Binary grey wolf optimizer models for profit based unit commitment of price-taking GENCO in electricity market

Accepted Manuscript Binary grey wolf optimizer models for profit based unit commitment of price-taking GENCO in electricity market Srikanth Reddy K, Lokesh Kumar Panwar, B.K. Panigrahi, Rajesh Kumar, Ameena Alsumaiti PII:

S2210-6502(16)30200-0

DOI:

https://doi.org/10.1016/j.swevo.2018.10.008

Reference:

SWEVO 457

To appear in:

Swarm and Evolutionary Computation BASE DATA

Received Date: 19 August 2016 Revised Date:

18 October 2018

Accepted Date: 19 October 2018

Please cite this article as: S. Reddy K, L.K. Panwar, B.K. Panigrahi, R. Kumar, A. Alsumaiti, Binary grey wolf optimizer models for profit based unit commitment of price-taking GENCO in electricity market, Swarm and Evolutionary Computation BASE DATA (2018), doi: https://doi.org/10.1016/ j.swevo.2018.10.008. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Binary Grey Wolf Optimizer Models for Profit Based Unit Commitment of Price-taking GENCO in Electricity Market

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Srikanth Reddy Ka*, Lokesh Kumar Panwarb, BK Panigrahia, Rajesh Kumarc, Ameena Alsumaitid a

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Department of Electrical Engineering, IIT Delhi, India. b Center for Energy studies, IIT Delhi, India c Department of Electrical Engineering, MNIT Jaipur, India d Department of Electrical and Computer Engineering, Khalifa University, Abu Dhabi, UAE.

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Abstract

The restructuration of electric power sector has renovated the power system operational planning. In the deregulated market, electricity is treated as an entity unlike in the traditional vertically integrated market model where it is treated as a service provided by the generation companies (GENCOs). The task of GENCO is to perform self-scheduling of available units so as to achieve maximum profit in restructured

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power sector. Therefore, the problem of profit based unit commitment (PBUC) in deregulated markets can be formulated as a commitment and generation allocation through self-scheduling procedure. The

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commitment schedule optimization, i.e whether the status of a thermal unit is to be changed to on or off state, is a binary problem. Thus, the PBUC problem requires binary optimization for commitment and real

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valued optimization for generation allocation. In this paper, three binary grey wolf optimizer (BGWO) models are presented to solve the profit based self-scheduling problem of GENCO. The BGWO models proposed in this paper differ with respect to the transformation function used to map real valued wolf position into a binary variable. The binary mapping of commitment status is carried out using sigmoid and tangent hyperbolic transfer functions. Also, in the sigmoidal transfer function, two binary transformation procedures, namely crossover and conventional sigmoidal transfer function, are presented. The effectiveness of the proposed models in improving the solution quality of PBUC problem is examined using two test systems, a 3 unit and a 10 unit test system. In addition, two cases of GENCO market

ACCEPTED MANUSCRIPT participation with and without reserve market participation are simulated. In the test case with reserve market participation, two commonly used reserve payment models are examined. Simulation results are presented and compared to existing approaches that have been used to solve the PBUC problem. The

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simulation results and statistical analysis demonstrate the improved solution quality (profit or fitness value) of the PBUC problem and statistical significance of the BGWO models with respect to solution quality obtained as compared to other established meta-heuristic approaches.

Profit base unit commitment (PBUC), Meta-heuristic optimization, Electricity market,

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Keywords:

Binary grey wolf optimizer (BGWO), Constrained optimization, Nature inspired optimization. ∗Corresponding

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author Email addresses: [email protected] (Srikanth Reddy K), [email protected] (Lokesh Kumar Panwar), [email protected] (B.K.Panigrahi), [email protected] (Rajesh Kumar) Preprint submitted to Swarm and Evolutionary Computation August 19, 2016

1. INTRODUCTION

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The electricity market structure around the globe has been transforming into deregulated nature from conventional vertically integrated model. As a result, the unit commitment and generation scheduling practice of the generation company (GENCO) changed from cost minimization to profit maximization [1].

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In this structure, GENCO acts as a price taking entity and schedules its available generation assets for

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various services as applicable, considering the generator’s operating and cost characteristics [2]. Therefore, the forecasted load and reserve requirement of the power system network provides the participation limit for GENCO. Therefore, in this self-scheduling procedure do not impose any obligations related to load/reserve fulfillment as in case of traditional/conventional market mechanism [3-4]. Thereupon, concurrent provisions to participate in various markets offering different services for a single GENCO were identified in the pursuit of profit maximization. However, earlier models have partially considered reserve costs and revenues [5]. Later, more appropriate and accurate models for market participation are developed and are adopted in this work [6].

ACCEPTED MANUSCRIPT The non-linear nature of the generation scheduling was intensified through the profit based unit commitment problem in comparison with traditional cost minimization problem. In PBUC, some of the constraints, like load and reserve satisfaction, are modeled as soft constraints, unlike cost minimization

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problem in which both the load constraint and reserve constraint are equality constraints. The constraints pertaining to the minimum up and down time and ramp rate remained same in PBUC as that of conventional Unit Commitment (UC) problem. Over the past decade, the problem of GENCO selfscheduling through PBUC attracted considerable interest. As a result, different optimization approaches are

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proposed to solve the PBUC problem. Earlier models incorporated reserve market participation of GENCO

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using forecasted energy and reserve price [7]. The Lagrangian multiplier adjustment approach is used to accompany the thermal unit scheduling [8]. However, the non-convex nature of the PBUC problem led to convergence related issues [16]. Later, the convergence issue is resolved using gradient based Lagrangian Relaxation (LR) approach. However, the gradient based solutions are vulnerable to local minima for nonconvex problems like PBUC. The local minima problem is solved using computationally intelligent and

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heuristic approaches like Genetic Algorithm (GA) in which softer demand constraints are modeled [3]. Thereafter, hybrid techniques like Lagrangian Relaxation-Evolutionary Programming (LR-EP) are proposed to suppress the issues with LR by using evolutionary programming for the Lagrangian multiplier

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update [9]. Similarly, Lagrangian Relaxation-Genetic Algorithm (LR-GA) is also proposed in which GA is used to update Lagrangian multipliers [10]. The evolutionary approaches are hindered by the fact the

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convergence of the algorithm is highly dependent on various factors such as mutation rate, crossover rate, parent selection method etc. Consequently, one has to perform multiple experimentations to find optimal parameters otherwise which the problem can encounter premature convergence. Later, the optimal allocation of energy and reserve capacities is carried out using Tabu-Search (TS) [11]. While, the TS approach scores in terms of excepting from local minima by allowing non-improving solution, it also requires optimal tuning of various parameters similar to GA. Thereafter, an integer based linear

ACCEPTED MANUSCRIPT programming approach is developed with superior performance as compared to LR [12]. However, the same endured the computational problems with increment in system dimension. Recent literature showed the application of various meta-heuristic approaches like Particle Swarm

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Optimization (PSO) [13], Muller Method (MM) [14], Nodal Ant Colony Optimization (NACO) [15], Parallel Nodal Ant Colony Optimization (PNACO) [16], Gravitational Search-Lagrangian Relaxation Based Artificial Neural Network (GSLRANN) [17], hybrid Lagrangian Relaxation- Invasive Weed Optimization (LR-IWO) [18], Tabu-Search-Evolutionary Particle Swarm Optimization (TS-EPSO) [19],

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Particle Swarm Optimization-Dynamic Programming (PSO-DP) [20], Variable Neighborhood TabuS

