# A hybrid multi-objective optimization method considering optimization problems in power distribution systems

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## Abstract

Various kinds of new engineering technologies have been studied to realize the low-carbon and sustainable power supply systems all over the world. In actual implementation of these technologies, mostly, there are multiple objectives with trade off relationships among each other, and also various constraints in the achievement of these objectives. Therefore, it should be essential to solve multi-objective optimization problems effectively in the applications of these new technologies in power systems. This paper proposes an improved method to realize multi-objective optimization for critical challenges in advanced power systems. To realize that, in an optimal dispersed generation installation problem, that is, one of effective measures for low-carbon power systems, various optimization methods and their combination methods are evaluated and a hybrid method for evolutionary algorithms was developed. The method can provide improved results compared with other state-of-the-art multi-objective optimization methods.

## Keywords

Dispersed generation Distribution system Evolutional strategy Multi-objective optimization Optimal power flow## 1 Introduction

As various environmental problems such as climate change due to global warming and air pollution have been real problems, the increasing penetration of renewable energy source (RES) generation which does not produce CO_{2} and other substances of concern is common critical challenge all over the world. Therefore, many RES generation systems have been installed in various countries and regions, and various subsidies and preferences also have been implemented.

Japan also plans a large number of RES generation system installation and the plan says that more than 20% of total electric power should be generated from RES and most of them should be from dispersed generations (DGs) located in electric power demand areas [1]. Although RES generation is an effective approach in the aspects of CO_{2} emission reduction, it should be difficult to achieve the grid-parity compared with current generation technologies due to high production cost, low generation efficiency and unstable power capacity etc., despite of many supports from above mentioned subsidies and preferences. The penetration rate of RES is not enough to consider aggressive targets for RES penetration. Therefore, it is important to maximize RES advantages and minimize its disadvantage for the realization of low- carbon society and optimization of multiple objectives with trade-off relationship each other, should be a critical challenge.

In this paper, an effective multiple objectives optimization method is discussed assuming that a distribution power system where many RES DGs are installed. In this discussion, power loss minimization and cost minimization by DG installation are defined as trade-off relation of multiple-objectives because RES DG can reduce CO_{2} emission itself and power loss reduction by DG can also contribute to CO_{2} reduction, while RES DG generation cost is expensive compared with conventional power price generally. Then, optimal solutions are derived by simulations using various state-of-the-art optimization techniques. Results are evaluated and discussed and then, a hybrid multi-objective optimization method combining existing effective multi-objective algorithms for novel power systems is proposed.

### 1.1 Optimization problems in power systems

In order to achieve various purposes and benefits for power systems, a large number of efforts have been provided to the area for effective optimization methods. In all areas of power systems, such as generation, transmission, distribution, and consumption, it is necessary to have appropriate plans to operate, thus it requires optimization tasks under given constraints. In addition, while optimization researches dealt with effective methods for various objectives are important, it should be necessary to evaluate the profitability of new technology implementations considering the shutdown of existing systems and the re-installation of new systems. Generally, the installation benefit and cost have a trade-off relationship and it means that future power systems should be optimized with multi-objectives including the profitability. Therefore, this paper focuses on effective optimization methods not only for the installation benefit of new technology but also for cost minimization.

### 1.2 Single-objective optimization problems

*= (*

**x***x*

_{1},

*x*

_{2},…,

*x*

_{ n }) is an

*n*-dimensional vector;

*f*(

*) is an objective function;*

**x***g*

_{ j }(

*) ≤ 0 are inequality constraints and*

**x***h*

_{ j }(

*) = 0 are equality constraints; functions*

**x***f*(

*),*

**x***g*

_{ j }(

*) and*

**x***h*

_{ j }(

*) are real-valued functions;*

**x***l*

_{ i }and

*k*

_{ i }are the upper and the lower bounds of

*x*

_{ i }, respectively.

- 1)
Particle swarm optimization (PSO)

Particle swarm optimization (PSO) is an evolutional computation technique which is inspired by a bird flocking, fish schooling and swarming theory, and utilizes particle swarms flying in problem space, called the hyperspace [3]. In the iteration process, each particle evolves into optimal or optimal approximation solution adjusting its velocity by the information of its best location and best neighbor location on its historical data. Because all particles share information of optimal solutions, PSO provides well convergence in optimal solutions. Therefore, PSO can be used also for various optimization problems in power systems.

