Optimal Transmission Switching for Reducing Market Power Cost

Transcription

1 Optima Transmission Switching for Reducing Market Power Cost Maria Aejandra Noriega Odor Degree project in Eectric Power Systems Second Leve, Stockhom, Sweden 2012 XR-EE-ES 2012:016

2 Optima Transmission Switching for Reducing Market Power Cost by Maria Aejandra Noriega Odor Supervised and Examined by Dr. Mohammad R. Hesamzadeh A thesis submitted to the Department of Eectrica Power Systems Schoo of Eectrica Engineering KTH Roya Institute of Technoogy in conformity with the requirements for the degree of Master of Science Stockhom, Sweden September 2012

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4 Abstract The conventiona transmission panning tends to focus excusivey on efficiency benefit, aowing cheaper remote generation to have priority dispatch over expensive oca generation (east cost approach). Because of this nowadays dereguated markets face the probem that their systems affect the competitiveness of payers, giving room for payers to exercise market power. The purpose of this study is to deveop a mathematica mode that quantifies the generation cost and reduces market power, by minimizing the socia cost and restraining producers from withhoding generation capacity. To do this, deterministic optima transmission switching is proposed, together with a Worst-Nash Equiibrium (WNE) optimization, to quantify the socia cost. This study considers the transmission switch formuation based on the DC Optima Power Fow (DCOPF) presented by Schmue S. Oren as a Mixed-Integer inear Program (MIP). This formuation empoys binary variabes to represent the state of the transmission ine. The effects of transmission switching with contingency anaysis are aso considered in the DCOPF formuation. To incude market power cost reduction in our probem, the socia cost of the system is modeed considering WNE, which maximizes the socia cost using inearization. The formuation incudes strategic generators that might choose i

5 to withhod some of their output and non-strategic generators. This under the condition that the profit of a portfoio with a strategic generator under Nash Equiibrium is aways greater than the profit of a portfoio where the offers are constant. A 14-node exampe system is studied where the efficiency benefits and competition benefits of transmission capacity by optima transmission switching are considered. The resuts demonstrate that the utiization of the proposed method increase economic benefit and improves competitiveness in the eectricity market. ii

6 Acknowedgments I woud ike to show my gratitude to Professor Mohammad R. Hesamzadeh for trusting me with this master degree project, his guidance and motivation during this eight months and heping me to earn how to think more critica. I aso ike to thanks Mahir Sarfati, Oga Gaand, Faeye Omoboboa and John Laury for their coaboration, suggestions and constructive critiques in a good working environment and friendship. I woud aso ike to give a specia thanks to my famiy; Jose A. Noriega, Bee Odor de Noriega and Pedro J. Noriega for inspiring me and being a roe mode in my professiona career and ife. Last but not east to Remco Ammeraan who without his support and encouragement I coud not succeed in this and many other enterprises. iii

7 List of Tabes 3.1 Generator data Load data Line data Optima dispatch for perfect competition Optima dispatch under Nash equiibrium (base case) Optima dispatch under Nash equiibrium (switched ine) Generation capacity offer to the market in the base case Power fow before/after switching Socia cost Generation capacity offer before/after switching Contingency anaysis ( SC & OTS under NE) iv

8 List of Figures 1.1 Withhoding strategy and price-quantity outcome [1] Withhoding strategy and producers surpus variation [2] Transfer of weath by the exercise of market power [1] Nash equiibrium found by two payers [3] Singe ine diagram of the 14 node exampe system Singe ine diagram of the 14 node exampe system after switching Impact of optima transmission switching in withhoding reduction Margina Cost Curve of u Margina Cost Curve of u Margina Cost Curve of u v

9 Contents Abstract i Acknowedgments iii List of Tabes iv List of Figures v Contents vi Nomencature 1 Chapter 1: Introduction Background Probem Definition Objective Resources/Toos used Chapter 2: Formuation of the Mathematica Mode Assumptions Mode Description Optima Transmission Switching vi

10 2.2.2 Worst-Nash Equiibrium Integrated Mode of Optima Transmission Switching and Worst-Nash Equiibrium Chapter 3: IEEE 14 Bus Test Case, Resuts and Discussion Modified IEEE 14 Node Exampe System Resuts and Discussion The Traditiona Mode for Perfect Competition vs. Nash Equiibria Mode The Proposed Mode for Reducing Market Power Cost Contingency Anaysis Chapter 4: Concusions and Future Work Concusion Future Work Bibiography 50 Appendix A: GAMS Code 57 vii

11 Nomencature Indices u i generator unit transmission ine node of the network n, m nodes between ine k p s c maximum generation capacity portfoio strategy contingency Sets U L N K O S set of units in the system set of transmission ines in the system set of nodes in the system set of constants set of portfoios in the system set of strategies 1

12 Mode parameters c u F F g d B n c cost minimum transmission capacity maximum transmission capacity maximum generation capacity demand admitance operation status of the ine during contingency, 0 the ine is contingency (disconnected)and 1 otherwise M C arbitrary big number incidence matrix Optimization mode variabes SC F g ĝ n π p π p,s p θ P x socia cost power fow in the transmission ine generation of each unit maximum capacity offer to the market operation status of the ine, 0 the ine is disconnected and 1 otherwise profit of portfoio, under Nash equiibrium profit of portfoio, without Nash equiibrium votage ange of node i oca margina price at node i behavior status of the generating unit, 0 for strategic units and 1 for the non-strategic units 2

