Optimal Planning for the Development of Power System in Respect to Distributed Generations Based on the Binary Dragonfly Algorithm

Abstract

With the increasing number of population and the rising demand for electricity, providing safe and secure energy to consumers is getting more and more important. Adding dispersed products to the distribution network is one of the key factors in achieving this goal. However, factors such as the amount of investment and the return on the investment on one side, and the power grid conditions, such as loss rates, voltage profiles, reliability, and maintenance costs, on the other hand, make it more vital to provide optimal annual planning methods concerning network development. Accordingly, in this paper, a multilevel method is presented for optimal network power expansion planning based on the binary dragonfly optimization algorithm, taking into account the distributed generation. The proposed objective function involves the minimization of the cost of investment, operation, repair, and the cost of reliability for the development of the network. The effectiveness of the proposed model to solve the multiyear network expansion planning problem is illustrated by applying them on the 33-bus distribution network and comparing the acquired results with the results of other solution methods such as GA, PSO, and TS.

1. Introduction

In traditional networks, the transmission and distribution unit was utilized only to transfer the energy from the production unit to consumers, but with an increase in the use of a distributed generation, a part of the production was also assigned to the distribution system. According to definition [1], the distributed generations (DGs) are small production units associated with the distribution network. There are also other definitions describing the DGs, for example, [2,3,4]. On the other hand, the use of DGs changes the features of the distribution network [5].

With the increase in the population and the demand for electricity load, the need to supply safe and secure energy for consumers is becoming necessary more than ever. Adding DGs to the distribution network is one of the key factors to achieve this aim, which has several benefits such as preventing a sudden change in prices in the restructured markets [6], shorter production times duration than in comparison with large power plants, emission reduction, etc. [7]. To maximize the use of DGs, they should be placed at the optimum size and location [8,9]. Solving the multiyear network expansion planning (MNEP) problem is difficult due to the existence of large variables and complex mathematical model [10]. Therefore, a comprehensive model to solve the problem is highly needed. The objective function of the problem could include different factors such as investment cost, operating and maintenance cost, and reliability cost. These factors could be integrated with the coefficients weight or be solved through multi-objective models such as the dynamic ant colony search algorithm [11,12]. Several methods are utilized to solve this problem; for instance, integer linear programming [13,14], nonlinear programming [15,16], and dynamic programming [17,18]. Due to the complexity of the problem and the large number of its variables, the use of metaheuristic to obtain the best objective function value has been widely considered. The genetic algorithm (GA), particle swarm optimization algorithm (PSO), simulated annealing algorithm (SA), and other algorithms [19,20,21,22] show a small fraction of the application of metaheuristic to solve the problem. Favuzza et al. believe that it would be better to place the DGs and improve the feeders simultaneously, rather than separately solving these two problems. In this research, an intelligent method is proposed in which the DGs are employed as the alternatives [11]. Ouyang et al. proposed a method to develop a distribution network along with the DGs. In this research, the problem is solved taking into account the growth of prices and the uncertainty of DGs [23]. Peker et al. proposed a two-step stochastic programming model for the reliability constrained power system expansion planning. The model has been applied on the IEEE 118-bus power system and IEEE reliability test system [24]. Sharma and Balachandra have provided a model for solving the problem considering the uncertainty of the renewable resources [25]. Ahmadigorji and Amjady used the binary whale optimization algorithm to solve the expansion planning of distribution networks based on the multiyear DG-incorporated framework [26]. Essallah et al. presented a new model for the sizing and placement of distributed generation in order to ensure the reduction in total power losses and voltage stability in the power grid [27]. Vita provided a new model based on the decision-making algorithm in order to optimize the size and placement of distributed generation [28].

The main contributions and innovations of this paper are considered as follows:

(a)

A multilevel optimization method for optimal network power expansion planning has been proposed.

(b)

A new version of binary dragonfly’s algorithm has been proposed.

(c)

Considering distributed generation as renewable energies in power expansion planning.

(d)

The proposed objective function involves the minimization of the cost of investment, operation, and repair, plus the cost of reliability for the development of the network.