DE, proposed by Storn and Price, is one of evolution strategies (ES) which is a stochastic direct search method and conducts multi-points search using populations. Although the control of mutation step size is required in ES algorithms, DE does not need to control the step size because it adopts a mathematical operation as its mutation using the weighted sum of the base vector and the difference vectors. As same as PSO, various improved algorithms have been proposed in DE. Adaptive DE is the collective term which shows subspecies of the standard DE algorithm targeting for convergence improvement, and various kinds of adaptive DE algorithms have been proposed. JADE is one of these algorithms and implements the mutation strategy, called the “DE/current-to-pbest” with optional archive and controls scaling factor and crossover rate in an adaptive manner [9]. *ε* constrained adaptive DE [10] was proposed to improve the scheme proposed in JADE and the *ε* constrained method, which was the, algorithm to convert unconstrained optimization method into constrained optimization method one using the *ε* level comparison, was applied.

### 1.3 Multi-objective optimization problems

*f*

_{ i }(

*) (*

**x***i*= 1,2,…,

*n*) are objective functions and

*C*(

*) is a constraint condition. In a single objective optimization problem, the best value is apparent because it is possible to be compared between two real numbers in size. Generally, there are trade-off relations among objectives in multi-objective optimization problems and thus optimal solution would not be a unique solution but multiple solutions or infinite population. Therefore, optimal solutions cannot improve the value of a certain objective function without degrading some values of the other objective functions would need to be searched, called the Pareto optimal solutions. If Pareto optimal solutions could be searched, the relation of objective functions would be clarified and better decision making could be made. The surface formed by Pareto optimal solutions is called the Pareto front and three aspects are considered to evaluate Pareto front in [11]: 1) the “convergence”, minimizing the distance from search results to the Pareto front; 2) “uniformity”, maintaining uniform solution distribution; 3) “extensity”, maximizing the extent of solutions following the Pareto front.*

**x**- 1)
PAES [18]

- 2)
Multi-objective PSO

- 3)
PESA, PESA-II and IPESA-II

PESA [16] is the multi-objective optimization algorithm integrating the ideas of strength Pareto evolutionary algorithm (SPEA) [19] and PAES [18], which are two major multi-objective optimization methods. PESA uses a population (archive) which stores an approximation to the Pareto front and an internal population which has candidate solutions same as SPEA, and also maintains hypergrid division which can trace the crowded factor of different areas in the archive as same as PAES. PESA-II [17] is the improved version of PESA and mating-selection processes were implemented not in individual based but in region based. Firstly, a hyperbox is selected and then an individual which is the result of evolutional operations is randomly selected from the hyperbox. IPESA-II [11], is an improved version of PESA-II with three improvements: the maintenance method of the archive, individual maintenance around boundary and the selection of the hyperbox by the crowded factor.

## 2 Effective optimization method in power distribution systems

Some preparation tasks for exploring effective optimization method in power distribution systems are conducted, including problem definition, procedure clarification and data preparation.

### 2.1 Definition of a problem

Many researches have been conducted to solve optimization problems in power system area. Major areas include power system planning and operation, environmentally constrained economic dispatch, state estimation and optimized power flow [20]. For the versatile and essential point of view, benefits or effects maximization of approaches for low-carbon power systems and cost minimization for these approaches should be fundamental trade-off relation objectives. In order to enhance existing power systems into new advanced systems, it must be necessary to achieve both new additional benefit provision to consumers and cost recovery of the investment at the same time. Therefore, this kind of problems is defined as the multi-objective problem in this paper.

The Author's group has conducted some other researches for optimal DG allocations for low-carbon society and it is found that inadequate DG installation would cause power loss increase while effective DG installation would contribute to the realization of very low power loss distribution systems [21]. Therefore, an optimal DG installation is one of critical problems for advanced low- carbon power systems considering most of RES generations, however the correct evaluation of these DGs installation should be the balance of various impacts including cost. In the view of above consideration, this paper deals with optimal DG installation problem considering power loss and cost minimizations. In particular, simulations of power loss and cost reduction by DG location and capacity for a distribution system model are executed and then are evaluated to find the optimal solutions.

### 2.2 Procedure clarification

Multi-objective optimization would be important in future power systems, because the enhancement of power systems would not be realized only by technical advantages, such as improvement of power supply stability with a large number of RES installations, but their cost effectiveness considering added-values would be essential. In most countries and regions, power systems are already one of social infrastructures and provide values to consumers with reasonable price. Therefore, consumers would not pay additional costs without significant additional values and then multi-objective optimization methods for evaluating the cost effectiveness of enhanced technologies should be critical.

