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This paper presents an application of GRADE Algorithm based approach along with PV analysis to solve multi objective optimization problem of minimizing real power losses, improving the voltage profile and hence enhancing the performance of power system. GRADE Algorithm is a hybrid technique combining genetic and differential evolution algorithms. Control variables considered are Generator bus voltages, MVAR at capacitor banks, transformer tap settings and reactive power generation at generator buses. The optimal values of the control variables are obtained by solving the multi objective optimization problem using GRADE Algorithm programmed using M coding in MATLAB platform. With the optimal setting for the control variables, Newton Raphson based power flow is performed for two test systems, viz, IEEE 30 bus system and IEEE 57 bus system for three loading conditions. Minimization of Real power loss and improvement of voltage profile obtained are compared with the results obtained using firefly and particle swarm optimization (PSO) techniques. Improvement of Loadability margin is established through PV curve plotted using continuation power flow with the real power load at the most affected bus as the bifurcation parameter. The simulated output shows improved results when compared to that of firefly and PSO techniques, in term of convergence time, reduction of real power loss, improvement of voltage profile and enhancement of loadability margin.

In Power systems planning and operation, voltage stability poses a major concern. Voltage instability or voltage collapse may come as a consequence of inadequate reactive power support from generators and transactions in transmission lines. Hence, understanding the concept of voltage stability and designing the prevention methodologies to mitigate the voltage instability is of great value to the utilities. Real Power versus Voltage (PV) analysis is useful for conceptual analysis of voltage stability and can be useful in specifying the active power margin.

Optimization of reactive power is proved to improve the voltage stability limits and to minimize system active power losses. Generator bus voltages, Transformer tap positions, the MVAR at the capacitor Banks and reactive power generation at generator buses are considered as the control variables. Many conventional methods used in VAR optimization are based on linear programming, nonlinear programming and quadratic programming method. The optimization of reactive power support to mitigate voltage collapse problem in power market systems was described using a sequential quadratic programming method [

The major drawbacks of the Conventional methods were that they were time consuming and they were not capable to solve complex problem with discrete variables. To overcome the disadvantages experienced in conventional methods, nature-inspired metaheuristic algorithms such as Genetic algorithm, colonial search algorithms and swarm intelligence techniques were proposed.

Comprehensive learning particle swarm optimization for reactive power dispatch [

Genetic algorithm [

Colonial search algorithms such as Ant colony and Binary Ant colony based optimization were also reported in reactive power optimization [

Differential Evolution (DE) algorithm was reported in various literatures [

Comparative Study of Firefly algorithm and Particle Swarm Optimization for Noisy Non-Linear Optimization Problems was reported in 2012 by Saibal K. Pal et al. [

In this paper, an algorithm namely GRADE algorithm is applied for solving multi objective optimization. The GRADE algorithm is based on a combination of genetic algorithm and differential evolution technique. The rest of this paper is organized as follows. In Section 2, a brief description of the formulation of the multi objective optimization problem along with equality and inequality constraints is presented. In Section 3, GRADE algorithm is presented in brief and the steps involved to solve multi objective optimisation problem is presented alongside a flowchart representing the same. In Section 4, the numerical results obtained during simulation for different loading conditions are explained in detail and analysis from the results is also presented. The conclusions are given in Section 5.

The main objective of multi objective optimization is to minimize the active power loss in the transmission network, which is defined as follows:

Another objective of this problem is to improve the voltage profile which is formulated mathematically as follows,

The overall objective function of the problem is thus formulated as follows,

where, P_{loss} = active power loss in the transmission network,

V_{max,spec} = is the maximum voltage specified for all the buses,

α and β are the penalty factors.

ConstraintsEquality Constraints. The equality constraints include the real and reactive power constraints which are given as follows:

1) Real Power Constraint

where, n = numbers of buses, except swing bus.

G_{ij} = mutual conductance between bus i and j.

B_{ij} = mutual susceptance between bus i and j.

q_{ij} = Load angle between bus i and j.

P_{i} = Real power injected into network at bus i.

V_{i}, V_{j} = Voltage magnitude at bus i, j.

2) Reactive Power Constraint

where, n = number of buses, except swing bus.

Q_{i} = Reactive power injected into network at bus i.

Inequality Constraints. The inequality constraints include the following,

1) Bus Voltage Magnitude Constraint

where, V_{i} = Voltage magnitude at bus i.

N_{B} = Total number of buses.

2) Generator Bus Reactive Power Constraint

where, Q_{Gi} = Reactive power generation at bus i.

