Binary Grey Wolf Optimizer for large scale unit commitment problem

Swarm and Evolutionary Computation xxx (xxxx) xxx–xxx

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Swarm and Evolutionary Computation journal homepage: www.elsevier.com/locate/swevo

Binary Grey Wolf Optimizer for large scale unit commitment problem ⁎

Lokesh Kumar Panwara, , Srikanth Reddy Kb, Ashu Vermaa, B.K. Panigrahib, Rajesh Kumarc a b c

Centre for Energy Studies, IIT Delhi, India Department of Electrical Engineering, IIT Delhi, India Department of Electrical Engineering, MNIT Jaipur, India

A R T I C L E I N F O

A BS T RAC T

Keywords: Unit Commitment Problem Heuristics Binary Grey Wolf Optimizer (BGWO) Constrained Optimization Power system optimization

The unit commitment problem belongs to the class of complex large scale, hard bound and constrained optimization problem involving operational planning of power system generation assets. This paper presents a heuristic binary approach to solve unit commitment problem (UC). The proposed approach applies Binary Grey Wolf Optimizer (BGWO) to determine the commitment schedule of UC problem. The grey wolf optimizer belongs to the class of bio-inspired heuristic optimization approaches and mimics the hierarchical and hunting principles of grey wolves. The binarization of GWO is owing to the UC problem characteristic binary/discrete search space. The binary string representation of BGWO is analogous to the commitment and de-committed status of thermal units constrained by minimum up/down times. Two models of Binary Grey Wolf Optimizer are presented to solve UC problem. The first approach includes upfront binarization of wolf update process towards the global best solution (s) followed by crossover operation. While, the second approach estimates continuous valued update of wolves towards global best solution(s) followed by sigmoid transformation. The LambdaIteration method to solve the convex economic load dispatch (ELD) problem. The constraint handling is carried out using the heuristic adjustment procedure. The BGWO models are experimented extensively using various well known illustrations from literature. In addition, the numerical experiments are also carried out for different circumstances of power system operation. The solution quality of BGWO are compared to existing classical as well as heuristic approaches to solve UC problem. The simulation results demonstrate the superior performance of BGWO in solving UC problem for small, medium and large scale systems successfully compared to other well established heuristic and binary approaches.

1. Introduction The unit commitment problem comprises the efficient utilization of generation resources in power system operational planning. The UC problem is a cost minimization problem which is often expressed as optimization problem associated with various types of constraints with respect to system as well as operation of generation units. The UC problem is a complex optimization problem associated by many constraints like load, reserve balance constraints, power generation bounds, minimum up and down time constraints, ramp rate constraints etc. The complexity of UC problem is greatly affected by system dimension and constant thrust to better the solution quality is indispensable. The earliest methods to solve the UC problem included classical optimization methods like mixed integer linear programming (MILP) [1], Dynamic programming (DP) [2], Priority list approach (PL) [3], branch and bound approaches (BB) [4]. Some of other approaches include dynamic programming with Lagrangian relaxation (DPLR) [5], ⁎

extended Lagrangian relaxation (ELR) [5], extended priority list (EPL) [6], semidefinite Programming (SDP) [7] etc. The list of advantages of classical methods lies in their simplest forms of representation and application, fast convergence and integer solutions etc. However, suffer from major drawbacks with poor solution quality (PL approach), problems with system dimensionality (dynamic and linear programming), exponentially increasing execution time with system dimension (branch and bound) etc. The same resulted in origin of several nature/ bio inspired evolutionary and heuristic approaches. The evolutionary and heuristic approaches are developed by mimicking nature phenomenon. The same are adapted to solve UC problem successfully. Some of the heuristic approaches include genetic algorithm (GA) which functions on the principles of natural selection and biological evolution of offspring of every generation [8]. Whereas, approaches like particle swarm optimization (PSO) [9], ant colony optimization (ACO) mimics the social behaviour and coordination among the population [10]. In the similar lines of inspiration from nature, many other optimization approaches like

Corresponding author. E-mail address: [email protected] (L.K. Panwar).

http://dx.doi.org/10.1016/j.swevo.2017.08.002 Received 17 June 2016; Received in revised form 31 May 2017; Accepted 11 August 2017 2210-6502/ © 2017 Elsevier B.V. All rights reserved.

Please cite this article as: Panwar, L.K., Swarm and Evolutionary Computation (2017), http://dx.doi.org/10.1016/j.swevo.2017.08.002

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TiMU Tih, off

Nomenclature

N H i h Fci γih SUih Pih SUihot SUicold TiMD

Number of units Total number of scheduling hours Thermal unit index (i = 1,2,3…. N ) Scheduling hour index (h = 1,2,3…. H ) Fuel cost function of i th unit Status bit (0 or 1) of i th unit for h th hour Start-up cost of i th unit for h th hour Scheduled power of i th unit for h th hour Hot start-up cost of i th unit Cold start-up cost of i th unit Minimum down time of i th unit

Tih, on Pimin Pimax Pdh Rsph RiDR RiUR

Minimum up time of i th unit Consecutive hours of de-committed state of i th unit going into h th hour Consecutive hours of committed state of i th unit going into h th hour Minimum generation limit of i th unit Maximum generation limit of i th unit System load for h th hour Spinning reserve requirement for h th hour Ramp down rate of i th unit Ramp down rate of i th unit

2. Problem formulation

evolutionary programming (EP) [11], simulated annealing (SA) [12], shuffled leaping frog approach (SFLA) [13], imperialistic competition algorithm (ICA) [14], etc., are applied to solve UC problem. Later, hybrid approaches are developed integrating the expedient properties of classical and heuristic approaches to solve the UC problem. Some of them are Lagrangian relaxation genetic algorithm (LRGA) [15], Lagrangian relaxation particle swarm optimization (LRPSO) [16], IPPDTM [17], hybrid harmony search random search approach (HHSRSA) [18] are used to improve the UC problem solution quality. Recently, the principles of quantum computing viz. uncertainty, superposition and interference are successfully applied to UC problem through evolutionary approaches. The applicability of quantum evolutionary approaches [19] improved the exploration and exploitation of heuristic approaches at comparatively lower population size with respect to other evolutionary approaches used to solve UC problem. The harmony search algorithm with hybridization with random search has been used to solve UC problem [30]. Also, the hybrid approaches of nature inspired such as PSO-GWO are investigated with application to UC problem [22]. The overview and insights of some other recent nature inspired approaches for solving UC problem can be found in literature review presented in [47]. Recently, Seyedali Mirjalili [21] proposed a meta-heuristic approach named grey wolf optimizer (GWO) mimicking the specific hierarchical and hunting behaviour of grey wolves (Canis lupus). The earlier models of GWO are economic load dispatch problem in power system. Later, the GWO integrated with PSO is used to solve UC problem of different dimensions [22]. The GWO is also used to solve many other industrial/research problems successfully [23]. Recently, Binary Grey Wolf Optimizer is developed and successfully applied for optimal feature selection purpose [24]. Motivated by the successful application of GWO & BGWO to industrial and research problems, this paper presents a Binary Grey Wolf Optimizer application to solve complex, non-linear and constrained UC problem. The presented approach improves the solution quality of traditional GWO to solve UC problem efficiently. The rest of the paper is organized as follows. The UC problem formulation and associated bounds, constraints are explained in Section 2. The principles of real valued grey wolf optimizer and Binary Grey Wolf Optimizer (BGWO) are presented in Section 3. Section 4 develops and describes the BGWO-UC approach. Section 5 presents the test system, parametric analysis and computational results for different dimensions of the test system. Comparison of proposed approach with the existing benchmarking algorithms in solving UC problem is also presented in Section 5. The performance and statistical significance of proposed BGWO models is also demonstrated in Section 5 using statistical tests. Finally, Section 6 concludes the paper with contributions.

