A sufficient stochastic framework based on U-transform to solve the practical reconfiguration strategy

Abstract

This paper proposes a new stochastic method based on u-transform (UT) to capture the uncertainties of the reconfiguration in a correlated environment. The proposed stochastic framework makes use of the idea covariance matrix to reach this goal. Also, in order to reach the maximum efficiency of the reconfiguration strategy, reliability indices are considered along with the traditional targets in the multi-objective framework. Therefore, the objective functions to be optimized are System Average Interruption Frequency Index (SAIFI), Average Energy Not Supplied (AENS), total active power losses and the total network costs. Since the problem investigated is a complex, nonlinear optimization problem, a new optimization method based on krill herd algorithm (KH) is introduced. In addition, a new modification method is proposed for the KH algorithm to improve its total search ability. The feasibility and performance of the proposed method are examined on the IEEE standard test system.

Keywords

U-transform multi-objective reconfiguration krill herd algorithm

Nomenclature

N _Cus

Total number of customers served

λ _i

Average failure rate of the ith component

U _i

Annual outage time of the ith component

La _i

Average load connected to the load point i in kW

State variables vector

N _br

Number of branches

R _i

Resistance of ith branch (Ω)

I _i

Current of ith branch (A)

Tie _i

Status of the ith tie switch

Sw _i

Status of the ith sectionalizing switch

P _Wind,I

Output power of the ith WT

C _Grid

Cost of power production by the grid

C _ENS

Cost of energy not supplied as the reliability cost

C _Wind

Cost of power production by WTs

Annual rates of benefit

Loading factor

| P_{ij}^{Line} |

Absolute active power flow over the distribution lines (kW)

P_{ij, max}^{Line}

Maximum transmission active power between the ith and the ith nodes (kW)

N _f

Number of feeders

P _i

Net injected active power component at the ith bus

Q _i

Net injected reactive power component at the ith bus

V _i

Amplitude of the voltage at the ith bus

θ _i,j

Angle of the branch admittance between the ith and jth buses

Y _i,j

Amplitude of the branch admittance between the ith and jth buses

V _max

Maximum value of voltage magnitudes of ith bus (Volt)

V _min

Minimum value of voltage magnitudes of ith bus (Volt)

g_i (X)

The ith objective function

|I_f, i|

Absolute current amplitude of the ith feeder

I_{f, i}^{max}

Maximum current amplitude of the ith feeder

W ^k

kth weighting factor of the sample point

P_xx/P_yy

Covariance of input variable X/output variable Y

PDF

Probability function of the uncertain variable

Mean value of PDF

Standard deviation of PDF

N _func

Number of objective functions

MF _gi(X)

