Optimal placement of distributed generation in radial networks considering reliability and cost indices

Abstract

In the current paper, the optimal placement merits of distribution generations (DGs) are investigated in the economical version of a grid-parallel mode. The proposed method is constructed based on a probabilistic approach which is called Point Estimate Method (PEM) to consider the uncertainty associated with the load demand prediction error along with the variation of cost of electricity and power production of DGs. The objective functions are the total active power losses, the total emission produced and the total cost related to the grid & DGs and cost of reliability. These functions are set to be optimized that result in utilizing an interactive fuzzy approach based on improved particle swarm optimization (IPSO) to let the decision maker apply his/her preferences to the system. It is apt to mention that the application of the interactive fuzzy approach is based on the result of conflicting behaviors of the investigated objective functions. Through using the Tai-Power distribution test system the feasibility and the efficiency of the proposed method have been shown.

Keywords

Distributed generation stochastic modeling point estimate method improved particle swarm optimization

Nomenclature

I _i

Active current of the ith line

N _d

Number of load variations, here 3

N _br

Number of branches

R _i

Resistance of the ith line

Length of time interval (hr)

P_{FC, j}^{t, k}

Expected active power produced by the jth PEM-FCPP at the kth interval

{LD}_{j}^{k}

Value of the jth load at kth interval

{CT}_{j}^{k}

Value of the jth cost factor in the economical model of PEM-FCPP at kth interval

N _tie

Number of tie switches

N _sw

Number of sectionalizing switches

N _load

Number of the active and reactive load values

N _price

Number of cost factor parameters with uncertainty

{\tilde{E}}_{Grid}^{t}

Expected value of emission produced by Grid

{\tilde{E}}_{FC, i}^{t}

Expected value of emission produced by ith PEM-FCPPs

{NOx}_{Grid}^{t}

Nitrogen oxide pollutants of Grid at interval t

SO 2_{Grid}^{t}

Sulfur oxide pollutants of Grid at interval t

{\tilde{P}}_{FC, i}^{t}

Expected value of power generated by ith PEM-FCPP

{\tilde{P}}_{sub}^{t}

Expected value of power generated by substation

price

Price of purchasing power from substation ($/kWh)

{\tilde{P}}_{a}

Expected value of the power for auxiliary devices

{\tilde{P}}_{H}^{t}

Expected value of the equivalent electric power for hydrogen production at interval t

P _th,t

Thermal power at interval t

{\tilde{L}}_{el, t}

Expected value of electrical load at interval t

{\tilde{L}}_{th, t}

Expected value of thermal load at interval t

η _t

Efficiency of FCPP at interval t

{\tilde{C}}_{n 1}

Expected value of the price of purchasing natural gas for PEM-FCPPs

{\tilde{C}}_{el, p}

Expected value of the tariff of purchasing electrical energy

{\tilde{C}}_{el, s}

Expected value of the tariff of selling electricity

{\tilde{C}}_{n 2}

Expected value of fuel cost for residential loads

{\tilde{C}}_{Hs}

Expected value of hydrogen selling price

\tilde{OM}

Expected value of operation and maintenance cost

{\tilde{C}}_{sub}^{t}

Expected cost of power generated by substation

{\tilde{C}}_{FC}^{t}

Expected cost of power generated by PEM-FCPP

P_{ij, max}^{Line, t}

Maximum active power flow between ith and jth bus at interval t

| {\tilde{P}}_{ij}^{Line, t} |

Expected value of active power between ith and jth bus at interval t

θ _ij

Angle of branch between the ith and jth buses

V _i

Amplitude of the voltage at the ith bus

δ _i

Angle of the voltage at the ith bus

Y _ij

Amplitude of admittance between the ith and jth buses

P _i

Net injected active power at the ith bus

| {\tilde{I}}_{f, i}^{t} |

Amplitude current of the ith feeder

I_{f, i}^{max}

Maximum current of the ith feeder

p _min,FC

Minimum active power of PEM-FCPP

p _max,FC

Minimum reactive power of PEM-FCPP

{\tilde{V}}_{i}^{t}

Expected value of voltage magnitude of the ith bus and interval t

V _max

Maximum value of voltage magnitude

V _min

Minimum value of voltage magnitude

{\tilde{P}}_{FC}^{t}

Expected electrical power produced at interval t minus the power for auxiliary devices