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Parallel Enhanced Particle Swarm Optimization with island model (VTS-PEPSO) [21], Lagrangian Relaxation-Particle Swarm Optimization (LR-PSO) [22], Parallel Artificial Bee Colony (PABC) [23], Multi Agent System (MAS) [24] based approach, Imperialist Competitive Algorithm (ICA) [25], Ant Colony Optimization (ACO) [26], Improved Pre Prepared Power Demand (IPPD) [27], Binary Fireworks Algorithm (BFWA) [37], etc., to solve the PBUC problem. The approaches have different pros and cons for

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application to high dimension problems such as UC and PBUC. For example, PSO based algorithms present a simple way of implementation with minimum parameter tuning. However, the vulnerability of falling into local minima hinders the application of PSO to problems with high dimension. The other class

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of algorithms called ant colony algorithms (NACO, PNACO, ACO) gains over other algorithms with inherent parallelism. However, ACO algorithms are prone to random decisions and uncertain convergence

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time. The physical phenomenon based algorithms such as Gravitational Search Algorithm (GSA) are also prone to fall into local optima and solution precision for computationally challenging problems. Therefore, the heuristic approaches are either simple to apply with local optimum vulnerability or need optimal parameter tuning for improved or best possible performance. The motivation behind this work is the successful implementation of meta-heuristic framework Binary Grey Wolf Optimizer [28] in solving several computationally challenging optimization problems such as PBUC in electricity market to maximize the GENCO profit. Some of the applications of GWO are

ACCEPTED MANUSCRIPT blackout risk prevention [29], optimal reactive power dispatch [30], solar maximum power point tracking (MPPT) [31], optimization of photonic crystal filter [32], etc. Recently, binary wolf optimizer has been developed and applied successfully to feature selection problem [33]. The application of GWO to the

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engineering problems is based on the advantages of the meta-heuristic approach GWO, such as its requirement for parameter tuning corresponding to a particular problem. This paper presents the application of various Binary Grey Wolf Optimization (BGWO) models to solve non-convex, computationally challenging, bounded and constrained PBUC problem of GENCO in power system.

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Rest of the paper is organized as follows: Section 2 introduces the PBUC problem of GENCO. Section

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3 outlines the theory of GWO and various BGWO models. The application and solution procedure of PBUC using BGWO is described in Section 4. The test system description, simulation conditions, parameter settings are explained in Section 5. The same section also presents the simulation results and discussion along with statistical significance of the proposed BGWO models. Finally section 6 concludes

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the paper with key findings of the work.

2. PROBLEM FORMULATION

In the self-scheduling problem such as PBUC, GENCO acts as price taker and schedules the available

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units to incur maximum profit. The different cash flows like GENCO costs and revenues in various market services i.e., energy and ancillary (reserve) markets, are illustrated along with GENCO objective in the

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following sub-sections. Nomenclature

A. Sets and Indices 



 

Hour of the scheduling day Conventional (coal fired) Generator(s) Time horizon (24 hours) Conventional generators set

ACCEPTED MANUSCRIPT B. Decision variables (, )

Energy market bid of   thermal unit for  hour

(, ) (, )

ON/OFF state of   thermal unit at  hour

C. Parameters System load at  hour



Lower generation limit for  thermal unit

 () 

Required system reserve for  hour

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()

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Upper generation limit for  thermal unit

(, )

Startup cost for   thermal unit at  hour

 ()

Cold startup cost of   thermal unit

 ()

Hot startup cost of   thermal unit

Minimum down time of   thermal unit



Time(hours) for which   thermal unit is OFF

 ()

Minimum up time of   thermal unit

! ()

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()

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 ()

 ()

Ramp down rate of   thermal unit

Time(hours) for which   thermal unit is ON

Ramp up rate of   thermal unit

" ()

Cold start hours of   thermal unit

()

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 ()

#()

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Reserve market bid of   thermal unit for  hour

Spot price of energy market at  hour

Reserve market price at  hour

ACCEPTED MANUSCRIPT $

Probability of contingency occurrence

!

Number of generators

%

Total number of hours

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2.1. Objective Function In contrast to the traditional or vertically integrated conventional power system operations [6], in deregulated markets, the objective of GENCO is to maximize profit by self-scheduling procedures. The

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objective of GENCO in a deregulated market will maximize the difference between total revenue (RV) and total cost (TC) as given by,

(1)

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&'(&)* +$,- = &'(. ( 0 − %)

Where, RV represents the total revenue that GENCO receives from market participation. The total revenue received by GENCO varies with respect to the payment methods pertaining to various market operating and clearing mechanisms in practice. Two of the widely used payment methods are discussed in detail as follows. The payment strategies for reserve allocated by generator in reserve market depend on the the

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amount of time during which, payment is made and type of payment. The payment for reserve capacity can be of two types, reserve capacity payments and energy payments. The reserve capacity payment is made to the generators irrespective of the reserve capacity’s deployment. The energy payment is made for the

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duration for which the actual generation from the reserve capacity is allocated.

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2.1.1. Reserve Payment Method A (payment for energy) In this payment method (method A), the allocated reserve is paid only for the actual time it is called upon to serve. Therefore, the reserve payment ($ () (, )) depends on the probability ($) with which it is called to serve. The following expression represents the total payment to GENCO participating in both energy and reserve market.

ACCEPTED MANUSCRIPT 

9:



9:

0 = 2 2345(, )6#()(, )78 + 2 2 $ () (, )(, ) ;< ;<

;< ;<

(2)

In general, for this method, the reserve price is much higher compared to that of spot energy market

function for this method is given by, 

9:



9:

+$,- = >2 2345(, )6#()(, )78 + 2 2 $ () (, )(, )? ;< ;<



;< ;<

9:

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price. This is to ensure that allocated reserve of GENCO makes considerable revenues. The total profit

(3)

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− >2 23(1 − $)4 A 5(, )6(, )7 + ${ A 5(, ) + (, )6(, )} + {(, )(, )}8? ;< ;<

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The first term denotes the revenue incurred by GENCO by scheduling generators in the energy market and the second term denotes the total cost (TC) of energy and reserve market schedules which again consists of two parts, fuel cost and startup cost. The shutdown costs, being a very small fraction of the startup cost, are often neglected. In (3),

A

, the fuel cost of   generator for  hour, can be given as follows. A 5(, )6

= ' (, )D + E (, ) + F

(4)

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In the above equation ' , E , F denote cost coefficients of   generator. In equation (3), (, ) is the startup cost, which can either be hot or cold startup cost, given by,  (),  (),

-  ( ) ≤  ( ) ≤ " ( ) +  () - " ( ) +  ( ) <  ()

(5)

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(, ) = G

2.1.2. Reserve Payment Method B (Payment for energy and capacity) In method B, the GENCO is entitled for two types of payments for the amount of the reserve capacity that is used, i.e., capacity payment and energy payment., Both of these payments are dependent on the reserve deployment probability. The GENCO will receive spot market price for the duration it generates electricity. On the other hand, the reserve price is paid for the amount of time in which the reserve capacity is not called to serve. Therefore, the total revenue can be expressed as follows.

ACCEPTED MANUSCRIPT 

9:



9:

0 = 2 2345(, )6#()(, )78 + 2 2{$#() (, ) + (1 − $) () (, )}(, ) ;< ;<

;< ;<

(6)

The first term of (6) represents the energy payments to GENCO, while the second term represents the

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reserve payments. In the reserve payment, the first term ( $#() (, )) represents the amount paid to GENCO reserve when called upon to serve, which is a function of the reserve deployment probability. For the rest of the time, GENCO receives capacity payment ((1 − $) () (, )) for the amount of the capacity



9:



9:

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scheduled for reserve. Thus, the total profit earned by GENCO in Method B is given as follows.