Membership function for ith objective function

Number of uncertain variables in the problem investigated

N _eq

Number of equality constraints

N _ueq

Number of inequality constraints

g_{i}^{min}

Minimum value of the ith objective function

g_{i}^{max}

Maximum value of the ith objective function

Scale parameter of Weibull PDF

τ _i

Weighting factor given to the ith objective function

N _p

Number of the Pareto solutions in the repository

V_{ind, i}^{m}

induced velocity of ith krill at the mth movement

V_{ind}^{max}

maximum induced velocity

V_{frg, i}^{m}

foraging velocity of ith krill at mth movement

α _ind,I

attractive/repulsive tendency factor

small positive number

N _p

population size

ρ, κ

empirical constant factors

V_{ind, i}^{m}

induced velocity of ith krill at the mth movement

V_{diff, i}^{m}

diffusion velocity of ith krill at mth movement

V_{r, i}^{m}

resultant velocity of ith krill at mth movement

W _i

weighting factor of ith individual

ω _ind/frg/diff

inertia of induction/ foraging/ diffusion motion

Iter _max

maximum iteration

u_j/l_j

upper/lower bound of jth control variable

1 Introduction

Distribution system as the last link between the generation and consumption plays a precious role in the power quality and reliability of the electrical services. Therefore, any improvement in the distribution system reliability can improve the quality of the electrical services greatly. One of the significant methods that can be a possible and available strategy for improving the reliability of the distribution systems is reconfiguration. Optimal reconfiguration is defined as the process of changing the topology of the distribution system using some remotely controlled switches called tie (normally open) and sectionalizing (normally closed) switch [1]. One significant limitation in the reconfiguration is the radiality of the network which should be preserved before and after the reconfiguration. Optimal reconfiguration can improve the distribution network from different aspects such as loss reduction, voltage improvement, load balance enhancement, reliability improvement, etc. [2]. Therefore, several researches have been implemented in the last years to assess different aspects of the reconfiguration strategy.

In the area of power loss reduction, approaches such as neural network [7], optimum flow pattern [8], graph theory [9], brute-force approach [10], heuristic techniques [11], expert systems [12], ant colony optimization algorithm [13, 14], hybrid simulated annealing algorithm and tabu search [15] are among the most significant techniques. With the purpose of investigating the effect of reconfiguration on the other targets, increasing the load balance [16] and minimization of the voltage deviation [17] are addressed in the literature. According to this short summery, much attention has been paid to the traditional and operational objective functions. In the area of reliability, reference [18] searched the effect of optimal reconfiguration on the system average interruption frequency index (SAIFI) and system average interruption duration index (SAIDI). While this work has assessed the reliability indices, but neglecting some significant targets such as power loss minimization is a big shortage. Genetic algorithm (GA) is used in [19] to solve the reconfiguration problem for improving the reliability of the system. Here again neglecting the effect of reconfiguration on the power losses and voltage of the buses is a deficiency. In addition, neglecting the uncertainty effects on the reconfiguration formulation is another issue that should be discussed.

According to the above discussion, this paper aims to solve the multi-objective reconfiguration problem considering both the traditional and reliability indices. The proposed framework will optimize the SAIFI and average energy not supplied (AENS) and power losses and total cost function. One of the new technologies that can affect the optimal reconfiguration is wind turbine (WT). The volatile and uncertain nature of the wind speed affects the output power of WT greatly. Moreover, the forecast error in the value of the active and reactive loads is another source of uncertainty that should be considered in the new analyses. In order to consider these uncertainties, this paper proposes a new stochastic method using the u-transform (UT). UT was first introduced in [16] to be used for modeling uncertainty in the power systems. The main feature of UT is its simple idea in modeling the correlated uncertainty using the P _xx covariance matrix. Because of the high complexities and nonlinearities of the problem investigated, a new optimization algorithm based on krill herd (KH) algorithm is proposed too. KH was first introduced in 2012 as a ne optimization evolutionary algorithm to solve the nonlinear problems using the behavior of krill [17]. Also a new modification method is designed to make the local and global searches of the KH more purposely. The performance of the proposed method is examined on the IEEE 32- bus test system. The rest of this paper is organized as follows: Section 2 describes the problem formulation. Section 3 explains the basics and theories of UT. The multi-objective framework based on fuzzy is described in Section 4. The KH optimization algorithm and its modification are explained in Section 5. The simulation results on the 32-bus test system are given in Section 6. Finally, the main concepts and conclusions are described in Section 7.

2 Multi-objective formulation of reconfiguration

2.1 Objective functions

The objective functions include both traditional and reliability targets as follows:

- Minimization of Average Energy Not Supplied (AENS):

AENS is a reliability index that deals with the energy and load and is calculated as follows: $\begin{matrix} ENS = \sum_{i = 1}^{N_{Cus}} {La}_{i} U_{i} \\ g_{1} (X) = AENS = \frac{ENS}{\sum_{i = 1}^{N_{Cus}} N_{i}} \end{matrix}$ (1)

In this formulation, X includes the optimal status of the tie and sectionalizing switches: $\begin{matrix} X = [\bar{Tie}, \bar{Sw}]; \\ \bar{Tie} = [{Tie}_{1}, {Tie}_{2}, {Tie}_{3}, \dots, {Tie}_{N_{tie}}] \\ \bar{Sw} = [{Sw}_{1}, {Sw}_{2}, {Sw}_{3}, \dots, {Sw}_{N_{sw}}]; \end{matrix}$ (2)

It should be noted that WT is a non-dispatchable device and thus is not considered as a control variable.

