Modified Grey Wolf Optimizer Used for Congestion ...Modified Grey Wolf Optimizer Used for Congestion Management in a Deregulated Power Systems 1Rajesh Retnamony and 2Jacob Raglend

Modified Grey Wolf Optimizer Used for Congestion

Management in a Deregulated Power Systems 1Rajesh Retnamony and 2Jacob Raglend

1Department of EEE, Noorul Islam University,

Kanya Kumari, Tamil Nadu, India. [email protected]

2School of Electrical Engineering, Vellore Institute of Technology,

Vellore, Tamil Nadu, India. [email protected]

Abstract

Congestion Management (CM) is one of the major technical issues in deregulated power systems. Optimal Power Flow (OPF) based CM method is considered in this work. To create congestion is to modify IEEE 14 bus system with 125% of overloading with worst line outage and the few bus voltages and line power flows are exceeds the limit. Congestion released by using FACTS (Flexible AC Transmission Systems) Devices such as SVC(Static VAR Compensator) and TCSC(Thyristor Controlled Series Compensator) devices. In CM the objective functions generation cost, real power loss, voltage deviation are minimized and bus voltages, line power flows maintains within limit. The single objective and multi-objective functions optimized using modified Grey wolf optimizer (mGWO). Four different mGWOcase results compared with Power System Analysis Toolbox (PSAT). The multi-objective weighted sum method provides best results as compared with other cases. Key Words:Congestion management, optimal power flow, modified grey wolf algorithm, flexible AC transmission systems.

International Journal of Pure and Applied MathematicsVolume 116 No. 22 2017, 315-326ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

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1. Introduction In a deregulated power system generation, transmission and distribution are controlled by separate companies. Main benefits from the deregulation are cheaper electricity, efficient capacity expansion planning, cost minimization, more choice and better service. Independent System Operator (ISO) who controls and maintains system as stable condition. The power transaction between the companies, overloaded condition and sudden line outage will create congestion in a transmission lines. In this OPF based CM generation cost, real power loss and voltage deviation are considered as objective function. Four different single and multi-objective cases optimized using mGWO with FACTS devices. Congestion management is used transmission switching with benders decomposition technique[1].

The objective functions social welfare maximization, real power and reactive power generation cost and LMP are solved using Price Responsive Demand Shifting (PRDS) bidding mechanism [2]. Generation cost, voltage profile improvement and FACTS cost function are considered as objective functions and optimized using Coordinated Aggregated-Based Particle Swarm Optimization algorithm (CAPSO) [3]. The objective functions social welfare, LMP and real power losses considered and UPFC device optimally located [4]. MOPSO algorithm used to optimize congestion management problem [5]. Multi-objective functions considered as congestion cost, voltage stability and transient stability solved using modified augmented e-constraint method [6]. Multi-objective Decentralized CM is used modified NSGA-II [7]. Multi-objective Fuzzy Evolutionary Programming (FEP) and NSGA-II used to optimize congestion management problem in IEEE 30 bus system [8]. L-index used for voltage stability enhancement [9]. The IEEE test bus system data taken from Matpower 4.1[10]. The location of the FACTS devices based on modal analysis [11]. Hybrid MOPSO used to optimize multi-objective problem and comparatively got the best result with other recent algorithms [12]. The control variables used in the optimal power flow problem is generator real power settings (PGi) and voltage settings (VGi), transformer tap settings (Ti), reactive power compensation setting (Qci), SVC reactive power and TCSC line reactance settings.

2. OPF Problem Formulation OPF is a nonlinear optimization problem the objective functions (f (x, u)) optimized using equality constraints (g (x, u)) and inequality constraints (h(x, u)), that is used to find the best control variables (u) and state variables (x). The general form of OPF problem can be expressed as follows:

Minimize: f (x, u) (1) Subject to: g (x, u) = 0 (2) h (x, u) ≤ 0 (3)

International Journal of Pure and Applied Mathematics Special Issue

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Problem Objectives Generation Cost Minimization

Minimization of generation cost (FG) in a power generation is expressed with the cost coefficients are ai, bi and ci,

FG =Min Gcost=� �aiPgi ² + biPgi + ci�NG

i=1 (4)