13 w q L action of generating unit u, reated to the capacity offer to the market quantity produce of an identica product Lagrangian optimization function Sackness variabes ρ i λ down λ up µ down u µ up u β down β up δ i α down α up ε down ε up Sackness variabe for the energy baance constraint Sackness variabe for the ower imit of the power fow constraint Sackness variabe for the upper imit of the power fow constraint Sackness variabe for the ower imit of the generation imit constraint Sackness variabe for the upper imit of the generation imit constraint Sackness variabe for the ower imit of the Kirchhoff s aw constraint Sackness variabe for the upper imit of the Kirchhoff s aw constraint Contingency sackness variabe for the energy baance constraint Contingency sackness variabe for the ower imit of the power fow constraint Contingency sackness variabe for the upper imit of the power fow constraint Contingency sackness variabe for the ower imit of the Kirchhoff s aw constraint Contingency sackness variabe for the upper imit of the Kirchhoff s aw constraint Binary variabes for Compementary conditions b ow λ b up λ b ow µ b up µ b ow β Binary variabe for the ower imit of the power fow constraint Binary variabe for the upper imit of the power fow constraint Binary variabe for the ower imit of the generation imit constraint Binary variabe for the upper imit of the generation imit constraint Binary variabe for the ower imit of the Kirchhoff s aw constraint 3

14 b up β b ow α b up α b ow ε b up ε Binary variabe for the upper imit of the Kirchhoff s aw constraint Contingency binary variabe for the ower imit of the power fow constraint Contingency binary variabe for the upper imit of the power fow constraint Contingency binary variabe for the ower imit of the Kirchhoff s aw constraint Contingency binary variabe for the upper imit of the Kirchhoff s aw constraint Outputs F g ĝ n power fow in the transmission ine generation of each unit maximum capacity offer to the market operation status of the ine, 0 the ine is disconnected and 1 otherwise 4

15 5 Chapter 1 Introduction 1.1 Background The eary eectricity industry was designed as vertica integration; where generation, transmission and distribution created a natura monopoy. This verticaintegrated market ed to a ack of incentives to improve the system and reduce generation and distribution costs. Over the ast two decades industry anaysts, reguators and poicy makers have focused on an economicay more efficient system. In December 1996 most of the European eectricity markets had changed their poicies to achieve a iberaized market, by aowing operating firms to compete freey in the energy market and prevent producers to no onger own the transmission and distribution network [4]. Liberaization decoupes generation and transmission sectors; aiming to increase the efficiency, reduce eectricity prices and create a more competitive market. Eectricity as a commodity presents specia characteristics that must be considered: production must aways meet the demand in a short period of time; demand is varying every second; and storage is very imited and expensive [5]. These considerations combined with an ineastic demand, transmission constrains and

16 1.1. BACKGROUND 6 market fragmentation are incentives for producers to abuse the market equiibrium and ater the eectricity price to increase their own profit. This abiity of abusing market equiibrium is known as market power. Market power is one of the main concerns of the dereguated markets nowadays; before iberaization vertica market power was exercised [2], now there is no more concern for that. Instead horizonta market power is a potentia probem [6]; when suppiers coude with other suppiers in the market to gain the abiity of atering the eectricity price. Evidence of horizonta market power abuse has been observed in severa operating eectricity markets, where the market price has increased up to 22% above competitive eve [7]. Perfect competition price can be set as a benchmark to evauate market power abuse, different indicators have been studied to quantify market power and its association with cost [8],[1]. There are different forms of market power that can be present; market dominance or oca market power (due to transmission congestion). Schoars have studied transmission congestion in the past to recognize the presence of market power in the system [9]. Market dominance concerns strategic producers that are arge enough to affect the price and maximize their profit. Strategies as Physica Withhoding and Economica Withhoding (bids) are impemented to ater the eectricity price through horizonta mergers or by cousion. Strategic investment poicies can aso be impemented to maintain market dominance or increase profit [10]. Physica Withhoding or Quantity Withhoding occurs when a suppier intentionay decides to offer ess generation capacity to the market than its maximum production capacity. Setting market price as the intersection between suppy and

17 1.1. BACKGROUND 7 demand curves [5], the quantity withhed ( Q w ) makes the suppy curve shift to the eft and a higher new price is set, increasing it from P to P e as in figure (1.1).[1] Figure 1.1: Withhoding strategy and price-quantity outcome [1]. The main consequence of market power exercise is an increase of the generation cost (consequenty eectricity price) to a higher vaue than the margina cost, usuay referred to as Market Power Cost [8]. This means an increase of profit for producers and a ower efficiency in the market. The efficiency in the market is measured by its Socia Wefare, the summation of producer s surpus and consumer s surpus [2]. The tota surpus of the system is the tota generation cost minus the variabe cost of production. In a perfect competition market, the tota surpus reaches its maximum and is considered to be an efficient market. In the case of market power, the socia wefare reduces which indicates a ower efficiency. In figure 1.2 an exampe where in perfect competition the producer generation is

18 1.1. BACKGROUND 8 60TWh and its surpus is 6000MSEK, under market power the generating unit can offer ess capacity (exampe 40TWh) unti it reaches the maximum surpus of the producer (7000MSEK). Figure 1.2: Withhoding strategy and producers surpus variation [2]. This eads to another important consequence of market power exercise, which is the transfer of weath from consumers to producers, as shown in figure 1.3. This transfer of weath under market power directy affects consumers. In genera, perfect competition is assumed in a iberaized market, where no market power is exercised and suppiers bid according to their margina cost. Nevertheess, in reaity, considering the specia characteristics of eectricity, it is impractica to rey on this assumption. Therefore market power shoud be considered in the iberaized market and its reguation is of extreme importance to increase efficiency and minimize Socia Cost.

