17

The simplified reformulation is shown as follows, X X max P robs [cs xst + Eξ ds (ξ)yst ] s∈S

(17a)

t∈T

s.t. xst = xn(s,t) ,

∀s ∈ S, t ∈ T , n ∈ N

{Constraints (16b)-(16u)},

(17b)

∀s ∈ S, t ∈ T , ξ ∈ Ξ

(17c)

where N is the set of nodes of the nodal based scenario tree, S represents all the scenarios, and T at stages. xst represents the here-and-now variables, specifically investment decisions, and yst represents wait-and-see variables, specifically operational variables. Constraints (17b) are nonanticipativity constraints that enforce all the variables that belonging to the same node should be equal. The non-anticipativity constraints bundle different scenarios to be one integrated problem. In Dantzig-Wolfe decomposition method, the stochastic structure is decomposed into a master problem and a set of subproblems. Each subproblem represents a scenario in the scenario tree. The Restricted Master Problem [RMP] incorporates the solutions from each subproblem to acquire the final optimal solution while enforcing the non-anticipativity. It is shown as follows,   X X X XX ˆ j ) [RMP]: max P robs · − ICst + ρjs Eξ (REV (18a) st,kξ s∈S

s.t.

X

t∈T

ρjst

·

j zˆst,ib

j∈Fs

t∈T

k

= αn(s,t),ib , ∀s ∈ S, t ∈ T , n ∈ N , i, b, : πst,ib

(18b)

j∈Fs

X

ρjs = 1, ∀s ∈ S, : πs0

j∈Fs ρjs , αn,ib

(18c)

≥ 0, ∀n, j ∈ Fs

(18d)

where Fs is the set of all feasible investment solutions for scenario s. πst,ib and πs0 are dual variables. The [RMP] calculates the overall optimal investment solution αn,ib from the convex combination of j the feasible investment solutions zˆst,ib that are acquired by solving subproblems for each scenario s ∈ S. ! " # X X X 0 [SPs ] : max P robs · Eξ (REVst,kξ ) − π ˆst,ib zst,ib − π ˆn (19a) t∈T

k

ib

s.t. {(16b) − (16u)}, ∀t, ∀k, ∀ω, ∀ξ

(19b)

The subproblems aim to evaluate the reduced costs of extreme points of the master problem. The most preferable reduced costs are determined to enter the basis. The master and sub problems are solved iteratively. At each iteration the [RMP] is updated by adding columns until convergence is achieved. The solution algorithm including Dantzig-Wolfe decomposition and linearization heuristics for decision-dependent probability, is summarized in Algorithm 1.

18

Algorithm 1 Solution algorithm for electricity investment planning ˆ1 = gini . 1: Initialize: i = 1, g ˆi − Zˆi−1 |/Zˆi > 1 do 2: while |Z 3: Compute P rob(ˆ gi ) ˆ 1 = x0 . 4: Initialize the [RMP] with UB=+∞, LB=−∞,j = 1, x 5: while (UB-LB)/LB> 2 do 6: Solve the [RMP] and update LB to be its optimal value ZRM P . ˆj . 7: Update the optimal investment decision x∗ = x 8: Solve the [SPs ], record Ptheir optimal values ZSPs ∀s ∈ S 9: Update U B = LB + s ZSPs . 10: Generate new columns ρjs and add them to [RMP], j ← j + 1. 11: end while ˆi . 12: Solve lower-level problems using information of x∗ to get optimal production level g 13: i←i+1 14: end while

5

Numerical Experiments and Computational Results

In this section, a series of numerical experiments are conducted and the results are analyzed. Our model and algorithms are tested using IEEE reliability testing systems. Table 6 shows the instances of each testing systems including the number of generators, the number of wind generators and the number of transmission lines. Table 6: IEEE reliability testing systems System