+$,- = >2 2345(, )6#()(, )78 + 2 2{(1 − $)#() + $ ()} (, )(, )? 

;< ;<

9:

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;< ;<

(7)

− >2 23(1 − $)4 A 5(, )6(, )7 + ${ A 5(, ) + (, )6(, )} + {(, )(, )}8? ;< ;<

Where, the second term of (7) represents the total cost of operation incurred by the GENCO for energy as

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well as reserve market participation [37].

2.2. Constraints

Similar to the UC problem, all the thermal unit constraints are applicable to PBUC problem as well.

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However, the system constraints of UC problem undergo small variations in the context of self-scheduling approach in PBUC problem. The details of system constraints and thermal unit’s operation constraints are

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explained in the following sections.

2.2.1. System constraints a. Load constraint

The load constraint in profit based unit commitment is different from the traditional unit commitment and suggests that the generators may or may not satisfy the whole load of the network and may generate less than or equal to the forecasted load requirements of the system.

ACCEPTED MANUSCRIPT 9:

2 (, ) ≤  ()

(8)

;<

b. Spinning reserve constraint

Similar to the load constraint, the GENCO does not have any obligation to abide by the reserve

[6]. 9:

2  (, )(, ) ≥  ()

(9)

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;<

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constraints, i.e., it can schedule its generators to supply less than or equal to the predicted system reserve

Where,  ()is the required reserve for  hour and  (, ) is the reserve available from   thermal unit

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at  hour, which is given by,

 (, ) = 5 − (, )6(, ) )

(10)

2.2.2. Thermal Unit Constraints

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 in equation (8) corresponds to the maximum capacity of   thermal unit.

Apart from system constraints, the PBUC is also associated with thermal unit constraints that need to be

dispatch (ED).

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satisfied during commitment, de-commitment and power allocation to generators according to the economic

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a. Generation limits

Considering the operation constraints of thermal generator and its auxiliary systems, actual generated power of any generator should be strictly within its upper and lower bounds. (11)  < (, ) <     where,  ,  are the minimum and maximum generation limits of  thermal unit respectively.

b. Minimum up/down times There exists a pre-defined time (in hours) between state transition (ON/OFF) of a thermal unit given as,

ACCEPTED MANUSCRIPT 51 − (,  + 1)6 ( ) ≤ ! ( ), -(, ) = 1 (,  + 1) ( ) ≤  ( ), -(, ) = 0

(12)

denotes the number of hours for which the   unit is in ON state. c. Ramp up/down rates

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Where,  () is the minimum time for which the unit should be kept ON once it is committed, ! ()

Similarly, the hourly variation of thermal unit power generation is limited by ramp up and down rates.

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, () < (, ) < , ()

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Where,

(13)

, () = max (5 6,  (,  − 1) −  ()) (14) , () = min (( ), (,  − 1) −  ()) In (14),  (),  () correspond to ramp down and up rates of the ith generator respectively, (,  − 1)

d. Initial states

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is the power generation of   plant during ( − 1) hour.

Information about ON/OFF status of thermal units prior to scheduling period is very important as it may

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influence cost parameters along with system and unit constraints.

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3. BINARY GREY WOLF OPTIMIZER (BGWO) 3.1. Overview of grey wolf algorithm (GWO) The grey wolf optimizer belongs to the class of bio-inspired heuristic search algorithms being used to solve computationally challenging, non-linear, constrained optimization problems. The GWO is motivated by the hierarchical and hunting behavior of grey wolves which often carry out their activities in a pack. The distinguishing features of grey wolves are explained as follows.

ACCEPTED MANUSCRIPT 3.1.1. Hierarchy in the grey wolf pack The wolf pack is divided into 4 groups based on the social hierarchy. The alpha (Q) wolf sits at the top of the wolf pack and is considered as the leader of the pack. The alpha wolf is also considered as the

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most intelligent among other wolves, but may not be the strongest physically. The decisions regarding movement and hunting are decided by the alpha wolf. The next level in the hierarchy consists of beta (R) wolf whose job is to make sure the decisions of the alpha wolf get implemented within the wolf pack. The beta wolf is the potential candidate to be the alpha wolf once the alpha expires. The lowest level of the wolf

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pack is occupied by omega (S) wolves. The omega wolves are subordinate to the remaining wolves of the pack and submit to the alpha, beta and delta wolves. The group of wolves which do not belong to

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Q, R 'TU S are classified as delta (V) wolves. The delta wolves subordinate to alpha and beta wolves. However, the delta wolves have the power to dictate the omega wolves. The delta wolves perform the safety and security actions of the pack. The delta wolves are also responsible for providing food and care

3.1.2. Encircling the prey

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for the entire wolf pack.

The foremost activity that the wolf pack performs when approaching the prey is to encircle it. The

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encircling procedure of the wolf around prey is mathematically modeled as follows: ur ur uuur uur D =| C. X p (k ) − X (k ) |

(15)

\X WX(Y + 1) = WXZ (Y) − X. [

(16)

ur ur where, the current iteration is represented by Y, A and C denote the coefficient vectors, position vector of

uuur uur the prey is denoted by X p (k ) and X represents the position vector of a grey wolf during the Y  iteration.

The coefficients X and X are expressed as follows.

X = 2'X. $X − 'X

(17)

ACCEPTED MANUSCRIPT ur ur C = 2.r2

(18)

r r r where, components of a reduces linearly from 2 to 0 as the iterative search process progresses, r1 , r 2 are

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random vectors uniformly distributed over [0, 1].

3.1.3. Hunting the prey

The hunting process is guided by the alpha wolf, which is aided by the beta and delta wolf in making hunting decisions. Thus, alpha, beta and delta wolves have better knowledge of the prey position compared

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to other wolves. Therefore, the first three best positions of the wolf pack are designated as the positions of

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the alpha, beta and delta wolves respectively. These positions are used by other wolves of the pack to update their respective positions on an iterative basis. The update process of the wolves in search of prey

ur ur uur uur Dα =| C1. X α − X |

(19)

ur ur uur uur D β =| C 2 . X β − X |

(20)

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(optimal solutions) is given as follows:

ur ur uur uur Dδ =| C 3 . X δ − X |

(21)

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The position of the remaining wolves with respect to the position of the best three wolves is estimated by

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using their respective distance from the alpha, beta and delta wolves (19-21) as follows: uur uur ur ur X 1 = X α − A1 .D α

(22)

uur uur ur ur X 2 = X β − A 2 .D β

(23)

uur uur ur ur X 3 = X δ − A 3 .D δ

(24)

The final position update of wolves can be expressed as follows. uur uur uur uur X1 + X 2 + X 3 X (k + 1) = 3

(25)

ACCEPTED MANUSCRIPT 3.2. Binary Grey Wolf Optimization The application of GWO for PBUC requires a conversion of real valued search space and variables to binary values through appropriate transformation. To this end, this paper presents four distinct approaches

result in different solution quality and convergence characteristics.

3.2.1. BGWO Model 1

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for GWO binary transformation The binary update process for the four approaches are different and may

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In the first model, a crossover operation is defined for the position update of the wolves in each iteration. This method updates the position of wolves with respect to the position of the alpha, beta and delta wolves

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prior to the integral update of wolf position based on the best three positions. The mathematical modeling of the crossover operation is formulated as follows [33].