Leading into the solution procedure, the formulation of objective and constraints of the UC problems are explained before. 2.1. Objective function The objective function of UC problem is modelled as a minimization problem of total cost which constitutes of fuel cost, start-up and shut down costs. 2.1.1. Fuel cost All the committed thermal units incur fuel costs due to the minimum power generation limits and the committed units are dispatched economically so as to reduce the overall fuel cost yet satisfying the system, thermal unit constraints. The fuel cost is expressed as a quadratic equation given by, 2

Fci(Pih ) = ai + bi(Pih )+ci(Pih ) ∀ h ∈ H ; i ∈ N

(1) th

where ai , biandci are the fuel cost coefficients of i unit. 2.1.2. Start-up cost The objective function also includes the start-up cost which is incurred two the boiler temperature changes as a consequence of commitment and de-commitment events. When returning to the commitment status (γih = 1) from de-commitment state (γih = 0), the start cost depends on number of hours the unit is in de-committed state. If the unit is in de-committed state for more than or equal to cold start hours (Ticold ) after minimum off time, cold start-up (SUicold ) cost is associated with its commitment event. However, if the units is in decommitted state after minimum off time but for less duration then cold start hours, then the hot start-up cost SUihot is associated with the commitment event. Therefore, the start-up cost applicable for i th thermal unit during h th hour is given by, hot ⎧ ⎪ SU if TiMD ≤ Tioff ≤ TiMD + Ticold i SUih = ⎨ ∀ h ∈ H; i ∈ N ⎪ cold if Tioff ≥ TiMD + Ticold ⎩ SUi

(2)

A. Shutdown cost:In this paper, shut down costs are neglected which are often modelled as constant values per de-commitment status of the unit.

Fig. 1. Hierarchy of grey wolf pack [21].

2

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Fig. 2. Representation of population structure of UC problem with BGWO.

F) Initial status: The status of generators in the ultimate/hour of previous day/scheduling horizon before the beginning of current scheduling horizon will reflect/affect parameters such as start-up costs, minimum up/down constraints etc.

Therefore, the total objective function is given by,

∑∑

TC =

h

Fci(Pih )γih

+

SUih(1−γih )γih

i

∀ h ∈ H ; i ∈ N ; γih∈{0, 1}

(3) 3. Grey Wolf optimizer (GWO)

2.2. Constraints

3.1. Overview of GWO

A) Generation limits: The actual generation of the units in committed state should comply with the generation limits as given by,

γihP

min i

h

≤ P γih ≤γihP i

The grey wolf more often lives and hunts in pack with an average of 5–12 wolves and follows certain hierarchy. The top of the hierarchy is occupied by alpha wolf which are called as leaders of the pack ( Fig. 1). This alpha wolf dictates the remaining wolves in tasks like hunting, moving, sleeping place etc. The alpha may not be always the strongest of the pack physically but intellectually it is considered as strongest of the pack. Often, the remaining wolves show their obedience to decisions taken by alpha by holding their tails down. The next level of hierarchy after alpha wolf is occupied by beta. The beta necessarily a subordinate of alpha and reinforces alpha’s decisions among other lower level wolves. It helps alpha in making decisions and is also the next candidate for alpha upon the expiration of latter. The last level of the hierarchy consists of omega wolves with no subordinates but to subordinate to other wolves and report them. These are least ranked in the system and are allowed to eat the remaining food that is left after all the other class of wolves are finished. If any of the wolf in the pack does not belong to either alpha, beta or omega, then it is called as delta wolf. The GWO employs particularly the hunting behaviour of the wolves which includes the following series of actions:

max

(4)

i

B) Load balance constraints: The system demand-supply balance constraint given by,

∑ Pihγih = Pdh;∀h ∈ H ; i ∈ N i

(5)

C) Spinning reserve: The hourly additional online capacity can be summed up as follows.

∑ γihPi

max

> Pdh + Rsph;∀h ∈ H ; i ∈ N

i

(6)

D) Minimum up/down time constraints: The time that should elapse between commitment and de-commitment events of units is predefined based on the reliability and satisfactory performance of particular unit. MU ⎧ 1, if 1 ≤ Tion , h −1 < Ti ⎪ MD γi = ⎨ 0, if 1 ≤ Tioff , h −1 < Ti ⎪ otherwise ⎩ 0 or 1,

• • •

h

(7)

As the initial step, they track, chase and approach the prey. Followed by approaching the pray, wolves encircle it and harass it Attacking the stationary prey after encircling it

3.2. Continuous valued GWO E) Ramp up/down rates: The ramp un/down rate constraints are given by,

γihP

h, min i

h


i

< γihP

(8)

The mathematical model of wolf behaviour involves ranking of solution quality based on the hierarchy of the wolves. The first, second and third best solutions are designated asalpha(α ), beta ( β ) and delta (δ ) respectively. The wolf position is updated by using the distance from the updated position of prey (as decided by best three solutions of α , β andδ ) of the wolves as follows.

(9)

→ ⎯ ⎯→ → ⎯ → ⎯ D = C . Xp (t ) − X (t )

(10)

→ ⎯ → ⎯ ⎯→ ⎯ → X (t +1) = Xp(t ) − A . D

(11)

h, max i

Where,

Pih, min = max (Pimin, Pih −1 − RiDR ) Pih, max = min(Pimax , Pih −1 + RiUR )

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Main Algorithm

Heuristic Adjustment

Start

Take the wolf position from main program

Initialization of the all values Initialize the all parameters of thermal units

k=1

Initialize the position of ‘NP’ wolves in search space

t=1

Calculate the fitness of each position using Economic Load Dispatch (ELD)

If t<=H

Return

Yes

Stop

Find the Up/Down time violation and repair using Up/Down time repair algorithm t=t+1

No

Heuristic Adjustment

Find the values of alpha (α), beta (β) and delta (δ) wolf

k<=MaxIter

No

Check the reserve and load constraint if violated then repair using spinning reserve repair algorithm

Yes

Update the position of wolfs using equation (24/26)

Run Unit De-commitment algorithm to remove the excessive reserve available

Generation of the Gaussian spark

Calculate the fitness of each position using Economic Load Dispatch (ELD)

Find the values of alpha (α), beta (β) and delta (δ) wolf

k=k+1

Fig. 3. Flowchart of UC problem solution using BGWO.

⎯ → ⎯ → Where the iteration number is denoted by t , A and C are coefficients, ⎯ → ⎯→ Xp denotes the vector of prey position, and X represents the vector of ⎯ → ⎯ → grey wolf position around prey. The A and C can be estimated as follows: ⎯ → A = 2→ a⎯ . r1⃗ − → a⎯

3.3. Binary Grey Wolf Optimizer (BGWO)

(12)

In the continuous valued grey wolf optimizer, the wolves update their position to a real value in the potential search space bounded by problem constraints. However, for certain problems like UC, the variables and search space is effectively limited to binary values [0, 1]. Therefore, mapping the real valued wolf position update to a binary value is an essential procedure The same can be realized through different approaches at different phases of optimization process. The two approaches employed in this study are explained as follows:

(13)

A) Binarization- Model 1 (BGWO1)

⎯ → ⎯r C = 2→ 2 ⎯r ∈[0,1] r1⃗ , → 2

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Fig. 4. Algorithm for minimum up/down time constraint.