Generator

Wind Generator

Transmission lines

3 bus 30 bus 57 bus 118 bus

6 9 7 54

3 4 2 10

3 41 80 186

Our model and algorithm are tested in a four-stage (T = 4) planning horizon with each stage spanning 5 years, i.e., 43800 generation hours at each stage. Unless specifically stated, we assume that the demand increases at the rate of 1% each year and the investment cost has an annual interest rate at 1% reflecting inflation. The converging gaps for linear heuristic 1 and for decomposition 2 in Algorithm 1 are set to be 1 × 10−3 and 5 × 10−3 , respectively. The endogenous uncertainty level (see Section 2) is set at 20%. Each stage is divided into four demand blocks as mentioned in Section 2. The demand level at each demand block is set as 0.95, 0.85, 0.75, 0.65, respectively. The wind capacity factor levels at each demand block are set as 0.55, 0.45, 0.35, 0.25. The other settings and data, such as the specs of generators, the demand amount, the local wind intensity factor, are acquired from IEEE reliability testing systems and mentioned in the later sections. All models and algorithms are programmed in C++ by calling the commercial MILP solver ILOG CPLEX 12.5. All experiments are implemented on a personal computer, which has quad Intel Core i7 processors with a CPU at 3.40 GHz and a RAM space of 8GB.

19

5.1

An Illustrative Example: Investment Analysis of IEEE 3 Bus System

Firstly our model is analyzed using a simple 3-bus testing system showing in Figure 6. This test network consists of three nodes and three transmission lines. At each node, there are a thermal generator already installed. The demand is also connected at each node. The entire network is divided into two wind zones that has different wind characters. Our model is applied to compute the optimal investment decisions on generator’s type, size, location and timing. 82 3. Wind Power Investment: A Static Approach

Figure 3.4: Static wind power investment. Three-node example: Network. Figure 6: IEEE 3 bus testing system

The data of all generators, both existing or invested, are listed in Table 7. Units G1, G2 and Datathermal pertaining to generation units are 1, provided 3.1. EachThe generaG3 are existing generator units at node 2 and in 3, Table respectively. generator W1, W2, W3 are wind generators that have not been constructed yet. The wind generator has zero fuel cost. tion unit is characterized by four production blocks with their associated capacThe investment budget2-5 is set at $40 Million. ities (columns in Table 3.1) and marginal costs (columns 6-9 in Table 3.1). Both the capacities and the marginal costs are considered fixed throughout the planning horizon. Data describing the peak demand at each node of the system are provided in Table 3.2. The peak demands given in Table 3.2 multiplied by the demand D factors Kn,o provide the demands at different nodes and for different operating

20

Table 7: Generator Data of IEEE 3 bus System Unit G1 G2 G3 W1 W2 W3

Type Thermal Thermal Thermal Wind Wind Wind

Location Node Node Node Node Node Node

1 2 3 1 2 3

Current Capacity [MW]

Max Capacity [MW]

150 150 100 0 0 0

200 200 150 100 100 100

Fuel Cost

O&M Cost

[$/MWh]

[$/MWh]

Investment Cost [$/MW]

31.67 64.16 39 0 0 0

26.24 8.34 13.29 15.26 15.26 15.26

150000 120000 184600 200000 200000 200000

The capacity factor represents the ratio of capacity that is effective and used for production. We assume all of the thermal generators has the same capacity factor of 0.85. On the other hand, the capacity factor of wind generator depends on wind conditions. The wind condition is both locationally and seasonally dependent (Baringo and Conejo 2011). The seasonal dependency is reflected by wind capacity factor levels for each demand block, mentioned in Section 2. The value of wind capacity factor levels are provided in Section 5. The locational dependency is related to the geographic conditions of specific power network. In this 3-bus system, we assume that the wind speed in the north wind zone is lower than the wind speed in south wind zone. Therefore, the capacity factor of wind generator at node 3 is higher than those in node 1 and 2. The capacity factor data is shown in Table 8 Table 8: Capacity Factor of IEEE 3 bus System Unit

Capacity Factor [p.u.]