19 1.1. BACKGROUND 9 Figure 1.3: Transfer of weath by the exercise of market power [1]. Traditionay, transmission switching anaysis is not considered as an economic optimization; most of the studies were based ony on system reiabiity criteria without any economic market anaysis, where transmission eements were treated as non-dispatchabe eements. Currenty there is an agreement to deveop a smarter eectrica grid which is stronger, more fexibe, more efficient and better to contro. Optima transmission switching has proven to be a reiabe method of incorporating more controabiity to the grid [11], and aso creating economic benefit comparing to other methods such as generation unit rescheduing or oad shedding [12]. Optima transmission switching has been compemented with other co-optimizations anciary services, generation topoogy and optima unit commitment scheduing to improve economic efficiency under perfect competition assumption [11],[13]. Under the same assumption in a iberaized eectricity market, effective optima transmission switching has recenty proved to be an important too to improve

20 1.1. BACKGROUND 10 overa market efficiency and reduce Socia Cost [11], [14], [15] considering contingency constraints to prove the system remains stabe and robust [16]. Previousy market power strategic behavior for economica withhoding has being mode using Nash-Cournot concepts [17], and physica withhoding has being reduce according to demand fexibiity [18]. The increase of Socia Wefare due to disconnection of a sma number of ines in the system has been reported in [19]. Author of [19] suggests that by co-optimizing topoogy and a simpe variation of the network by Transmission System Operators (TSO), market power can be reduced. An impact in market power reguation by transmission augmentation and additiona transmission capacity has presented evidence of market power cost by reguating strategic behaviour [20],[21]. In the current eectricity market structure, it is important to aim for a more efficient network, a smarter grid that can be more controabe [22]. A fair equiibrium for producers and consumers where there is no transfer of weath must be procured. The starting point of this thesis is the work carried out by [14] and [16] to formuate a transmission switching optimization probem to reduce Socia Cost, and the work presented by [23] to incude Extreme Nash-Equiibria concepts to mode the strategic behavior of generating units (Market Power), where inearization techniques are impemented to formuate the probem as a singe-stage mixed-integer inear program probem. This thesis proposes an optima transmission switching mode to minimize Market Power Cost and therefore increase the competition and efficiency in the eectricity market and reduce cost for society.

21 1.2. PROBLEM DEFINITION Probem Definition Perfect competition has been a widey accepted assumption in eectricity market studies. However if it is considered that the present dereguated eectricity markets affect the competitiveness of payers, by big producers being abe to exercise market power, the main chaenge of the coming environment for eectricity market is reducing the cost associated with this behavior. The idea of this project is to deveop a mathematica mode in order to: (1) expose how market power can increase socia cost by withhoding generation capacity, and (2) how optima network configuration can reduce this cost and increase the competition of payers by manipuating the connectivity of the transmission ines. 1.3 Objective To accompish this aim, reducing market power cost, two modes are studied and integrated. The first mode is based on the work made by [14] and [16], that considers traditiona schemes of the eectricity market with the assumption of perfect competition between payers, which is not aways the case. To have a better understanding of the rea nature of the eectricity market, imperfect competition must be incuded. The second mode introduces market power behavior, using Game Theory concepts (Nash equiibrium) to mode strategic units in the system, that can choose to withhod capacity offered to the eectricity market. Both modes are integrated to have a system under market power conditions where transmission ines can be optimay switched to reduce the market power cost.

22 1.4. RESOURCES/TOOLS USED Resources/Toos used The optimization probem is executed using the Genera Agebraic Modeing System (GAMS) patform. The deterministic optimization probem is soved with the CPLEX sover in GAMS. The incidence matrices and other sets of parameters are created in MATLAB and transferred to GAMS by using Comma-Separated Vaues (CSV) fies. The code is run on a computer with Inte Xeon E5345 with a 2.33 GHz cocking frequency and 16 GB of RAM. Linear programming reaxation is empoyed to sove the optimization probem in GAMS.

23 13 Chapter 2 Formuation of the Mathematica Mode Nowadays it is important for the eectricity market reguator to minimize any form of market power and to offer the consumer ower cost of generation. This thesis proposes the use of optima transmission switching by Transmission System Operators to minimize market power behaviour of producers and ower market power cost. To minimize the Socia Cost under market power, a DC Optima Power Fow (DCOPF) mode of the system [14] is considered. This thesis presents an eectricity market mode that contempates a market power formuation and a DCOPF formuation with optima transmission switching, it aso incudes contingency anaysis formuation. Both formuations are integrated in the mode as a bi-eve optimization probem [24]. This chapter expains the formuation foowed to integrate two modes. The first assuming perfect competition (DCOPF mode) with transmission switching and the second considering market power behavior, based on the substitution of the equations of the DCOPF mode by its equivaent optimaity conditions, Karush-Kuhn-Tucker (KKT) conditions.

24 2.1. ASSUMPTIONS Assumptions Foowing assumptions where made to simpified the mode: Perfect information; No capacity imitations, units have enough instaed capacity to suppy the oad; The oad is not price sensitive; There are no osses in the system, N i=1 G = N i=1 D; A ines are avaiabe in the initia conditions; A generators are avaiabe; Non-strategic units offer 100% of the maximum capacity to the market; Each portfoio (owner) has ony one strategic unit; Margina pricing: price is set by the maximum production cost. 2.2 Mode Description A inearized bi-eve optimization probem is proposed. First the DC Optima Power Fow (DCOPF) mode (with transmission switching) and Worst-Nash Equiibrium (WNE) mode are formuated independenty and in detai in section and section respectivey. Finay the proposed mode is formuated in section 2.2.3, integrating the modes described in section and section as a inearized bi-eve optimization using Karush-Kuhn-Tucker (KKT) conditions.