^`ab = Crossover ( χ1 , χ2 , χ3 ) _

(26)

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= Crossover ( χ1 , χ2 , χ3 ) represents a suitable crossover operation between χ1 , χ 2 , χ3 which Where, ^`ab _ are derived from the movement of wolves in the direction of the alpha (α ) , beta ( β ) and delta (δ ) wolves

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transformations:

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respectively. The update process of the binary vectors χ1 , χ 2 , χ3 is determined by the following

 1 if ( χαD + qαD ) ≥ 1 χ = 0 otherwise D 1

(27)

where, χ αD represents the α wolf position in dimension D and qαD denotes the binary step given by, 1 if pαD ≥ n qαD =  0 otherwise

(28)

ACCEPTED MANUSCRIPT Where, n is uniformly distributed random number in [0,1] and pαD represents the continuous update step in D dimension estimated using sigmoid transformation as given by, pαD =

1 1 + exp(−10*( A1D DαD − 0.5)

(29)

 1 if ( χ βD + qβD ) ≥ 1

χ 2D = 

1 if pβD ≥ n qβ =  0 otherwise D

1 1 + exp(−10*( A1D DβD − 0.5)

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pβD =

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0 otherwise

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Similarly, the sigmoid transformation for beta and delta wolves is expressed from (30) to (35).

 1 if ( χδD + qδD ) ≥ 1

χ 3D = 

0 otherwise

(30)

(31)

(32)

(33)

(34)

1 1 + exp(−10*( A1D DδD − 0.5)

(35)

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1 if pδD ≥ n qδD =  0 otherwise

pδD =

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by,

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Upon the estimation of χ1D , χ 2D , χ 3D the wolf position update operation through crossover operation is given

1  D  χ1 if n < 3  1 2  D χ =  χ 2D if ≤ n ≤ 3 3  D  χ 3 otherwise  

(36)

where, χ1D , χ 2D , χ 3D represent binary values corresponding to the first, second and third best fitness values,

χ D denotes the output of dimension D, n is a uniformly distributed random number over the range [0,1].

ACCEPTED MANUSCRIPT 3.2.2. BGWO Model 2 This model uses a tangent hyperbola transformation based binary position update. The formulation for

{

}

uur 1 if T X (k + 1) > n χ Dk +1 =  0 otherwise

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tangent hyperbola transformation based update process is given as follows:

(37)

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where, n denotes a random number which is randomly distributed over [0, 1] and T { X (k + 1)} is the tangent hyperbola transformation of real valued update to binary value is given as follows. uur

uur e µ X ( k +1) − 1 T X (k + 1) = µ uuXr ( k +1) e +1

}

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{

(38)

where, c is a coefficient that takes integer values to define the shape of tangent hyperbolic function.

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3.2.3. BGWO Model 3

This model uses the sigmoid transformation for binary position update. The formulation for the sigmoid transformation update is presented as follows.

1 if S { X (k + 1)} > n 0 otherwise

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γ Dk +1 = 

(39)

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where, n is a uniformly distributed random number over range [0, 1] and S { X (k + 1)} is the sigmoid transformation function as given by, 1

S { X ( k + 1)} = sigmoid ( X ( k + 1)) = 1+ e

Where X α , X β , X δ represent the updated real positions of α , β , δ .

 X α + X β + X δ −10  3 

   − 0.5  

(40)

ACCEPTED MANUSCRIPT 4. BGWO Implementation to UC Problem

The binary valued GWO models presented in this work are used to determine the optimal commitment schedule of the thermal units. The generation scheduling of the committed units is handled by using the

4.1 Representation of Binary variables of UC Problem

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Lambda iteration technique [37].

The commitment status of   generator of the d  wolf during  hour of Y  iteration is

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,g  represented by χ,g generator of d  wolf during ,f . Thus, χ,f = 1 denotes the committed status of the 

 hour of Y  iteration and vice versa for χ,g ,f = 0. The application of BGWO to solve UC problem is

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started by initializing the “NP” wolves with each wolf having a dimension of ! − Eh −  matrix. Where,

! denotes the number of thermal units and  represents the total number of scheduling hours of UC

problem. Therefore, for Y ℎ iteration of d ℎ wolf, the representation of binary variables representing the

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<,g j,g commitment status of N thermal units for H hours will range from i<,f to i9,f .

Figure. 1 Representation of population structure of UC problem with BGWO

The commitment matrix of all units of d  wolf over a 24 hour scheduling horizon during the Y 

iteration is denoted by kgf . The representation of the population structure is shown in Figure 1. The fuel cost being a quadratic function, solution can be obtained through Economic Load Dispatch (ELD) process.

ACCEPTED MANUSCRIPT The step by step procedure for the application of BGWO to solve PBUC problem is explained in the following sections.

4.2 Binary grey wolf position initialization

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The initialization of the wolves in potential search space (binary) is a random process. Within the range of sigmoid transformation function, a real valued random number ( Rnst ) is created. The random number Rnst is

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used in (37) and (40) for binary transformation. Thereafter, another uniformly distributed N rst random number between [0, 1] is generated. Further, the initial binary valued position of the grey wolves are

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updated as follows:

0 if S ( Rnst )orT ( Rnst ) > N rst

χit,,1w = 

(41)

 1 otherwise

The positions of wolves corresponding to the first three best fitness values of the objective function are

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assigned to alpha, beta and delta wolves respectively. The fitness values can be estimated using Lambda iteration technique for the committed thermal units during initialization process (Eq. 41). 4.3 Binary position update

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The BGWO models presented in this paper ((26) to (40)) differ with respect to the binary position update process . While any of the BGWO MODEL 1, BGWO MODEL 2, and BGWO MODEL 3 can be used for

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binary transformation, the conversion using BGWO MODEL 3 is explained as a sample case. The following expression explains the binary transformation of updated real valued wolf position. 1 if S { X (k + 1)} > R 0 otherwise

χ it,,wk +1 = 

(42)

where, R is a uniformly distributed random number between 0 and 1, S { X (k + 1)} is the sigmoid transformation of real valued wolf position as given by (40).

ACCEPTED MANUSCRIPT 4.4 Binary position update for alpha, beta and delta wolves The position of best three wolves is also updated in every iteration. The position update for alpha, beta and delta wolves is carried out by comparing the best fitness value of every iteration, with the existing position

l,g kga< , - (kga< f ) ≤ (k ) kl,ga< =m f kl,g , ,ℎ*$d)*

l,g ga< o,g kga< f , - (k ) ≤ (kf ) ≤ 5k 6

ko,g , ,ℎ*$d)*

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ko,ga< =m

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and fitness values given as follows:

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l,g o,g ga< p,g kga< f , - 4 (k ) and 5k 67 ≤ (kf ) ≤ 5k 6 kp,ga< = m p,g k , ,ℎ*$d)*

4.5 Thermal unit power schedules

(43)

(44)

(45)

The unit commitment schedules obtained from BGWO are supplied to Lambda iteration technique to obtain

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the optimal generation schedules of committed thermal units. The vector of generation schedules is represented by P(k ) = [ P1k , P2k ....Pwk ] . Where, each member of P ( k ) is of dimension N-by-H matrix and

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represents the optimal generation schedules of committed units as follows,

 p1,1,wk  1,k p k Pw =  2,w K  1,k  pN ,w

p1,2,wk

K

p2,2,wk K

K K

pN2,,kw K

p1,Hw,k   p2,Hw,k  K   pNH,,wk 

(46)

4.6 Termination criteria

The predefined iteration number is used as a reference to terminate the solution process. Therefore, the BGWO will stop at the instant the iteration count reaches a pre-defined maximum number of iterations. Figure 2 shows the flow chart of the proposed BGWO approach for solving UC problem. The search procedure is started by initiating all the problem related parameters followed by initialization of random

ACCEPTED MANUSCRIPT wolf positions bounded by the search space limits. For the randomly initialized solution sets, fitness value is evaluated using Lambda iteration technique based economic load dispatch (ELD) [7] and alpha, beta and delta wolves are designated in the population. The binary position of all omega wolves is updated using

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(36-40) and the updated wolf positions are verified and repaired for any possible constraint violation using heuristic adjustment procedure [19]. The updated wolf positions are compared to alpha, beta and delta wolves to update the positions using (43) to (45). Finally, the iteration number is checked so as to decide

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whether the next iteration is to be performed or the search process is to be ended.