This model employs a crossover strategy which is stochastic in nature. The wolves update their position according to the following equation.

ϒi k+1 = Crossover (γ1, γ2, γ3),

Fig. 5. Algorithm for reserve constraint satisfaction.

(14) Where, γβD represents the vector of β wolf position in D dimension

Where, the ϒi k+1 = Crossover (γ1, γ2, γ3), represents a suitable crossover operation among γ1, γ2, γ3 which are binary values effected by the movement of wolves in the direction of best three positions of wolves namely alpha(α ), beta(β ) and delta(δ ) respectively. The update process of binary vectors γ1, γ2, γ3 is determined by the following transformations.

⎧ 1 if (γ D + q D ) ≥ 1 α α γ1D = ⎨ ⎩0 otherwise

and qβD denotes the binary step given by,

⎧ 1 if p D ≥ r β qβD = ⎨ ⎩ 0 otherwise ⎪

⎪

(15)

pβD =

γαD

Where, represents the vector of α wolf position in D dimension and qαD denotes the binary step given by,

⎧ 1 if p D ≥ r α qαD = ⎨ ⎩ 0 otherwise

1 1 + exp( − 10*(A1D DβD − 0.5)

A1D ,

(20)

DβD

belongs to continuous valued grey wolf optimizer Where, and are estimated from (12), (10) for D dimension The update operation of γ3D is also carried out in the similar fashion as given by,

(16)

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

1 pαD = 1 + exp( − 10*(A1D DαD − 0.5)

(19)

Where, r is uniformly distributed random number ∈[0, 1] and pβD represents the continuous update step in D dimension estimated using sigmoid transformation as given by,

⎧ 1 if (γ D + q D ) ≥ 1 δ δ γ3D = ⎨ ⎩0 otherwise

(21)

γδD

represents the vector of δ wolf position in D dimension where, and qδD denotes the binary step given by,

(17)

Where, A1D , DαD belongs to continuous valued grey wolf optimizer and are estimated from (12), (10) for D dimension Similarly, γ2D can be updated as follows,

⎧ 1 if p D ≥ r δ qδD = ⎨ ⎩ 0 otherwise

⎧ 1 if (γ D + q D ) ≥ 1 β β γ1D = ⎨ ⎩0 otherwise

where, r is uniformly distributed random number ∈[0, 1] and pδD represents the continuous update step in D dimension estimated using sigmoid transformation as given by,

⎪

⎪

(18) 5

(22)

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2248500

600

2248000 500

400

2247000 2246500

300

2246000 200

Mean time (sec.)

Total cost ($)

2247500

2245500 Mean average cost ($)

2245000 2244500

100

0 5

10

15

20

25

30

35

40

45

50

55

Population size Fig. 7. Variation of total cost and execution time with population size.

The second model employs a direct sigmoid transformation of updated continuous valued function to arrive at binary values as given by,

⎧ 1 if S{X (k + 1)}> r γDk+1 = ⎨ ⎩0 otherwise

(25)

Where, r is the 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 (ϒ(k + 1)) = 1+

⎧⎛ X + X + X ⎞ ⎫ −10⎨⎜ 1 2 3 ⎟ −0.5⎬ e ⎩⎝ 3 ⎠ ⎭

(26)

where, X1, X2 , X3 are real valued position updates of wolf position with respect to the positions of best three wolves (α , β , δ ) of the pack. Fig. 6. Algorithm for de-commitment procedure.

1 pδD = 1 + exp( − 10*(A1D DδD − 0.5)

4. BGWO Implementation to UC problem The BGWO in this paper is used to find the optimal commitment schedule of thermal units. The power generation scheduling among the committed units through economic load dispatch is evaluated using Lambda-iteration technique. The generalized representation of UC problem is shown in Fig. 2.

(23)

where, A1D , DδD belongs to continuous valued grey wolf optimizer and are estimated from (12), (10) for D dimension. Upon the estimation of γ1D, γ2D, γ3D the wolf position update operation through cross over operation is given by, 1 ⎧γ D if r < 3 ⎪1 ⎪ γ D = ⎨ γ2D if 1 ≤ r ≤ 2 3 3 ⎪ ⎪ γ D otherwise ⎩3

A) Representation of Binary variables of UC Problem:The commitment status of i th generator of wth wolf during t th hour of k th iteration is represented byγit,,wk . Thus, the commitment of i th generator for t th is confirmed by assigning “1” and vice versa for de-commitment. The BGWO is adapted to UC problem by assigning each wolf with a dimension of N-by-H matrix. Where, N and H represent the number of units and number of scheduling hours respectively. Hence, the status bits of wth wolf in a population of NP range from γ1,1,wk to γNH,,wk for k th iteration. The commitment matrix of all units of

(24)

where, γ1D, γ2D, γ3D represent the first, second and third best binary values in dimension D, γ D denotes the output of dimension D, r is a uniformly distributed random number over range [0.1]. B) Binarization-Model 2 (BGWO2) Table 1 Effect of population size on operational cost and mean computational time. Population

Best cost ($)

Mean cost ($)

Worst cost ($)

Mean time

Standard deviation (%)

5 10 15 20 25 30 35 40 45 50 55

2,247,472.885 2,246,561.534 2,246,305.835 2,246,155.426 2,246,034.335 2,245,795.973 2,245,462.311 2,245,366.781 2,245,195.54 2,245,188 2,245,179

2,248,158.293 2,247,465.041 2,246,998.242 2,246,648.633 2,246,451.574 2,246,274.076 2,246,126.048 2,246,017.991 2,245,938.252 2,245,911 2,245,901

2,248,966.053 2,247,922.831 2,247,706.541 2,247,119.906 2,247,217.468 2,246,816.112 2,247,113.193 2,246,896.702 2,246,883.871 2,246,872 2,246,861

47.62476968 95.44879995 144.3136118 191.1416483 239.9412435 283.8151838 334.2757163 384.0085725 430.2869865 479.0560583 527.486265

0.017447527 0.019286349 0.014819079 0.012964035 0.015926715 0.011029745 0.019884921 0.019624868 0.02254337 0.022050963 0.021294126

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Table 2 Unit characteristics of 4 unit system.