G1, G2, G3 0.85 W1, W2 0.4* W3 0.5* ∗: The average value of capacity factors over four demand blocks As discussed in Section 2, the uncertain demand follows normal distribution with the mean (D) and standard deviation (Std), provided in Table 9. We assume the demand increases at the rate of 2% each year. Table 9: Demand of IEEE 3 bus System Demand

D [MW]

Std [MW]

D1 D2 D3

120 100 100

20 20 20

The transmission line data are provided in Table 10. We test our model under two transmission conditions: uncongested and congested. The uncongested network’s transmission line has enough capacity to transmit generated power. On the other hand, in the congested network, the transmission limit capacity is limited. The constraints of transmission limitation will be binding in the solution process. 21

Table 10: Transmission line data of IEEE 3 bus System Line

From node

To node

S [p.u.]

f max uncongested [MW]

f max congested [MW]

1 2 3

1 1 2

2 3 3

5 5 5

100 100 100

30 30 30

The uncongested result is shown in Table 11. Note that Table 11 only illustrates the investment decision solutions for one of the uncertain scenario. The result shows that the investment involves on both thermal and wind generators. The wind generator is more preferable for investors due to its low-cost advantage. We notice that node 3 has the most investment due to the wind intensity in node 3 is larger than node 1 and 2. This result indicates that the investment decision provided by our model takes both sizing and siting into consideration. We also notice that the investment decision covers all available stages (our settings prohibits the investment on Stage 4) to advocate the growing demand and to minimize the construction cost. This result shows that it is necessary to consider the multistage framework for long-term investment planning. Table 11: Result for uncongested network of IEEE 3 bus system Unit G1 G2 G3 W1 W2 W3

Investment Decision [MW] Stage 1 Stage 2 Stage 3 Stage 4 11 0 0 0 0 0 26 0 0 0 0 0 0 1 0 0 0 0 0 0 73 27 0 0

Expected Profit ($)

2.95E+08

Table 12: Result for congested network of IEEE 3 bus system Unit G1 G2 G3 W1 W2 W3

Investment Decision [MW] Stage 1 Stage 2 Stage 3 Stage 4 0 0 5 0 0 0 44 0 0 0 0 0 0 0 0 0 6 0 0 0 72 10 0 0

Expected Profit ($)

2.79E+08

The investment decisions of congested network is recorded in Table 12. Comparing to results from uncongested network, the congested case has less wind investment in node 3, even though the wind power has advantage at node 3. This is because the transmission line limits the power flow within the network, and becomes the bottleneck. The investment decision has adjust the investment decisions to make sure the investment decision is feasible. This compromise is also reflected by the decrease of total expected profit comparing to the uncongested case. 22

5.2

Decomposition Approach Analysis

In Section 4, we present a hybrid transformation decomposition solution approach. The effectiveness of the solution approach is presented in this section. However, due to the limitation of off-shelf solver, we are not able to get solutions until a series of transformation is conducted. As a result, this analysis can only compare between two solvable versions of our model: Equation (16) and Equation (18)-(19). Therefore, the following case study focuses on the effect of decomposition to examine the effect of reducing the computational complexity of our model. In this case study, we conduct computations on different sizes of electricity systems to compare the performance of proposed decomposition algorithm to the one from directly solving the problem (16). The computational results for different test cases are recorded in Table 13. The column “Bus” includes IEEE 3 bus, 30 bus, 57 bus and 118 bus testing systems. The column “Budget” includes the investment budget limits in unit of $Million. The investment budget is setting from $10 Million to $ 50 Million for each test case. Table 13: Computation time: Direct vs Decomposition Bus

Budget (Million $)

Time (sec) Direct Colgen

10 3.43 20 5.41 30 5.40 3 40 22.78 50 36.52 10 9.70 20 539.33 30 44427.14 30 40 79116.53 50 (12%) 20 1350.19 30 (4.92%) 57 40 (4.34%) 50 (4.34%) 10 3124.69 20 7118.54 30 60107.64 118 40 (4.15%) 50 (15.27%) Note: The dash in the table indicates