Real Power Loss Minimization

Due to the transaction between generator node and demand node, the real power and reactive power losses produced in the transmission lines. Our aim is to minimize the losses in the power system network. The objective is to minimize the real power loss (FPL) in the transmission line is expressed as

FPL= Min Ploss = � Gij�Vi² + Vj² − 2ViVj cos(δi− δj�nl

i=1 (5)

Where Vi, Vj → Voltage Magnitude at Bus i and Bus j Gij → Conductance in the Line i-j δi ,δj → Voltage Angle in the Bus i and Bus j nl → Total Numbers of Transmission Lines

Voltage Deviation Minimization

The voltage gap between the reference voltage and load bus voltage is less means the voltage deviation get minimized. It can be mathematically expressed as.

FVD =Min VD=� ��Vi − Viref ��

NL

i=1 (6)

Where, 𝑉𝑉i𝑟𝑟𝑒𝑒𝑓𝑓 is specified reference voltage at bus i, which is taken as 1.0 p.u. NL- no. of load buses.

Problem Constraints

The list of equality constraints and inequality constraints are below. Equality Constraints

Equality constraints of the given objective functions are PGi–PDi= Vi� Vj[Gij cos(δi − δNB

j=1 j) + Bij sin(δi − δj)] (7)

QGi–QDi= Vi� Vj[Gij cos(δi− δNBj=1 j) + Bij sin(δi − δj)] (8)

PGi, QGi→ real power & reactive power generations at bus i. PDj,QDi→ real power and reactive power demands at bus i. Gij →Conductance in between the line i-j Bij → Susceptance in between the line i-j i → 1 to NB, NB →total numbers of bus.


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Inequality Constraints

Inequality constraints of the given objective functions are PGi

min ≤ PGi ≤ PGimax

,i= 1,2,….,NG (9) VGi

min ≤ VGi ≤ VGimax

, i= 1,2,….,NG (10) Ti

min ≤ Ti ≤ Timax

, i= 1,2,….,NT (11) Qci

min ≤ Qci ≤ Qcimax

, i= 1,2,….,NC (12) VPQi

min ≤ VPQi ≤ VPQimax ,i= 1,2,….,NPQ (13)

QGimin ≤ QGi ≤ QGi

max .i= 1,2,….,NG (14) SLk

min ≤ SLk ≤ SLkmax ,i= 1,2,….,NE (15)

Reactive power constraint of SVC -200MVAR <Qsvci< 200MVAR i ∈ NSVC (16)

Reactance constraint of TCSC -0.5XL <XTCSCi< 0.5XC i ∈ NTCSC (17)

Where Xij = Xline + XTCSC Xline→ transmission line reactance,

XTCSC→ TCSC reactance in transmission line, Xij→ reactance between the bus I and j

The real power flow in a transmission lines is maintained below limit. 𝑃𝑃𝑖𝑖𝑖𝑖 ≤ 𝑃𝑃𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 ; i=1,2,….NB; j=1,2,….NB; (18)

Where, Pij real power flow at branch i-j. 𝑃𝑃𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 maximum power flow limit(MW) Pij = Vi [Gij(Vi – Vjcosϴij) - BijVjsinϴij] (19)

Voltage Level at a load bus is maintained within a specific upper and lower limit (0.9-1.1)pu, determined by the operator. 𝑉𝑉𝑖𝑖𝑚𝑚𝑖𝑖𝑚𝑚 ≤ 𝑉𝑉𝑖𝑖 ≤ 𝑉𝑉𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 ; i=1,2,….NPQ (20)

The control variables used in the OPF problem is generator real power settings (PGi) and voltage settings(VGi), transformer tap settings (Ti),reactive power compensation setting(Qci) and FACTS device settings (Qsvci,XTCSCi) .𝑉𝑉PQ𝑖𝑖 voltage at PQ bus, 𝑆𝑆𝐿𝐿𝑘𝑘 is kth line apparent power. max and min represents maximum and minimum control variables value. Total Numbers of Generators (NG), Transformers (NT), Switchable VAR Sources (NC), and PQ Buses (NPQ).