Pimax

Unit

1 2 3 4

Pimin (MW)

a

b

c

($/MWh)

($/MW2h)

TiMU (h)

SUihot ($)

SUicold ($)

Initial status

($/h)

TiMD (h)

Cs

(MW)

(h)

(h)

300 250 80 60

75 60 25 20

648.74 585.62 213 252

16.83 16.95 20.74 23.6

0.0021 0.0042 0.0018 0.0034

5 5 4 1

4 3 2 1

500 170 150 0

1100 400 350 0.02

5 5 4 0

8 8 −5 −6

⎯→ ⎯ ⎯→ ⎯→ ⎯ ⎯→ ⎯ ⎯ X2 = | Xβ − A2 . Dβ |

Table 3 Commitment and generation schedule of 4 unit system. Commitment Schedule

Generation schedule

Hour

U1

U2

U3

U4

U1

U2

U3

U4

1 2 3 4 5 6 7 8

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

0 1 1 1 0 0 0 0

0 0 0 0 0 0 0 0

300 300 300 300 276.1901 196.1901 202.8567 300

149.9991 204.9991 250 214.9991 123.8093 83.8093 87.1426 199.9991

0 25 49.9992 25 0 0 0 0

0 0 0 0 0 0 0 0

⎯→ ⎯ ⎯→ ⎯→ ⎯ ⎯→ ⎯ ⎯ X3 = | Xδ − A3 . Dδ | ⎯⎯⎯→ ⎯→ ⎯ ⎯→ ⎯ where, Xα , Xβ , Xδ represents the first three best solutions, ⎯→ ⎯ ⎯→ ⎯ ⎯→ ⎯ ⎯→ ⎯ ⎯→ ⎯ ⎯→ ⎯ A1 , A2 , A3 are estimated using (12) Dα , Dβ , Dδ are estimated using (10). The binary position update of grey wolf is carried out using the real valued updates in (26) and is given by,

γit, ,wk +1

⎧ ⎧ ⎯⎯⎯→ ⎯⎯⎯→ ⎯⎯⎯→ ⎫ ⎪ 1 if S ⎨ Xα + Xβ + Xδ ⎬> R 3 =⎨ ⎩ ⎭ ⎪ ⎩0 otherwise

(29)

where, R is a uniformly distributed random number between 0 and ⎧ ⎯⎯⎯X→α + ⎯⎯⎯X→β + ⎯⎯⎯X→δ ⎫ ⎬ is the sigmoid transformation of real valued wolf 1, S ⎨ 3 ⎩ ⎭ position as given by,

wth wolf over 24 scheduling horizon during k th iteration is denoted byΥkw . The conventional numerical techniques can be used to solve ED problem as the fuel cost is approximately represented by a quadratic equation. The detailed procedure for deciding ON/OFF schedule of units is explained in preceding sections. B) Binary grey wolf position initialization: The initialization ((k = 0) ) of wolves is assigned on random fashion to start with. A real valued random number Nrsr is generated within the range of sigmoid function considered. The sigmoid transformation is applied to the random number to arrive at probability/value in the range [0, 1]. Another random number Nrur is generator with is uniformly distributed between 0 and 1. Thereafter, the initial binary valued position of the grey wolves are updated as follows. ⎧ 0 S (N sr ) > N ur r r γit,,1w = ⎨ ⎩ 1 otherwise

(28)

⎯→ ⎯ S{ X (k + 1)} =

1 1+

⎧ ⎫ ⎛ ⎯⎯⎯⎯→ ⎯⎯⎯→ ⎯⎯⎯→ ⎞ ⎪ Xα + Xβ + Xδ ⎪ ⎟ −0.5⎬ −10⎨⎜⎜ ⎟ 3 ⎪ ⎪ ⎠ ⎭ e ⎩⎝

(30)

D) Binary valued solution for UC schedule: The commitment and decommitment status of units are represented by set of solutions ϒ kw ϒ(k ) = {ϒ1k , ϒ2k ...... ϒ kw} of which each set consists of N-by-H matrix. The binary variables observed in BGWO can be mapped to UC schedule ϒ kw as given by,

⎡ γ1, k γ 2, k ⎢ 1, w 1, w ⎢ 1, k 2, k ϒ kw = ⎢ γ2, w γ2, w … … ⎢ 1, k 2, k ⎣⎢ γN , w γN , w

(27)

For the next iteration, the positions of α , β , δ are assigned using first, second and third best fitness values obtained for initialized wolf potions. The same can be evaluated by executing Lambdaiteration procedure for the committed units for the first iteration. C) Binary grey wolf position update: The position of wolf in potential binary search space is updated based on the real valued update of α , β , δ wolves. The real valued position update of α , β , δ is given as follows.

… γ1,Hw, k ⎤ ⎥ … γ2,Hw, k ⎥ … …⎥ ⎥ … γNH,,wk ⎦⎥

(31)

E) Unit output continuous valued variables: The unit commitment schedules obtained in (31) above are supplied to Lambda-iteration technique to obtain 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 represents optimal generation schedules of committed units as follows,

⎯ ⎯→ ⎯→ ⎯ ⎯⎯⎯→ ⎯→ ⎯ X1 = | Xα − A1 . Dα |

Table 4 Performance comparison of proposed BGWO for 4 unit test system. Generation cost ($)

Iteration time (s)

Method

Best

Mean

Worst

Best

Mean

Worst

Change (%)

Improved Lagrangian Relaxation (ILR) [27] B. SMP [26] A.SMP [26] Lagrangian Relaxation and PSO (LRPSO) [27] Binary Differential Evolution (BDE) [28] Genetic Algorithm (GA) [29] Hybrid HS and Random Search Algorithm [30] BGWO

75,231.9 74,812 74,812 74,808 74,676 74,675 74,476 73,933.1

NA 74,877 74,877 NA NA NA 74,476 73,933.1

NA 75,166 75,166 NA NA NA 74,476 73,933.1

–

–

–

– – – 20.68 3.455

– – – 22.86 –

– – – 22.97 –

1.726 1.174 1.174 1.169 0.994 0.993 0.728 0

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Fig. 8. Convergence characteristics of different test systems for BGWO2 & QEA. (a) 10 unit case (b) 20 unit case (c) 40 unit case (d) 60 unit case (e) 80 unit case (f) 100 unit case.

⎡ p 1, k p 2, k ⎢ 1, w 1, w 2, k ⎢ 1, k Pwk = ⎢ p2, w p2, w … … ⎢ 1, k 2, k ⎣⎢ pN , w pN , w

… p1,Hw, k ⎤ ⎥ … p2,Hw, k ⎥ … … ⎥ ⎥ … pNH,,wk ⎦⎥

F) The proposed approach considers number of iterations as termination criteria. Therefore, the BGWO stops as soon as the iteration reaches pre-defined maximum number of iterations. Fig. 3 shows the flow chart of proposed BGWO approach for solving UC problem.

(32)

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MATLAB 2010b environment on system with i5 INTEL processor, 16 GB RAM, Microsoft operating system. The simulation results for proposed approach are presented and discussed for various test systems. Later the comparison of proposed approach with other benchmark approaches for solving UC problem is presented to demonstrate the effectiveness of BGWO to solve UC problem. Population size determination: The optimal population for carrying out the numerical experiments is evaluated using simulation studies and observations of time/solution quality as a function of the population size. The numerical observation against the population variation are summarized in Table 1. The simulations are performed for 40 unit test case as a trade-off between small (10 unit) and large (100 unit) scale systems. The total cost and execution time are inversely related to each other for variation in population (Fig. 7). The reduction of total cost with respect to population has reduced progressively. Whereas, the execution time witnessed a linear variation along with population. The optimal population of 30 is selected based on the saturation of cost reduction at higher population size yet resulting in higher execution time. Apart from population size, all the other parameters of the algorithm are evaluated along the execution. The range of parameter a in (12) is set to 0–2, which is adopted from literature upon the observation of its optimality among all the ranges under study.

