Profit-maximization generation maintenance scheduling through bi-level programming

Accepted Manuscript

Profit-maximization Generation Maintenance Scheduling through Bi-level Programming Peyman Mazidi, Yaser Tohidi, Andres Ramos, Miguel A. Sanz-Bobi PII: DOI: Reference:

S0377-2217(17)30635-5 10.1016/j.ejor.2017.07.008 EOR 14558

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

5 August 2016 7 May 2017 3 July 2017

Please cite this article as: Peyman Mazidi, Yaser Tohidi, Andres Ramos, Miguel A. Sanz-Bobi, Profitmaximization Generation Maintenance Scheduling through Bi-level Programming, European Journal of Operational Research (2017), doi: 10.1016/j.ejor.2017.07.008

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Highlights • Bi-level problem for generation maintenance scheduling in deregulated power system • Nash equilibrium is obtained as the solution of the non-cooperative game • Coordination process among system operator and generation companies

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• Analysis of strategic behaviors of generation companies in maintenance scheduling

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Profit-maximization Generation Maintenance Scheduling through Bi-level Programming Peyman Mazidia,b,∗, Yaser Tohidia , Andres Ramosb , Miguel A. Sanz-Bobib Royal Institute of Technology, Electric Power and Energy Systems (EPE) Department, Stockholm 10044, Sweden b Comillas Pontifical University, Research Institute for Technology (IIT), Madrid 28015, Spain

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a KTH

Abstract

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This paper addresses the generation maintenance scheduling (GMS) dilemma in a deregulated power system. At first, under a centralized cost minimization framework, a GMS is formulated and set as the benchmark (cost-minimization GMS). Then, the cost-minimization is changed into a profit-maximization problem of generation companies (GENCOs) and the GMS is developed as a bi-level optimization. Karush–Kuhn–Tucker conditions are applied to transform the bi-level into a single-level mixed-integer linear problem and subsequently, Nash equilibrium is obtained as the final solution for the GMS under a deregulated market (profit-maximization GMS). Moreover, to incorporate reliability and economic regulatory constraints, two rescheduling signals (incentive and penalty) are considered as coordination processes among GENCOs and independent system operators. These signals are based on energy-not-supplied and operation cost, and ensure that the result of profit-maximization GMS is in the given reliability and social cost limits, respectively. These limits are obtained from the cost-minimization GMS. Lastly, the model is evaluated on a test system. The results demonstrate applicability and challenges in GMS problems.

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Keywords: Maintenance, Scheduling, OR in Energy, Coordination Process, Deregulated Power System

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1. Introduction

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There are two main maintenance categories: corrective and preventive. Corrective maintenance denotes remedial actions performed after a failure in order to restore the operation back to its previous operating state. Preventive maintenance refers to actions carried out to maintain operability of an asset at an acceptable level. Generation maintenance scheduling (GMS) problem in power systems is to arrange a plan for generation units to take preventive maintenance actions in order to decrease their probability of failure. In addition, generation and maintenance scheduling is commonly performed for short-, medium- and long-term horizons ranging from hours to years (Shahidehpour and Marwali, 2000; Ghazvini et al., 2013). In a vertically integrated (regulated) power system, there is generally one stakeholder that acts as a system operator and decides on the maintenance decisions for generation companies (GENCOs). In this type of power system, the objective is to minimize the (social and operational) costs within acceptable reliability limits. (Froger et al., 2014). In a liberalized (deregulated) power system, there are two main stakeholders with different interests. An independent system operator (ISO) that is responsible for operating the power system by setting regulations on social welfare and reliability of the system. And GENCOs that seek to maximize their profit by considering cost and profit aspects. Therefore, organizing such a plan in a liberalized power system is a difficult and complex conundrum with conflicting objectives for ISO and GENCOs. To account for various objectives, optimization techniques have gained popularity by addressing multiple criteria in GMS in power systems. One important aspect here is to link these objectives. Optimization techniques have the flexibility to allow for coordination of different objectives which this coordination assists to fulfill the requirements of different actors in the system (Chattopadhyay, 2004). Hence, GMS in a liberalized power system will be beneficial to all the players by including all three aspects of cost, reliability and profit. ∗ Corresponding author.

Peyman Mazidi, KTH Royal Institute of Technology, Osquldasvg 6, Stockholm 10044, Sweden; E-mail: [email protected]

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In a regulated power system, solving a GMS problem with the objective of minimizing costs results in a perfectly competitive market scenario, which is used as the benchmark in this paper. However, by emergence of competition through electricity market in a deregulated power system, game theoretic approaches are gaining attention. For instance, to find an equilibrium point in profit for all the GENCOs participating in the market, obtaining a Nash equilibrium point strategy would be beneficial. Where specific studies in accordance with this paper are presented in the following paragraphs, a thorough review on the research performed on maintenance scheduling can be found in Yamayee (1982); Kralj and Petrovic (1988); Dahal (2004); Khalid and Ioannis (2012); Froger et al. (2014) and Froger et al. (2016). Yamayee (1982) provided explanations of the terminologies and the maintenance scheduling problem. The survey was carried out with the view of checking the objective functions, deterministic or stochastic analysis and optimization techniques. The approaches were cost- and reliability-oriented. The cost-oriented approaches were computationally infeasible, whereas the reliability-oriented approaches were too simplistic. Kralj and Petrovic (1988) evaluated the main characteristics of the maintenance scheduling problem with respect to objectives and constraints. For the system to benefit most from these aspects, a combination of economic and reliability views was suggested. The method should be able to provide optimal or near-to-optimal solution and a link between the operational researchers and the power system planners would bring more benefit for all. Dahal (2004) addressed the transition from regulated to deregulated power systems. In other words, the maintenance scheduling which traditionally was performed by a power system operator, is no longer centrally controlled and this could endanger the system’s reliability criterion. Therefore, the traditional approaches are not ideal for the new environment and should be updated. Khalid and Ioannis (2012) reviewed the techniques applied in solving maintenance scheduling including heuristics and mathematical techniques. They mention that where heuristics may not lead to an optimal solution for complex systems, mathematical techniques may fail due to time-limits. It was also mentioned that a multi-stage approach can assist as it breaks down the main problem into several sub-problems. Froger et al. deliver a thorough review on the maintenance scheduling field from various perspectives (Froger et al., 2014, 2016). For instance, they reviewed studies with objectives of cost and reliability in power systems with regulated structure. Next, they studied problems in power systems with deregulated structure and added research works that consider uncertainty. After going through the different techniques applied in solving the maintenance scheduling problem, they concluded that considering solely cost or reliability objectives is not suitable for the new deregulated structure. Hence, they propose approaches based on maximization of profit of GENCOs while coordinating the decisions of different agents in the system; this becomes a game theoretic problem. They raise several aspects that still require to be looked into, such as load uncertainty, coordination between GENCOs and the ISO and integration of renewable sources with their stochastic nature. The literature review in this paper covers state-of-the-art studies in three areas of: 1) bi-level programming, 2) coordination process and 3) game-theory which are related to GMS. 1.1. Bi-level programming in GMS