Line Loading (LL)

Line loading minimization is to optimize the power flow of each line within a limit and minimize the line overloading objective function. It is used to minimize the power flow gap between actual values and limit value and expressed as:

Min LL=� �Pij (t)− Pijmax �²NL

i=1 (21)


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Here NL total number of lines

LL- Line loading Pij Power flow in each lines

PijmaxMaximum power flow limit in each line N-1 Contingency Analysis in Power System Using Severity Index (SI)

Severity Index used to find out the worst line outage based on the N-1 contingency analysis, that can be expressed as

SI =� � PkPk

max �2movl

k=1

(22)

Where

Pk = Power flow in line k (MW)

Pkmax = Power flowmax limit in line k (MW)

Ovl is the set of overloaded lines, m as a weight coefficient.

3. Proposed Approach for Congestion Management

Here the two methods used to solve the CM problem that is conventional method and non-conventional method. The MATLAB-PSAT used for conventional method and the intelligent technique like mGWO used to optimize the OPF problem.

MATLAB - Power System Analysis Toolbox (PSAT)

PSAT is a MATLAB toolbox for electric power system analysis and control13. It includes power flow, continuation power flow, opf, small signal stability analysis and time domain analysis. The PSAT is a user friendly tool and all the operations done by using graphical user interfaces with Simulink library. The IEEE 14 bus PSAT Simulink model is shown in the Figure.1.

Figure 1: IEEE 14 Bus PSAT Simulink Model


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Grey Wolf Optimizer (GWO) Source of Inspiration

GWO is a typical swarm-intelligence algorithm which is inspired from the leadership hierarchy and hunting mechanism of grey wolves in nature14. In the hierarchy of GWO, alpha (𝛼𝛼) is considered the most dominating member among the group leader and decision maker. The rest of the subordinates to 𝛼𝛼are beta (𝛽𝛽) and delta (𝛿𝛿) which help to control the majority of wolves in the hierarchy that are considered as omega (𝜔𝜔). The 𝜔𝜔wolves are of lowest ranking in the hierarchy. The mathematical model of hunting mechanism of grey wolves consists of the following:

1. Tracking, chasing, and approaching the prey. 2. Pursuing, encircling, and harassing the prey until it stops moving. 3. Attacking the prey.

Social Hierarchy

While modelling the GWO social hierarchy, the fittest solution in the grey wolves is alpha (α). The second best solution is beta (β), the third best is delta (δ) and the rest is omega (ω) wolves. In the GWO algorithm the hunting mechanism is done by α, β, and δ wolves, the remaining ω wolves followed by these three wolves.

Encircling Prey

The position vector of prey Xp��⃗ , position vector of grey wolf X��⃗ and the coefficient vectors A��⃗ and C ��⃗ and current iteration t now the mathematical model of grey wolves encircling prey during the hunt can be written as

D��⃗ = �C ��⃗ . Xp��⃗ (t) − X��⃗ (t)� (23) X��⃗ (t + 1) = �Xp��⃗ (t) − A��⃗ . D��⃗ � (24)

The A��⃗ and C ��⃗ can be calculate as; A��⃗ = 2a�⃗ . r1��⃗ − a�⃗ (25) C�⃗ = 2 . r2��⃗ (26)

Where a�⃗ values in the iterations are normally decreased from 2 to 0, random vectors r1 and r2 are in [0, 1].

Hunting

Grey wolfs hunting behavior is modeled with α, β, and δ wolves knows the knowledge about the prey position and updated their position, the equations are shown below.

Dα��⃗ = �C1��⃗ . Xα��⃗ − X��⃗ � (27) Dβ��⃗ = �C2��⃗ . Xβ��⃗ − X��⃗ � (28) Dδ��⃗ = �C3��⃗ . Xδ��⃗ − X��⃗ � (29) X1��⃗ = Xα��⃗ − A1��⃗ . (Dα��⃗ ) (30) X2��⃗ = Xβ��⃗ − A2��⃗ . (Dβ��⃗ ) (31)


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X3��⃗ = Xδ��⃗ − A3��⃗ . (Dδ��⃗ ) (32)

X��⃗ (t + 1) = X1��⃗ +X2��⃗ +X3��⃗

3 (33)

Attacking the Prey

In GWO algorithm when the hunting process finished by attack the prey. It is stated mathematically to decrease the vector value from 2 to 0. If |A| <1 the grey wolves force to attack the prey.