Table 5 Numerical results of BGWO approaches for different test systems. Units

10

20

40

60

80

100

Approach

QEA BGWO1 BGWO2 QEA BGWO1 BGWO2 QEA BGWO1 BGWO2 QEA BGWO1 BGWO2 QEA BGWO1 BGWO2 QEA BGWO1 BGWO2

Cost ($) Best

Average

Worst

Std. Deviation

563,936.3 563,976.64 563,937.31 1,123,404 1,125,546.4 1,123,297 2,245,520 2,252,475.1 2,244,701 3,364,972 3,368,934.4 3,362,515.2 4,489,856 4,495,173.4 4,483,381.1 5,608,527 5,628,975 5,604,145.9

563,980.04 564,378.58 563,937.31 1,123,455.9 1,126,126.3 1,124,215 2,245,675.4 2,257,866.7 2,245,144.9 3,366,000.2 3,375,220.5 3,366,488.3 4,492,184.7 4,506,362.2 4,486,675.6 5,609,920.5 5,637,659 5,607,031.1

564,017.6 565,518.14 563,937.31 1,123,505 1,127,393.2 1,124,379.2 2,245,933 2,263,333 2,246,020.6 3,367,247 3,384,305.8 3,367,143.5 4,493,418 4,513,026.3 4,488,568.4 5,611,208 5,643,899.3 5,607,722.7

18.97852 0.1067645 0 36.9198 0.0557585 0.0016277 164.8325 0.1536076 0.0135283 927.0687 0.144654 0.0192798 956.229 0.1094866 0.0122751 1042.705 0.0813199 0.0104419

G) Constraint handling: The heuristic approach of handling constraints based on rule based mechanism is adopted in this work [19]. The possible violation of constraints like minimum up-down times during initialization as well as update procedures has to be handled appropriately. Therefore, the heuristic approach based techniques are used to avoid the occurrence of infeasible solutions, thereby improving the solution quality. Apart from up/down time constraints, excessive reserve allocation may also prove costly. Therefore, the proposed approach used unit rule based de-commitment algorithm to deal with reserve constraints.

5.1. Performance of BGWO for Test system 1 This test system consists of 4 thermal units, whose characteristics are presented in Table 2. The schedule horizon is 8 h [26]. The mean execution time of the system with 30 wolf population is 3.455 s. The commitment and generation schedules of 4 unit system using BGWO are produced in Table 3. The solution quality and execution time of 4 unit UC problem using BGWO is compared with other existing approaches (Table 4). The same demonstrates the superior performance of BGWO over existing approaches. The maximum and minimum improvements in solution quality due to BGWO compared to the approaches is 1.725% and 0.728% respectively.

4.2. Constraint repair The heuristic approach of handling constraints based on rule based mechanism is adopted in this work [19]. In the process of random initialization and update process of evolutionary algorithms like BGWO, often the state transition of units may violate minimum-up down time constraints.

5.2. Performance of BGWO for Test system 2 The total cost convergence characteristics of BGWO for different test systems are shown in Fig. 8. The comparison of convergence of BGWO is made with existing QEA approach and the results shows considerable improvement in solution quality at comparable convergence speed for lower systems. The numerical results for both the models are presented in Table 5. The variation in best cost attainted in different rails of BGWO for various test systems is shown in Fig. 9. The variation is plotted along with mean best cost and it can be observed that the deviation from mean value is lower in 10 and 100 unit case compared to all other test systems. The difference between highest and lowest best costs among different trails is shown in Fig. 10. It can be observed that, the difference increased as a function of test system size.

A) Minimum up/down constraints: The commitment and De-commitment event of unit must abide the up/down constraints of the same. This paper uses rule based heuristic adjustment process to tackle the up/down time violation as shown in Fig. 4. B) Spinning reserve and load satisfaction repair: The load and spinning reserve satisfaction constraints are enforced to guarantee the reliability of the system. In case of these constraint violation, the de-committed units are committed until the demand and spinning reserve constraints are satisfied. The detailed procedure of load and reserve constraint repair strategy is explained in Fig. 5 as follows. C) De-commitment algorithm under excessive spinning reserve: The satisfaction of up/down time constraint and spinning reserve constraint may end up in committing extra thermal units thereby resulting in residual spinning reserve which may increase the operational cost unnecessarily. This can be avoided by implementing De-commitment algorithm as explained in Fig. 6.

5.2.1. Comparison of proposed approaches with various other approaches The comparison with other algorithms for various test systems from 10 to 100 units is presented in Table 4. The comparison of both the proposed approaches with other approaches reveals the effectiveness of BGWO with respect to GA [8], EP [11], SA [12], SFLA [13], GSA [38], GVNS [40], ICA [14], IPSO [36], QEA [20], HHSRSA [30], PSO-GWO [22] etc., for all test systems from 10 to 100 units. However, no particular inferences can be drawn from comparison for computational times owing to the differences in computational facilities of different approaches. The comparison of BGWO approaches with other approaches for small scale systems of test system 2 is shown in Table 6.

5. Numerical results and discussion The proposed BGWO is modelled to solve UC problem with test systems of different dimensions starting from 10 to 100 units for 24 h scheduling horizon [25]. The simulation studies are performed in 9

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Fig. 9. Variation of best cost among different trails (a) 10 unit case (b) 20 unit case (c) 40 unit case (d) 60 unit case (e) 80 unit case (f) 100 unit case.

10

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0.07

comparison of BGWO against other existing algorithms is presented in Table 11, which demonstrates the superior performance of BGWO to solve UC problem under different operating conditions.

0.06

Difference (%)

0.05

5.3. Statistical analysis 0.04

The statistical tests are often used to compare the performance and demonstrate the statistical significance of approaches for a given problem [48]. Statistical tests such as Friedman test, Wilcoxon test etc., are used in recent studies to compare the effectiveness of different approaches in solving UC problem [49]. The test ranks, test statistic and p-value (frequentist inference) for null hypothesis rejection are used as performance metrics/indices for comparing various approaches. This paper considers Friedman (aligned and non-aligned) ranks, Wilcoxon test and Quade test for performance comparison of proposed approaches with the existing binary approaches such as BPSO, QEA, QBPSO, BFWA etc. The independent solution samples required for performance comparison using these tests are obtained by running respective algorithms for same number of independent trails (30 trails each).

0.03 0.02 0.01 0 10

20

40

60

80

100

Units Fig. 10. Percentage difference between highest and lowest best cost.

The statistical analysis of mean and standard deviation of proposed approach confirms the effectiveness of BGWO2 over BGWO1 and other existing approaches. The commitment and economic dispatch schedule for 10 unit and 24 h horizon using proposed BGWO approach is shown in Table 7. The effect of system size on computational time is shown in Fig. 11. The quadratic progression of execution time with system size can is observed. The performance comparison of medium (40, 60 units) and large scale (80, 100 units) systems is presented in Tables 8, 9 respectively. The effectiveness of proposed BGWO approach for solving UC is also evaluated for different operating conditions i.e. 5% spinning reserve. The unit commitment schedule using BGWO under 5% spinning reserve condition is given in Table 10. The performance

5.3.1. Friedman test Friedman test provides a non-weighted and non-aligned ranking of comparison for solutions/independent samples of various approaches. The Friedman ranks (non-aligned) and p-values for BGWO and other approaches are presented in Table 12. The individual test system ranking and overall ranking shows the superiority of BGWO 2 over BGWO 1 and other binary, quantum approaches. The p-values in all three cases are well below 0.05 (significance level) demonstrates the statistical significance of proposed approaches when compared to other existing binary approaches. The increasing p-values from 10 unit test

Table 6 Comparison of BGWO with other approaches for small scale (10 and 20 unit) systems. 10 unit system