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Due to the inherent structure of maintenance scheduling problems, many authors have utilized multi-level approaches to tackle this issue (peng Zhan et al., 2014; Geetha and Swarup, 2009; Ebrahimipour et al., 2015). Mainly in a two-level GMS problem, the upper-level schedules maintenance where the lower-level manages other considered criteria. While objective of the upper-level ranges from profit-maximization to risk-reduction (Barot and Bhattacharya, 2008; Bisanovic et al., 2011; Wu et al., 2008; Conejo et al., 2005), objective of the lower-level normally includes social welfare-maximization and operation cost-minimization (Badri and Niazi, 2012; Abirami et al., 2014; Pandzic et al., 2013; Cant, 2008). However, due to the complexity of the problem, many simplifications are considered, e.g., disregarding transmission constraints, assuming given electricity prices, neglecting stochasticity of demand and renewable sources. This work on the other hand, includes these aspects and assesses their impacts. In addition, to represent the impact of variability of renewable resources, the generators’ capacity constraint and ramping rates 1 which are generally considered in economic dispatch problems 2 are also included (Ekpenyong et al., 2012). In this regard, several case studies are defined and analyzed in subsection 3.2; Case1 is considered as the benchmark model 1 Upward/downward

ramp rates are rates at which the generators can increase/decrease their output power, respectively.

2 Economic dispatch refers to the determination of the output of the generators so that the balance between electricity demand and the generation

supply is maintained at the lowest cost.

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in 3.2.1, Case2 is the model that considers the profit of the GENCOs in 3.2.2, renewable wind power is considered in Case4 in 3.2.4 and Case8 in 3.2.8 and demand stochasticity is considered in Case9 in 3.2.9. Please notice that the generators’ ramp rates and transmission constraints are considered in all of the cases. 1.2. Coordination in Profit-maximization GMS

1.3. GMS representation in Market Equilibrium

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While the objective of GENCOs in this GMS is to maximize their profits, the system reliability and economic levels should be maintained at acceptable levels. This requires a coordination procedure to attain an optimal solution that satisfies both of these objectives. In Geetha and Swarup (2009) and Elyas et al. (2013), at first, GENCOs’ problem for finding maintenance schedules is solved, then, through an iterative process, a rescheduling signal is issued. The signal promotes alternative decisions for GENCOs so that system requirements are met. Various forms of reliability and economic signals are assessed in Conejo et al. (2005); Eshraghnia et al. (2006); Geetha and Swarup (2009); Moro and Ramos (1999); Baillo et al. (2001); Elyas et al. (2013) and peng Zhan et al. (2014). However, GMS can also benefit from a comparative study of these signals. In this paper, two rescheduling signals are defined and issued to the GENCOs. System’s reliability limit is based on energy-not-supplied (ENS) and economic limit is based on operation costs (OC). ENS and OC signals ensure that the result of the profit-maximization GMS is in the given reliability and operation cost limits, respectively. In this paper, these reliability and economic limits are obtained from the defined cost-minimization GMS. Moreover, incorporation of these two signals into one signal is proposed and compared with each of them individually. In this regard, Case5 in 3.2.5 considers solely the ENS-based rescheduling signal, Case6 in 3.2.6 considers solely the OC-based rescheduling signal and Case7 in 3.2.7 integrates both rescheduling signals into the model.

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In a deregulated power system where each GENCO aims to maximize its profit, the resulting GMS is the Nash equilibrium (NE) of the simultaneous-move game which is played by the GENCOs considering all of their strategies (Gabriel et al., 2013). To this goal, researchers adopted game theory for GMS (Kim et al., 2005; Jia et al., 2007; Min et al., 2013). For instance, Kim et al. (2005) and Jia et al. (2007) perform studies where the market-clearing price is forecasted from the marginal price of each generation unit, demand stochasticity and renewable sources are neglected, and rescheduling signal is not considered. Min et al. (2013) consider a reliability rescheduling signal through penalty and incentive where market clearing price is predicted from the marginal price of generators. However, demand stochasticity and integration of renewable energy resources are also disregarded. Moreover, the study of impact of considering portfolios, e.g., GENCOs with multiple generation units, is also neglected. Portfolio analysis provides insight into evaluating the strategic cooperation of generation units. In this regard, different portfolio cases are defined and studied in this paper. For instance, in Case1 and Case2 each generator belongs to one GENCO, in Case3 in 3.2.4 generators 1, 4 and 5 belong to GENCO1 and generators 2 and 3 belong to GENCO2. This paper studies strategic behavior of GENCOs in a GMS problem. Two bi-level formulations for GMS problem are provided as cost-minimization and profit-maximization GMS. Where the aim of cost-minimization GMS is minimizing operation cost of the system without accounting for profit of GENCOs, the goal of profit-maximization GMS is to maximize the profit of GENCOs satisfying at the same time the reliability and economic targets imposed by the ISO. These defined bi-level problems are transformed into a mixed-integer linear problem (MILP) through Karush– Kuhn–Tucker (KKT) conditions. By assuming that ISO considers reliability and economic limits in the system, impact of these two types of limits are analyzed through rescheduling signals. In addition, variations in renewable energy resources as well as demand stochasticity are introduced and their impacts are explored. Since NE is obtained as the final solution of the GMS problem among the GENCOs, their strategic behavior where each GENCO individually attempts to maximize their profit is exemplified. The overall diagram of the relation among the GENCOs and ISO is displayed in Fig. 1. The rest of the paper is organized as follows: Section 2 derives the formulations for cost-minimization and profitmaximization GMS problems. Section 3 provides the details for the test system as well as considered scenarios in order to assess the model. Section 4 discusses the observed strategic behavior and influences of the agents in the system. Finally, Section 5 summarizes the study performed in this paper.