N_{g} = Number of generator buses.

3) Reactive Power Source Capacity Constraints

where, Q_{Ci} = Reactive power generated by i^{th} capacitor bank.

N_{C} = No. of capacitor banks.

4) Transformer Tap Position Constraints:

where, T_{k} = Tap setting of transformer at branch k.

N_{T} = No. of tap-setting transformer branches.

The GRADE algorithm is a combination of genetic algorithm and differential evolution technique. The algorithmic scheme of GRADE algorithm is really alike to that of the genetic algorithm except that it uses the simplified differential operator like the differential evolution technique. The parameters used in GRADE Algorithm are given in

GRADE algorithm uses 3 genetic operators which include mutation, crossing and selection.

Mutation operator

Mutation operator is applied to the parental population, thus producing a new population of offsprings. From the unit interval, a random number p is generated for each parent P. One offspring O is created by mutation for a parent P if p is smaller than radioactivity. In such a case, the new random point RP is generated inside a given domain and new offspring O is created on a random position on the line connecting the parent P and the random point RP. This operator creates each time different number of offsprings, but in average this number should converge to population_size * radioactivity. Radioactivity is a control parameter of GRADE algorithm defining the part of offsprings created by mutation.

Crossover Operator

Crossover operator is designated to create such a number of new offsprings, that the total number of offsprings n_Offsprings will be the same as parents n_Parents (population will be doubled). To create an offspring, two members P1 and P2 of parental population are randomly chosen. Then the vector of their difference is computed, multiplied by cross_rate and added to the better one between P1 and P2. Cross_rate is a number each time randomly generated from the interval (0; cross_limit). cross_limit is another control parameter of GRADE algorithm.

Selection Operator

Operator selection should select new population from parents and offsprings or more precisely, it eliminates chosen offsprings and parents, until the complete population has its initial size. Each time when one member is rejected, best members are selected for next generation and the worse of them is discarded. This selection process has two advantages: It ensures that the best member will survive to the next generation, even very bad member has a possibility to survive and certain diversity of population remains.

The GRADE algorithm suffers from serious disadvantage that it tends to form clusters. In order to overcome this disadvantage a niching strategy called CERAF strategy is employed. It produces areas of higher level of “radioactivity” in the neighborhood of all previously found local extremes by increasing the mutation probability (i.e. ceraf radioacitivity) in these areas many times. Parameters used to implement CERAF strategy and the corresponding values employed are given in

Parameter | Description Used Value | |
---|---|---|

pop_rate | Control the size of population | 10 |

Radioactivity | Control the number of offsprings created by mutation | 0.2 |

cross_limit | Control the distance of offsprings from its better parent created by crossing | 1.0 |

Parameter | Description Used Value | ||
---|---|---|---|

RAD | Control the radius of the radioactivity area | 0.25 | RAD |

deact_rate | Control the decreasing in size of radioactive area | 0.995 | deact_rate |

Quiet | Control number of generations before new local extreme is marked | 100 | Quiet |

The GRADE algorithm used for searching an optimal solution for multi objective optimisation is given in

Step 1: Read the power flow data, set the minimum and maximum value of control variable and initiate transformer tap positions.

Step 2: Generate the initial population in random manner and assign the objective function value to all chromosomes in the population. The size of the population is then defined as the number of variables of objective function multiplied by parameter pop rate.

Step 3: Several new chromosomes are created using the mutation operators―the mutation and the local mutation (their total number depends on the value of a parameter called radioactivity―it gives the mutation probability).

Step 4: Create another set of new chromosomes using the simplified differential operator; thus doubling the population.

Step 5: Assign objective function values to all newly created chromosomes.

Step 6: Apply CERAF strategy.

Step 7: Apply selection operator to the double-sized population, thus decreasing the amount of individuals to its original value.

Step 8: Perform load flow analysis.

Step 9: Steps 3 - 7 are repeated until the variables are within their limits.

Step 10: Stopping criteria are checked, if satisfied the search process stops and displays the result, else proceed to the next iteration.

The effectiveness of GRADE algorithm based optimization technique is tested in IEEE 30-bus and 57-bus test systems and the results are compared with the results obtained using firefly and Particle Swarm Optimization algorithms. The proposed algorithm is developed in MATLAB 7 and run on a PC with INTEL i5 processor of 4GB RAM. For implementing GRADE technique, 30 trials each for different loading conditions are performed in the above mentioned test systems

The standard IEEE 30-bus test system [

The reactive power generation limits for the IEEE 30-bus system are listed in

Different loading conditions are considered for multi objective optimization. The normal loaded condition has a load of 2.834 p.u and two other loading conditions of which one is light loaded and the other heavy loaded when compared to that of the normal loaded condition are considered.