The structure of this study is summarized as follows: The mathematical formulation for multiyear network expansion planning is provided in Section 2. Section 3 presents the dragonfly optimization algorithm. The simulation and results are denoted in Section 4. The conclusion is summarized in Section 5.

2. Mathematical Formulation for Multiyear Network Expansion Planning

2.1. Problem Assumptions

In the present paper, each distributed generator could produce separately. The cost of investment and maintenance has been considered too.

2.2. Objective Function

In this study, the objective function is defined in a way that the cost of investment, operation and repair, along with the reliability cost are minimized in the presence of the DGs and its impact on the network. The main goal is to provide the best schedule for installing the DGs and network development within a specified time. The first phase of the objective function is the cost of investment including the installation cost of DGs and reinforcement of the feeder. The second phase is the operating cost, which includes the cost of purchasing power from the upstream grid, the operating cost of the DGs, the maintenance cost, and the cost of losses. The third phase of the objective function is the reliability value consisting of the unsupplied energy of the consumer. Finally, the mathematical model of the objective function is shown as follows:

$$MinF=IC+OC+RC$$

(1)

where IC and OC are the investment cost and operation cost, respectively. The mathematical formulations are as follows:

$$IC={\displaystyle \sum _{i=1}^{N_y}{\displaystyle \sum _{j=1}^{N_b}\gamma \left(i\right)\times \left\{\begin{array}{l}\left[In{v}^{DG}\times I{C}_j\times k_{g,j}\times \left(u\left(i-Y_j^{DG}+1\right)-u\left(t-Y_j^{DG}\right)\right)\right]+\\ \left[F{R}_j\times F{L}_j\times k_{r,j}\times \left(u\left(i-Y_j^{rein}+1\right)-u\left(t-Y_j^{rein}\right)\right)\right]\end{array}\right\}}}$$

(2)

$$OC={\displaystyle \sum _{i=1}^{N_y}{\displaystyle \sum _{j=1}^{N_b}\gamma \left(i\right)\times \left\{\begin{array}{l}{\displaystyle \sum _{ls=1}^{N_{ls}}\left[{\pi }_{i,ls}\times P{U}_{i,ls}\times T{D}_{ls}\right]}+\\ {\displaystyle \sum _{i=1}^{N_y}{\displaystyle \sum _{J=1}^{N_b}\left[Op{e}^{DG}\times P{G}_{i,j,ls}\times {TD}_{ls}\right]}}\end{array}\right\}}}$$

(3)

Moreover, the converting function γ(t) to return the net present value of future cost is computed as follows:

$$\gamma \left(t\right)=\frac{1}{{\left(1+Dr\right)}^t}$$

(4)

where Dr is the discount rate, Inv^DG is the investment cost of the DG, and IC is the installation capacity. K_g,i and k_r,i are binary variables to represent the installation decision of DG at bus i and the decision for reinforcement of the feeder section before bus i, respectively. U(t) is the unit step function ($\mathrm{if}t>0\to U\left(t\right)=1,\mathrm{O}\mathrm{therwise}U\left(t\right)=0$). FR is the reinforcement cost of the feeder section. π_i,ls is the electricity price for purchasing power from the upstream grid at load level ls of year i. PU_i,ls is the generated active power from the upstream grid at load level ls of year i. TD_ls is the time duration of load level ls. Ope^DG is the operation and maintenance cost of the DG. PG_i,j,ls is the generated active power of DG in bus j and load level ls of year i. N_y, N_b, and N_ls are the number of years of the planning period, number of load buses, and number of load levels, respectively.

$${\pi }_{i,ls}={\left(1+IR\right)}^i\times {\pi }_{0,ls}$$

(5)

where ${\pi }_{0,ls}$ is the electricity price for purchasing power from the upstream grid at load level ls of year 0.

$$P{U}_{i,ls}=Deman{d}_{i,ls}+P_{i,ls}^{loss}-{\displaystyle \sum _{j=1}^{N_b}P{G}_{j,i,ls}}$$