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Figure 1: Overall diagram from the relationships among the decision variables

2. Problem Formulation 2.1. Cost-minimization GMS

t,g

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t

zt,g − T g = 0

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zt,g − Z ≤ 0

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Cost-minimization GMS is considered as a maintenance scheduling problem where one entity, i.e., ISO, decides about both maintenance scheduling and operation of the system. Both of these tasks are carried out by the objective of minimizing the operation cost of the system. Formulation of cost-minimization GMS through a bi-level optimization problem (BLP) is represented as follows:  XXh i min ρ s ( fg αg qt,g,s + βg qt,g,s + og ) + Cg zt,g (1a)

∀t

(1c)

∀t > Nt , g

(1d) (1e)

∀t, i, s

(1g)

qt+1,g,s − qt,g,s − Qg ≤ 0 : λ2t,g,s

∀t < Nt , g, s

(1h)

∀t < Nt , g, s

(1i)

(θt,i,s − θt, j,s ) − Fi, j ≤ 0 : λ4t,i, j,s Xi, j

∀t, i, j | j ∈ Θ(i), s

(1j)

qt,g,s − (1 − zt,g )Qt,g ≤ 0 : λ5t,g,s

∀t, g, s

(1k)

q,θ

t,g,s

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qt,g,s +

g∈Υ(i)

X (θt,i,s − θt, j,s ) + Dt,i,s = 0 : λ1t,i,s Xi, j j∈Θ(i)

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qt,g,s − qt+1,g,s − Qg ≤ 0 : λ3t,g,s

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(1b)

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zt+1,g − zt,g − zt+Tg ,g ≤ 0 Xh i ρ s ( fg αg qt,g,s + βg qt,g,s + og ) min

∀g

− qt,g,s ≤ 0 : λ6t,g,s

∀t, g, s

(1f)

(1l)

The upper-level, (1a)-(1e), schedules maintenance while minimizing the total operation and maintenance costs of the system and the lower-level, (1f)-(1l), finds the optimal dispatch of the system given the maintenance schedules found in the upper-level. The first term in (1a) is fuel cost of generation units, the second term is variable operation cost, the third term is fixed operation cost and the last term is cost of maintenance actions where qt,g,s results from the lower-level and is constant in the upper-level. Equation (1b) declares the duration of maintenance actions for each generator and (1c) limits the maximum number of generators that can be under maintenance simultaneously. Constraint (1e) links the maintenance start time with its corresponding duration while (1d) maintains the values of zt,g within the predefined limits in (1e) in the time period of the problem.

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(1b) − (1e), (1g) − (1l)

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Objective function of the lower-level, (1f), is the same as objective function of the upper-level minus the maintenance cost. Note that zt,g is constrained in the lower-level. Equation (1g) is the power balance equation at each bus (node). Equations (1h) and (1i) define the upward and downward ramping limits of each generator and (1j) confirms that the flows in the transmission lines remain in the capacity limit. Finally, (1k) and (1l) apply the upper and lower limits of the produced power in each of the generation units. In the BLP of (1a)-(1l), the commitment status of the units are considered in two main categories, “unavailable” and “available”. Units are unavailable (zt,g = 1) whenever they undergo maintenance by enforcing equation (1k) and the cost Cg zt,g is incurred, (Ekpenyong et al., 2012). On the other hand, the available units (zt,g = 0) are divided into “in production” and “idle”. Whenever a unit is in production (qt,g,s > 0), it endures the costs fg αg qt,g,s + βg qt,g,s and all units have an idle constant cost of og over the study horizon for being available, regardless of their operational status. Moreover, the upper- and lower-level objective functions of this BLP are collinear and have a Pareto Optimal solution (Marcotte and Savard, 1991). This results in transformation of the derived BLP into a single-level optimization problem (SLP) as in (2):  XXh i min ρ s ( fg αg qt,g,s + βg qt,g,s + og ) + Cg zt,g (2a) The above SLP is considered as the benchmark and it assumes that ISO solves GMS with the objective of minimizing operation and maintenance costs. Note that this SLP is a MILP and can be solved by available commercial software. 2.2. Profit-maximization GMS

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In profit-maximization GMS, GENCOs decide about maintenance scheduling with the objective of maximizing their profit while ISO dispatches the system and minimizes the operation cost. These two objectives contradict each other and the resulting BLP cannot be simply merged as an SLP (as in 2.1) and requires additional attention.

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2.2.1. GMS for each GENCO In profit-maximization GMS, each GENCO at the upper-level problem aims to maximize its profit while ISO seeks minimizing the operational costs at the lower-level. These two objectives oppose each other as correspondingly one increases electricity prices whereas the other one decreases them. In this paper, profit of each GENCO is defined as subtracting costs of power production and maintenance of generators from revenue obtained by selling power, i.e., multiplication of price and power. GMS of each GENCO is formulated as follows:  X Xh i max ρ s (µt,i,s qt,g,s − fg αg qt,g,s − βg qt,g,s − og ) − zt,gCg (3a) z

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µt,i,s

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s

∀t, i, s ∀t, g < Ξ(p)

(1b) − (1e) Xh i min ρ s ( fg αg qt,g,s + βg qt,g,s + og ) q,θ

(3b) (3c) (3d) (3e)

t,g,s

(1g) − (1l)

(3f)

where µt,i,s is the locational marginal price at time t, bus i and stochastic scenario s, and obtained through (3b) using the Lagrange multiplier (dual variable, λ1t,i,s ) of the power balance equation in (1g). The symbol “i ← g” indicates the bus i in which generator unit g is located. At each iteration, the decisions of other GENCOs are considered as fixed, (3c).

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t,g,s

Stationarity conditions are written as follows:

∂L(q, θ, λ) = ρ s ( fg αg + βg ) − λ1t,i,s − λ2t,g,s + λ3t,g,s + λ5t,g,s − λ6t,g,s = 0 ∂q i←g

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2.2.2. Karush–Kuhn–Tucker Conditions To handle the developed conflicting BLP, KKT approach is applied and the BLP transforms into an SLP. In order to do this, KKT conditions are derived from the lower-level problem of (1f)-(1l): stationarity (gradient of Lagrangian with respect to lower-level variables), complementary slackness, primal constraints (original inequalities) and dual constraints (Chinneck, 2015). Forming the Lagrangian function (L) provides:   X Xh X (θt,i,s − θt, j,s ) i X + Dt,i,s L(q, θ, λ) = ρ s ( fg αg qt,g,s + βg qt,g,s + og ) + λ1t,i,s −qt,g,s + Xi, j t,g,s t,i,s g∈Υ(i) j∈Θ(i)  (θ − θ )  X X X i i h h t,i,s t, j,s λ4t,i, j,s − Fi, j + λ2t,g,s qt+1,g,s − qt,g,s − Qg + λ3t,g,s qt,g,s − qt+1,g,s − Qg + Xi, j t
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∂L(q, θ, λ) = ρ s ( fg αg + βg ) − λ1t,i,s + λ2t−1,g,s − λ2t,g,s − λ3t−1,g,s + λ3t,g,s + λ5t,g,s − λ6t,g,s = 0 ∂q i←g ∂L(q, θ, λ) = ρ s ( fg αg + βg ) − λ1t,i,s + λ2t−1,g,s − λ3t−1,g,s + λ5t,g,s − λ6t,g,s = 0 ∂q i←g X (λ1t,i,s − λ1t, j,s ) X (λ4t,i, j,s − λ4t, j,i,s ) ∂L(q, θ, λ) = + =0 ∂θ Xi, j Xi, j j∈Θ(i) j∈Θ(i)