ΔP _u

Upper limit ramp rate

ΔP _D

Lower limit ramp rate

Conversion factor (kg of hydrogen/kW of electric power)

v _cell

Cell operating voltage v_cell = 0.6

PLR _t

Part load ratio at interval t

r _TE,t

Thermal energy to electrical energy ratio

Probabilistic solution set of output variables

S _i

ith probabilistic output variable

Input set of uncertain variables

σ _zl

Standard deviation of z_l

z _l

Uncertain input random variable

f _zl

Probability density function

ξ _l,k

Standard location of z_l

μ _zl

Mean value of the input random variable z_l

E (S_{i}^{k})

kth moment of ith output random variable

λ _l,3

Skewness coefficient of z_l

ω _l,i

Weighting factors for estimated locations of input random variable z_l

z _{l,1, z_l,2}

Estimated locations of input random variable z_l

f_{i}^{min}

Lowest limit of ith objective function

f_{i}^{max}

Highest limit of ith objective function

μ _fi(X)

Membership function for ith objective function

μ _ref,i

The ith reference membership value

W _i

weighting factor of ith individual

1 Introduction

Lately, the renewable energy power sources like Fuel Cell Power Plants (FCPPs), wind power plants, photovoltaics, etc. have been widely used within the most popular research areas. Meanwhile, since FCPPs have especial aspects, they have attracted the attention of researchers. These especial characteristics that can be exemplified as the high efficiency and the low noise [1], extreme convenience and high portability [2], straightforward structure and straightforward operation [3], ability to follow the load changes [4], fast start up [5], high reliability and cleanness [6, 7] get them more appealing than any other types of electrical power sources. Among different types of FCPPs, PEM-FCPP is the most vulnerable choice used to supply residential loads especially in the voltage level of distribution. This merit comes from its low temperature (80– 100 C°), rapid start up, extremely low emission, and very low noise [8, 9]. In addition, the efficiency of PEM-FCPP can be increased to the higher value by the help of hydrogen production strategy. The obtained hydrogen is either sold to the customers to gain profit or it is used in case that the high load level occurs [10]. Economically speaking, hydrogen based efficiency improvement is optimal and thoroughly mandatory [11, 12]. In the new competitive electric power markets, the whole performance of the PEM-FCPP can be improved by utilizing the hydrogen production capacity along with recovering the wasted thermal energy during the electrical generation time. To reach a better and frugal improvement, many works are implemented on the PEM-FCPPs to enable it considering the effect of both the thermal recovery and the hydrogen production in the evaluations.

Recently, researchers have centered their attention round the DFR problem and viewed it as a suitable method to improve the total condition of the system. In [13], by taking the best advantages of the fuzzy theory, Das proposes an appropriate method of solving Multi objective DFR (MDFR) problem. In [14], a novel heuristic based method has been proposed by Gomes et al. to find the best structure for the network. Moreover, in [15], a hybrid algorithm is proposed by Ahuja et al. based on an evolutionary algorithm, ant colony algorithm, and artificial immune systems to respond the DFR problem. Besides, Morton and Mareels have suggested a brute-force technique to measure the best radial configuration which can minimize the total active power losses in [16]. In [17], Niknam has suggested a new two – phase method. This method first changes the Multi-Objective Optimization (MOP) problem into a single objective problem and then resolve MDFR problem. Notwithstanding, the major existing problem in the present and all mentioned works is the unwilling use of deterministic calculations to solve DFR problem. The effect of some uncertainty resources is far beyond being neglected. The reason justifies this is that these uncertainty resources like the load forecast error and the price variations can cause a wrong optimum solution for DFR problem. Indeed, the large integration of the renewable energy sources has made many challenging issues concerning their operation and management. Thus, the new complicated power systems which take the best advantages of different types of renewable energy sources are more vulnerable to be influenced by the uncertainty than previous systems [18]. Indeed, although very many different deterministic methods have been suggested to investigate and solve the MDFR problem, no method can properly solve the mentioned problem by taking the effect of uncertainty into its account. Notwithstanding, until now PEM-FCPPs have been investigated without considering the impacts of the grid. It is proper to mention that, new complex power systems with the uncertainties raising values can eclipse the performance of any Micro Grid such that the optimal performance gets deteriorated.