(6)

where Demand_i,ls is the total load demand of the distribution network at load level ls of year i. The reliability cost (RC) is calculated through Equation (7) [26]:

$$RC={\displaystyle \sum _{i=1}^{N_y}\gamma \left(i\right)\times VOL{L}_i\times EEN{S}_i}$$

(7)

$$VOL{L}_i={\left(1+IR\right)}^i\times VOL{L}_0$$

(8)

$$EEN{S}_i={\displaystyle \sum _{ls=1}^{N_{ls}}\frac{{TD}_{ls}}{8760}}\times \left\{{\displaystyle \sum _{j=1}^{N_b}{\sigma }_j\times FLj\times \left[S{P}_{j,i,ls}+TP{P}_{j,i,ls}\right]\times T_{rs}+U{P}_{j,i,ls}\times T_{rp}}\right\}$$

(9)

where EENS_i is the expected energy not supplied of the distribution network in year i. VOLL₀ is the value of lost load in the base year. σ_j is the average failure rate of the feeder section located before bus i. FL_j is the length of the feeder section located before bus i. SP is the set of loads which can be restored by DG. UP is the set of loads which cannot be restored by DG. TPP is the set of loads which are shed after the fault occurrence on the ith feeder section and can be restored by the upstream grid. T_rs is the required time to detect the fault location and isolating in the faulted feeder section. T_rp is the required time to repair the faulted feeder section.

2.3. Problem Constraints

The nodal power balance for all buses should be satisfied by the following equations:

$$\begin{array}{l}P{U}_{i,ls}={\displaystyle \sum _{j=1}^{N_t}\left|V_{1,i,ls}\right|\left|V_{j,i,ls}\right|}\left|A{D}_{1,j}\right|\cos \left({\delta }_{1,i,ls}-{\delta }_{j,i,ls}-v_{1j}\right)\\ \forall ls\in N_{ls},{\forall }_i\in N_y\end{array}$$

(10)

$$\begin{array}{l}Q{U}_{i,ls}={\displaystyle \sum _{j=1}^{N_t}\left|V_{1,i,ls}\right|\left|V_{j,i,ls}\right|}\left|A{D}_{1,j}\right|\sin \left({\delta }_{1,i,ls}-{\delta }_{j,i,ls}-v_{1j}\right)\\ \forall ls\in N_{ls},\forall i\in N_y\end{array}$$

(11)

$$\begin{array}{l}P{G}_{b,i,ls}-P{D}_{b,i,ls}={\displaystyle \sum _{j=1}^{N_t}\left|V_{b,i,ls}\right|\left|V_{j,i,ls}\right|}\left|A{D}_{b,j}\right|\cos \left({\delta }_{b,i,ls}-{\delta }_{j,i,ls}-v_{bj}\right)\\ \forall b\in N_b,\forall ls\in N_{ls},\forall i\in N_y\end{array}$$

(12)

$$\begin{array}{l}Q{G}_{b,i,ls}-Q{D}_{b,i,ls}={\displaystyle \sum _{j=1}^{N_t}\left|V_{b,i,ls}\right|\left|V_{j,i,ls}\right|}\left|A{D}_{b,j}\right|\sin \left({\delta }_{b,i,ls}-{\delta }_{j,i,ls}-v_{bj}\right)\\ \forall b\in N_b,\forall ls\in N_{ls},\forall i\in N_y\end{array}$$

(13)

where Equations (10) and (11) demonstrate the constraints of power balance in the reference bus. Equations (12) and (13) illustrate the constraints of power balance for different load buses (from 1 to N).

2.3.1. Voltage Constraint

The voltage magnitude of each bus should be within the secure and admissible operating range:

$$\begin{array}{l}{V}_{\min }\le \left|V_{j,i,ls}\right|\le V_{\max }\\ V_{\max }=1.05\\ V_{\min }=0.95\end{array}$$

(14)

In order to increase the performance of the optimization algorithm, the voltage constraint has been added to the objective function as a penalty part. The new version of the objective function is calculated as the following equation:

$$\begin{array}{l}MinF=IC+OC+RC\\ +{\displaystyle \sum _{j=1}^{N_b}\max \left(V_j-1.05,0\right)\times 100}\\ +{\displaystyle \sum _{j=1}^{N_b}\max \left(0.95-V_j,0\right)\times 100}\end{array}$$

(15)

where V_j is the voltage of jth bus.