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And from complementarity conditions:   λ2t,g,s qt+1,g,s − qt,g,s − Qg = 0   λ3t,g,s qt,g,s − qt+1,g,s − Qg = 0  (θ − θ )  t,i,s t, j,s λ4t,i, j,s − Fi, j = 0 Xi, j   λ5t,g,s qt,g,s − (1 − zt,g )Qt,g = 0 λ6t,g,s (−qt,g,s ) = 0

∀t = 1, g, s

(5)

∀t ∈ (1, Nt ), g, s

(6)

∀t = Nt , g, s

(7)

∀t, i, s

(8)

∀t < Nt , g, s

(9)

∀t < Nt , g, s

(10)

∀t, i, j ∈ Θ(i), s

(11)

∀t, g, s

(12)

∀t, g, s

(13)

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The complementarity conditions (9)-(13) include nonlinearities. One way to linearize these nonlinearities is by defining new additional binary variables through disjunctive constraints (Fortuny-Amat and McCarl, 1981). Since one of the goals of this paper is to obtain a NE solution for the GMS problem, these new binary variables extremely complicate the problem and a solution may not be reachable. Since the lower-level problem, (1f)-(1l), is convex, as the unit commitment decisions (start-up and shut-down) are not considered and zt,g,s in (1k) acts as a parameter, these complementarity conditions are substituted by strong duality equation as follows:   X X X X  ρ s ( fg αg qt,g,s + βg qt,g,s + og ) = ρ s og + λ1t,i,s Dt,i,s − λ2t,g,s Qg + λ3t,g,s Qg t,g,s

−

X

t,i, j∈Θ(i),s

λ4t,i, j,s Fi, j −

X t,g,s

t,g,s

t,i,s

t
λ5t,g,s (1 − zt,g )Qt,g

Now, the BLP has been converted into an SLP in the following form:  X Xh i max ρ s (µt,i,s qt,g,s − fg αg qt,g,s − βg qt,g,s − og ) − zt,gCg z,q,θ

t,g∈Ξ(p)

(14)

(15a)

s

(1b) − (1e), (1g) − (1l), (3b), (3c), (5) − (8), (14)

(15b)

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The obtained SLP has two nonlinear terms which can be linearized. One nonlinearity is in the last term of the strong duality (14), λ5t,g,s (1 − zt,g ), and the other one is in the objective function in (15a), λ1t,i,s qt,g,s , which comes from (3b). In order to linearize λ5t,g,s (1 − zt,g ), at first, this term is substituted by a new variable as (Williams, 2013): λ0t,g,s = λ5t,g,s (1 − zt,g )

(16)

Afterwards, three following constraints need to be added to (15) as the equivalent of the nonlinear term in order to ensure correct transformation:

λ5t,g,s

∀t, g, s

(17)

∀t, g, s

(18)

∀t, g, s

(19)

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λ5t,g,s

λ0t,g,s ≤ (1 − zt,g )M1   − λ0t,g,s ≤ 1 − (1 − zt,g ) M1   − λ0t,g,s ≥ − 1 − (1 − zt,g ) M1

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where M1 is a large positive constant (“bigM”) (Fortuny-Amat and McCarl, 1981). To linearize λ1t,i,s qt,g,s , at first, stationarity conditions (5)-(7) are multiplied by qt,g,s and are summed for each portfolio: X  X h i 1 − q1,g,s λ21,g,s + q1,g,s λ31,g,s + q1,g,s λ51,g,s − q1,g,s λ61,g,s ρ s ( λ1t,i,s qt,g,s − fg αg qt,g,s − βg qt,g,s ) = ρ s i←g g∈Ξ(p),s t,g∈Ξ(p),s X   qt,g,s λ2t−1,g,s − qt,g,s λ2t,g,s + qt,g,s λ3t,g,s − qt,g,s λ3t−1,g,s + qt,g,s λ5t,g,s − qt,g,s λ6t,g,s + t∈(1,Nt )

+ qNt ,g,s λ2Nt −1,g,s − qNt ,g,s λ3Nt −1,g,s + qNt ,g,s λ5Nt ,g,s − qNt ,g,s λ6Nt ,g,s



(20)

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It should be noted here that to create the terms required for the linearization, (20) is divided into three time periods of t = 1, t ∈ (1, Nt ) and t = Nt . Moreover, after the multiplication, the terms ρ s ( fg αg + βg ) − λ1t,i,s in (5)-(7) are moved to the other side of the equality and then factored by ρ s . Next, through complementarity slackness conditions (9) and (10) (re-written again in three time periods), and substituting (3b), (13), (12) and (16) into (20), the right-hand side renders in: X X X h   i λ0t,g,s Qt,g (21) λ2t,g,s Qg + λ3t,g,s Qg + ρ s (µt,i,s qt,g,s − fg αg qt,g,s − βg qt,g,s ) = t,g∈Ξ(p),s

t
t,g∈Ξ(p),s

t
t,g∈Ξ(p)

µt,i,s =

1 λ1t,i,s ρ s i←g

AC

zt,g = z0 X zt,g − T g = 0 t

X g

zt,g − Z ≤ 0

zt,g = 0 zt+1,g − zt,g − zt+Tg ,g ≤ 0 X X (θt,i,s − θt, j,s ) − + Dt,i,s = 0 : λ1t,i,s qt,g,s + Xi, j g∈Υ(i) j∈Θ(i) qt+1,g,s − qt,g,s − Qg ≤ 0 : λ2t,g,s

(22a)

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which forms the linear equivalent of the nonlinear terms in the objective function (15a). Using (21), (15) transforms into the final linearized SLP with GENCOs’ portfolio and stochastic terms: X hX X h i i max ρ s (λ2t,g,s Qg + λ3t,g,s Qg ) + ρ s (λ0t,g,s Qt,g − og ) − zt,gCg ∀t, i, s

(22b)

∀t, g < Ξ(p)

(22c)

∀g

(22d)

∀t

(22e)

∀t > Nt , g

(22f)

∀t < Nt , g

(22g)

∀t, i, s

(22h)

∀t < Nt , g, s

(22i)

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qt,g,s − qt+1,g,s − Qg ≤ 0 : λ3t,g,s