Therefore, the present study investigates the simultaneous effect of the PEM-FCPP operation management and the optimal MDFR in a stochastic environment. The PEM-FCPP economical version encompasses both the thermal recovery and the hydrogen production. This makes the PEM-FCPP improve its overall efficiency. It is notable that some important uncertainties and prices are considered within these evaluations such as the uncertainty of the active and reactive load consumptions as well as the uncertainty of the price spent to buy natural gas for PEM-FCPP, fuel cost of residential loads, tariff to purchase electricity, tariff to sell electricity, hydrogen selling, operation and maintenance cost and the purchasing power cost from the grid. Within the domain of uncertainty analysis, the Monte Carlo Simulation (MCS) approach is highly widespread and convenient whereas it is completely time-consuming. This weak point gets this approach need a lot of memory. Thus, in the present study, a simple and flexible technique which is known as a point estimate method (PEM) is applied to adequately solve the Stochastic DFR (SDFR) problem considering the effect of PEM-FCPPs [19]. This results in handling the uncertainties of the active and reactive load as well as the price through the 2m PEM [20]. Moreover, the completion of PEM-FCPPs all features is attained by introducing the hydrogen production and thermal load supplement in the economical version of PEM-FCPPs. Consequently, the complete version of PEM-FCPPs would simultaneously result in the reduction of the system total cost and the enhancement of PEM-FCPPs efficiencies. The assessed objective functions are known as [21] 1) the complete losses of active power 2) the system complete emission and 3) the total cost of both the grid and FCPPs. Therefore, by considering all aspects of a desired method the present suggested one is a complex type of the multi-objective optimization problem with equality and inequality constraints [21 –24]. In addition, an interactive fuzzy satisfying method is presented to attain the best satisfying solution among non-inferior solutions in the MOP. This method makes finding the “most-preferred” solution easy and allows the decision maker to adjust the reference membership values for each of the objective functions. In order to improve the whole performance of the algorithm which is a meta-heuristic population-based optimization algorithm inspired from the stochastic behaviour of the fish swarms in the optimization applications, a novel adaptation phase is suggested and added to the original algorithm [24 –27]. At the end, the simulation results are assessed in four different scenarios which distinctly represent the effect of the thermal recovery and the hydrogen production on the optimal operating management of PEM-FCPPs. Simulation results show clearly that the total efficiency of the PEM-FCPP is enhanced effectively by considering the thermal and hydrogen energy in evaluations. In short, the main contributions of the paper can be summarized as follows: 1) Optimal Placement of Distributed Generations in the radial grids, 2) Consideration effect of thermal recovery and hydrogen production capability of PEM-FCPP in the grid-connected mode, 3) introduction of a new stochastic framework for optimal operation management of problem considering different uncertainty resources, 4) introduction of a new optimization algorithm based on improved particle swarm optimization (PSO) to solve the proposed problem suitably.

2 Problem formulation

3.1 Objective functions

The following three objective functions have shaped the studied stochastic problem. It is proper to mention that, the superscript ∼ represents the expectation value of each variable.

- The total energy losses are minimized through:

In the follows the total energy loss is minimized: $\tilde{f_{1}} (X) = P_{loss} = \sum_{t = 1}^{Nd} t (\sum_{i = 1}^{N_{br}} R_{i} \times {| {\tilde{I^{t}}}_{i} |}^{2})$ (1)

In the above formulation, the control vector X consists of the optimal status of the Tie switches, sectionalizing switches, output power of the FCPPs, the value of load after thermal compensation and the cost factor value in the economical model of PEM-FCPP at kth interval.

- Minimization of the voltage deviation:

By reducing the maximum voltage deviation of the buses from the nominal value, the voltage profile will be improved: $f_{2} (X) = d_{voltage} (X) = max [| 1 - V_{min} | & | 1 - V_{max} |]$ (2) where V_min and V_max are the minimum and maximum values of bus voltages, respectively.

- Minimization of the total cost of DGs and reliability: The total cost includes the cost of the grid and the cost of the PEM-FCPPs DG and the cost of reliability. The grid cost and the DG costs are evaluated as follows: $\begin{matrix} {\tilde{C}}_{sub}^{t} = \tilde{price} \times t \times \sum_{j} {\tilde{P}}_{sub}^{t} \\ {\tilde{C}}_{FC} = {\tilde{C}}_{n 1} \times t \sum_{j} (\frac{{\tilde{P}}_{FC}^{t} + {\tilde{P}}_{a}^{t} + {\tilde{P}}_{Hj}^{t}}{η_{j}}) + \\ {\tilde{C}}_{el, p} \times t \sum_{j} max ({\tilde{L}}_{el, j}^{t} - {\tilde{P}}_{FC}^{t}, 0) - \\ {\tilde{C}}_{el, s} \times t \sum_{j} max ({\tilde{P}}_{FC}^{t} - {\tilde{L}}_{el, j}^{t}, 0) + \\ {\tilde{C}}_{n 2} \times t \sum_{j} max ({\tilde{L}}_{th, j}^{t} - {\tilde{P}}_{th, j}^{t}, 0) + \\ \tilde{OM} + {\tilde{C}}_{Hs} t \sum_{j} {\tilde{P}}_{H, j}^{t} F \end{matrix}$ (3)