- 1)
Distribution model

- 2)
Effective single-objective optimization method in power systems

For the target distribution system with allocated DGs, major single objective optimization algorithms should be used to understand their effectiveness for optimization problems in power systems. As mentioned in above, recently, the population-based descent method has received many attention, thus DE and PSO are selected as base single optimization algorithms in this paper.

- 3)
Enhancement of the effective single-objective optimization method for multi-objective optimization problems in power systems

Multi-objective optimization algorithm is considered on the enhancement of the effective single -objective optimization algorithm selected in the previous step. Considering the enhancement of the effective single objective optimization method, various hybrid approaches using proven major multi-objective algorithms should be discussed. In case that Pareto front would be evaluated on the three aspects: “convergence”, “uniformity” and “extensity”, it was assumed that each multi-objective optimization method had specific ranges in which high quality Pareto front was provided by pre-conducted basic researches and simulations. Therefore, it is expected that the hybrid approach of proven multi-objective algorithms can provide an effective Pareto front for the multi-objective optimization problems in future power systems. Then, the algorithm which finds the best Pareto front in the aspects of “convergence”, “uniformity” and “extensity” will be selected as the best multi-objective optimization algorithm.

### 2.3 Data preparation

- 1)
Distribution system model

*P*= 4.4239(p.u.) and reactive power

*Q*= 3.1053(p.u.) for the total load of

*P*= 4.2300(p.u.) and

*Q*= 2.8870(p.u.). Therefore, total power loss is calculated as

*P*

_{loss}= 0.1939(p.u.) and

*Q*

_{loss}= 0.2183(p.u.) and power loss rate for injected power at slack bus are

*P*: 4.383%,

*Q*: 7.030%, respectively. Parameters for the system model such as branch attributes, load at bases were also referred to [22].

- 2)
Constraint definition for optimization problem

- a.
The number of installation DGs is 2, 3 and 4.

- b.
DG would be installed at one of the buses.

- c.
One DG would be installed per one part where the load would be installed in the same range.

Capacity constraint for each DG and slack bus

Element | max | min | max | min |
---|---|---|---|---|

DG | 4.0 | 0.0 | 2.0 | 0.0 |

Slack bus | 6.0 | 1.0 | 6.0 | 1.0 |

- 3)
Cost related data

Installed DGs and cost parameters

Number of DGs | 2 | 3 | 4 | |
---|---|---|---|---|

DG-1 | DG location | 7 | 5 | 5 |

Fixed/Variable cost for | 0.0/0.5 | 0.0/0.5 | 0.0/0.5 | |

Fixed/Variable cost for | 0.0/0.4 | 0.0/0.4 | 0.0/0.4 | |

DG-2 | DG location | 16 | 13 | 13 |

Fixed/Variable cost for | 0.0/0.5 | 0.0/0.5 | 0.0/0.5 | |

Fixed/Variable cost for | 0.0/0.4 | 0.0/0.4 | 0.0/0.4 | |

DG-3 | DG location | – | 18 | 18 |

Fixed/Variable cost for | – | 0.0/0.5 | 0.0/0.5 | |

Fixed/Variable cost for | – | 0.0/0.4 | 0.0/0.4 | |

DG-4 | DG location | – | – | 55 |

Fixed/Variable cost for | – | – | 0.0/0.5 | |

Fixed/Variable cost for | – | – | 0.0/0.4 | |

Slack power | Fixed/Variable cost for | 0.0/0.3 | 0.0/0.3 | 0.0/0.3 |

Fixed/Variable cost for | 0.0/0.0 | 0.0/0.0 | 0.0/0.0 |

- 4)
Calculation result of power flow by interior point method

Results of OPF by interior point method

No. of DGs | 2 | 3 | 4 | |
---|---|---|---|---|

DG-1 | Location | 7 | 5 | 5 |

Capacity ( | 1.983168 | 1.157973 | 0.854512 | |

Capacity ( | 1.022116 | 0.447595 | 0.244404 | |

DG-2 | Location | 16 | 13 | 13 |

Capacity ( | 1.261669 | 1.393307 | 1.295070 | |

Capacity ( | 0.882551 | 0.976911 | 0.910799 | |

DG-3 | Location | – | 18 | 18 |

Capacity ( | – | 0.688242 | 0.688242 | |

Capacity ( | – | 0.473893 | 0.473893 | |

DG-4 | Location | – | – | 55 |

Capacity ( | – | – | 0.400341 | |

Capacity ( | – | – | 0.267712 | |

Slack power | Capacity ( | 1.000000 | 1.000000 | 1.000000 |

Capacity ( | 1.000000 | 1.000000 | 1.000000 | |

Minimum power loss ( | 0.01484 | 0.00952 | 0.00817 |

## 3 Simulation of single-objective optimization

For the exploring effective methods for optimization problems in power systems, the effective single-objective optimization problem is considered.