2.3.2. Thermal Capacity of Feeders

The feeders’ power must be less than their rated capacity:

$$\begin{array}{l}\left|S_{j,i,ls}\right|\le \left(1+k_{r,j}\times u\left(i-Y_j^{rein}+1\right)\right)\times \left|S_j^M\right|\\ \forall b\in N_b,\forall ls\in N_{ls},\forall i\in N_y\end{array}$$

(16)

In this study, the feeders’ reinforcement is supposed as doubling their thermal capacity. The total capacity of distributed generation on the grid should not exceed its maximum capacity:

$${\displaystyle \sum _{j=1}^{N_b}\left[k_{g,j}\times I{C}_j\times u\left(i-Y_j^{DG}+1\right)\right]\le IL\times Deman{d}_i}$$

(17)

where IL is a factor that determines the capacity constraints of the DGs per year.

3. Dragonfly Optimization Algorithm

In this paper, the dragonfly algorithm (DA) is utilized to solve the MNEP problem. The dragonfly is an insect of the dragonfly or Odonata. This insect is claimed to be one of the oldest winged insects. The fastest insect is Odonata, which is a kind of Australian dragonfly. Dragonflies have special conditions and special predator organs in terms of larvae and puberty, as well as the predatory features, and spend most of their life in a neonatal period [29]. In static groups, the dragonflies form a small group to cover a small area for hunting their own food [30].

However, in dynamic groups, a large number of dragonflies form a large group and go a long way in hunting [31]. The shape of the legs of this insect is in the form of a basket to make hunting easier, and the mouthpieces of the insect are also very powerfully rodent. Dragonflies are fast hunters that hunt their prey in the air. They have exceptionally high vision powers and can track the hunter’s insects and easily hunt with their blubber legs. Mirjalili introduced the dragonfly algorithm in 2016 [32]. The group behavior of dragonflies consists of three primitive principles [33]: (I) Separation, which implies the static collision avoidance of the neighbourhood individuals. (II) Alignment, which represents the velocity matching of neighbourhood individuals. (III) Cohesion, which refers to the tendency of neighbourhood individuals towards the center of the mass.

Figure 1 indicates the main factors in position updating of individuals in swarms. Each of these five factors is mathematically formulated as follows:

The separation equation:

$$S_i=-{\displaystyle \sum _{n=1}^N{X}_i-X_n}$$

(18)

where X_i and X_n are the position of the current individual and the position nth neighbouring individual, respectively. N is the number of neighbouring individuals.

The alignment equation:

$$A_i=\frac{{\displaystyle \sum _{n=1}^N{V}_n}}{N}$$

(19)

where V_n presents the velocity of nth neighbouring individual.

The cohesion equation:

$$C_i=\frac{{\displaystyle \sum _{n=1}^N{X}_n}}{N}-X_i$$

(20)

The attraction towards a food source equation:

$$F_i=X^{+}-X$$

(21)

$X^{+}$ is the position of the food source.

The distraction outwards of an enemy equation:

$$E_i=X^{-}+X$$

(22)

$X^{-}$ is the position of the enemy.

In this paper, it is assumed that the behaviour of the dragonflies is a combination of these five corrective patterns.

$$\Delta X_{t+1}=\left(s{S}_i+a{A}_i+c{C}_i+f{F}_i+e{E}_i\right)+w\Delta X_t$$

(23)

where s, a, c, f, e are weight coefficients of the five main factors (separation, alignment, cohesion, attraction food, and the position of enemy). In addition, S_i, A_i, C_i, F_i, E_i are the separation, alignment, cohesion, food source, and position of enemy of ith individual, respectively. w is the inertia weight, and t is the iteration counter.