In light loaded condition, the load is reduced by 50% of the normal load in all load buses and in heavy loaded condition, the load is increased by 50% of the normal load in all load buses.

Bus No | 1 | 2 | 5 | 8 | 11 | 13 |
---|---|---|---|---|---|---|

0 | −40 | −40 | −10 | −6 | −6 | |

10 | 50 | 40 | 40 | 24 | 24 |

1.1 | 0.9 | 1.05 | 0.95 | 1.05 | 0.95 |

Under light loaded condition the load is reduced to 1.4170 p.u and the base case loss is obtained as 0.037765 p.u. Under normal loaded condition the load is 2.834 p.u and the base case loss is obtained as 0.17557 p.u. Under heavily loaded condition the load is 4.2510 p.u and the base case loss is obtained as 0.4495 p.u. A comparison of fitness value for various loading condition is provided in

From

After 30 trials the real power losses obtained by reactive power optimization using GRADE algorithm is presented in

The optimal values of the control variables after optimization for three loading conditions are shown in

Parameter | Lightly loaded condition | Normal loaded condition | Heavily loaded condition | ||||||
---|---|---|---|---|---|---|---|---|---|

Optimization Technique | Firefly | PSO | GRADE | Firefly | PSO | GRADE | Firefly | PSO | GRADE |

Fitness Value | 0.12732 | 0.043762 | 0.043761 | 0.43534 | 0.20071 | 0.20069 | 1.4924 | 0.54832 | 0.54832 |

Loading Condition | Lightly loaded condition | Normal loaded condition | Heavily loaded condition | ||||||
---|---|---|---|---|---|---|---|---|---|

Optimization Technique | Firefly | PSO | GRADE | Firefly | PSO | GRADE | Firefly | PSO | GRADE |

P_{loss}(p.u) | 0.035772 | 0.034768 | 0.034256 | 0.17476 | 0.16953 | 0.16939 | 0.43357 | 0.4339 | 0.4311 |

Loading Condition | Lightly loaded condition | Normal loaded condition | Heavily loaded condition | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Optimization Technique | Worst | Best | Mean | Std. Deviation | Worst | Best | Mean | Std. Deviation | Worst | Best | Mean | Std. Deviation |

P_{loss}(p.u) | 3.8002 | 3.4756 | 3.5657 | 0.10135 | 19.954 | 16.939 | 17.7497 | 0.672046 | 43.683 | 43.41 | 43.2472 | 0.3058 |

Control variables | Lightly loaded condition | Normal loaded condition | Heavily loaded condition |
---|---|---|---|

V_{1} | 1.1 | 1.1 | 1.1 |

V_{2} | 1.1 | 1.1 | 1.1 |

V_{5} | 1.1 | 1.1 | 1.0890 |

V_{8} | 1.1 | 1.1 | 1.1 |

V_{11} | 1.0999 | 1.1 | 1.1 |

V_{13} | 1.1 | 1.1 | 1.0767 |

Q_{C10} | 0.27648 | 0.50 | 0.20 |

Q_{C24} | 0.048992 | 0.213704 | 0.20 |

T_{1} | 1.0003 | 0.970824 | 1.0484 |

T_{2} | 1.05 | 1.0157 | 0.95 |

T_{3} | 0.95 | 0.95 | 0.95 |

T_{4} | 0.97338 | 0.95 | 0.9759 |

Q_{1} | 0.2806 | 0.0499289 | 0.10 |

Q_{2} | −0.36843 | 0.10 | −0.40 |

Q_{5} | −0.39967 | −0.40 | −0.096311 |

Q_{8} | 0.049558 | −0.007805 | −0.10 |

Q_{11} | −0.027887 | 0.239964 | 0.24 |

Q_{13} | −0.06 | 0.141589 | −0.06 |

From ^{th} bus of the IEEE 30 bus system is found to be the weakest bus from power flow results and hence voltage at 30^{th} bus is compared to establish the effectiveness of GRADE Algorithm is improving the voltage profile.

It is noted that, from Figures 3-5 in all the loading conditions voltage profile improvement is optimum when controllers are tuned using GRADE Algorithm.