Search for the Prey (Exploration)

The grey wolves are search for a prey based on the position of α, β, and δ. They diverged from each other wolves to search for the prey and converged to attack the prey. If |A| >1 grey wolves force to search the better prey..

Modified Grey Wolf Optimizer

The mGWO discussed by Nitinmittal15, In GWO the value of (a) decreases linearly from 2 to 0 .By update the value of (a) in a search space and update equation as follows:

a=2(1- t2

T2) (34)

Where T indicates the maximum number of iterations and t is the current iteration. Using this exponential decay function, the numbers of iterations used for exploration and exploitation are 70% and 30%, respectively.

The GWO parameters taken in the problem is population size (N) 25, Maximum iteration (T) 100 and total number of runs 1.

The Pseudo Code of mGWO Algorithm 1. Generate initial search agents Xi (i=1, 2,…., n) 2. Initialize the vector’s a, A and C 3. Estimate the fitness value of each hunt agent Xα =the best hunt agent Xβ =the second best hunt agent Xδ =the third best hunt agent 4. Iter=1 5. repeat 6. for i=1: Xs (grey wolf pack size) Renew the location of the current hunt agent using Equation (7). End for 7. Estimate the fitness value of all hunt agents 8. Update the value of Xα, Xβ, Xδ 9. Update the vectors a, A and C 10. Iter=Iter+1 11. until Iter>= maximum number of iterations {Stopping criteria} 12. output Xα End


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4. Congestion Management in Modified IEEE14 Bus System and Results Discussion

In this research work IEEE14 bus includes 14 numbers of buses, 20 transmission lines, 3 Transformers, 11 load buses, 1 shunt capacitor, real power load 259MW and reactive power load 81.4MVAr, 16 numbers of control variables consider for optimization. This work is done using MATLAB 7.9 (R2009) with necessary coding. Totally seven different cases worked out those cases are described below:

Base Case: IEEE14 test bus system is with no modifications. Congested Case: Modified IEEE14 test bus by 125% overloading with worst line outage. PSAT Case: Modified IEEE14 bus system power flow run with SVC and TCSC devices using PSAT.

Case 1: Single objective optimization as generation cost minimization, F1 =Min Gcost.

Case 2: Single objective optimization as real power loss minimization, F2 =Min Ploss .

Case 3: Single objective optimization as voltage deviation minimization, F3 =Min VD.

Case 4: Multi-Objective optimization by weighted sum method,multi-objectives are generation cost, real power loss and voltage deviation minimization.

F4= w1.FG+ w2.FPl+w3.FVD (34) Where, wi(i=1,2….n) is a weighted factor for ith objective function.

The power flow run by using PSAT toolbox for base case and congested case. In congested case the worst line outage is line 2(between bus 1and 2) with high SI index value of 5.82. In this case the objective functions are high as compared with base case and few bus voltages at bus 5and line power flows at line 1,3,4,5 and 10 not within limit. The locations of FACTS devices find using modal analysis [11]. The SVC device located at bus 5 and TCSC device located at line 1(between bus 1 and 2). In PSAT case the FACTS devices located in the PSAT Simulink model and the power flow will be run, the objective functions minimized and the bus voltages and line power flow improved but not maintains a limit using this conventional method. In a Case 1-4 OPF run by using mGWO with FACTS devices. The control variable and results comparison for all the seven cases is shown in Table 1. In this result shows that congested case generation cost value optimized in the case 4 is 11590($/h) to 9684($/h). Similarly the remaining objective functions Ploss, Qloss, voltage deviation and line loading values are also optimized as 35.91MW to 10.73MW, 107.86MVAr to 34.79MVAr, 0.615pu to 0.190pu, 67309 to 23686 respectively. The


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comparative bus voltage at all cases shown is at Figure. 2 and comparative line power flow at all cases shown is at Figure. 3. In this Figure 3 shows that the power flows in congested case is not in limit and using PSAT won’t get the best solutions. By rescheduling the generators using mGWO method and the voltages and power flows maintains with in limit and objective functions minimized. Case 4 provide a best result as compared with all the other cases. The iterative solutions of case 1 to 4 is shown in Figure.3.The voltage and power flow violations shown in Table. 2.