25 2.2. MODEL DESCRIPTION Optima Transmission Switching The aim of this mode is to minimize the generation cost of the system using optima transmission switching, subject to physica constraints and Kirchhof aws for power fow. DC approximations are used in the system, so the mode can be defined as a inear probem. This mode is base under the assumption of perfect competition. The objective function and constrains of the optimization probem are base on a DC Optima Power Fow. Objective function and constraints are modified to incude the switching of the transmission ines and the cost associated. The objective function of this optimization probem and its constraints are stated beow [14]: Minimize c u g u + L c (1 n ) (2.1a) u U =1 Subject to: 0 g u g u u U (2.1b) F n F F n L (2.1c) g i + d i + F C,i = 0 i N, N (2.1d) B (θ n θ m ) F + (1 n ) M 0 L (2.1e) B (θ n θ m ) F (1 n ) M 0 L (2.1f) This formuation incudes a new eement: n, which is a binary variabe, and it represents the connection status of each transmission ine in the system. When a ine is connected, n = 1 and when it has been removed or opened, n = 0.

26 2.2. MODEL DESCRIPTION 16 The objective function (2.1a) incudes the cost of switching a transmission ine. The constraints of the optimization probem are: The generation imits of each unit (2.1b); The power fow imits of the ine (2.1c). A ines are modeed with the same maximum transmission capacity. The binary variabe n enabes the maximum capacity of the ine when the ine is connected and imits it to zero when the ine is disconnected. The energy baance constraint (2.1d). Kirchhoff s aw (2.1e),(2.1f), the mathematica function of this two equation is to accentuate the fact that a disconnected ine has no power fow. If the ine is connected, this variabe wi be zero and the formua represents the connection of the ine between nodes n and m. The parameter M is an arbitrary arge number, and it is introduced to infuence how the system perceives the network configuration. If a transmission ine is connected (n=1) Kirchhoff Law represents the connection of the ine between the nodes and the anges difference between θ m and θ n wi encourage the power fow through the ine. When the ine is disconnected (n=0) these two equations wi no onger represent this ange difference between nodes (ony a numerica affirmation) and competes the formuation to represent the disconnection of the ine. This formuation wi ensure no fow in an opened ine [14].

27 2.2. MODEL DESCRIPTION 17 Contingency Anaysis To ensure the system is abe to sustain its reiabiity, contingency anaysis is considered in the formuation. Additiona constrains are added to the previous (2.1) to incude the contingency status of the system: F c n n c F c F c n n c L (2.2a) g i + d i + F c C,i = 0 i N, N (2.2b) B (θ nc θ mc ) F c + (2 n n c ) M 0 L (2.2c) B (θ nc θ mc ) F c (2 n n c ) M 0 L (2.2d) A new eement n c (binary variabe) is incuded in this constrains (2.2) to represent the connection status of each transmission ine in the system during contingency [16]. Contingency is defined as the oss of one singe eement in the system, in this case a transmission ine. When ine is contingency, n c = 0 (for ine ) and when is not contingency, n c = 1. Equations (2.1)-(2.2) formuate the mode for DCOPF with Optima Transmission Switching (OTS) and Contingency Anaysis Worst-Nash Equiibrium Assuming that generating units can behave strategicay, choosing to withhod some of their outputs, we consider that market power is exercised by a suppier

28 2.2. MODEL DESCRIPTION 18 (owner of a strategic unit) when it is intentionay withhoding its production capacity. This action wi increase the cost of generation in the system and increase the profit of the unit. Perfect competition conditions for eectricity market studies, where a units are non-strategic units (are assumed to bid competitivey) is no onger an assumption, since some units are considered to be strategic in this mode. Game theory concepts are used in this project to mode the strategic decision of the generating units, in particuar Cournot Competition and Nash Equiibrium concepts. The Cournot Competition concepts considered for this project incude: (1) Few payers (suppiers) and many strategies. (2) Strategies are define as the quantities (q) the payers produce of an identica product. (3) The cost of production is c u q (with a constant margina cost). (4) Payers aime to maximize their profit [25]. A strategy profie S p is a Nash Equiibrium if for each individua payer, its choice s is the best response to the other payers choices. Figure 2.1 shows the response curve of two payers; BR1(R2) refects the best responses of payer 1 (R1) according to payer 2 responses (R2), the same for BR2(R1). The strategy chosen by the each payer where they intersect (R1 and R2 respectivey) is the the Nash Equiibrium response for each of them. Some considerations under Nash Equiibrium are: (1) Under Nash Equiibrium there is no individua incentive of each payer to deviate from s if a actions of a payers are hod constant. Ony under Nash Equiibrium are no regrets for a payers. (2) Everybody in the game beieves that everybody ese wi aso pay their particuar part in the Nash Equiibrium. (3) There can be many Nash Equiibriums for a probem. Probems can converge to a bad equiibrium for payers base on decreasing the profits, and can aso converge to a good equiibrium for the payers

29 2.2. MODEL DESCRIPTION 19 base on increasing profits [26], [27]. Figure 2.1: Nash equiibrium found by two payers [3]. When a payers reach their own strategy and thereby have maximized their own profit (π p ), the Nash Equiibrium has been reached [28]. In this study suppiers (producers) wi act as payers trying to find the Nash equiibrium to maximize their profit. Market power in the system is mode as the Worst-Nash Equiibrium (WNE), defined as the Nash Equiibrium which maximizes the generation cost (Socia Cost). There wi be strategic units and non-strategic units in the formuation. For this mode, each producer that owns an strategic unit is define as to posses a portfoio for each strategic unit it owns. For simpicity it is assumed that each owner that behaves strategicay may have maximum one portfoio. Then there wi be as many portfoios as strategic units are. Where the strategic unit is abe to exercise strategies considering Nash equiibrium. The aim of this mode (2.3) is to maximize the generation cost (Socia Cost) of the system an obtain the Worst-Nash Equiibrium, subjected to the two main constraints. These constraints are:

30 2.2. MODEL DESCRIPTION 20 Non-strategic units wi offer their maximum capacity to the system and strategic units can choose from a finite set of actions how much they wi offer to the market. The profit of a portfoio under Nash Equiibria (π p ) is higher than the profit of a portfoio that does not considers Nash Equiibria for its dispatch (π p,s p ). Maximize Subject to: c u g u u U (2.3a) ĝ u = K g u x uk w k + w 0 g u u U (2.3b) k=1 π p π p,s p u U (2.3c) Strategic and non-strategic units u are part of the set of generating units U, have a variabe cost of production c u and their production under Nash Equiibrium is g u. Then the generation cost or Socia Cost (SC) under Nash Equiibrium is define (2.3a). The offered capacity (ĝ u ) of a strategic unit is determined by profit maximization. In (2.3b) g u is the maximum capacity of the generating unit u, the binary variabe x uk indicates whether the unit is strategic x uk = 0 or nonstrategic x uk = 1. The percentage of maximum capacity that a strategic unit wi offer is defined by w 0. And the withhed capacity is defined by w k so that K k=1 w k = 1 w 0. So when the unit is non-strategic K k=0 w k = 1 and the unit wi offer its maximum capacity (2.4). Each strategic unit is assumed to have potentiay 4 eves of output, therefore each portfoio wi have 4 different possibe

31 2.2. MODEL DESCRIPTION 21 strategies; since a portfoio has ony one strategic unit and no non-strategic units. g u w 0 if x uk = 0 ĝ u = K g u w k if x uk = 1 k=0 (2.4) It is stated before and again in (2.3c) that the profit of a portfoio under Nash Equiibrium π u is greater than the profit of the same portfoio rather than when it is not considering the equiibrium strategies (πp p,s ). We said that the same portfoio does not considers Nash equiibrium when we hod the offers of every other portfoio constant, and repace the offer of this portfoio with another strategic combination s S p [28]. If there were more than one strategic unit for portfoio, the profit of it woud be the sum of the individua profits of each strategic units (π u ) as: π p = u O(p) π u For the proposed mode the profit of a portfoio p, π p, is the same as the profit of a unit u, π u. The profit of a singe unit can be defined as: π u = (P i c u )g u, where P i is the oca margina price at the node i, c u is the variabe cost of the unit and g u is its respectivey dispatch. This equation can aso be expressed as π u = µ up u ĝ u, where µ up u is the Lagrange mutipier of the constraint of each unit so it cannot exceed its offer. The expression for the profit of a unit π u is not inear, so to inearize this expression two considerations are made [23]:

32 2.2. MODEL DESCRIPTION 22 (1) If X and Y are positive variabes in an optimization probem, b is a binary variabe and M is a arge number, the non-inear equation XY = 0 can be rewritten: X b + Y (1 b) = 0 (2.5a) Can be written as two simpe inear constrains: M(1 b) X M(1 b) Mb Y Mb (2.5b) (2.5c) (2) Considering if λ is a Lagrange mutipier and x is a decision variabe, the mutipication of this two variabes is not inear, supposing that x takes a set of finite vaues x = m j=1 a jb j + a 0 for constants a 0,..., a m and m binary variabes b 1,..., b m. Introducing a new variabe c j that satisfies: c j λb j = 0. This expression is of the form of (2.5a) and can be written as two simpe inear constrains (2.5b),(2.5c). Then the expression for the profit of a unit π u is inearize introducing a variabe z uk = µ up u x uk, where x uk is a binary variabe. The equation z uk is now inearized as three equations [23]: z uk µ up u M (1 x uk ) (2.6a) z uk µ up u M (1 x uk ) (2.6b) z uk Mx uk (2.6c)

33 2.2. MODEL DESCRIPTION 23 Introducing inearized equations (2.6) to the definition of the profit of a portfoio π p : π p = K k=1 k u z uk w k + µ up u w 0 ku (2.7) The mode to obtain the Worst-Nash Equiibrium is formuated in (2.3)-(2.7). These equations are inearized, so the mode can be defined as a inear probem Integrated Mode of Optima Transmission Switching and Worst- Nash Equiibrium Nowadays it is important for the eectricity market reguator to minimize any form of market power and to offer the consumers ower cost of generation. This thesis proposes the use of optima transmission switching impemented by Transmission System Operators to minimize market power behaviour of producers and ower market power cost. A inearized bi-eve optimization probem is proposed, WNE formuation and OTS formuation are integrated as a set of primary conditions and secondary conditions respectivey. This integration is based on the substitution of the OTS formuation by its equivaent optima conditions, aso known as Karush- Kuhn-Tucker (KKT) conditions [29],[30]. The primary conditions are defined by the same formuation presented for the WNE mode (2.3)-(2.7). KKT conditions are deveoped for the DCOPF with Optima Transmission Switching and Contingency Anaysis mode, these conditions wi be defined as the secondary conditions of our integrated mode. The inner dispatch probem minimizes the socia cost, subject to constrains as in (2.1)-(2.2), except for (2.1b)

34 2.2. MODEL DESCRIPTION 24 which from now on wi be redefine as 0 g u ĝ u (2.8b) to incude the strategic units behavior. The secondary conditions are defined as: Maximize u U c u g u + L c (1 n ) (2.8a) =1 Subject to: 0 g u ĝ u u U (2.8b) F n F F n L (2.8c) g i + d i + F C,i = 0 i N (2.8d) B (θ n θ m ) F + (1 n ) M 0 L (2.8e) B (θ n θ m ) F (1 n ) M 0 L (2.8f) F c n n c F c F c n n c L (2.8g) g i + d i + F c C,i = 0 i N (2.8h) B (θ nc θ mc ) F c + (2 n n c ) M 0 L (2.8i) B (θ nc θ mc ) F c (2 n n c ) M 0 L (2.8j) First the Lagrangian function of the DCOPF with Optima Transmission Switching and Contingency Anaysis of (2.8) is defined as (2.9).Where ρ i, λ down, λ up, µ down u, µ up u, β down, β up, δ i, α down of the probem constraints., α up, ε down and ε up are the Lagrange mutipiers