The result of continuation power flow analysis before and after optimization for different loading conditions

is presented. As the 30^{th} bus of the IEEE-30 bus system is found to be the weakest bus, real power at bus number 30 is considered as load parameter in continuation power flow. Under various loading conditions the PV curve is obtained and the comparison of the PV curve before and after optimization is done.

Under Light loaded condition the curve is as shown in

Under normal loaded condition the curves are Superimposed for cases before and after optimization and are as shown in

Under heavy loaded condition the curves are as shown in

A comparasion of loadability margin for three loading conditions before and after optimization using GRADE algorithm is furnished in

Sl. No. | Loading conditions | Loadability margin(p.u) | |
---|---|---|---|

Before optimisation | After optimisation | ||

1 | Light loaded | 0.508603 | 0.569762 |

2 | Normal loaded | 0.481380 | 0.563519 |

3 | Heavy loaded | 0.449522 | 0.536189 |

From

The effectiveness of GRADE algorithm is minimizing the real power losses, improving the voltage profile and enhancing the loadability limit is tested using second test system, viz, IEEE-57 Bus system. The IEEE 57-bus systems [

The reactive power generation limits for the IEEE 57-bus system are listed in

Different loading conditions are considered for multi objective optimization. The normal loaded condition has a load of 12.5080 p.u and two other loading conditions of which one is light loaded and the other heavy loaded when compared to that of the normal loaded condition is considered. In light loaded condition the total load is reduced by 50% of the normal load and in heavy loaded condition the total load is increased by 50% of the base case as in test system1. The GRADE algorithm was tested for the multi objective optimization problem using MATLAB 7 programming and is run for 30 trials each for different loading conditions in INTEL i5 processor.

Under light loaded condition the load is 6.2540 p.u and the base case loss is obtained as 0.243750 p.u. Under normal loaded condition the load is 12.5080 p.u and the base case loss is obtained as 0.278638 p.u. Under heavy loaded condition the load is 18.7620 p.u and the base case loss is obtained as 1.581204 p.u.

To establish the effectiveness of GRADE algorithm, multi objective optimization for the test system is performed using firefly and particle swarm optimization techniques. Fitness value and real power loss obtained using the above three techniques are compared and tabulated in

From

Control Variable are set as per the values obtained by solving multi objective optimization problem using GRADE algorithm and from ^{st} bus of IEEE-57 bus system is found to be the weakest bus. Hence voltage at 31^{st} bus is observed before and after optimization under the three loading conditions to check the effectiveness of GRADE Algorithm. The results are presented in Figures 10-12 respectively.

From Figures 10-12 it is understood that voltage profile at bus no 31 improved considerably when controllers are turned as per the values obtained using GRADE Algorithm.

The result of continuation power flow analysis before and after optimization for different loading conditions is presented. Real power at bus number 31^{st} is considered as the bifurcation parameter for continuation power flow. PV curve is plotted for the three loading conditions, as in the previous test case.

Under Light loaded condition, the super imposed PV curves before and after optimization are shown in

Under normal loaded condition the PV curves are obtained before and after optimization and it is presented in

Bus No | 1 | 2 | 3 | 6 | 8 | 9 | 12 |
---|---|---|---|---|---|---|---|

0 | −40 | −40 | −40 | −10 | −6 | −6 | |

10 | 50 | 50 | 40 | 40 | 24 | 24 |

1.1 | 0.9 | 1.05 | 0.95 | 1.05 | 0.95 |

Parameter | Lightly loaded condition | Normal loaded condition | Heavily loaded condition | ||||||
---|---|---|---|---|---|---|---|---|---|

Optimization Technique | Firefly | PSO | GRADE | Firefly | PSO | GRADE | Firefly | PSO | GRADE |

Fitness Value | 0.42732 | 0.2801 | 0.2797 | 1.093 | 0.3362 | 0.3359 | 4.4924 | 2.0835 | 2.0819 |

Loading Condition | Lightly loaded condition | Normal loaded condition | Heavily loaded condition | ||||||
---|---|---|---|---|---|---|---|---|---|

Optimization Technique | Firefly | PSO | GRADE | Firefly | PSO | GRADE | Firefly | PSO | GRADE |

P_{loss}(p.u) | 0.19762 | 0.189052 | 0.185047 | 0.27863 | 0.249934 | 0.243697 | 1.30134 | 1.2899 | 1.284536 |

Control variables | Lightly loaded condition | Normal loaded condition | Heavily loaded condition | Control variables | Lightly loaded condition | Normal loaded condition | Heavily loaded condition |
---|---|---|---|---|---|---|---|