(22j)

(θt,i,s − θt, j,s ) − Fi, j ≤ 0 : λ4t,i, j,s Xi, j

∀t < Nt , g, s ∀t, i, j | j ∈ Θ(i), s

(22k)

qt,g,s − (1 − zt,g )Qt,g ≤ 0 : λ5t,g,s

∀t, g, s

(22l)

ρ s ( fg αg + βg ) − λ1t,i,s − λ2t,g,s + λ3t,g,s + λ5t,g,s − λ6t,g,s = 0

− qt,g,s ≤ 0 : λ6t,g,s

∀t, g, s

(22m)

∀t = 1, g, s

(22n)

ρ s ( fg αg + βg ) − λ1t,i,s + λ2t−1,g,s − λ2t,g,s − λ3t−1,g,s + λ3t,g,s + λ5t,g,s − λ6t,g,s = 0

∀t ∈ (1, Nt ), g, s

(22o)

ρ s ( fg αg + βg ) − λ1t,i,s + λ2t−1,g,s − λ3t−1,g,s + λ5t,g,s − λ6t,g,s = 0

∀t = Nt , g, s

(22p)

i←g

i←g

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X (λ1t,i,s − λ1t, j,s ) X (λ4t,i, j,s − λ4t, j,i,s ) + =0 ∀t, i, s Xi, j Xi, j j∈Θ(i) j∈Θ(i)  X  X X X  ρ s ( fg αg qt,g,s + og + βg qt,g,s ) = ρ s og + λ1t,i,s Dt,i,s − λ2t,g,s Qg + λ3t,g,s Qg −

X

t,i, j∈Θ(i),s

λ4t,i, j,s Fi, j −

X t,g,s

t,g,s

t,i,s

λ5t,g,s (1 − zt,g )Qt,g

t
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λ0t,g,s ≤ (1 − zt,g )M1   λ5t,g,s − λ0t,g,s ≤ 1 − (1 − zt,g ) M1   λ5t,g,s − λ0t,g,s ≥ − 1 − (1 − zt,g ) M1

(22q)

(22r)

∀t, g, s

(22s)

∀t, g, s

(22t)

∀t, g, s

(22u)

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The above problem is a MILP which can be solved with available commercial software.

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2.2.3. Nash Equilibrium Each GENCO seeks maximizing its profit individually by abiding with the constraints imposed by regulatory bodies. Such situation constructs a multiple-leaders single-follower game (Gabriel et al., 2012). In this game, ISO is the follower and GENCOs are the leaders which are assumed to compete non-cooperatively. The players are considered to have complete information which means they know the strategy space of one another in the game. In this regard, NE solution concept is defined as a set of GENCOs’ strategies where no GENCO is able to increase its profit by individually deviating from that strategy. Hence, decisions of all GENCOs are in a Nash equilibrium. In this context, GENCOs’ strategies are their maintenance decisions. To formulate this equilibrium point, NE strategy vector W is defined as (23). In order to obtain such condition, the solution needs to result in:

Π p (w∗p

|

W = {w∗1 , w∗2 , ..., w∗N p }

w∗−p )

≥ Π p (w p |

w∗−p )

(23) ∀p, w p

(24)

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where Π p is profit of GENCO p as objective function in (22). NE is mathematically obtained by iteratively solving (22) for each GENCO until a point where no GENCO tries to change its strategy which conceptually is the result of simultaneous play of the game. In this problem, there is possibility that multiple Nash equilibria points exist. In this paper, the first Nash equilibrium found is considered. To find the other Nash equilibria, if there are any, the solution can be repeated by removing the already found Nash equilibrium and starting from a different initial point. 2.3. Rescheduling Signal The derived SLP in (2), i.e. cost-minimization GMS, solves GMS from ISO’s perspective with the objective of minimizing operation and maintenance costs. The OC and ENS outcomes of this GMS are the reference for ISO’s regulatory decisions. In this paper, multiplication of a weighting factor into the ENS and OC obtained from costminimization GMS are considered as ISO’s regulatory limits in profit-maximization GMS. The OC and ENS outcomes

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of the profit-maximization GMS may violate these regulatory limits and when they are violated, a rescheduling signal, consisting of two terms of penalty and incentive, is issued to each GENCO in order to motivate them to modify their decisions. In this paper, two types of penalty and incentive (based on OC and ENS) for rescheduling signals are defined. Penalty- and incentive-type1 contemplate the limit for ENS (ΨENS , fENS ) and penalty- and incentive-type2 consider the limit for OC (ΨOC , fOC ). It should be noted that these signals are calculated at the end of each obtained NE and thus, act as a parameter in the next iteration. The formulation for penalty-type1 is as follows: vENS (ENS NE,k−1 − ENS CM ) Ng

∀k > 1

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ΨENS ,k =

(25)

where ENS NE,k−1 is the resultant ENS of the NE in previous iteration for profit-maximization GMS, ENS CM is the resultant ENS for the cost-minimization GMS, and vENS is the weighting factor of reliability penalty signal estimated for each network configuration. The same formulation can be derived for ΨOC , fENS and fOC . A binary variable c is defined where 0 and 1 correspondingly mean receiving incentive or penalty. To obtain the value of c for each GENCO at every iteration, additional constraints and terms are required as follows: 1 − zt,g = b1k,g

g∈Ξ(p)

ak − 1 ≥ − c ≥ ak

X

∀t =

0 tk,g

− T g + 1, g, k

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0 ∀t = tk,g , g, k

[b1k,g + b2k,g ]M2

g∈Ξ(p)

(26) (27)

∀k

(28)

∀k

(29)

∀k

(30)

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0 where tk,g is the last hour generator g was under maintenance in iteration k, and M2 is a sufficiently enough large number (“bigM”). If the reliability or economic limits are violated at iteration k, a rescheduling signal is created and (26)-(30) prepare the penalty or incentive in the current iteration based on the recorded maintenance decisions in previous iterations. For instance, if the signals are violated and the maintenance decision in iteration k is the same as a maintenance decision in any of the past iterations, c becomes 1 and a penalty is issued. In case the signals are violated and the maintenance decision in iteration k is not the same as any of the maintenance decisions in the past iterations, c becomes 0 and an incentive is issued. The goal here is to encourage GENCOs to bring the ENS and OC lower than the ISO’s limits by changing their maintenance schedules. In this regard, at each iteration k, constraint (26) records the previous iterations’ times when the maintenance of generator g finished; in other words, the last time periods where zt,g was equal to 1. Similarly, constraint (27) records the first time when the zt,g was equal to 1, i.e., when the maintenance started. Then at iteration k, if generator g decides to choose the same maintenance decision as any of the previous iterations, this would result in the same start and end maintenance times. Such behavior by the generator triggers binary variables of b1 and b2 to become 0. Constraints (28) and (29) ensure that the binary variable ak turns 1 whenever a generator repeats its maintenance decision. In such cases, constraint (30) enforces the binary variable c to order the penalty signal. If b1 and b2 are 1 for each iteration, the constrains on ak , (28) and (29), are relaxed enforcing ak to be 0. Consequently, c becomes 0 where it offers incentive to GENCOs in order to motivate them to choose a new maintenance schedule. In order to summarize the profit-maximization model, the steps are illustrated in Fig. 2. Finally, the complete SLP formulation of profit-maximization GMS of GENCOs with penalty and incentive is as follows: X hX X i i h ρ s (λ0t,g,s Qt,g − og ) − zt,gCg − c ΨENS + (1 − c) fENS (31a) max ρ s (λ2t,g,s Qg + λ3t,g,s Qg ) + z,q,θ