35 2.2. MODEL DESCRIPTION 25 L = u U c u g u + L c (1 n ) =1 + ρ i (g i + d i + F C,i ) L =1 U u L =1 L =1 ) λ down (F n F + ) µ down u (g u g u + U u L =1 µ up u ) λ up (F F n (g u g u ) ( ) β down B (θ n θ m ) + F (1 n ) M ( ) β up B (θ n θ m ) F (1 n ) M + δ i (g i + d i + F c C,i ) L =1 L =1 L =1 L =1 ) α (F down c n n c F c ) α up (F c F c n n c ( ) ε down B (θ nc θ mc ) + F c (2 n n c ) M ( ) ε up B (θ nc θ mc ) F c (2 n n c ) M (2.9) The Dua feasibiity conditions are: ρ i, δ i free i N (2.10a) λ down, λ up 0 L (2.10b)

36 2.2. MODEL DESCRIPTION 26 µ down u, µ up u 0 u U (2.10c) β down, β up 0 L (2.10d) α down, α up 0 L (2.10e) ε down, ε up 0 L (2.10f) The Lagrangian function is formuated by setting each side of the constraints ess/equa to zero, separating the ower boundary from the upper boundary conditions. Each of this conditions are mutipied by a Lagrange mutipier respectivey. Finay, the objective function is added to the sum of these constraints as in (2.9). The process is known as Lagrange reaxation, where these changes an unconstrain probem of minimization is obtain from a constrain probem with equaity and inequaity restrictions [29]. Now KKT conditions are derive from the Lagrangian. The stationary conditions derived from the Lagrangian are defined as: L = c u + ρ i µ down u + µ up u + δ i = 0 (2.11a) g u L L L θ i = =1 β down (C i, ) T B =1 β up (C i, ) T B = 0 (2.11b) L = ρ i λ down + λ up + β down β up = 0 (2.11c) F L L L θ ic = =1 ε down (C i, ) T B =1 ε up (C i, ) T B = 0 (2.11d) L = δ i α down + α up + ε down ε up = 0 (2.11e) F c The stationary conditions (2.11) are defined as the partia derivatives of the Lagrange function in respect to the variabes of the secondary conditions and equa

37 2.2. MODEL DESCRIPTION 27 to zero. These variabes are g u, θ i, F,θ ic and F c. Then compementary sackness conditions derived from the Lagrange function: L =1 L =1 U u U u L =1 L =1 L =1 L =1 L =1 L =1 ) λ down (F n F = 0 L (2.12a) ) λ up (F F n = 0 L (2.12b) ) µ down u (g u g u = 0 u U (2.12c) ) µ up u (g u g u = 0 u U (2.12d) ( ) β down B (θ n θ m ) + F (1 n ) M = 0 L (2.12e) ( ) β up B (θ n θ m ) F (1 n ) M = 0 L (2.12f) ) α (F down c n n c F c = 0 L (2.12g) ) α up (F c F c n n c = 0 L (2.12h) ( ) ε down B (θ nc θ mc ) + F c (2 n n c ) M = 0 L (2.12i) ( ) ε up B (θ nc θ mc ) F c (2 n n c ) M = 0 L (2.12j) These compementary conditions ony consider the constraints that are defined as ess/equa than zero. They are the ower boundary and upper boundary of each constrain mutipied by their respective Lagrange mutipier and equa to zero.

38 2.2. MODEL DESCRIPTION 28 Conditions in (2.12) are no-inear. To guaranty that the KKT conditions sove the probem and obtain the goba optima soution, these equations are reformuated as a Mix Integer Linear Programming probem (MILP) by introducing binary variabes to inearize the probem as was estate above in (2.5). A set of two inear inear equations represent each equation of the compementary sackness condition: M ( 1 b ow λ M ( ) b ow λ λ ow ) ) (F n F M ( ) 1 b ow λ M ( ) b ow λ (2.13a) (2.13b) ( M ( M 1 b up λ ) b up λ ) ) ( (F F n M ( ) M λ up b up λ 1 b up λ ) (2.13c) (2.13d) M ( 1 b ow µ u M ( ) b ow µ u µ ow u M ( 1 b up µ u M ( ) b up µ u µ up u ) ) (g u g u M ( ) 1 b ow µ M ( ) b ow µ u ) ) (g u g u M ( 1 b up µ M ( ) b up µ u u u ) (2.13e) (2.13f) (2.13g) (2.13h) ( ) ( M 1 b ow β ) ( B (θ n θ m ) + F (1 n ) M M 1 b ow β ) ( M b ow β ) β ow ( M b ow β ) (2.13i) (2.13j)