20 unit system

Approach

Best ($)

Average ($)

Worst ($)

Std. (%)

Avg. Time (Sec)

Best ($)

Average ($)

Worst ($)

Std. (%)

Avg. Time (Sec)

LR [8] GA [8] EP [11] LRGA [15] SA [12] ICGA [31] MA [32] IPPDTM [17] GRASP [33] LMBSI [34] DPLR [5] GACC [35] LRPSO [16] ELR [5] EPL [6] IPSO [36] BFWA [25] IBPSO [37] SFLA [13] ICA [14] IQEA [19] QEA [20] SDP [7] QBPSO [18] HHSRSA [30] PSO-GWO [22] RM [39] GVNS [40] hGADE/r1 [46] hGADE/cur1 [46] Enh-hGADE [46] BGWO1 BGWO2

565,825 565,825 564,551 564,800 565,828 566,404 566,686 563,977 565,825 563,977 564,049 563,977 565,869 563,977 563,977 563,954 563,977 563,977 564,769 563,938 563,977 563,938 563,938 563,977 563,937.68 565,210.2 563,977 563,938 563,938 563,959 563,938 563,976.6 563,937.3

565,825 – 565,352 564,800 565,988 566,404 566,787 563,977 565,825 563,977 564,049 564,791.5 565,869 563,977 563,977 564,162 564,018 564,155 564,769 563,938 563,977 563,969 563,938 563,977 563,965.3 – – – 564,044 564,088 563,997 564,378.58 563,937.3

565,825 570,032 566,231 564,800 566,260 566,404 567,022 563,977 565,825 563,977 564,049 565,606 565,869 563,977 563,977 564,579 564,855 565,312 564,769 563,938 563977 564,672 563,938 563,977 563,995.3 – – – 564,283 564,350 564,261 565,518.1 563,937.3

– 0.74 0.3 – 0.08 – 0.06 – – – – 0.0857 – – – 0.11 – 0.0253 – – 0 0.13 – 0 – – – – 0.06 0.07 0.06 0.106 0

– 221 100 518 3 7.4 61 0.52 17 7.3 108 85 42 4 – – 65.42 27

1,130,660 1,126,243 1,125,494 1,122,622 1,126,251 1,127,244 1,128,192 – 565,825 1,123,990 1,128,098 1,125,516 565,869 1,130,660 1,124,369 1,125,279 1,124,858 1,125,216 1,123,261 1,124,274 1123,890 1,123,607 1,124,357 1,123,297 1,124,889 – 1,123,990 1,123,297 1,123,386 1,123,426 1,123,386 1,125,546.4 1,123,297

1,130,660 – 1,127,257

1,130,660 1,132,059 1,129,793

– 0.52 0.38

1,127,955 – 1,128,213 – – – – 1,127,153 – – – – 1,124,941 1,125,448 – – 1,124,320 1,124,689 – 1,123,981 1,124,912.8 – – – 1,124,436 1,124,502 1,124,262 1,126126.3 1,124,215

1,129,112 – 1,128,403 – – – – 1,128,790 – – – 1,127,643 1,125,087 1,125,730 – – 1,124,504 1,125,715 – 1,124,294 1,124,951.5 – – – 1,125,045 1,125,076 1,124,939 1,127,393.2 1,124,379

0.25 – 0.02 – – – – 0.1004 – – – 0.21 – 0.0155 – – 0.05 0.19 – 0.09 – – – – 0.15 0.15 0.13 0.055 0.0016

– 733 340 1147 17 22.4 113 – 571 15.24 299 225 91 16 – – 106.03 55

15 19 25.41 18 16.83 – 1.15 0.23 24 24 26 64.19 66.15

11

42 28 63.94 50 – – 2.14 2.46 51 48 56 80.47 87.533

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Table 7 Unit commitment schedule and reserve available for 10 unit system using BGWO2. Hour

Unit1

Unit2

Unit3

Unit4

Unit5

Unit6

Unit7

Unit8

Unit9

Unit10

Load

Reserve

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

455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455

245 295 370 455 390 360 410 455 455 455 455 455 455 455 455 310 260 360 455 455 455 455 425 345

0 0 0 0 0 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 0 0 0

0 0 0 0 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 0 0 0

0 0 25 40 25 25 25 30 85 162 162 162 162 85 30 25 25 25 30 162 85 145 0 0

0 0 0 0 0 0 0 0 20 33 73 80 33 20 0 0 0 0 0 33 20 20 20 0

0 0 0 0 0 0 0 0 25 25 25 25 25 25 0 0 0 0 0 25 25 25 0 0

0 0 0 0 0 0 0 0 0 10 10 43 10 0 0 0 0 0 0 10 0 0 0 0

0 0 0 0 0 0 0 0 0 0 10 10 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0

700 750 850 950 1000 1100 1150 1200 1300 1400 1450 1500 1400 1300 1200 1050 1000 1100 1200 1400 1300 1100 900 800

210 160 222 122 202 232 182 132 197 152 157 162 152 197 132 282 332 232 132 152 197 137 90 110

900

5.3.3. Quade test Quade test can be used to for performance comparison among different approaches, including the block differences that go unidentified in the Friedman as well as Friedman Aligned ranks test, where all the algorithms are given/assigned with same importance. The algorithms/approaches are weighted in the Quade test based on block difference and can help in establishing the statistical significance of a particular algorithm within the group of algorithms considered for study [48]. The p-values and ranks of different approaches estimated using weighted ranks (using simple non-aligned ranks with exclusion of block differences) method are presented in Table 14. The ranks obtained using Quade test are comparable to the Friedman test (nonaligned) and the superior performance of BGWO 2 over other approaches is also reflected in Quade ranks. The p-values of Quade rank are considerably lower when compared to other two tests (Friedman and Friedman aligned ranks test). The same reveals the higher significant differences between BGWO and other approaches when block differences are included through weighted ranks.

800

Execution time (Sec.)

700 600 500 400 300 200 100 0 10

20

40

60

80

100

Untis Fig. 11. Time complexity of BGWO approach across system dimension.

system to 100 unit test system signifies a higher statistical significance of BGWO approaches for larger test systems compared to smaller test systems.

5.3.4. Wilcoxon pairwise compassion The Friedman, Friedman aligned ranks and Quade test provide the justification that proposed BGWO has significant differences with the existing approaches. However, the comparison of differences in the whole group of independent samples obtained for all the approaches may mask any hidden similarities between proposed approach and the existing approaches. Therefore, the pairwise comparison is performed on different approaches across various test systems as presented in Tables 15–17. In each comparison, the instances with no significance differences (at a significance level of 0.05) are highlighted. It can be observed that, except 10 unit systems (only BGWO 1), there exist significance differences between the proposed BGWO variants and other approaches. Also, the BGWO 2 which has shown superior performance when compared to all other approaches is significantly different (at 0.05 level of significance) with respect to each of the algorithm/approach for all the test systems.

5.3.2. Friedman aligned ranks test The simple/non-aligned ranks of Friedman test provides a fair intra-set comparison between various approaches. However, the same fails in considering the relative performance of each algorithm with respect to the average performance of all the approaches under consideration. The Friedman Aligned ranks test provides the comparison of any given approach against the average performance of all the approaches under study. The Friedman aligned ranks and corresponding statistical significance of BGWO against other approaches is presented in Table 13. Similar to the Friedman ranks, BGWO 2 achieved superior performance over BGWO 1 as well as other approaches. On the other hand, the p-values of Friedman aligned ranks test are lower compared to their Friedman (non-aligned) test counterparts. The same reveals a higher significance differences between proposed and existing approaches at larger scale with respect to the relative/mean performance.