- Minimization of System Average Interruption Frequency Index (SAIFI):

SAIFI is a significant reliability index that tries to reduce the average failures of the system which can affect the consumers and is calculated as below: $g_{2} (X) = SAIFI = \frac{\sum_{i = 1}^{N_{Cus}} λ_{i} N_{i}}{\sum_{i = 1}^{N_{Cus}} N_{i}}$ (3)

- Minimization of total active power losses: The most basic target of using reconfiguration was to minimize the feeder resistive losses as follows: $g_{3} (X) = P_{loss} (X) = \sum_{i = 1}^{N_{br}} R_{i} \times {| I_{i} |}^{2}$ (4)

- Minimization of the total cost:

From the market view, the significant objective is to minimize the total network costs that incorporate the cost of power produced by the grid, cost of power produced by wind power sources and cost of ENS as follows: $g_{4} (X) = Cost = {Price}_{Grid} + {Price}_{Wind} + {Price}_{ENS}$ (5) $\begin{matrix} {Price}_{ENS} = C_{ENS} \times ENS \\ {Price}_{Grid} = C_{Grid} \times P_{sub} \end{matrix}$ (6) The cost associated with the WT is calculated as follows: ${Price}_{Wind} = \sum_{i = 1}^{N_{Wind}} C_{Wind, i} = a + b \times P_{Wind, i}$ (7) $\begin{matrix} a = \frac{Capitalcost ($ / kW) * Capacity (kW) * Gr}{Life time (Year) * 365 * 24 * LF} \\ b = Fuel cost ($ / kWh) + Operation & MaintenanceCost ($ / kWh) \end{matrix}$ (8)

2.2 Limits and constraints

-maximum power flow capacity of each feeder is calculated as follows: $| P_{ij}^{Line} | < P_{ij, max}^{Line}$ (9)

-power equality constraints are the active and reactive power balance in each bus as follows: $\begin{matrix} P_{i} = \sum_{i = 1}^{N_{bus}} V_{i} V_{j} Y_{ij} cos (θ_{ij} - δ_{i} + δ_{j}) \\ Q_{i} = \sum_{i = 1}^{N_{bus}} V_{i} V_{j} Y_{ij} sin (θ_{ij} - δ_{i} + δ_{j}) \end{matrix}$ (10)

- maximum and minimum voltage level on the buses: $V_{min} \leq V_{i} \leq V_{max}$ (11)

- maximum power flow capacity of the main feeders: $| I_{f, i} | \leq I_{f, i}^{max}; i = 1, 2, \dots, N_{f}$ (12)

-keeping the radial topology of the network. The topology of the distribution grid is radial and should be kept after the reconfiguration. Therefore, each time a loop is formed, a switch in that loop is opened to make it radial.

3 Probabilistic load flow using the U-transform

With the adventure of new technologies as well as the enlarging size of the electrical systems, more

uncertainty is injected in the operation and management strategies [18, 19]. Therefore, the transition from the deterministic framework to the stochastic framework is an evitable issue [20]. In this regard, this paper proposes a ne method based on UT to handle the uncertainties of the problem. Simplicity, easy coding and ability of modeling the correlated environment are some of the main characteristics of this technique. In the UT method, each uncertain parameter is replaced by its appropriate probability density function (PDF). Then, the UT method solves the stochastic problem with n number of uncertain variables by converting it to 2n+1 number of deterministic problems. The mean value of the uncertain parameter is supposed as μ and the standard deviation and the correlation of the parameters are modeled in the covariance P _xx. If the nonlinear problem is Y = f (X), the target is to reach the mean μ _y and covariance matrix P _yy of the output y.