Table 1: Control Variables and Objective Functions Comparison for IEEE14

Bus System Control Variables

Minimum Maximum Base Case

Congested case

PSAT Case

Case 1 Case 2 Case 3 Case 4 (Multi-Objective Case)

P1(MW) 0 332.40 232.60 319.66 301.64 149.53 139.67 146.15 90.03 P2(MW) 0 140 40 40 40 31.25 72.59 28.82 112.17 P3(MW) 0 100 0 0 0 74.87 51.73 77.40 2.04 P6(MW) 0 100 0 0 0 40.14 52.58 44.57 100.00 P8(MW) 0 100 0 0 0 39.50 21.19 31.10 30.25 VG1(pu) 0.95 1.1 1.06 1.06 1.060 1.012 0.990 1.0151 1.06 VG2(pu) 0.95 1.1 1.045 1.045 1.036 0.978 0.961 0.9969 1.0233 VG3(pu) 0.95 1.1 1.01 1.01 1.049 1.038 1.022 1.0508 0.9772 VG6(pu) 0.95 1.1 1.07 1.07 1.019 1.005 0.981 1.0368 1.0388 VG8(pu) 0.95 1.1 1.09 1.09 1.01 0.9723 1.0273 1.0023 1.0379 T8(pu) 0.9 1.1 0.978 0.978 0.978 0.9181 0.9 0.9 0.9375 T9(pu) 0.9 1.1 0.969 0.969 0.969 0.9354 0.9183 0.9082 0.9236 T10(pu) 0.9 1.1 0.932 0.932 0.932 1.0798 1.0733 0.9459 0.9231 Q14(MVAr) 0 20 0.19 0.19 0.19 6.3E-15 7.6E-15 9.89E-15 3.79E-15 QSVC5(pu) -2 2 0.415 0.328 0.401 0.284 0.412 XTCSC2(pu) -0.2XL 0.8XC 0.0473 0.0218 0.0312 0.0416 0.0467 G cost($/h) 7780 11590 10748 10738 10105 10541 9684 PLoss(MW) 13.6 35.91 17.89 11.54 14.01 13.18 10.73 Qloss(MVAr) 28.74 107.86 70.33 43.21 50.77 47.16 34.79 VD(pu) 0.576 0.615 0.566 0.252 0.346 0.205 0.190 Line Loading 19074 67309 53382 30817 26850 24053 23686

Figure 2: IEEE 14 Bus Voltage Magnitude Comparison for PSAT and mGWO

Figure 3: IEEE 14 Line Loading Comparison for PSAT and mGWO Cases


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Table 2: IEEE 14 Bus Violated Voltage and Power Results Comparison Voltage And Power Flow

Limit Base Case Congested case PSAT Case Case 1 Case 2 Case 3 Case 4 (Multi-Objective Case)

V5(pu) Range (0.9-1.1)

1.015 0.891 1.044 0.960 0.940 0.981 0.983

Line 1(MW) 150 146.88 319.66 301.64 149.53 139.67 146.14 90.03 Line 3(MW) 85 73.25 105.31 104.35 51.13 67.81 43.85 84.00 Line 4(MW) 85 55.48 104.56 103.24 51.18 60.35 52.44 48.87 Line 5(MW) 85 42.15 99.38 98.57 47.07 53.10 47.75 37.67 Line 10(MW) 50 50.03 59.43 53.45 17.67 11.20 15.62 19.94

Figure 4: IEEE 14 Bus System Case1 to 4 Objective Function Vs Iteration

Using mGWo

5. Conclusion and Future Work In a deregulated power systems congestion management is one of the major technical issue. The OPF based congestion management is considered here and the results are compared with both conventional and nonconventional methods with FACTS devices. Conventional methods not provide the optimized results. The mGWO algorithm (non-conventional method) four different cases are given the good results. In that cases multi-objective optimization using mGWO delivers a best optimized result in an IEEE 14bus system. Generation cost, real power loss and voltage deviation optimized and the bus voltages and power flow in line are within limit. In future the congestion management will be done with different recent intelligent techniques in a higher order IEEE test bus systems.

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Modified Grey Wolf Optimizer Used for Congestion ...Modified Grey Wolf Optimizer Used for Congestion Management in a Deregulated Power Systems 1Rajesh Retnamony and 2Jacob Raglend