To improve the performance of the algorithm, the position of dragonflies is updated based on the following equation:

$$X_{t+1}=X_t+L\left(d\right)\times X_t$$

(24)

where t and d are the iteration and the dimension of the position vectors, respectively. The dragonflies are required to fly around the search space using a random walk called Levy flight [32]. L is the Levy flight that can be calculated as follows:

$$L\left(x\right)=0.01\times \frac{r_1\times \sigma }{\left|r_2\right|\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.}$$

(25)

where r₁, r₂ are two random numbers in [0, 1], β is a constant, and σ is calculated as follows:

$$\sigma ={\left(\frac{\Gamma \left(1+\beta \right)\times \sin \left(\raisebox{1ex}{$\pi \beta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}{\Gamma \left(\raisebox{1ex}{$1+\beta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\times \beta \times 2^{\left(\raisebox{1ex}{$\beta -1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.},\Gamma \left(x\right)=\left(x-1\right)!$$

(26)

The dragonfly algorithm flowchart is presented in Figure 2 [34].

Binary Version of the Dragonfly Algorithm

In continuous search spaces of the dragonfly algorithm, the search agents of DA get a new position by adding the step vectors. However, in the binary space, search agents can only obtain either 0 or 1.

The reference [35] concluded that the best way to solve this problem is to use the conversion function. Reference [36] showed that the v-shaped transfer functions are better than the s-shaped transfer functions in terms of exploitation. The following function has been used in this study:

$$T\left(\Delta X\right)=\left|\frac{\Delta X}{\sqrt{\Delta X^2+1}}\right|$$

(27)

The search position is updated as follows:

$$X_{t+1}=\{\begin{array}{l}-X_{t}r<T\left(\Delta X_{t+1}\right)\\ X_{t}r\ge T\left(\Delta X_{t+1}\right)\end{array}$$

(28)

where r is a random number in the interval [0, 1].

4. Simulation and Results

The proposed optimization method is tested on a 33-bus IEEE-standard network to solve the multiyear network expansion planning problem. The single-line diagram of this network is shown in Figure 3. The network consists of 32 load buses, a reference bus, and 32 lines. The data of this network is visible in

Table 1. The data of the IEEE 33-bus distribution network [37].

Number of Buses		R (ohm)	X (ohm)	P (kW)	Q (kVAR)
From Bus i	To Bus j
1	2	0.0922	0.047	100	60
2	3	0.493	0.2511	90	40
3	4	0.366	0.1864	120	80
4	5	0.3811	0.1941	60	30
5	6	0.819	0.707	60	20
6	7	0.1872	0.6188	200	100
7	8	0.7114	0.2351	200	100
8	9	1.03	0.74	60	20
9	10	1.044	0.74	60	20
10	11	0.1966	0.065	45	30
11	12	0.3744	0.1238	60	35
12	13	1.468	1.155	60	35
13	14	0.5416	0.7129	120	80
14	15	0.591	0.526	60	10
15	16	0.7463	0.545	60	20
16	17	1.289	1.721	60	20
17	18	0.732	0.574	90	40
2	19	0.164	0.1565	90	40
19	20	1.5042	1.3554	90	40
20	21	0.4095	0.4784	90	40
21	22	0.7089	0.9373	90	40
3	23	0.4512	0.3083	90	50
23	24	0.898	0.7091	420	200
24	25	0.896	0.7011	420	200
6	26	0.203	0.1034	60	25
26	27	0.2842	0.1447	60	25
27	28	1.059	0.9337	60	20
28	29	0.8042	0.7006	120	70
29	30	0.5075	0.2585	200	600
30	31	0.9744	0.963	150	70
31	32	0.3105	0.3619	210	100
32	33	0.341	0.5302	60	40

. The active and reactive load of the network are 3715 kW and 2300 kVAR, respectively.

In addition, the annual growth of the network 0.05 is assumed.

Table 2. The simulation data [26].