t
µt,i,s =

1 λ1t,i,s ρ s i←g

zt,g = z0

t,g∈Ξ(p)

s

∀t, i, s

(31b)

∀t, g < Ξ(p)

(31c)

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X

zt,g − T g = 0

∀g

(31d)

t

zt,g − Z ≤ 0

∀t

(31e)

g

∀t > Nt , g

(31f)

X

zt,g = 0

∀t < Nt , g

(31g)

∀t, i, s

(31h)

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qt+1,g,s − qt,g,s − Qg ≤ 0 : λ2t,g,s

∀t < Nt , g, s

qt,g,s − qt+1,g,s − Qg ≤ 0 : λ3t,g,s

(θt,i,s − θt, j,s ) − Fi, j ≤ 0 : λ4t,i, j,s Xi, j

qt,g,s − (1 − zt,g )Qt,g ≤ 0 : λ5t,g,s − qt,g,s ≤ 0 : λ6t,g,s

∀t < Nt , g, s

(31j)

∀t, i, j | j ∈ Θ(i), s

(31k)

∀t, g, s

(31l)

∀t, g, s

ρ s ( fg αg + βg ) − λ1t,i,s − λ2t,g,s + λ3t,g,s + λ5t,g,s − λ6t,g,s = 0

(31i)

(31m) (31n)

ρ s ( fg αg + βg ) − λ1t,i,s + λ2t−1,g,s − λ2t,g,s − λ3t−1,g,s + λ3t,g,s + λ5t,g,s − λ6t,g,s = 0

∀t ∈ (1, Nt ), g, s

(31o)

ρ s ( fg αg + βg ) − λ1t,i,s + λ2t−1,g,s − λ3t−1,g,s + λ5t,g,s − λ6t,g,s = 0

∀t = Nt , g, s

(31p)

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∀t = 1, g, s

i←g

i←g

i←g

−

X

t,i, j∈Θ(i),s

λ4t,i, j,s Fi, j −

X t,g,s

t,g,s

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1 − zt,g = b2k,g X ak − 1 ≤ [b1k,g + b2k,g ]M2 g∈Ξ(p)

X

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ak − 1 ≥ − c ≥ ak

[b1k,g + b2k,g ]M2

g∈Ξ(p)

t,i,s

(31q)

t
λ5t,g,s (1 − zt,g )Qt,g

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t,g,s

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(31r)

∀t, g, s

(31s)

∀t, g, s

(31t)

∀t, g, s

(31u)

∀t = ∀t =

0 tk,g , g, k 0 tk,g − T g

(31v) + 1, g, k

(31w)

∀k

(31x)

∀k

(31y)

∀k

(31z)

where (31) can be similarly written for OC. The original contributions of the proposed model with respect to previous approaches are summarized as following: the prices are considered endogenous to the model as compared to the previous studies where the prices are predicated and given. In addition, transmission constraints, generation units’ ramp rates, stochasticity of the demand and variability in renewable sources are taken into account. Moreover, to introduce regulations’ limits from system’s perspective into the model, two new signals are formulated. The first signal addresses the reliability perspective (ENS) and the second signal attends to the economic (social cost) perspective (OC). Based on these two signals, two

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Figure 2: Summarized flowchart of the proposed profit-maximization GMS model

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incentive and penalty terms are created and issued to GENCOs in case their results cause violations of the reliability and economic limitations. Finally, to find the outcome that satisfies both of the imposed restrictions by the ISO and deliver an optimum profit solution for all GENCOs, a game theoretic Nash equilibrium solution is obtained.

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3. Case Study 3.1. Test System

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The models (2), (22) and (31) are coded in GAMS v24.4.6 (Rosenthal, 2016) and solved using CPLEX v12.6.2 (IBM, 2016) on a machine running with 16 GB of RAM and four Intel i7 2.60 GHz processors. A 5-bus network (Fig. 3) is defined to demonstrate applicability of the developed GMS model. Buses can be considered as electrical nodes in the electric transmission system, that the incoming and outgoing flows of power should be in balance at any moment, and are denoted by letter B. Generators that produce the electric power in the system are represented by the letter G and are connected to the buses. The generators have constraints such as maximum producible power and upward/downward ramp rates. Similar to the generators, one bulk of demand is considered at each bus and is symbolized by the letter D. Finally, the transmission lines that connect the buses together are indicated by the letter L and have transmission capacity constraints. Details of the generators are provided in Table 1. The inductance of transmission lines L1, L2, L3, L4, L5, L6 and L7 are correspondingly 0.100, 0.280, 0.200, 0.075, 0.300, 0.095 and 0.200. The capacities of all lines are set to 60 MW. In addition, the maximum number of generators that can go under maintenance simultaneously is set to 3, Z = 3, and the required number of maintenance actions for all generators is assumed to be 1 over the time period of study, Nt . Although the adopted time unit is hour, the model is generic and can be expanded to the desired time horizon in hourly basis. 3.2. Cases In this paper, nine main cases (C1 − C9) are designed to address multiple criteria that were mentioned in the Introduction section and identified as scientific gaps in this field. Thus, these nine cases focus on strategic maintenance scheduling of generation units under portfolio operations, addition of demand stochasticity and variations of renewable sources, and change in the behavior of the GENCOs when faced with the rescheduling signals received from ISO.