4 Multi-objective optimization based on fuzzy

Generally, a multi-objective optimization problem can be shown as follows: $\begin{matrix} min G = [g_{1} (X), g_{2} (X), \dots, g_{N_{func}} (X)]^{T} \\ s . t . \\ q_{i} (X) < 0 i = 1, 2, \dots, N_{ueq} \\ h_{i} (X) = 0 i = 1, 2, \dots, N_{eq} \end{matrix}$ (13)

Mathematically, a multi-objective optimization problem can have many optimal solutions each of them called a Pareto optimal solution. In order to understand the Pareto optimal solution, suppose X ₁ and X ₂ are two possible solutions in the problem search space Ω. The solution X ₂ dominates solution X ₁ if the following two conditions are satisfied [21 –24]: $\begin{matrix} 1) \forall j \in {1, 2, \dots, N_{func}}, g_{j} (X_{2}) \leq g_{j} (X_{1}) \\ 2) \exists k \in {1, 2, \dots, N_{func}}, g_{k} (X_{2}) < g_{k} (X_{1}) \end{matrix}$ (14)

In order to find the set of Pareto solutions in the problem, all the objective functions should be unified in the same framework. Here we make use of the membership function idea with trapezoidal shape asfollows: ${MF}_{g_{i}} (X) = {\begin{matrix} 1 & for & g_{i} (X) \leq g_{i}^{min} \\ 0 & for & g_{i} (X) \geq g_{i}^{max} \\ (g_{i}^{max} - g_{i} (X)) / (g_{i}^{max} - g_{i}^{min}) & for & g_{i}^{min} \leq g_{i} (X) \leq g_{i}^{max} \end{matrix}$ (15)

Here $g_{i}^{min}$ and $g_{i}^{max}$ are calculated by the single objective optimization of ith objective function, separately. All the Pareto points are stored in the repository and sorted using the below equation: $N_{MF} (j) = \frac{\sum_{i = 1}^{N_{func}} τ_{i} \times {MF}_{gi} (X_{j})}{\sum_{j = 1}^{N_{p}} \sum_{i = 1}^{N_{func}} τ_{i} \times {MF}_{gi} (X_{j})}$ (16)

5 Optimization approach based on modified krill herd (MKH)

KH algorithm was introduced in 2012 by Gandomi for modeling the krill herd searches for the food using the evolutionary behavior of the population. The main ideas of the KH in comparison with honey bee mating optimization [25 –28], particle swarm optimization (PSO) [29, 30], teacher learning algorithm [31], cuckoo search algorithm [32] and firefly algorithm [34, 35] are easy implementation, fast convergence and few adjusting parameters. After generation of the krill population and sorting the krill based on the scoring function, the best krill is stored as X ^b. Then the position of the krill population is updated using three concepts: 1) induction, 2) foraging and 3) random diffusion.

- Induction movement: This movement simulates the influence of different krill on each other. It is clear that the most effect is induced by the surrounding krill. In this manner, the velocity of ith krill at mth iteration is updated as follows [17]: $V_{ind, i}^{m} = α_{ind, i} V_{ind, i}^{max} + ω_{ind} V_{ind, i}^{m - 1}$ (17) $\begin{matrix} α_{ind, i} = \sum_{j = 1}^{N_{s}} [\frac{f_{i} - f_{j}}{f_{w} - f_{b}} \times \frac{X_{i} - X_{j}}{| X_{i} - X_{j} | + ɛ}] + \\ 2 [rand (.) + \frac{i}{{Iter}_{max}}] f_{i}^{b} X_{i}^{b} \end{matrix}$ (18)

- Foraging movement: This movement simulates the effort of the krill for finding food based on their last experiences and the position of the food as follows [17]: $\begin{matrix} V_{frg, i}^{m} & = 0.02 [2 (1 - \frac{i}{{Iter}_{max}}) f_{i} \frac{\sum_{i = 1}^{N_{s}} \frac{X_{i}}{f_{i}}}{\sum_{i = 1}^{N_{s}} \frac{1}{f_{i}}} \\ + f_{i}^{b} X_{i}^{b}] + ω_{frg} V_{frg, i}^{m - 1} \end{matrix}$ (19)

- Diffusion movement: This movement simulates a random distributed movement for the krill as follows: $V_{diff, i}^{m} = υ \times ω_{diff}$ (20)

Here υ is distributed uniformly [−1, 1].