### 3.1 Application of proven optimization algorithms

### 3.2 Simulation results of single-objective algorithms

- 1)
PSO

- 2)
DE

Predominant values are converged with around 20 iterations for 2 DGs, around 40 iterations for 3 DGs and around 50 iterations for 4 DGs. In addition, convergence rates to optimal values for all 3 patterns are fast compared with other three algorithms, and “ADE-OPF” provides the best performance among 4 algorithms in our simulation results.

### 3.3 Discussion of simulation results

- 1)
All 4 utilized algorithms can provide good performance for the convergence of predominant solutions and it is confirmed that “CFA-PSO-OPF” and “ADE-OPF”, which are subspecies of original OPF and DE algorithms, respectively, provide better performance compared with original algorithms.

- 2)
“ADE-OPF” is one of adaptive DE algorithms with adequately the highest performance.

- 3)
Because the objective function “Power loss minimization by DGs” is a constrained optimization problem, it is confirmed that these four algorithms can be used for constrained optimization problems.

## 4 Multi-objective optimization problems in power systems

By the enhancement of the effective single objective optimization method, an effective multi-objective optimization method is considered for the evaluation of future advanced power system.

### 4.1 Enhancement for the application of single objective optimization problem

The enhancement of “ADE-OPF” which has confirmed its effectiveness for single-objective optimization problem in power systems was considered to be applied for multi-objective problems.

In the consideration, a methodology which manages multi-objective space efficiently and finds a good approximation set of the Pareto front should be required. Therefore, the utilization of the archive method in the PAES [18], which is called the “PAES-Archive method” in this paper, was used for the effective management of solutions generated in multi-objective space by “ADE-OPF”.

### 4.2 Challenges in hybrid approach of optimization methods

- 1)
Utilization of the adaptive grid in IPESA-II

- 2)
Pareto front creation with the hybrid approach of “ADE-OPF” and “Adaptive-Grid method”

- a.
The mating-selecting method utilized in IPESA-II to create solutions in multi-objective space could not find Pareto front solutions effectively if enough solutions would not exist in the space.

- b.
If solutions in adaptive grid in the multi-objective space would be smaller than 2, “ADE-OPF” would be utilized to create solutions in the space, otherwise mating-selection would be utilized.

- 3)
Optimization testing using H-ADE-IPESA-II method

Using H-ADE-IPESA-II, Pareto front for the multi-objective optimization problem was able to be found effectively.

From the result, H-ADE-IPESA-II which is hybrid approach of Adaptive DE and IPESA-II is one of effective methods for constrained multi-objective optimization problems in power systems.

### 4.3 Discussion of multi-objective optimization results

- 1)
Optimization of power loss and cost by the installation of 3 DGs considering variable cost

- 2)
Optimization of power loss and cost by the installation of 3 DGs considering both fixed and variable cost

With respect to objective function for cost, both fixed and variable costs are considered and the Pareto front is created.

- 3)
Optimization of power loss and cost by the installation of 3 DGs considering discrete DG capacity

With respect to objective function for loss minimization, discrete DG capacity settings are considerable. Therefore, the Pareto front using discrete DG capacity settings are also considered.

## 5 Conclusions

In many regions and countries, huge investment has been made for electric power, and power systems are one of important social infrastructures. Because most people in the regions and countries can adequately use electricity, the cost reduction is one of major requirements recently. It is necessary to enhance current power systems into advance clean and sustainable systems considering current some environmental issues. Therefore, utility companies are required to deal with such benefit and cost optimization problems with trade-off relationships in their objectives.

This paper provides effective tools for multi-objective optimization problems which are essential conditions for problems in power systems, and thus many researches considering new tool development have been conducted.

Because most of multi-objective optimization tools are prepared for non-constrained problems, it is difficult to have adequate results for constrained problems using most of these methods in convergence, uniformity and extensity. In order to solve that, adequate results can be provided by the hybrid approach between adaptive DE method and IPESA-II. Although this paper dealt with only two objective optimization problems, it is assumed that the approach can be applied to more than 3 objectives optimization problems and these problems are conducted in future work. In addition, test data are used in all simulations in this paper, but real multi-objective optimization problems should have additional constraints related to electrical and economical aspects. Therefore, results of the Pareto front provided by the proposed approach and actual decisions made in some actual projects are also needed to be compared in the future.

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