Optimal Direct Colgen

Relative Difference

281.01 9.52E+07 9.59E+07 0.74% 32.79 2.75E+08 2.75E+08 0.01% 121.73 3.72E+08 3.72E+08 0.06% 58.39 4.55E+08 4.55E+08 0.18% 222.16 5.27E+08 5.27E+08 0.11% 946.07 9.20E+07 9.16E+07 0.51% 4117.52 1.43E+08 1.43E+08 0.32% 34953.79 1.97E+08 1.98E+08 0.55% 58875.85 2.21E+08 2.20E+08 0.55% (1.56%) – – – 802.06 9.67E+07 9.66E+07 0.11% 1035.38 – 1.48E+08 – 959.62 – 1.56E+08 – 1910.91 – 1.57E+08 N/A 1385.84 4.94E+07 4.94E+07 0.00% 3572.46 9.87E+07 9.87E+07 0.00% 44034.73 1.40E+08 1.40E+08 0.13% (2.19%) – – – (11.88%) – – – that no feasible solution is obtained within 80000 sec.

The column “Optimal” records the optimal values acquired by each method. The last column “Relative Difference” refers to the the optimal value between the direct solve method and proposed Dantzig-Wolfe decomposition method. From Table 13, we notice that the relative difference between the optimal of two methods is always less the 1%. This indicates that the Dantzig-Wolfe decomposition method can provide equally accurate results as directly solving it by the solver. The column “Time” records the computation time consumed by different methods. The column “Direct” refers to the results acquired by directly calling CPLEX to solve the problem of (16). The column “Decomposition” refers to the computational results from using Dantzig-Wolfe decompostion algorithm. The percentage in parenthesis represents the relative gap when the 80000s time 23

limit is reached. We notice that for small systems, i.e., 3 bus, the direct method takes less time than using decomposition. For medium size system, i.e., 30 bus, there is no significant difference between two approaches in terms of computation time. When it comes to larger systems, i.e. 57 bus and 118 bus, the Dantzig-Wolfe decomposition method is able to solve the problem in a shorter time than directly calling CPLEX. For some test cases, when both methods cannot acquire the optimal within the time limit, the decomposition approach provides a better gap than direct method. This is because for small size network, the size of branch-and-bound tree is small and can be handled by CPLEX. As the system size increases, the size of branch-and-bound tree grows exponentially which leads to the increasing computation time. On the other hand, the decomposition method breaks down the large size problem (stochastic problem) into small size subproblems (deterministic problem). From Table 5 of Section 4.4, we can observe that the time difference between solving stochastic problem and deterministic problem is greater for larger electricity systems. This means the time difference between direct calling CPLEX and solving each iteration in the decomposition approach is larger for larger electricity systems. We also observe that there is no significant increase of the iteration number. Therefore, for large electricity systems, the decomposition approach takes less computation time than directly calling CPLEX.

5.3

Investment Budgets Analysis

From the result in Table 13, we can also analyze the optimal expansion decisions for different investment budgets. We notice that as the investment budget increases, the computation also takes longer time. This is because the problems with larger investment budget has larger feasible solution region, thus the binary variables’ branch-and-bound tree has more nodes that usually leads to a longer solution process. Moreover, as expected, the optimal profit is increasing as the investment budget is getting larger.

5.4

Decision-Dependent Analysis

In this section, we compare the decision-dependent stochastic programming approach over the tradition stochastic programming approach. In many stochastic programming literatures, the effect of stochastic programming is often evaluated via the value of stochastic programming solution (VSS). The VSS compares the optimal values of the stochastic program (here and now solution) and the expected result of using the expected value solution (EEV). Using the similar idea of VSS, we compute the value of decision-dependent stochastic programming solution (VDDSS), which evaluates by how much does the solution of traditional stochastic program fall short of the optimality in this decision-dependent model. The VDDSS, as defined in (Zhan et al. 2017b), is calculated by first acquiring the optimal solution from a traditional stochastic model. Then, this solution is plugged into the decision-dependent formulation and the objective function value is then acquired. Finally, the VDDSS is calculated as the difference between the optimal objective value from decision-dependent approach and the objective value by using the traditional stochastic model solution in the decision-dependent model. The computation is conducted on 3 Bus testing system, where the investment budget is set at $10 Million.