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ACCEPTED MANUSCRIPT

Figure 2. Flowchart of UC problem solution using BGWO

4.7 Constraint handling

The heuristic approach of handling constraints, based on rule based mechanism is adopted in this work [19]. The possible violation of constraints such as, minimum up-down times during initialization as well as update procedures, has to be handled appropriately. In evolutionary algorithms like BGWO, the

ACCEPTED MANUSCRIPT state transition of units may violate the minimum up and down time constraints during the processes of random initialization and update. Therefore, heuristic approach based techniques are used to avoid the occurrence of infeasible solutions and thereby improving the solution quality. The procedure for minimum

i =1

if A(i, t ) = 1

if ON h (i ) < M DT (i )

A(i, t ) = 0

A(i, t ) = 1

A(i, t ) = 0

if A(i, t − 1) = 1

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if A(i, t − 1) = 0

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up-down time constraint handling using heuristic adjustment is explained in Figure 3.

if OFFh (i ) < M UT (i )

A(i, t ) = 1

A(i, t ) = 0

A(i, t ) = 1

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if i = NG

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Figure 3. Algorithm for minimum up-down time constraint

5. Simulation Results and Discussion

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This section presents the simulation results and the statistical analysis of various test cases to demonstrate the performance of BGWO models for PBUC problem. The simulations for solving PBUC problem using BGWO are carried out in MATLAB R2012b environment operating on Mac OS X version 10.9.1 and 2.7 GHz processor. Two test systems with 3 and 10 units with various test cases of scheduling and reserve payment methods are considered. Most of the parameters of BGWO are self-derivable in the course of the search process. Other parameters such as population number and maximum iterations are set based on test case. For Case 1, the number of wolves (population) and the number of iterations have been

ACCEPTED MANUSCRIPT set to 30 and 200 respectively. However, considering the complexity of Case 2 (energy and reserve scheduling) compared to Case 1 (energy only scheduling), the maximum number of iterations are increased to 1000. In BGWO models presented in this paper, only Model 2 requires the parameter setting of c and it

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is set to be “2”. 5.1. Test system 1: 3 Unit Test system

The test system description of the 3 unit test system with cost coefficients and operating parameter

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constraints is given in Table 1. A total scheduling horizon of 12 hours is considered and the forecasted demand, reserve and associated prices in spot and reserve markets are given in Table 2. The commitment

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and generation schedule of the 3 unit system for energy and reserve market allocation according to Method B is given in Table 3. The energy market only participation case and reserve allocation with payment Method A of Case 2 are not considered due to unavailability of comparison from literature. The performance comparison of the proposed BGWO Model 3 with other existing approaches like LR-EP [9], Muller Method [34], PBUC+ Hamiltonian Economic Dispatch (HED) [35]. The performance of BGWO is

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compared to other approaches for $ = 0.005 and () = 0.1 ∗ #() The total profit achieved using BGWO approach is $ 9074.19 for the PBUC (Method B) which is higher as compared to the Muller Method ($ 9030.5) [34], PBUC+HED ($ 8973.3) [35] and traditional UC ($4048.8) [35].

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Table 1: Test system 1 parameters

Unit

suvw t

sutx t

Unit1

600

100

450

400

100

200

50

Unit3

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Unit2

min up time (h)

vt ($/{)

3

3

10

500

0.002

400

3

3

8

300

0.0025

300

3

3

6

100

0.005

startup cost ($) min down time (h)

|t ($/}~{) t ($/}~{€ )

Table 2. Energy, reserve and price forecasts for test system 1

Hour

demand forecast (MW)

Reserve forecast ($)

Spot price forecast ($)

Reserve price forecast ($)

Hour

demand forecast (MW)

Reserve forecast ($)

Spot price forecast ($)

Reserve price forecast ($)

1

170

20

10.55

31.65

7

1100

100

11.3

33.9

2

250

25

10.35

31.05

8

800

80

10.65

31.95

3

400

40

9

27

9

650

65

10.35

31.05

ACCEPTED MANUSCRIPT 4

520

55

9.45

28.35

10

330

35

11.2

33.6

5

700

70

10

30

11

400

40

10.75

32.25

6

1050

95

11.25

33.75

12

550

55

10.6

31.8

Table 3. GENCO scheduling in energy and reserve markets for test system 1

Unit 1 0 0 0 0 0 0 0 0 0 0 0 0

Reserve (MW) Unit 2 Unit 3 0 20 0 0 0 0 0 0 0.0023 0 0 0 0 0 0 0 0 0 35 0 40 0 50 0

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Energy (MW) Unit 2 Unit 3 0 170 0 200 0 200 0 200 399.99 200 400 200 400 200 400 200 400 200 130 200 200 200 350 200

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Unit 1 0 0 0 0 0 0 0 0 0 0 0 0

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hour 1 2 3 4 5 6 7 8 9 10 11 12

5.2 Test System 2: 10 unit test system

Test system 2 is a 10 unit system with 24 hour scheduling time horizon. The cost parameters, operating

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characteristics and constraints of test system 2 are given in Table 4. The load demand, reserve forecasts and associated prices are presented in Table 5.

Table 4: Unit cost and operating characteristics of test system 2 Unit 2

Unit 3

Unit 4

Unit 5

Unit 6

Unit 7

Unit 8

Unit 9

Unit 10

455

130

130

162

80

85

55

55

55

150

20

20

25

20

25

10

10

10

1000

970

700

680

450

370

480

660

665

670

16.19

17.26

16.60

16.50

19.70

22.26

27.74

25.92

27.27

27.79

8

8

5

5

6

3

3

1

1

1

0.00048

0.00031

0.002

0.00211

0.00398

0.00712

0.00079

0.00413

0.00222

0.00173

Min down time (h)

8

8

5

5

6

3

3

1

1

1

Hot start cost ($)

4500

5000

550

560

900

170

260

30

30

30

Cold start cost ($)

9000

10,000

1100

1120

1800

340

520

60

60

60

Cold start hours (h)

5

5

4

4

4

2

0

0

0

0

Initial status(h)

8

8

-5

-5

-6

-3

-3

-1

-1

-1

 ' ($/ℎ)

E ($/ℎ) Min. up time (h) F ($/ℎD )

150

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455

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Unit 1

ACCEPTED MANUSCRIPT The proposed algorithm is tested for PBUC Method A, with $ = 0.05 and () = 5 ∗ #(),

for PBUC Method B: with $ = 0.005 and () = 0.1 ∗ #() [6]. The performance of the proposed

algorithm is compared with respect to many of the benchmark optimization techniques developed to solve