Parameters	Value	Unit
InvDG	318	$/kW
OpeDG	0.036	$/kWh
FRj	6000	$/km
IL	0.4	-
IR	10	%
Dr	12	%
$N_y$	4	Years
Population size	200	-
Max_iter	100	-
Power facor	0.9	-
σi	0.2	fr/yr/km
Vmin	0.95	p.u.
Vmax	1.05	p.u.
Trs	1	hr
Trp	4	hrs

reports the required data for simulation, which are used from [26]. For this network, the three-level load is assumed based on which size of the load, the duration of load level, and the price of electricity at each level are shown in

Table 3. The details of the load duration curve [26].

Load Level	Percentage of Annual Peak Load (%)	Time Duration (hrs)	Ρ0,ls ($/MWh)
Heavy (ls = 1)	100	1500	70
Medium (ls = 2)	70	5000	50
Light (ls = 3)	50	2260	35

. The problem is solved by the proposed method and the details of the best solution are presented in

Table 4. The results of the best solution based on the proposed method.

Objective function value (M$)	4.7403
Feeders’ reinforcement cost	0.006584
DG investment cost	0.300479
Purchased power cost	2.308
DG operation cost	1.8586
Reliability cost (M$)	0.04806
Optimal DG capacity and location for every year of the planning period	First year	100 kW at bus 16 200 kW at bus 17 400 kW at bus 31 100 kW at bus 32
	Second year	100 kW at bus 6 400 kW at bus 11 200 kW at bus 15 100 kW at bus 20
	Third year	100 kW at bus 3 100 kW at bus 26
	Fourth year	100 kW at bus 22
Feeders’ reinforcement pattern for every year of the planning period	First year	---
	Second year	Feeders 1, 3, 29
	Third year	Feeders 5, 10, 11, 16, 26
	Fourth year	Feeder 32

. In this scenario, four DGs in the first and second years, two DGs in the third year, and one DG in the fourth year are installed.

Figure 4 represents the average network voltage before (without DG) and after (with DG) simulation. According to Figure 4, the voltage improvement rate could be clearly seen. As shown in the diagram, the base year of the program is also below the rated voltage (0.95) and, in order to improve the network voltage in the first year, it is necessary to add the DGs to the network. The voltage in the fourth year, due to the annual increase in load, gets the worst value, with an average of less than 0.93. Nevertheless, at the end of the planning, the network would no longer have this problem and therefore the voltage would increase significantly.

The problem solution has a great impact on the reliability of the network.

Table 5. The EENS_t values for each year of the planning period.

EENSt (kWh/year)	Without DG				With DG
	GA	PSO	TS	Proposed Method	GA	PSO	TS	Proposed Method
EENS1	19,201	21,251	29,234	18,191	17,692	18,490	28,221	12,847.9
EENS2	21,298	23,390	31,402	19,101	16,721	18,432	27,172	11,910.6
EENS3	23,295	24,607	32,521	20,056.1	16,734	17,456	26,639	11,724.7
EENS4	23,971	24,806	32,902	21,058.9	15,892	16,786	26,015	11,579.8

indicates the size of the EENS index before and after solving the problem (without DG and with DG). As it is shown in the Table, the expected reduction in unsupplied energy is significant. In the fourth year, this amount has reached from 21,058.9 to 11,579.8. The run times of each optimization method for 33-bus test systems are indicated in

Table 6. The run times of the proposed method and other optimization methods to solve the multiyear network expansion planning problem on 33-bus distribution networks.

Optimization Methods	Run Time for 33-Bus Test System (s)
GA	2452
PSO	1126
TS	1989
Proposed method	1490

.

5. Conclusions

According to some issues such as the amount of investment and the rate of return investment, and the power network conditions such as losses, voltage profiles, reliability, and maintenance, providing optimal annual planning methods for the development of the power system is essential. In this paper, a multilevel method is presented for optimal network power development planning based on the binary dragonfly optimization algorithm considering the DGs. The proposed objective function involves minimizing the cost of investment, operating and repairs, and the reliability cost for the network development by considering the presence of DGs. Additionally, the results indicate that considering DG decreases the EENS of the 33-bus test system by 29.37%, 37.64%, 41.45%, and 45.01% in the first, second, third, and fourth years, respectively. The proposed optimization method has been analyzed on a standard network, which has significantly improved the results compared with other benchmark methods such as GA, PSO, and TS.