V_{1} | 1.08623 | 1.1 | 1.1 | T7 | 1.04543 | 0.95 | 0.989527 |

V_{2} | 1.09163 | 1.1 | 1.1 | T8 | 0.988768 | 0.95 | 0.95 |

V_{3} | 1.09716 | 1.1 | 1.1 | T9 | 0.95 | 1.05 | 0.950728 |

V_{6} | 1.1 | 1.1 | 1.1 | T10 | 1.02535 | 0.95 | 0.95 |

V_{8} | 1.1 | 1.1 | 1.1 | T11 | 0.968509 | 0.96518 | 0.95 |

V_{9} | 1.07875 | 1.1 | 1.1 | T12 | 0.981064 | 0.95 | 1.01125 |

V_{12} | 1.07661 | 1.1 | 1.1 | T13 | 0.964782 | 1.05 | 0.995869 |

Q_{C18} | −2.86067 | 20 | 20 | T14 | 1.00633 | 0.95 | 0.990177 |

Q_{C25} | −0.455466 | 20 | 20 | T15 | 0.989047 | 0.95 | 1.05 |

Qc_{53} | 0.204621 | 13.1967 | 20 | Qg_{1} | 2.79599 | 0 | 3.61023 |

T_{1} | 1.04844 | 0.95 | 0.95 | Qg_{2} | −17.3942 | −40 | 50 |

T_{2} | 1.04991 | 0.95016 | 0.95 | Qg_{3} | 50 | −40 | 15.4129 |

T_{3} | 0.96976 | 0.95 | 0.95 | Qg_{6} | 20.2413 | −18.1918 | −28.3228 |

T_{4} | 0.995547 | 1.05 | 0.955142 | Qg_{8} | 34.9479 | −10 | 20.2949 |

T5 | 0.973679 | 0.95 | 1.00837 | Qg_{9} | 14.3681 | 24 | −6 |

T6 | 0.95073 | 0.95 | 0.95 | Qg_{12} | 6.1866 | 24 | −6 |

Under heavy loaded condition the total load is increased by 50% of normal load in all load buses. The PV curves are obtained before and after optimization and presented in

A comparasion of loadability margin under three loading conditions before and after optimization obtained using GRADE algorithm is furnished in

From

From

Sl. No. | Loading conditions | Loadability margin(p.u) | |
---|---|---|---|

Before optimisation | After optimisation | ||

1 | Light loaded | 0.284611 | 0.295366 |

2 | Normal loaded | 0.232494 | 0.324625 |

3 | Heavy loaded | 0.449522 | 0.536189 |

IEEE 30 Bus system | IEEE 57 Bus system | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Firefly | PSO | GRADE | Firefly | PSO | GRADE | |||||||||||||

L | N | H | L | N | H | L | N | H | L | N | H | L | N | H | L | N | H | |

10 | 8 | 15 | 12 | 7 | 13 | 5 | 4 | 8 | 19 | 15 | 25 | 21 | 11 | 23 | 15 | 8 | 16 | |

L―Light loaded condition; N―Normal loaded Condition; H―Heavy loaded condition.

A GRADE algorithm based approach is presented along with PV analysis to solve multi objective optimization problem of minimizing real power losses and improving the voltage profile and hence enchancing the Performance of power systems. Real and Reactive power losses are considered as equality constraints. Inequality constraints comprised of generator bus voltages, transformer tap settings, reactive power ratings at the capacitor banks and reactive power generation at generator buses. The GRADE Algorithm based optimization approach is developed using M coding in MATLAB plat form. To illustrate the effectiveness of the GRADE Algorithm based approach, studies are performed in two test systems, viz, IEEE 30 bus system and IEEE 57-bus system for three loading conditions. Results obtained using GRADE Algorithm are compared with the results obtained using firefly algorithm and particle swarm optimization technique. In all the three loading conditions tested, GRADE Algorithm based optimization approach yielded reduced real power loss and improved voltage profile. It is also observed through PV curve using continuation power flow that loadability margin increased considerably when control variable values are tuned using GRADE Algorithm based approach. Hence it is concluded that, the GRADE algorithm performs better than the firefly and particle swarm optimization techniques, in terms of convergence time, reduction in real power losses, improving voltage profile and enchancing the load ability margin of power systems.

G. Kannan,D. Padma Subramaniam,Solai Manokar, (2016) Application of Grade Algorithm Based Approach along with PV Analysis for Enhancement of Power System Performance. Circuits and Systems,07,3354-3370. doi: 10.4236/cs.2016.710286