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PBUC problem for three cases. Case 1 corresponds to scheduling without reserve allocation. Scheduling of generators allocating energy and reserve corresponding to the forecasted energy and reserve at their respective prices is considered in Case 2. Whereas, Case 3 deal with traditional unit commitment in which

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10% of load is always considered as reserve. Table 5: System and Market Parameters Load (MW)

Reserve (MW)

Energy

Hour

Load (MW)

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Hour

Reserve (MW)

Price ($/MWh) 700

70

22.15

2

750

75

22

3

850

85

23.1

4

950

95

22.65

5

1000

100

23.25

6

1100

110

7

1150

115

8

1200

120

9

1300

130

10

1400

140

11

1450

12

1500

Price($/MWh)

13

1400

140

24.6

13

1300

130

24.5

13

1200

120

22.5

13

1100

110

22.3

13

1050

105

22.25

22.95

13

1000

100

22.05

22.5

13

1200

120

22.2

22.15

13

1400

140

22.65

22.8

13

1300

130

23.1

29.35

13

1100

110

22.95

145

30.15

13

900

90

22.75

150

31.65

13

800

80

22.55

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1

Energy

5.2.1 Case 1: Scheduling in energy market only In this case, the GENCO acts as a price taker and schedules the available units for the energy market only. Since, Case 1 does not have any provision for the reserve market participation, the payment method for reserve becomes irrelevant. The simulation of Case 1 using BGWO models is performed for 30 identical trials. The solution quality and best profit obtained using BGWO models is compared to various existing approaches and is presented in Table 6. It can be observed that, Model 1 is out performed by recent heuristic approaches as well as the other BGWO models presented in this paper. The same can be attributed

ACCEPTED MANUSCRIPT to the randomness introduced in various steps such as binary update stage of alpha, beta and delta wolves (28),(31) and (34) as well as the crossover function. On the other hand, Model 2 and Model 3 have shown a better performance compared to Model 1 with respect to the solution quality and best profit. The

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convergence characteristics of various BGWO models for average and best profit are presented in Figure 4. The commitment and scheduling for Case 1 of test system 2 is shown in Table A1.

Profit ($)

Improvement in profit ($)

TS-RP[11]

101086.00

TS-IRP[11]

103261.00

6416.16 4241.16

Muller Method [14]

103296.00

ACO [26]

103890.00

105549.00

1953.16

105873.80

1628.36

105873.00

1629.16

105878.00

1624.16

105942.00

1560.16

106850.69

651.47

104961.95

2540.21

107212.69

289.47

107502.16

0

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Approach

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Table 6 Performance comparison of BGWO Models to solve PBUC for Case 1

IPPD [27] NACO[15] VTS-PEPSO[21] PABC[23] PNACO[16] BFWA [37] BGWO Model 2

3612.16

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BGWO Model 3

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BGWO Model 1

4206.16

Figure 4. Convergence characteristics of BGWO Models for Case 1

ACCEPTED MANUSCRIPT 5.2.2 Case 2: Scheduling in energy and reserve markets- Payment method A In this case, GENCO schedule in both the energy and the reserve market is considered. The minimum generation constraints of the thermal units implicate the reserve participation of any thermal unit with energy market but not vice versa. This case is examined for two payment methods explained in earlier

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sections. The scheduling operation of Case 2-Method A is performed with $ = 0.05 and () = 5 ∗

#(). The performance comparison of the proposed BGWO models for Case 2-Method A with other

existing approaches is presented in Table 7. The proposed BGWO Model 3 performed better compared to

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Model 1 and Model 2. While, Models 1 and 2 performed better than other approaches, except BFSA,

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Model 3 has also outperformed the recent heuristic BFSA approach application to PBUC problem. The convergence characteristics (Figure 5) of the best profit occurrence shows the same order of merit among various proposed models as that of Case I. However, the average profit convergence characteristic of Model 1 is worse compared to Model 2 and 3whereas, the best profit characteristic of Model 1 is better compared to that of Model 2. The slow convergence of BGWO MODEL 1 can be attributed to the

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exploration of new solutions with randomness (mutations) in update process of alpha, beta and delta wolves (Eqs.28, 31, 34) in everyiteration and also the randomness introduced by the crossover operation (Eq. 36). Although the average fitness value of BGWO MODEL 2 is lower than that of BGWO MODEL 3,

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the convergence is faster compared to both BGWO MODEL 1 and BGWO MODEL 3. Therefore, BGWO

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MODEL 2 with tangent hyperbolic transformation provides a tradeoff between fitness value and speed of convergence. The commitment and generation schedules of GENCO in energy and reserve markets for test system 2, Case 2-Method A are given in Table A2. Table 7. Performance comparison of BGWO Test system 2-Case 2-Method A, NR: Not Reported Improvement w.r.t best profit ($)

Approach

Best profit ($)

Mean profit ($)

Worst profit ($)

Time (Sec.)

Std. Dev. (%)

GSLRANN[17]*

60330

-

-

-

-

Pl [36]

-

109825

NR

4.79

NR

-

DP [36]

NR

111230

NR

2671.39

NR

-

50878.3

ACCEPTED MANUSCRIPT -

NR

109525

NR

17.81

NR

GA [36]

109814

109525

109213

21.24

NR

1394.3

BABC [36]

110286

110036

109776

47.52

NR

922.3

BFSA [36]

110908

110664

110412

14.1

NR

300.3

BGWO Model 2

110551.86

108773.196

107368.131

35.47

1.19366791

656.44

BGWO Model 1

110225.59

109484.083

108702.691

28.73

0.400143305

982.71

BGWO Model 3

111208.3

111019.723

110872.876

37.86

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LR [36]

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0.051363463

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Figure 5. Convergence characteristics of BGWO models for Payment Method B (case 2) of test system 2

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5.2.3 Case 2: Scheduling in energy and reserve markets- Payment method B In this payment method, the amount of capacity allocated or scheduled in reserve market will receive both capacity and energy payments depending upon the probability of deployment ($). The simulation trials are

performed with $ = 0.005 and () = 0.01 ∗ #(). The convergence characteristics of average and best profit obtained using BGWO models are plotted in Figure 6. It can be observed that, Model 3

performed better compared to the other models. Though the best cost convergence of Model 3 is better compared to Model 2, the average fitness value corresponding to Model 2 is superior compared to both

0

ACCEPTED MANUSCRIPT Model 1 and Model 3. The commitment and schedules of GENCO in reserve and energy corresponding to best fit are presented in Table A3. The performance comparison of proposed BGWO models to solve PBUC problem are compared to existing approaches and are presented in Table 8. From the same, it can be

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observed that the solution quality of BGWO Model 3 is superior compared to Model 2 and Model 1. Also, BGWO-Model 3 performed better compared to approaches like NACO [15], TS-EPSO [19], LR [8], GA [3], LR-IWO [18], Hybrid LR-EP [19], PSO [13], LR-GA [10], PSO-DP [20], ICA [25], MAS [24] etc., which is demonstrated by the superior solution quality. The better performance of BGWO Model 3

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compared to other two models in terms of best and average solution quality can be attributed to the fact

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that, the exploitation (reduced randomness or mutation) at higher iterations is superior compared to BGWO Model 1 (high exploration rate due to crossover functionality) and BGWO Model 2 (low exploitation rate contributed by tangent hyperbolic function). The modified sigmoidal function considered in BGWO Model 3 attains a better balance between exploration rate (during initial iterations) and exploitation rate (during end iterations). While, the improvement in solution quality of PBUC problem for BGWO Model 3 with

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respect to other binary variants in the literature can be attributed to both the efficient search methodology

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of GWO as well as the balanced exploitation and exploration properties of the modified sigmoidal function.