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Figure 3: The utilized 5-Bus test system

Table 1: Operational and maintenance information of generators

Qt,g

Qg ,Qg

αg

fg

og

βg

Tg

Cg

G1 G2 G3 G4 G5 G6

85 85 85 85 85 80a

75 75 75 75 75 80

0.1 0.3 0.5 0.7 0.9 0

1 3 5 7 9 0

100 300 500 700 900 100

1 3 5 7 9 9

1 2 3 4 5 3

10 15 20 25 30 10

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Set based on Fig. (5)

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Generator

160

Bus1 Bus2 Bus3 Bus4 Bus5

Demand (MW)

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120

100

80

60

40

20 5

10

15

20

25

30

35

Time period (h) Figure 4: Load profile in the test system

40

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50

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Table 2: Defined case studies

a b

Type

Portfolio

Rescheduling Signal

Wind

Model

Cost-Mina Prof-Maxb Prof-Max Prof-Max Prof-Max Prof-Max Prof-Max Prof-Max Prof-Max

1 1 2 3 2 2 2 4 2

ENS OC ENS & OC -

X X -

(2) (22) (22) (22) (31) (31) (31) (22) (22)

Cost-Minimization Profit-Maximization Table 3: Defined portfolios Portfolio1

Portfolio2

Portfolio3

Portfolio4

G1 G2 G3 G4 G5

G1, G4, G5 G2, G3 -

G1, G4, G5 G2, G3 G6 -

G1, G4, G5 G2, G3, G6 -

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GENCO1 GENCO2 GENCO3 GENCO4 GENCO5

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C1 C2 C3 C4 C5 C6 C7 C8 C9

Table 2 and Table 3 show a summary of case studies and the considered portfolios for each case. The distribution of the demand at each bus in the test system during the 52 time periods is illustrated in Fig. 4.

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3.2.1. Case1 C1 is considered as the benchmark and the current state-of-the-art algorithm which is created within the costminimization framework. In this case all the decisions are made by one entity, ISO.

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3.2.2. Case2 C2 is the proposed formulation where the objective is for the GENCOs to maximize their profit without considering the newly defined rescheduling signals. Both of these cases apply Portfolio1 which considers that each GENCO has one generation unit, Table 3.

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3.2.3. Case3 C3 utilizes the same profit-maximization formulation as in C2. C3 adds the portfolio weight in the approach and provides insight into the analysis of strategic combined operation of the generation units.

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3.2.4. Case4 C4 introduces the wind power integration (G6 with wind-scenario1) for the first time and it considers that the wind farm is an independent player in the market, GENCO3, Table 3. This case provides the possibility to analyze how much decisions of a single renewable player can affect the outcomes of other players. 3.2.5. Case5 C5 considers that a reliability criterion imposes a limit and the rescheduling signal (penalty/incentive) is issued based on whether this reliability limit is violated. 3.2.6. Case6 C6 performs the same analysis as in C5 while the reliability signal is substituted by an economic one.

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Scenario1 Scenario2 Scenario3 Scenario4

70

50 40 30 20 10 0 5

10

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25

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Figure 5: Wind profiles for the test system

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3.2.7. Case7 C7 incorporates both reliability and economic limits and a rescheduling signal is issued if either of these two limitations are violated. The limits for the rescheduling signal on whether it should be issued are set to 1.32 and 7.6 times of OC and ENS from cost-minimization GMS, respectively, and are adopted by the ISO based on the prioritized preferences. The values of vENS and vOC are correspondingly considered to be 1000 and 0.8 for the test system as they produce large enough penalty/incentive terms in the process.

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3.2.8. Case8 In C8, impacts of variations in wind power is included by defining four distinct wind scenarios as displayed in Fig. 5. Wind-scenario1 (C8.1) is assumed to show a normal behavior (without sudden extreme changes in wind) and mean available power of 38 MW at each time period. Wind-scenario2 (C8.2) and Wind-scenario3 (C8.3) have the lowest and highest average available wind power of 29 and 43 MW, respectively. Finally, Wind-scenario4 (C8.4), despite having an average available wind power of 36 MW, exhibits some extreme behaviors such as falling from maximum to zero.

4. Results

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3.2.9. Case9 Demand stochasticity is considered only in C9 and variations range from -4 to +4 standard deviations through nine load levels. The probability of each load level is illustrated in Fig. 6.

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Table 4 presents outcomes of all cases in terms of OC, ENS and average electricity price over time period of the study. It should be reminded that the solutions are affected by the initialization and therefore one initial point has been considered in all case studies, (3c). The problem can be easily scaled up; however, existence of a Nash equilibrium cannot be guaranteed for any case. The maximum number of iterations to reach an answer in the cases was 30. The numbers of iterations are dependent on the coefficients of the network as well as the considered level of reliability and economic limitations. An observation from Table 4 is the distinct difference between C1 and C2 - C9. This shows that strategic behavior of GENCOs affects the outcomes significantly, thus, any analysis based on C1 would hold rather large simplifying assumption. Analogously, comparing C1 with the rest of the cases through Fig. 7, it can be seen that GENCO1 schedules the maintenance for its expensive generators (G4 and G5) simultaneously in order to increase the electricity prices. In such situations, the cheap generator of GENCO1 (G1) will produce at maximum capacity and largely increases the profit. Fig. 8 shows the overall load, prices and maintenance schedules at the market equilibrium point

Figure 6: Stochasticity defined for demand

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Table 4: Short summary of cases

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OC (M$)

ENS (MW)

Avg. Price ($/MWh)

C1 C2 C3 C4 C5 C6 C7 C8.1 C8.2 C8.3 C8.4 C9

3.375 3.557 4.489 4.150 4.424 4.401 4.415 4.154 4.264 4.039 4.274 4.529

173 372 1390 992 1319 1294 1309 996 1113 869 1126 1433

25 29 27 21 26 26 26 21 22 22 23 23

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for C2. In this case, the Nash solution is obtained where all generators act as an individual company, Portfolio1, Table 3. By comparing C3 with C5 - C7 it can be seen how the network would benefit by incorporating penalty and incentive signals. Since in C3, there is neither penalty nor incentive for GENCOs discrediting economic and reliability limitations, the results are completely in favor of them with high OC and ENS. In addition, comparison of C2 and C3 shows OC and ENS increase significantly when the generators join each other (Portfolio1 versus Portfolio2). A comparison of C5 and C6 shows that when a rescheduling signal is issued solely based on violation of reliability limit, this results in higher OC and ENS than the case that the signal is issued solely based on economic limit. This exhibits that in this test system, a signal based on the economic limit outperforms the one based on the reliability limit in having lower ENS and OC. C7 demonstrates a case where ISO requires both the economic and reliability limits explicitly. As seen in Table 4, C7 results in lower ENS and OC values than C5. In case the priorities of economic and reliability limits vary, one could modify the weights accordingly. Integration of renewable sources (C8.1 − C8.4) have shown to affect the network by driving ENS, OC and prices down, Fig. 9. The wind power in this system accounts for 16% of the installed capacity, G6 in Table 1. To analyze possible impacts of wind in the network, four wind scenarios are defined and tested. Average, low and high availability of wind are considered through scenario1, scenario2 and scenario3 correspondingly and are evaluated through C8.1, C8.2 and C8.3, respectively. Table 4 indicates that the average and high wind power affect average prices less than

