Essay: Investigation of Multi-Objective Optimal Power Flow Problem in Power System

Abstract: In this paper Sub-Population Based Particle Swarm Optimization technique is proposed for Multi-Objective Optimal Power Flow problem. This technique is used to optimize the objectives of cost, loss and voltage stability index and maximum loadability in off line mode. Various optimization techniques are applied to solve the power flow problem in the literature. In this paper new technique is proposed to optimize the objectives in multi-mode and analysis the power flow problem in detail. This proposed approach has been evaluated on the standard IEEE 30 bus test system. The results are simulated and validated.
Keywords’ Cost, Loss, Voltage stability index, Loadability, Optimization, SPPSO
I. INTRODUCTION
Power system is a large electrical network used for generation, transmission and distribution of electrical power. Electrical Power system consists of large number of components such as generators, conductors, capacitors, reactors, Transformers, power electronic converters, protective devices etc. The difference between the other network based markets from electricity market is the transmission network. Transmission network is mastered by power flow equations. These power flow equations are non-linear and for large system it is complicated. In the recent years, as a result of globalization, the demand of electric power has been increased rapidly. Power engineers require special tools to optimally analyze, monitor, and control different aspects of power systems operation and planning. Most of these tools are properly formulated as some sort of optimization problems. Hence in order to operate the power system efficiently optimization is implemented in power system with respect to several objectives and constraints.
The optimal power flow (OPF) is the backbone tool that has been extensively researched since its first introduction in the early 1960s. The purpose of OPF is to find the optimal settings of a given power system network that optimize a certain objective function while satisfying its power flow equations, system security, and equipment operating limits [1-2]. Different control variables are manipulated to achieve an optimal network setting based on the problem formulation [3].
To decrease the generation cost, loss reduction is an effective method. Stability of power system should be in acceptance level. Increment of loading sometime leads to instability. Stability margin can be maintained with minimal load increment. It is possible to predict the maximum loadability in order to maintain the stability, amount of reactive load increased before the system starts. Maximum loadability is based on the equality and inequality constraints [4]. By improving the maximum loadability in weak bus, voltage stability margin could be enhanced.
To estimate the maximum loadability various techniques are reported in [5-8]. Voltage stability and thermal limit also included in maximum loadability [9].The application of line-based voltage stability index is known as fast voltage stability index (FVSI). In FVSI Problem solving process simpler with less mathematical burden. It performed well, compare to conventional voltage stability analysis (VSA).
Many mathematical techniques such as quadratic programming [10],linear programming[11], and inerter point method [12] have been applied to solve the optimal power flow problem. To avoid the drawbacks in conventional, evolutionary methods used to solve the OPF problem. The application of artificial intelligence technologies utilised to estimate the maximum loadability as other search technique.
In this work, voltage stability Index is incorporated to assess the voltage stability condition, generation cost, loss is minimized. To obtain the load ability in maximum FVSI technique is used. The above is tested and implemented in IEEE30 test bus system.
A brief introduction has been provided in this section for the existing optimal power flow system. The rest of the paper is arranged as follows. In Section II, the optimal power flow problem is formulated and discussed. In Section III, the basic concepts of PSO & SPSPO are explained. Section IV provides the details simulation of proposed algorithm is tested and the results are presented. Finally concluding remarks appear in Section V.
II. PROBLEM STATEMENT
The Optimal power flow problem is to minimize fuel cost, transmission loss and voltage stability index and maximize the loadability, which satisfying equality and inequality constraints.
A. Problem Objectives
1) Minimization of fuel cost
The total fuel cost can be expressed as
(1)
(2)
(3)
ai, bi, ci are the cost coefficients of the ith generator and is the real power output of the ith generator.
2) Minimization of transmission loss
(4)
and are bus voltage amplitude at the two ends of the kth line .gk is the conductance of the kth line. n is the number of transmission line.
3) Maximum loadability
(5)
is line impedance, is line reactance, is voltage at the sending end, is the reactive power at the receiving end.