39 2.2. MODEL DESCRIPTION 29 ( M 1 b up β ( ) M b up β M ( 1 b ow α ) β up M ( ) b ow α α ow M ( 1 b up α M ( ) b up α α up M ( 1 b ow ε M ( ) b ow ε ε ow ) ( M ( 1 b up ε ( ) ( B (θ n θ m ) F (1 n ) M M ( M b up β ) ) (F c n n c F c ) M ( 1 b ow α M ( ) b ow α ) (F c F c n n c ) M ( 1 b up α M ( ) b up α ) ) 1 b up β ) (2.13k) (2.13) (2.13m) (2.13n) (2.13o) (2.13p) ) ( ) B (θ nc θ mc ) + F c (2 n n c ) M M ( ) 1 b ow ε M ( ) b ow ε (2.13q) (2.13r) ) B (θ nc θ mc ) F c (2 n n c ) M M ( ) 1 b up ε M ( ) b up ε ε up M ( ) b up ε (2.13s) (2.13t) Recaing that the primary conditions are defined by the same formuation presented for the WNE mode (2.3)-(2.7), the integration of the secondary conditions to the primary conditions in the bi-eve probem is now compete by defining the secondary conditions as (2.8), (2.11) and (2.13). The compete optimization probem is formuated in (2.14), where the objective function is to find the the maximum Socia Cost under Nash equiibrium and minimize it by impementing Optima Transmission Switching. The overa probem is a Mixed-Integer Linear Program:

40 2.2. MODEL DESCRIPTION 30 Maximize (2.8a) (2.14a) Subject to: (2.3b) (2.7), (2.8b) (2.8j), (2.11), (2.13) (2.14b)

41 31 Chapter 3 IEEE 14 Bus Test Case, Resuts and Discussion For a better understanding the IEEE fourteen node exampe system is modified and used to iustrate the proposed mode. Data from the system was obtained from the University of Washington Power System Test Case Archive [31]. Both transmission switching in perfect competition and under market power are appied to the modified IEEE 14-node exampe system and studied in detai. The deterministic optimization probem is soved with the CPLEX sover in Genera Agebraic Modeing System (GAMS) patform [32]. The incidence matrices and other sets of parameters are created in MATLAB and transferred to GAMS by using Coma-Separated Vaues (CSV) fies. The code is run on a computer with Inte Xeon E5345 with a 2.33 GHz cocking frequency and 16 GB of RAM. Linear programming reaxation is empoyed to sove the optimization probem in GAMS. 3.1 Modified IEEE 14 Node Exampe System The modified IEEE 14-node exampe system is iustrated in figure 3.1, this exampe system has fourteen nodes and twenty power ines. The generation costs are assumed to be inear, resistance of the ine and shunt capacitance are zero,

42 3.2. RESULTS AND DISCUSSION 32 maximum transmission capacity of the ines is set to 20MW, osses and reactive power are ignored. The exampe system in figure 3.1 is modified to have three portfoios, each with one strategic unit that has four possibe eves of output. The strategic units are u 3, u 5 and u 6 (shadow units) and they beong to Portfoio 1, Portfoio 2 and Portfoio 3 respectivey. The characteristics of the generators is presented in tabe 3.1, tabe 3.2 shows the oad data and transmission ines data is presented in tabe 3.3. Modifications from the IEEE test system are made for the generators data, aso synchronous condensers (bus 3, bus 6 and bus 8) are not considered for this study. Tabe 3.1: Generator data. ID Bus Unit Size [MW] Cost of generation [$/MWh] u u u u u u u u Resuts and Discussion The Traditiona Mode for Perfect Competition vs. Nash Equiibria Mode The traditiona mode for perfect competition is a Mixed Integer Linear Programming probem presented in (2.1) as Optima Power Fow formuation, coded using

43 3.2. RESULTS AND DISCUSSION 33 G6 BUS 13 BUS 14 BUS 12 BUS 11 BUS 10 BUS 6 G7 BUS 9 BUS 5 BUS 1 BUS 4 BUS 2 BUS 3 Figure 3.1: Singe ine diagram of the 14 node exampe system. G3 the GAMS patform and soved with the CPLEX sover. The optima generation dispatch for the 14-node exampe system in perfect competition is shown in tabe 3.4. Transmission switching under perfect competition (A transmission eement open) wi resut in SC increase, therefor the optima soution is no transmission eement open (minimum SC). When Nash equiibrium is considered in the exampe system with no transmission switching in the ines, the resuts are referred to as Base Case study. The MILP programming dispatch resuts are obtained and

44 3.2. RESULTS AND DISCUSSION 34 Tabe 3.2: Load data. Bus Tota Demand [MW] presented in tabe 3.5. Under traditiona perfect competition, u 3 offers its true margina cost to the market. The same unit under Nash equiibrium withhods about 93.75% of its rea capacity. The Base Case where units are aowed to have strategic behavior by withhoding capacity, shows a drastic increase in socia cost from $20688 MWh to $60117 MWh, which is amost 290%. This study shows the impact that strategicay behaving generators can cause and how important it is to contro its disposition to physica withhoding. It aso provides with a better understanding of how important it is to consider market power when economic considerations are studied in the system and socia cost is to be minimized.

45 3.2. RESULTS AND DISCUSSION 35 Tabe 3.3: Line data. ID From Bus To Bus X ID [Ω] L L L L L L L L L L L L L L L L L L L L The Proposed Mode for Reducing Market Power Cost In contrast with the Base Case for the Nash Equiibrium mode, the proposed mode for reducing market power cost (2.14) appied to the 14-node exampe system can be used to obtain an optima panning for switching severa ines in the system (ines 4-9 and 13-14). After opening the transmission ines suggested by the program, the system s network is shown as in figure 3.2.