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5

5.5

x 10

VDDSS($)

5 4.5 4

VDDSS($)

3.5 3 2.5 2 1.5 1 0.5 0

0

10

20 30 Cost Uncertainty Level (%)

40

50

Figure 7: The VDDSS on 3 Bus System Figure 7 shows that as cost uncertainty changes from 0% to 50%, the VDDSS increases dramatically. When the cost variation level is equal to zero, we observe that VDDSS is also equal to zero. This is because both decision-dependent and traditional stochastic reduce to the same deterministic model. When uncertainty level is greater than zero, the VDDSS is always positive which by definition means that the decision-dependent stochastic approach outperforms the traditional stochastic approach. We also observe that VDDSS curve has an overall increasing trend as the uncertainty level increases. This implies that the decision-dependent approach shows greater advantage over traditional stochastic approach while experiencing higher volatility. From the above observation, we can conclude that the decision-dependent stochastic approach can better represents the power system expansion planning problem than traditional stochastic approach. It is important to take into account the decision-dependent approach for long-term investment planning problems.

6

Conclusions

In this paper, a bilevel multistage decision-dependent stochastic programming model is proposed to solve for the long-term power generation investment expansion planing problem considering the market framework. This model seeks for the optimal, timing, sizing and siting for both thermal and wind power units to be built to maximize the expected profit for a profit-oriented power investor. The proposed formulation is based on the bilevel framework that includes an upperlevel stochastic expansion planning problem and a collection of lower-level problems that solves for economic dispatch. In the proposed model, the decision-dependency is included in the stochastic 25

approach. The formulation of the decision-dependent probability distribution is based on the cost economic scale theory in electricity systems. The bilevel structure is reformulated to a single level problem by taking advantage of the KKT optimal conditions. To further resolve the computational challenges and accelerate the calculation process, a hybrid solution approach is developed, including linear transformation of the profit term, linearization heuristics for decision-dependent probability and implementation for Dantzig-Wolfe decomposition. Extensive case studies are conducted based on IEEE reliability test systems. The study on a 3 bus electricity system shows that the multistage framework is able to advocate the growing demand of each stage and to minimize the construction cost. The comparison between the solutions for uncongested and congested network shows our model is able to take both sizing and siting into consideration. The study of computation time and optimal value demonstrates that the decomposition approach can better handle large systems. Finally, the numerical experiments of VDDSS show that it is important to take the decisiondependent approach for long-term investment planning problems with non-stationary transition probabilities. Future research tasks include adding the transmission expansion decisions to our model and employing parallel computation algorithms to the decomposition approach to further reduce computation time.

Acknowledgments This work is in part supported by National Science Foundation through Grant CMMI-1355939, and the AFRL Mathematical Modeling and Optimization Institute. The authors would also like to thank the reviewers and Editors for their helpful suggestions and comments.

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Biographies Yiduo Zhan is a senior operations research analyst in Monsanto Company. He received the B.S. degree in physics from the University of Science and Technology of China, Hefei, China, in 2010, and the M.S. degree in modeling and simulation in 2013 and Ph.D. degree in Operations Research and Industrial Engineering and Management Systems in 2017 from the University of Central Florida, Orlando, FL, USA. His research interests lie within the area of multistage optimization in the energy system. Qipeng P. Zheng is an Assistant Professor with the Department of Industrial Engineering and Management Systems, University of Central Florida, Orlando, FL, USA. He received the Bachelors degree in automation from North China University of Technology, Beijing, China, in 2001, the Masters degree in automation from Tsinghua University, Beijing, in 2005, and the Ph.D. degree in industrial and systems engineering from the University of Florida, Gainesville, FL, USA, in 2010. His research area is optimization, network science, machine learning and applications especially in management systems, energy and power systems, sustainability, and transportation planning under uncertainty. He is the Co-Editor-In-Chief of Springer journal, Energy Systems.

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