B. Problem constraints
1) Inequality constraints
(6)
LFij’ Line flow limits (7)
(8)
(9)
(10)
(11)
2) Equality constraints
(12)
(13)
Where i=1,2′..nb. nb is the number of buses, Qc and Qd are the generator and demand reactive power respectively, Gij and Bij are the transfer conductance and susceptance between bus i and j respectively, Vi and Vj are the voltage magnitudes at bus i and j respectively, Ti transformer tap settings, ??i and ??j are the voltage angle at bus i and j respectively.
C. Maximum Loadability Determination
FVSI values calculated using the load flow solution to search for maximum loadability point of a particular load bus and continued for all the load buses. Calculating FVSI involves running the load flow program using Newton Raphson method,evaluating FVSI value for every lines in the system,increasing the reactive power loading gradually at a load bus until FVSI value is 0.9.
III. CONCEPTS OF PARTICLE SWARM OPTIMIZATION
Particle Swarm Optimization (PSO) is an evolutionary algorithm that is used to find optimal (or near optimal) solutions to numerical and qualitative problems. Particle Swarm Optimization was developed by James Kennedy and Russell Eberhart in 1995 based on flocking behavior seen in many species of birds. The concept of the PSO is that , at each time step, the particles changes its velocity of (accelerating) and moves toward its pbest and gbest locations (local version of PSO). Acceleration is weighted by a random term, with separate random numbers being generated for acceleration toward pbest and gbest locations. In the past several years, PSO has been successfully applied in many research and application areas. It is demonstrated that PSO gets better results in a faster, cheaper way compared with other methods.
The parameters of individual best and global best are in equ.14 and equ.15.
(14)
(15)
The velocity and the positions are updated in the each iteration. The updating of velocity and updating of position using the equ.16 and equ.17.
(16)
(17)
Where, current velocity of individual i at iteration k, modified velocity of individual i at iteration k+1, current position of individual i at iteration, inertia weight parameter, c1,c2 acceleration factors, random numbers between 0 and 1, best position of individual i until iteration k, best position of the group until iteration k
The search mechanism of the PSO using the modified velocity and position of the individual i based on (16) and (17) is illustrated in Fig.1.
Fig.1: Search mechanism of PSO
A. Sub-Population Based Particle Swarm Optimization
In Sub-Population Based Particle Swarm Optimization (SPPSO), particles are grouped and search the best from the individual group. Similarly search the best in all the groups. Then find the overall best position. Fig.2. shows the main procedure for sub-population based particle swarm optimization technique.
Fig.2: Flow chart of SPPSO
(18)
C3 acceleration factors, random numbers between 0 and 1, is the global best of the group, is the best in the overall group. This SPPSO gives fast and better result compare to PSO.
B. Sub-Population Based Particle Swarm Optimization for Optimal Power flow
The following steps are followed to obtain the multi-objective optimal power flow using SPPSO.
Step 1: The data and control variables are initialized.
Step 2: The positions and velocities are generated.
Step 3: Calculate the fitness function using Optimal Power Flow.
Step 4: The position and velocities are updated using formulae.
Step 5: Calculate the objectives.
IV. SIMULATION AND NUMERICAL RESULTS
IEEE 30-bus test system used to validate the performance of the proposed method. The 30-bus IEEE test system has 41 transmission lines, six generators and four transformers (T6’9, T6’10, T4’12 and T27 ‘ 28). The lower and upper voltage magnitudes and transformer tap limits are considered between 0.9 and 1.1 p.u. The parameters required for implementation of the proposed PSO algorithm are adjusted by 200 times running of this algorithm. These parameters involve maximum iteration and number of population which are set on 200 and 100, respectively.