ACCEPTED MANUSCRIPT Figure 6. Convergence characteristics of BGWO models for Method B (case 2) of test system 2 Table 8. Performance comparison of proposed BGWO models for Case 2 (payment Method B) of test system 2 Improvement in profit ($)

Muller Method[14]

103280.00

5467.42

GSLRANN [17]*

59959.416

48788.004

NACO[15]

105565.00

3182.42

TS-EPSO[19 ]

106878.00

PNACO [16]*

107045.35

LR-IWO [18]*

107789.27

LR[8]

107870.05

GA[3]

107885.80

Hybrid LR-EP [9]

107,838.57

PSO [13]

107,838.57

LR-PSO [22]

107,852.32

LRGA [10]

107920.80

PSO-DP[20]*

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Profit ($)

1869.42 1702.07 958.15 877.37 861.62 908.85

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Method

908.85 895.1

826.62

108044.8125

702.6075

107,682.00

1065.42

107,715.00

1032.42

108083.91

663.51

BFWA [37]

108357.88

389.54

BGWO Model 1

107379.213

1368.207

BGWO Model 2

108408.556

338.864

ICA [25] CICA 3–Logistic map [25]

BGWO Model 3

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MAS [24]*

108747.42

0

* Verified using the schedules reported in articles

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5.2.4 Statistical analysis

To demonstrate the performance and statistical significance of the proposed approaches, BGWO Model 1,

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BGWO Model 2 and BGWO Model 3 are statistically compared within themselves as well as with other binary approaches using standard statistical tests. The other binary approaches considered for the comparison are Binary Particle Swarm Optimization (BPSO), Quantum Inspired Binary Particle Swarm Optimization (QBPSO), and Binary Fireworks Algorithm (BFWA). It is to be noted that the independent samples required for the tests are obtained by running identical trials of the above mentioned approaches with same number of population, iteration count etc. This is to ensure fair comparison of the algorithms to the maximum possible extent. The tests considered for assessing the performance of the proposed models

ACCEPTED MANUSCRIPT are Friedman Test and Wilcoxon Test [38], which are widely used to perform statistical analysis for similar problems such as unit commitment (UC) [39, 40]. Table 9. Friedman test results at 0.05 significance level w.r.t solution quality for proposed BGWO variants and other

Method

Rank P-value 1

Case 2-Method A P-value 2

Rank

5.93

6

QBPSO

4.8

4.26

BFWA

2.53

4.6 6.32E-13

2.11E-06

P-value 2

Rank

P-value 1 P-value 2

5.93

4.267

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BPSO

P-value 1

Case 2- Method B

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Case 1

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binary variants for different test cases

1.80E-13

2.8

1.12E-05

1.24E-12 1.28E-05

2.66

4.667

BGWO Model 2 1.2

2.46

1.53

BGWO Model 3 2.4

1

1.8

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BGWO Model 1 4.13

The Friedman test was conducted first among the BGWO variants themselves and other approaches to compare the solution quality. The tests are conducted using 30 independent samples obtained through

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identical simulation runs of all the approaches considered. The test statistics for three test conditions (Case 1, Case 2-Method A, Case 2- Method B) are reported in Table 9 for comparison. The P-value 1 reported in

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Table 9 is estimated for proposed BGWO variants as well as other binary variants,while, P-value 2 reported in Table 9 corresponds to the samples obtained for BGWO variants only. It can be observed that for all the test cases, the differences between the proposed BGWO are significantly different among themselves as well as other binary variants (under 0.05 significance level). The high value of p-value 2 compared to pvalue 1 shows the significant differences of higher degree between BGWO models and other models when compared to significant differences between BGWO themselves. The overall rank estimation through Friedman test across all the test cases for the proposed BGWO and other binary approaches is presented in

ACCEPTED MANUSCRIPT Table 10. The same reflects the superior performance of all the BGWO approaches except BGWO Model 1 with respect to other binary approaches used in solving the PBUC problem. It is observed that only BFWA approach attained better rank compared to one of the proposed BGWO approaches i.e., BGWO Model 1.

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The deterioration of BGWO Model 1 can be attributed to the randomness introduced in crossover operation as explained in (Eq. 36). The comparison of various binary approaches and proposed BGWO variants with respect to solution quality is presented in Figure 7. It can be observed from the comparison that, BGWO Model 1 performed better compared to BPSO and QBPSO but not BFWA for all the tests cases. While, for

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Case 2-Method A, all the proposed approaches including BGWO Model 1 performed superior compared to

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other binary approaches with respect to best and mean cost. In addition, the BGWO Model 2 and BGWO Model 3 of the proposed approaches performed better compared to all other approaches with respect to solution quality.

Table 10. Overall rankings of BGWO and other binary approaches using Friedman test across all the test cases

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Approach

Over all rank 17.86667

QBPSO

13.33333

BFWA

9.933333

BGWO Model 1

11.46667

BGWO Model 2

5.2

BGWO Model 3

5.2

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BPSO

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ACCEPTED MANUSCRIPT

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Figure 7. Comparison of BGWO variants and other binary approaches with respect to profit under different test cases (a) Case 1 (b) Case 2-Method A (c) Case 2-Method B

It can be observed from the overall rank estimation that, both BGWO Model 2 and BGWO Model

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3, outperform other approaches including BGWO Model 1. Also both performed equally well with respect to solution quality across different test cases. Therefore, to investigate the superiority of BGWO Model 2 and BGWO Model 3, the Wilcoxon test, a pairwise comparison based statistical test is performed between BGWO Model 2 and BGWO Model 3. The results of Wilcoxon test for various test cases are reported in Table 11. The p-values of the comparison confirms the statistical superiority of BGWO Model 3 over BGWO Model 2 at 0.05 significance level, except Case 2-Method B. It is also evident from the signed

ACCEPTED MANUSCRIPT ranks of Case 1 and Case 2-Method A that, BGWO Model 3 outperforms BGWO Model 2 and is statistically superior to BGWO Model 2. Table 11. Wilcoxon signed rank test results for different test cases with pairwise comparisons between BGWO

Case 1

Case 2-Method A

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Model 2 and BGWO Model 3 for test system 2 Case 2-Method B

R-

P-value

R+

R-

P-value

119

1

0.0026

120

0

6.10E-05

R+

R-

P-value

36

84

0.5614

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R+

6. Conclusion and Future Scope

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This paper presents a successful implementation of new binary grey wolf optimization (BGWO) models for solving the PBUC problem. The BGWO models are used to optimize energy and reserve market allocations from available capacity. Two cases of market participation with and without reserve market participation are examined for various payment methods for reserve in later case. The simulation results confirm the effectiveness of BGWO-Model 3 compared to the other two BGWO models and other established models

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for various test cases. Model 1 is comparatively less effective than Model 2 and Model 3. While Model 2 and Model 3 of the proposed BGWO showed superior performance compared to other approaches. However, the performance of the proposed BGWO models differs from one to another. The average profit

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convergence of Model 2 at times outperforms Model 3 average profit convergence characteristic. Though,

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Model 3 resulted in highest best profit in all the cases. The superior performance of BGWO models as compared to existing approaches with respect to solution quality and statistical significance is also backed by the statistical test results and analysis. The possible future extensions of the work can include hybridization of BGWO with real coded heuristic approaches for allocating power dispatch of committed units in the UC problem. Also, the proposed BGWO can be extended in future with real coded hybrid approaches to solve combined economic and emission dispatch (CEED) problem in power system operation planning.