According the above three movements, the position

The original KH algorithm is also equipped with the crossover and mutation operators from the genetic algorithm. Therefore, after that the positions of the krill are updated using the above equations, the crossover and mutation operators are used to improve the position of the krill once more.

As mentioned before in the introduction section, we have provided a new modification method for the KH that can improve its total search ability in both local and global searches. The proposed modification method is a local search around each solution in the krill population. Considering X _old as the previous position of the krill, the below equation is used to update the position of each krill: $X_{new} = X_{old} + ɛ A^{k}$ (21)

The above equation simulates a random search in the local areas of each krill.

6 Simulation results

In order to see the performance of the proposed method, Baran and Wu 32-bus distribution test system containing two feeders, five looping branches, five tie switches and 32 sectionalizing switches are employed. The voltage level of the network is 12.66 kV [21]. For better comparison, the initial restive losses of the network are 202.67 (kW). The single line diagram of the test system is depicted in Fig. 1 wherein the tie switches are shown by dotted lines.

This test system supplies 5058.25 (kW) and 2547.32 (kVar) loads, respectively. In order to see the WT on the network, four WTs with the capacity of 300 kW are located on buses 4, 11, 17 and 22. The reliability data of the test system are taken from [22]. Table 1 shows the data of the customers on each bus. The number of krill is chosen to be 25 with the maximum iteration of 200.

In order to see the performance of MKH algorithm, the first part of the simulations is in the deterministic framework neglecting the uncertainty. Table 2 shows the results of single-objective optimization of the power losses function. Here the WTs are not considered yet. According to these results, the proposed MKH algorithm could reach the global optimal solution that is found by the well-known methods in the area.

In Table 3, the simulation results of optimization for other targets are given. For better comparison, the results of optimization by particle swarm optimization (PSO) and GA are also given in the table. According to these results, the proposed method has shown better performance than the PSO and GA. The optimal switching pattern is also shown in Table 3. From these results, the usefulness of reconfiguration for improving the reliability of the distribution system is clear.

In order to see the effect of WT on the objective functions, Table 4 shows the results of single-objective optimization of the objectives considering WT in the network. By comparing the results of Table 4 with the results of Tables 2 and 3, it is deuced that the use of WT can improve the network from both the operation and reliability points of view. Of course, cost function can not be compared since this function can not be calculated without WT.

In the second part of the simulation, the results of multi-objective optimization are discussed. As mentioned before, the idea of Pareto points are employed here. For better comparison, the three-dimensional plot of the Pareto points are shown in Figs. 2 to 4. In each of these figures, one of the objective functions is neglected. For example in Fig. 3, the Pareto points are plot for multi-objective optimization neglecting SAIFI. From these figures, the conflicting behavior of the objective functions can be deduced easily. The results of single-objective optimization are also shown in these figures. Each of these points can be assumed as a promising optimal solution for the problem in hand. In other word, these figures show that a suitable framework is required to optimally choose from the set of Paretopoints.

7 Conclusion

This article proposed a new stochastic framework based on MKH to solve the optimal reconfiguration in the smart grids. The proposed method can capture the uncertainties of the random parameters such as output power of WT and value of the active and reactive loads suitably. The simulation results on the IEEE 32-bus test system showed the positive effect of reconfiguration on the reliability indices such as AENS and SAIFI. Also, the usefulness of reconfiguration strategy for reducing the total network losses and costs can be deduced too. From the optimization view, the necessity of considering all targets in the same framework was sensed since their behavior can be in conflict with each other. The superiority of the MKH over the other traditional algorithms such as PSO, GA and original KH was seen too.

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