46 3.2. RESULTS AND DISCUSSION 36 Tabe 3.4: Optima dispatch for perfect competition. u g [MW] Cost [$/h] u u u u u u u u Tota Tabe 3.5: Optima dispatch under Nash equiibrium (base case). u g [MW] Cost [$/h] u u u u u u u u Tota

47 3.2. RESULTS AND DISCUSSION 37 Tabe 3.6: Optima dispatch under Nash equiibrium (switched ine). u g [MW] Cost [$/h] u u u u u u u u Tota Tabe 3.7: Generation capacity offer to the market in the base case. u g [MW] ĝ [MW] u u u u u u u u The new topoogy of the system changes the power fow of the ines as shown in tabe 3.8. It shows an increase of 233.3% in congested ines. When a faut occurs in one of the transmission ines that are congested before switching, the proposed mode shows the system remains stabe and transmission ines are optimay switched off to minimize the socia cost. The resuts show a significanty ower Socia Cost compared to the Base Case,

48 3.2. RESULTS AND DISCUSSION 38 Tabe 3.8: Power fow before/after switching. F L [MW] ID Before After L L L L L L L L L L L L L L L L L L L L

49 3.2. RESULTS AND DISCUSSION 39 G6 BUS 13 BUS 14 BUS 12 BUS 11 BUS 10 BUS 6 G7 BUS 9 BUS 5 BUS 1 BUS 4 BUS 2 BUS 3 Figure 3.2: Singe ine diagram of the 14 node exampe system after switching. G3 tabe 3.9, with a reduction of 70.45%. This proposed mode encourages strategic units (u 3, u 5, u 6 ) to be more competitive and offer more capacity to the market, as it is shown in tabe Hereby the tota withhed capacity of the system has been reduced from 70.4% to 55.8% (figure 3.3) of the instaed capacity in strategic generators, which is an increase of 51.43MW capacity offered to the market. The strategic units u 3 and u 6 are the major contributors to reducing the Socia Cost by

50 3.2. RESULTS AND DISCUSSION 40 Tabe 3.9: Socia cost. # ines switched Open Lines SC [$/MWh] Bus 4-9, Bus Tabe 3.10: Generation capacity offer before/after switching. u g [MW] ĝ u [MW] before after u u u u u u u u ceary reducing their withhed strategy, bidding more competitivey and offering a capacity coser to their maximum as is shown in figure 3.4 and figure 3.5. Optima Transmission Switching reduces the tota withhoding, but ooking cosey at tabe 3.10 we can see that u 5 instead of reducing its own withhoding, the proposed mode has given it incentive to exercise more market power (figure 3.6); nevertheess the tota Socia Cost sti decreases and hence society benefits from it. The switching of these ines (ines 4-9 and 13-14) jointy give the highest socia cost reduction. If there is a imitation on the number of ines that are aowed to switch and it is ower than the optima soution, the Socia Cost wi sti be reduced and ower than the Base Case socia cost, but wi be higher than the

51 3.2. RESULTS AND DISCUSSION 41 Figure 3.3: Impact of optima transmission switching in withhoding reduction. optima soution. It has been proven that Optima Transmission Switching can be used as one of many poicies to increase Competition Benefit by minimizing market power cost. These resuts show the high impact of switching a sma number of ines in the system on the tota operationa cost. It is important to notice that the proposed method is made for the systems that were designed or expanded by ony taking into consideration the excess demand in the system, therefore the Socia Cost are not considered during panning, which are most of the systems in operation today Contingency Anaysis To ensure the system is abe to sustain its reiabiity under the proposed mode for reducing Market Power cost, contingency anaysis is considered in formuation (2.14). To represent the connection status of each transmission ine during contingency, the parameter n c (binary variabe) is introduced in the formuation. The variabe tests each ine as a contingency eement and evauate its effects on

52 3.2. RESULTS AND DISCUSSION Variabe Operationa Cost [$/MWh] Capacity offered [MW] Capacity offer before switching Capacity offer after switching Max. Production Capacity Figure 3.4: Margina Cost Curve of u 3. the system reiabiity incuding Optima Transmission Switching to reduce market power cost. When contingency is defined for a transmission ined, a new network topoogy is set. For this topoogy the program suggests an optima panning for switching severa ines in order to minimize the socia cost. The proposed mode considering contingency anaysis is appied to the 14-node exampe system. Contingency for each ine is studied and resuts are presented in tabe For most of the contingency cases transmission ines are optimay switched off to minimize the socia cost, for these contingency cases the system remains stabe. During the contingency anaysis the Socia Cost variate for different cases, some of them present an increase up to 9% of the socia cost (for contingencies in ine 2, ine 3 and ine 5 respectivey) compared to the socia cost for optima transmission

53 3.2. RESULTS AND DISCUSSION Variabe Operationa Cost [$/MWh] Capacity offered [MW] Capacity offer before switching Capacity offer after switching Max. Production Capacity Figure 3.5: Margina Cost Curve of u 6. switching under Nash Equiibria when there is no contingency ($17764 MWh). Observing tabe 3.11 the optima transmission ines switched to reduce SC under contingency for the feasibe resuts are mainy the same as the ones proposed in section Then for these contingency scenarios it can be concuded that the system can sustain its reiabiity under the proposed mode for reducing socia cost. Nevertheess this concusions can not be generaized for a scenarios, tabe 3.11 shows infeasibe resuts for severa contingency conditions (35% of the contingency cases are infeasibe). An infeasibe soution stands for: there is no soution that fufis a the constrains, the probem has no soution [5]. These infeasibiities do not necessariy refect the stabiity of the system. There are two main reasons to which these infeasibiity resuts can be attributed: (1) There is no Nash Equiibrium found in the system, there is no strategy that can be arranged between the

54 3.2. RESULTS AND DISCUSSION Variabe Operationa Cost [$/MWh] Capacity offer before switching Capacity offer after switching Max. Production Capacity Capacity offered [MW] Figure 3.6: Margina Cost Curve of u 5. payers (producers). (2) The mathematica mode impemented is too simpe to reach the soution. Therefore taking into account a these considerations, the proposed mode (optima transmission switching to reduce market power cost) presents promising resuts to sustain the reiabiity of the system after a faut has occur.