6. Conclusion and future scope This paper presents two binary grey wolf (BGWO) models to solve UC problem. The objective of UC problem is formulated as a cost 12

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Table 8 Comparison of BGWO with other approaches for medium scale (40 and 60 unit) systems. 40 unit system

60 unit system

Approach

Best ($)

Average ($)

Worst ($)

Std. (%)

Avg. Time (Sec)

Best ($)

Average ($)

Worst ($)

Std. (%)

Avg. Time (Sec)

MA [32] LR [8] GA [8] EP [11] LRGA [15] SA [12] ICGA [31] IPPDTM [17] GRASP [33] LMBSI [34] DPLR [5] GACC [35] LRPSO [16] EPL [6] IPSO [36] BFWA [25] IBPSO [37] SFLA [13] ICA [34] IQEA [19] QEA [20] SDP [7] HHSRSA [30] hGADE/r1 [46] hGADE/cur1 [46] Enh-hGADE [46] BGWO1 BGWO2

2,249,589 2,258,503 2,251,911 2,249,093 2,242,178 2,250,063 2,254,123 2,247,162 2,259,340 2,243,708 2,256,195 2,249,715 2,251,116 2,246,508 2,248,163 2,248,228 2,248,581 2,246,005 2,247,078 2,245,151 2,245,557 2,243,328 2,248,508 2,243,724 2,243,522 2,243,522 2,252,475 2,244,701

2,249,589 2,258,503 – 2,252,612 2,242,178 2,252,125 2,254,123 2,247,162 2,259,340 2,243,708 2,256,195 2,253,270 2,251,116 2,246,508 – 2,248,572 2,248,875 2,246,005 2,247,078 2,246,026 2,246,728 – 2,248,653 2,245,582 2,245,321 2,245,020 2,257,866 2,245,145

2,249,589 2,258,503 2,259,706 2,256,085 2,242,178 2,254,539 2,254,123 2,247,162 2,259,340 2,243,708 2,256,195 2,256,824 2,251,116 2,246,508 2,252,117 2,248,645 2,249,302 2,246,005 2,247,078 2,246,701 2,248,296 – 2,248,757 2,247,130 2,246,540 2,246,487 2,263,333 2,246,021

0 0 0.35 0.31 0 0.2 0 0 0 0 0 0.08457 0 0 0.18 – 0.011561 0 0 0.07 0.12 – – 0.15 0.13 0.13 0.15360 0.00345

217 – 2697 1176 2165 88 58.3 6.49 1511 27 1200 614 213 – – 238.02 110 150 151 132 43 157.73 179.666 137 123 147 169.24 153.5

3,370,595 3,394,066 3,376,625 3,371,611 3,371,079 – 3,378108 3,366874 3,383,184 3,362,918 3,384,293 3,375,065 3,376,407 3,366,210 3,370,979 3,367,445 3,367,865 3,368,257 3,371,722 3,365,003 3,366,676 3,363,031 – 3,363,470 3,362,908 3,362,908 3,368,934 3362515

3,370,820 3,394,066 – 3,376,255 3,371,080 – 3,378,108 3,366,874 3,383,184 3,362,918 3,384,293 3,378,976 3,376,407 3,366,210 – 3,367,828 3,368,278 3,368,257 3,371,722 3,365,667 3,368,220 – – 3,365,587 3,364,841 3,364,538 3,375,221 3,366,488

3,371,272 3,394,066 3,384,252 3,381,012 3,371,081 – 3,378,108 3,366,874 3,383,184 3,362,918 3,384,293 3,382,886 3,376,407 3,366,210 3,379,125 3,367,974 3,368,779 3,368,257 3371722 3,366,223 3,372,007 – – 3,368,196 3,367,820 3,367,820 3,384,306 3,367,144

0 – 0.23 0.28 0 – 0 0 0 0 0 0 0 0 0.24 0 0.009708 0 0 0.04 0.16 – – 0.14 0.15 0.14 0.14465 0.00293

576 – 5840 2267 2414 – 117.3 11.39 2638 40 3199 1085 360 – – 422.29 172 280 366 273 54 260.7 – 277 307 326 281.2 268.2

Table 9 Comparison of BGWO with other approaches for large scale (80 and 100 unit) systems. 80 unit system

100 unit system

Approach

Best ($)

Average ($)

Worst ($)

Std. (%)

Avg. Time (Sec)

Best ($)

Average ($)

Worst ($)

Std. (%)

Avg. Time (Sec)

MA [32] LR [8] GA [8] EP [11] LRGA [15] SA [12] ICGA [5] IPPDTM [17] GRASP [33] LMBSI [34] DPLR [5] GACC [35] LRPSO [16] EPL [6] IPSO [36] BFWA [25] IBPSO [37] SFLA [13] ICA [34] IQEA [19] QEA [20] SDP [7] hGADE/r1 [46] hGADE/cur1 [46] Enh-hGADE [46] BGWO1 BGWO2

4,494,214 4,526,022 4,504,933 4,498,479 4,501,844 4,498,076 4,498,943 4,490,208 4,525,934 4,483,593 4,512,391 4,505,614 4,496,717 4,489,322 4,495,032 4,491,284 4,491,083 4,503,928 4,497,919 4,486,963 4,488,470 4,484,365 4,486,180 4,485,160 4,485,160 4,495,173 4,483,381

4,494,378 4,526,022 – 4,505,536 4,501,844 4,501,156 – 4,490,208 – 4,483,593 4,512,391 4,516,731 – 4,489,322 – 4,492,550 4,491,681 4,503,928 4,497,919 4,487,985 4,490,128 – 4,489,500 4,487,968 4,487,293 4,506,362 4,486,676

4,494,439 4,526,022 4,510,129 4,512,739 4,501,844 4,503,987 – 4,490,208 – 4,483,593 4,512,391 4,527,847 – 4,489,322 4,508,943 4,493,036 4,492,686 4,503,928 4,497,919 4,489,286 4,492,839 – 4,498,651 4,494,013 4,489,114 4,513,026 4,488,568

0 0 0.12 0.32 0 0.13 – 0 – 0 0 0.0947 – 0 0.31 – 0.0120 0 0 0.05 0.1 – 0.28 0.2 0.09 0.1094 0.0122

664 – 10,036 3584 3383 405 176 31.23 3308 54 8447 1975 543 – – 676.53 235 690 994 453 66 353 368 343 404 473.4 469.6

5,616,314 5,657,277 5,627,437 5,623,885 5,613,127 5,617,876 5,630,838 5,609,782

5,616,699 5,657,277 – 5,633,800 5,613,127 5,624,301 –

5,616,900 5,657,277 5,637,914 5,639,148 5,613,127 5,628,506 –

0.01 0 0.19 0.27 0 0.19 –

– 5,602,844 5,640,488 5,636,522 – 5,608,440 – 5,612,422 5,611,181 5,624,526 5,617,913 5,607,561 5,611,797 – 5,610,074 5,610,336 5,607,487 5,637,659 5,607,031

– 5,602,844 5,640,488 5,646,529 – 5,608,440 5,628,506 5,612,790 5,612,265 5,624,526 5,617,913 5,608,525 5,613,220 – 5,620,236 5,620,839 5,612,131 5,643,899 5,607,723