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Figure 7: Maintenance schedules for C1, C3 − C9

Figure 8: Electricity prices and maintenance schedules for C2

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C8.1 C8.2 C8.3 C8.4

35 30 25 20 15 10 5 0 5

10

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Time period (h)

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Figure 9: Electricity prices for the case 8

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ENS and OC. From Fig. 7, it can be seen that maintenance actions of expensive generators are scheduled at the same time and all on the time periods with the highest demand levels. C4 investigates a case where a new renewable generator (G6 with wind-scenario1) enters the market as an individual company in comparison with the case that it joins one of the GENCOs, GENCO2 as in C8.1. Interestingly, the results of C8.1 shows that both GENCOs 1 and 2 gain more profit if G6 is acquired by GENCO2. The profit of GENCO1 increases by 0.6% and the profit of GENCO2 increases by 29%. Furthermore, following extreme changes in the wind power, scenario4, two main observations are made. One is a sudden large increase in availability of wind power, scenario4 in Fig. 5 and at time t = 6, which has resulted in no reduction in prices. The other one is a sharp reduction at time t = 30 in wind power. This decrease made same reflection on the prices. It should be noted that at both of these times, t = 6 and t = 30, the demand levels have similarly low levels. In addition, comparison between C8.1 and C8.4 displays that sudden extreme behaviors of wind affect the system’s average price, ENS and OC even more than a normal low wind availability. Table 5 displays the profit of GENCOs. At first, it can be seen that in the final schedules of maintenance, GENCOs find solutions where none of them receives any penalty. Since the incentives are for satisfying the constraints set by ISO, they are taken from the social welfare to achieve these ENS and OC targets. In C5 where the rescheduling signal is issued from violation of reliability limit, both GENCOs gain more profit than the case that the rescheduling signal is issued from violation of economic limit. C6 and C7 result in the same profit since the economic limit plays the key role by setting more strict restrictions on maintenance decisions. An important remark is the change of the leading GENCO through cases embedded with wind renewable source. When availability of wind is average and high, C8.1 and C8.3 respectively, GENCO2 earns more profit. And when the availability of wind is low, C8.2, GENCO1 secures more profit. This is a noteworthy outcome which demonstrates how the change in availability of wind can switch the position of prominent GENCOs. Note that there is no rescheduling signal (penalty or incentive) in C4. All the considered nine case studies were solved in less than 10 minutes with C9 requiring the highest computational time. Since the proposed model includes disjunctive constraints in the form of “bigM”, it requires high computational power and proper adjustment of the “bigM” (Lumbreras et al., 2014; Lumbreras and Ramos, 2016). Several difficulties may rise when larger test systems are considered. Some of these difficulties are: • A large scale problem will take very long time to be solved, if possible at all, with current configurations. • In a large scale model, it will also be extremely difficult, if possible at all, to assess strategic behaviors of players as the number of decisions variables influencing one another increases significantly. Thus, the proposed model is suitable to gain insight on the strategic operations and learn about the outcomes. • A Nash Equilibrium solution does not exist for every system.

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Table 5: Profit and incentive of each GENCO for all cases (M$)

GENCO1

GENCO3

Profit

Incentive

Profit

Incentive

Profit

3.323 1.482 3.099 2.858 2.858 1.491 1.932 2.078 2.162 3.061

0.249 0.179 0.444 -

2.495 1.245 2.339 2.263 2.263 1.602 1.859 2.146 1.956 2.318

0.246 0.179 0.444 -

0.367 -

• Use of snapshots 3 • Use of dimensionality reduction techniques • Use of decomposition techniques

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On these points, three main proposals can be applied here:

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GENCO2

5. Conclusion

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These three methods of solving large scale problems are gaining attention as the systems are becoming more integrated. Due to diffusion of smart grids, the number of variables in the system increases significantly and it becomes impossible to account for all of them in the analysis. For instance, in the first proposal, this paper benefits from a very simplified version of utilizing four snapshots of different profiles of wind power. Each profile can be formed in order to assess one aspect of a real situation. There are several research works in similar fields which are applying snapshots in their calculations (Fitiwi et al., 2015; Ploussard et al., 2016). Similarly, there are studies trying to address large scale systems through different reduction techniques which build a smaller equivalent of the large scale system (e.g. less number of generators, shorter time horizon) (Hamon et al., 2014; Shayesteh et al., 2015). Finally, application of decomposition techniques (e.g. Benders’ Decomposition (Bean, 1992; Codato and Fischetti, 2006)) could be applied here which may require additional simplifications. Therefore, future studies can further analyze the model through the mentioned points to overcome such obstacles and evaluate their outcomes.

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This paper derives formulation for solving generation maintenance scheduling of non-cooperative generation companies (GENCOs) through game theory. The problem is addressed by developing a bi-level optimization problem and transforming it into a single-level problem using Karush–Kuhn–Tucker optimality conditions. Afterwards, two rescheduling signals are defined to coordinate the conflicting situation between GENCOs and an independent system operator. These signals favor in two different aspects of energy-not-supplied (ENS) and operation cost (OC) limits required by the system operator. Therefore, a combination of these signals is also presented to propose a trade-off among them. Moreover, wind variations as well as demand stochasticity are considered. Multiple case studies have been defined in order to address the improvements of the proposed model over the current available approaches in the field of generation maintenance scheduling. These cases include renewable resources integration degree, locational marginal prices and strategic portfolio behavior as well as the aforementioned reliability and economic considerations. The results indicate that strategic behavior of GENCOs significantly affects the outcomes such as prices, OC and ENS. Additionally, the results show that if solely one of the signals of OC or ENS are considered, the system

3 A snapshot can be considered as a profile of e.g. demand or wind that represents all the operating situations that are relevant to the considered problem.

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could still be further away from its optimum operation point. Moreover, portfolio analysis displayed that a wind power plant joining one of the two portfolios can increase profit of all, rather than entering the market independently. Finally, the analysis of different wind scenarios shows how change in the availability of wind power can switch the prominent GENCOs in profit. Acknowledgment

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The authors would like to acknowledge the Erasmus Mundus Joint Doctorate (EMJD) Fellowship in Sustainable Energy Technologies and Strategies (SETS) program funded by the European Commission’s Directorate-General for Education and Culture which provided the opportunity for carrying out this work. The authors would also like to thank the anonymous reviewers for their insightful comments and suggestions. References

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