Table I and Fig. 3 shows the generation cost of IEEE 30 bus test system. Table I shows the generation cost of Particle Swarm Optimization (PSO) Improved Particle Swarm Optimization (IPSO) and Sub-Population Based Particle Swarm Optimization (SPPSO) are 802.105$/hr, 801.89$/hr and 800.72$/hr. It is clear that SPPSO gives better result compare to other reported techniques. Fig. 3 shows the generation cost convergence of IEEE 30 test bus system with number of generations for Particle Swarm Optimization (PSO) and Sub-Population Based Particle Swarm Optimization (SPPSO)
TABLE I. BEST GENERATION COST OF DIFFERENT ALGORITHMS
PSO IPSO SPPSO
Pg1(MW) 177.126 177.0431 176.7174
Pg2(MW) 49.172 49.209 48.8569
Pg3(MW) 21.413 21.5135 21.4719
Pg4(MW) 22.538 22.648 21.6423
Pg5(MW) 10.621 10.4146 12.0878
Pg6(MW) 12 12 12.00
Cost($/hr) 802.105 801.97 800.72
Fig.3. Generation Cost convergence of IEEE 30 bus test system
Table II and Fig. 4 shows the transmission loss of IEEE 30 bus test system. Table II shows the transmission loss of Particle Swarm Optimization (PSO), Improved Particle Swarm Optimization (IPSO) and Sub-Population Based Particle Swarm Optimization (SPPSO) are 13.58 MW, 13.39MW and 8.9 MW. It shows that SPPSO gives minimum of transmission loss compare to other computational algorithm. Fig. 4 shows the transmission loss convergence of IEEE 30 test bus system with number of generations of Particle Swarm Optimization (PSO) and Sub-Population Based Particle Swarm Optimization (SPPSO).
TABLE II. BEST TRANSMISSION LOSS OF DIFFERENT ALGORITHMS
PSO[14] IPSO[14] SPPSO
Loss(MW) 13.58 13.39 8.9
Fig.4. Transmission loss convergence of IEEE 14 bus test system
Fig. 5 shows the graph of Fast Voltage Stability Index of IEEE 30 bus test system. By injecting reactive power to all buses of IEEE 30 bus test system up to the value of 0.9, it is easy to identify that Bus 29 is the lowest loaded bus. Similarly, Bus 30 and Bus 26 are also less loaded buses next to Bus 29. Table III shows the Maximum of Reactive power injection of a bus and Fast Voltage Stability Index of the corresponding bus. Bus 29, Bus 30 and Bus 26 has the Maximum loadability values of 36.9754MW, 33.9586MW and 30.9821MW respectively.
Fig.5. Fast Voltage Stability Index of IEEE 30 bus test system
TABLE III. BEST MAXIMUM LOADABILITY OF IEEE 30 BUS TEST SYSTEM
Bus no. Bus 26 Bus 29 Bus 30
Qmax
(MVAR) 30.9821 36.9754 33.9586
FVSI 0.9755 0.9969 0.9857
Table IV shows the control variables related to multi-objective problem for Particle Swarm Optimization (PSO) Improved Particle Swarm Optimization (IPSO) and Sub-Population Based Particle Swarm Optimization (SPPSO). The bold numbers shows that reduction of cost, loss and fast voltage stability index of SPPSO. It indicates that the SPPSO gives better result compare to other techniques.
TABLE IV. CONTROL VARIABLES RELATED TO IEEE 30 BUS TEST SYSTEMS
PSO[14] IPSO[14] SPPSO
Pg1, MW 177.126 177.0431 176.7174
Pg2, MW 49.172 49.209 48.8569
Pg3, MW 21.413 21.5135 21.4719
Pg4, MW 22.538 22.648 21.6423
Pg5,MW 10.621 10.4146 12.0878
Pg6,MW 12 12 12.00
Vg1 1.05 1.045 1.045
Vg2 1.0442 1.043 1.043
Vg3 1.446 0.998 0.998
Vg4 1.0408 1.009 1.009
Vg5 0.9601 1.014 1.014
Vg6 1.05 1.047 1.047
T6-9,pu 1.01 1.012 1.012
T6-10,pu 0.99 0.971 0.971
T4-12,pu 1.01 1.023 1.023
T27-28,pu 1.02 1.014 1.014
Cost($/hr) 802.105 801.97 800.72
Loss(MW) 13.58 13.39 8.9
V. CONCLUSION
This proposed approach has been evaluated on the standard IEEE 30 bus test system. The proposed approach utilizes the fast convergence of particle swarm optimization to set optimal solution of the control variables. Different objective functions have been considered for optimal power flow to minimize the fuel cost, power transmission losses, voltage stability index and maximize load ability. The proposed approach has been tested and examined with different objectives to demonstrate its effectiveness and robustness.
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