– 0 0 0.1012 – 0 0.24 – 0.01554 0 0 0.04 0.07 – 0.27 0.26 0.13 0.08131 0.01044

1338 – 15,733 6120 4045 696 342.2 46.55 4392 73 12,437 3547 730 – – 1043.47 295 1430 1376 710 80 392 397 451

13

5,602,844 5,640,488 5,626,514 5,608,440 5,619,284 5,610,954 5,610,293 5,624,526 5,617,913 5,606,022 5,609,550 5,602,538 5,604,787 5,606,075 5,604,787 5,628,975 5,604,146

836.54 822.23

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Table 10 Commitment and generation scheduling under 5% spinning reserve. Hour

U1

U2

U3

U4

U5

U6

U7

U8

U9

U10

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

455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455

244.9992 294.9993 394.9992 364.9992 389.9991 455 409.9992 455 455 455 455 455 455 455 455 309.9991 259.9993 359.9991 455 455 455 455 419.9992 344.9991

0 0 0 0 0 0 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 0 0 0

0 0 0 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 0 0 0

0 0 0 0 25 59.999 25 29.999 109.999 162 162 162 162 104.999 29.999 25 25 25 29.999 162 104.999 162 25 0

0 0 0 0 0 0 0 0 20 42.999 80 80 42.999 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 25 25 25 25 25 0 0 0 0 0 25 25 27.9991 0 0

0 0 0 0 0 0 0 0 0 0 12.999 52.999 0 0 0 0 0 0 0 42.999 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 10 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

natured problems in power systems such as profit based unit commitment (PBUC). Other directions of possible 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. Some other interesting future works may be listed as follows.

Table 11 Performance comparison of BGWO for 5% spinning reserve condition. Method

Best cost ($)

Average cost ($)

Worst cost ($)

BPSO [41] GA [41] APSO [42] BP [42] TSGB [43] IPSO [44] Hybrid PSO-SQP [45] B.SMP [26] Hybrid HS-Random Search algorithm [30] BGWO

565,804 570,781 561,586 565,450 560,263.92 558,114.8 568,032.3 558,844.76 557,905.643

566,992 574,280 – – – – – 558,937.24 558,267.2

567,251 576,791 – – – – – 559,155 558,682

556,840.099

558,026.296

558,358.3

•

•

minimization optimization problem associated with bound, equality and inequality constraints. The first of the two Binarization models uses binary update upfront, followed by crossover operation. Whereas the second one employs real valued update of wolves followed by sigmoid transformation for binary update. The two models are tested on various test systems operating under different conditions. The simulation results comparison with other existing approaches demonstrates the superiority of BGWO approaches tin solving UC problem efficiently. The proposed BGWO can be extended to solve other binary

•

The BGWO approach proposed in this paper has been tested for single objective function (cost minimization of UC problem). However, the same can be used to develop multi-objective problems involving UC framework [50]. The other objectives of UC problem such as emission reduction and reliability maximization can form multi-objective UC (MOUC) problem for future studies [51]. Apart from binary natured problems such as UC and MOUC, the real valued hybrid variants of GWO can be developed in future with application to real time problems across various fields of engineering [52,53]. The extension of real valued and combinational variants of the proposed BGWO and GWO frameworks for other multi-objective problems [54] may be considered for the future studies.

Table 12 Performance comparison of various approaches for different test systems using Friedman test. Approach

BPSO QEA QBPSO BFWA BGWO 1 BGWO 2

10 unit system

40 unit system

100 unit system

Overall rank

Rank

p-value

Rank

p-value

Rank

p-value

3.666667 5.733333 5.133333 2.6 2.666667 1.2

3.26E−12

3.866667 5.466667 5.533333 2.933333 2.2 1

4.17E−14

4 5.533333 5.466667 2.133333 2.866667 1

6.08E−14

14

11.53333 16.73333 16.13333 7.666667 7.733333 3.2

Swarm and Evolutionary Computation xxx (xxxx) xxx–xxx

L.K. Panwar et al.

Table 13 Performance comparison of various approaches for different test systems using Friedman aligned ranks test. Approach

10 unit system Rank

BPSO QEA QBPSO BFWA BGWO 1 BGWO 2

40 80.3333 68.6 24.4 41.3333 18.3333

40 unit system p-value

p-value

Rank

9.04E−13

100 unit system

51.8667 74.7333 76.2667 32 30.0667 8.0667

3.82E−16

Rank 53.1333 76.8667 74 29.1333 31.8667 8

Overall rank p-value

6.54E−16

145 231.9333 218.8667 85.5333 103.2667 34.4

Table 14 Performance comparison of various approaches for different test systems using Quade test. 10 unit system Rank BPSO QEA QBPSO BFWA BGWO 1 BGWO 2

3.75 5.9 4.975 2.641667 2.516667 1.216667

40 unit system p-value

1.67E−15

100 unit system p-value

Rank 3.883333 5.4375 5.5625 2.925 2.191667 1

0

Rank 4 5.591667 5.408333 2.133333 2.866667 1

Overall rank p-value

0

11.63333 16.92917 15.94583 7.7 7.575 3.216667

Table 15 Wilcoxon pairwise comparison for 10 unit system.

BPSO QEA QBPSO BFWA BGWO 1 BGWO 2

BPSO

QEA

QBPSO

BFWA

BGWO 1

BGWO 2

1 6.10E−05 0.000122 6.10E−05 0.488708 6.10E−05

6.10E−05 1 0.005371 6.10E−05 0.000183 6.10E−05

0.000122 0.005371 1 6.10E−05 6.10E−05 6.10E−05

6.10E−05 6.10E−05 6.10E−05 1 0.406067 6.10E−05

0.488708 0.000183 6.10E−05 0.406067 1 0.003906

6.10E−05 6.10E−05 6.10E−05 6.10E−05 0.003906 1

BPSO

QEA

QBPSO

BFWA

BGWO 1

BGWO 2

1 6.10E−05 6.10E−05 0.000183 0.000183 6.10E−05

6.10E−05 1 0.910156 6.10E−05 6.10E−05 6.10E−05

6.10E−05 0.910156 1 6.10E−05 6.10E−05 6.10E−05

0.000183 6.10E−05 6.10E−05 1 0.008362 6.10E−05

0.000183 6.10E−05 6.10E−05 0.008362 1 6.10E−05

6.10E−05 6.10E−05 6.10E−05 6.10E−05 6.10E−05 1

BPSO

QEA

QBPSO

BFWA

BGWO 1

BGWO 2

1 6.10E−05 6.10E−05 6.10E−05 6.10E−05 6.10E−05

6.10E−05 1 0.719727 6.10E−05 6.10E−05 6.10E−05

6.10E−05 0.719727 1 6.10E−05 6.10E−05 6.10E−05

6.10E−05 6.10E−05 6.10E−05 1 0.00116 6.10E−05

6.10E−05 6.10E−05 6.10E−05 0.00116 1 6.10E−05

6.10E−05 6.10E−05 6.10E−05 6.10E−05 6.10E−05 1

Table 16 Wilcoxon pairwise comparison for 40 unit system.

BPSO QEA QBPSO BFWA BGWO 1 BGWO 2

Table 17 Wilcoxon pairwise comparison for 100 unit system.

BPSO QEA QBPSO BFWA BGWO 1 BGWO 2

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