ACCEPTED MANUSCRIPT Appendix Table A1: Commitment and scheduling of GENCO for Case 1 using BGWO (Model 3) for test system 2 U3 0 0 0 0 0 0 0 0 130 130 130 130 130 130 130 0 0 0 0 0 0 0 0 0

U4 0 0 0 0 130 130 130 130 130 130 130 130 130 130 130 130 0 0 0 0 0 0 0 0

U5 0 0 0 0 0 0 0 0 130 162 162 162 162 130 0 0 0 0 0 0 0 0 0 0

U6 0 0 0 0 0 0 0 0 0 68 80 80 0 0 0 0 0 0 0 0 0 0 0 0

U7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

U8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

U9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

U10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total ($)

Cost ($) 13683.1 14554.5 16301.9 17353.3 20072.8 20214.0 20214.0 20214.0 29084.0 29108.2 29048.0 29048.0 26851.6 26184.0 23105.8 20214.0 17353.3 17353.3 17353.3 17353.3 17353.3 17353.3 17177.9 15427.4 491976.1

Revenue($) 15505.0 16500.0 19635.0 20611.5 23250.0 23868.0 23400.0 23036.0 29640.0 41090.0 42571.8 44689.8 32767.2 31850.0 26325.0 23192.0 20247.5 20065.5 20202.0 20611.5 21021.0 20884.5 20475.0 18040.0 599478.3

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U2 245 295 395 455 415 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 445 345

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U1 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Profit ($) 1821.9 1945.5 3333.1 3258.2 3177.2 3654.0 3186.0 2822.0 556.0 11981.8 13523.8 15641.8 5915.6 5666.0 3219.2 2978.0 2894.2 2712.2 2848.7 3258.2 3667.7 3531.2 3297.1 2612.6 107502.2

Table A 2. Commitment and schedules of GENCO under Payment Method A for Case 2 of test system 2

0.0/0

0.0/0

0.0/0

0.0/0

13743.6

Revenue Profit ($) ($) 15892.6 2149.0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

14619.3

16912.5

2293.2

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

16353.7

19981.5

3627.8

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

17353.3

20611.5

3258.2

130.0/0

25.0/35

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

21570.9

23831.2

2260.4

130.0/0

60.0/102

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

21962.8

25830.2

3867.4

130.0/0

110.0/52

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

22930.9

26167.5

3236.6

130.0/0

130.0/0

30.0/120

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

25371.4

27244.5

1873.1

130.0/0

130.0/0

130.0/32

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

26215.7

29822.4

3606.7

130.0/0

130.0/0

162.0/0

68.0/12

0.0/0

0.0/0

0.0/0

0.0/0

29121.6

41178.0

12056.4

130.0/0

130.0/0

162.0/0

80.0/0

0.0/0

0.0/0

0.0/0

0.0/0

29048.0

42571.8

13523.8

455/0

130.0/0

130.0/0

162.0/0

80.0/0

0.0/0

0.0/0

0.0/0

0.0/0

29048.0

44689.8

15641.8

455/0

130.0/0

130.0/0

162.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

26851.6

32767.2

5915.6

455/0

130.0/0

130.0/0

130.0/32

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

26215.7

32046.0

5830.3

455/0

455/0

130.0/0

130.0/0

30.0/120

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

24271.4

27675.0

3403.6

455/0

440.0/15

0.0/0

130.0/0

25.0/90

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

20999.1

24000.4

3001.3

17

455/0

415.0/40

0.0/0

130.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

19547.3

22472.5

2925.2

18

455/0

455.0/0

0.0/0

130.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

20214.0

22932.0

2718.0

19

455/0

455/0

0.0/0

130.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

20214.0

23088.0

2874.0

20

455/0

455/0

0.0/0

130.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

20214.0

23556.0

3342.0

21

455/0

455/0

0.0/0

130.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

20214.0

24024.0

3810.0

22

455/0

455/0

0.0/0

130.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

20214.0

23868.0

3654.0

23

455/0

445.0/10/0

0.0/0

130.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

17186.5

20531.9

3345.3

24

455/0

345.0/80

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

0.0/0

15496.6

18491.0

2994.4

Unit 2

Unit 3

Unit 4

Unit 5

Unit 6

Unit 7

Unit 8

Unit 9

Unit 10

Cost ($)

1

455/0

245.0/70

0.0/0

0.0/0

0.0/0

0.0/0

2

455/0

295.0/75

0.0/0

3

455/0

395.0/60

0.0/0

4

455/0

455.0/0

0.0/0

5

455/0

390.0/65

0.0/0

6

455/0

455/0

0.0/0

7

455/0

455/0

0.0/0

8

455/0

455/0

9

455/0

455/0

10

455/0

455/0

11

455/0

455/0

12

455/0

13

455/0

14

455/0

15 16

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Unit 1

Total ($) 518977.3 630185.6 111208.3

ACCEPTED MANUSCRIPT Table A3. Commitment and schedules of GENCO under Payment Method B for Case 2 of test system 2 Unit 3 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 130.0/0 130.0/0 130.0/0 130.0/0 130.0/0 130.0/0 130.0/0 130.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0

Unit 4 0.0/0 0.0/0 0.0/0 0.0/0 130.0/0 130.0/0 130.0/0 130.0/0 130.0/0 130.0/0 130.0/0 130.0/0 130.0/0 130.0/0 130.0/0 130.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0

Unit 5 Unit 6 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 25.0/35.0 0.0/0 60.0/102.0 0.0/0 110.0/52.0 0.0/0 30.0/120.0 0.0/0 130.0/32.0 0.0/0 162.0/0 68.0/12.0 162.0/0 80.0/0 162.0/0 80.0/0 162.0/0 0.0/0 130.0/32.0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0

References

Unit 7 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0

Unit 8 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0

Unit 9 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0 0.0/0

Unit 10 Total cost ($) Revenue ($) 0.0/0 13689.18 15667.03 0.0/0 14560.98 16672.4 0.0/0 16307.07 19779.8 0.0/0 17353.3 20611.5 0.0/0 21489.11 23492.9 0.0/0 21870.54 25489.6 0.0/0 22884.29 25997.2 0.0/0 25262.45 26857.7 0.0/0 26187.19 29716.2 0.0/0 29109.55 41126.8 0.0/0 29047.98 42571.8 0.0/0 29047.98 44689.8 0.0/0 26851.61 32767.2 0.0/0 26187.19 31931.9 0.0/0 23105.76 26325 0.0/0 20213.96 23192 0.0/0 17353.3 20247.5 0.0/0 17353.3 20065.5 0.0/0 17353.3 20202 0.0/0 17353.3 20611.5 0.0/0 17353.3 21021 0.0/0 17353.3 20884.5 0.0/0 17178.77 20498.77 0.0/0 15434.33 18228.52 Total ($) 499901 608648.5

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Unit 2 245.0/70.0 295.0/75.0 395.0/60.0 455.0/0 390.0/65.0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 445.0/10.0 345.0/80.0

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Unit 1 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0 455.0/0

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Hour 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Profit ($) 1977.8 2111.4 3472.7 3258.2 2003.8 3619.0 3112.9 1595.3 3529.0 12017.2 13523.8 15641.8 5915.5 5744.7 3219.2 2978.0 2894.2 2712.2 2848.7 3258.2 3667.7 3531.2 3320.001 2794.184 108747.4

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