In this formulation, each term is as follows: the first one is the PEM-FCPP fuel cost per day, the second is the cost of the electrical energy purchased daily while the load demand more than what the PEM-FCPP produces, the third term is the daily profit of selling the extra produced electrical energy, the forth one is the buying expense of gas per day when PEM-FCPP is unable of supplying the residential loads completely, the fifth is the operation and maintenance cost of PEM-FCPP and the sixth or the last term is the daily profit obtained from selling the PEM-FCPP generated hydrogen.

Finally, the cost of reliability is associated with the cost of interruption in the loads as follows: ${Cost}_{Rel} = \sum_{i = 1}^{N_{br}} {ECOST}_{i} = \sum_{i = 1}^{N_{br}} L_{a_{(i)}} C_{i} λ_{i}$ (4) where λ_i is the failure rate of ith line; L_a (i) is the average load value connected to ith bus. Also C_i is the cost of interruption of the ith bus in ($/kW) which is a nonlinear function of the interruption duration time. Considering all mentioned costs and profits, the total cost can be calculated as follows: ${\tilde{f}}_{3} (X) = Cost = \sum_{t = 1}^{N_{d}} \sum_{i = 1}^{N_{FC}} {\tilde{C}}_{FC, i}^{t} + {\tilde{C}}_{sub}^{t}$ (5)

2.2 Limits and constraints

- Distribution lines’ constraints: each line can not pass its maximal thermal power flow limit as follows: $| {\tilde{P}}_{ij}^{Line, t} | < P_{ij, max}^{Line, t}$ (6)

- The distribution power flow equations demonstrate the following Constraints. This is a general limitation in each power system problem that considers the main system. $\begin{matrix} {\tilde{P}}_{i} = \sum_{i = 1}^{N_{bus}} {\tilde{V}}_{i} {\tilde{V}}_{j} Y_{ij} cos ({\tilde{θ}}_{ij} - {\tilde{δ}}_{i} + {\tilde{δ}}_{j}) \\ {\tilde{Q}}_{i} = \sum_{i = 1}^{N_{bus}} {\tilde{V}}_{i} {\tilde{V}}_{j} Y_{ij} sin ({\tilde{θ}}_{ij} - {\tilde{δ}}_{i} + {\tilde{δ}}_{j}) \end{matrix}$ (7)

It is worthy to mention that the above equation is considered as the load flow equation applied as an equality constraint.

- Considering the radial aspect of the network: The distribution systems are usually made radial due to many benefits obtained by the radial structure of the networks. Therefore, the radial characteristic of the system should minutely be preserved during the optimization process. Indeed, within the network, each loop formed by the use of the existing sectionalizing and tie switches is opened to preserve the radial aspect of the network.

- The constraints of the feeder current [20]: $| {\tilde{I}}_{f, i}^{t} | \leq I_{f, i}^{max}; i = 1, 2, \dots, N_{f}$ (8)

- The limitation of the PEM-FCPP active power production: Each FCPP can produce power to its maximum capacity: $p_{min, FC} \leq {\tilde{p}}_{FC, j}^{t} \leq p_{max, FC}$ (9)

- Constraints of the bus voltage: The voltage level of each bus should be preserved in the pre-defined ranges: $V_{min} \leq {\tilde{V}}_{i}^{t} \leq V_{max}$ (10)

- Constraints of the ramp rate: The rate of power increase/decrease by each FCPP is limited as follows: ${\tilde{P}}_{FC}^{t} - {\tilde{P}}_{FC}^{t - 1} \leq Δ P_{u}$ (11) ${\tilde{P}}_{FC}^{t - 1} - {\tilde{P}}_{FC}^{t} \leq Δ P_{D}$ (12)

3 The economical model of DG with thermal recovery ability

Here, by considering all efficient parameters of the market, grid and FCPPs, an economical PEM-FCPP model is suggested (as shown in Equation 3). The main yardsticks taken into account in proposing such optimal model are mentioned as below.

In some daily time intervals, supplying customers by the electrical energy produced by the PEM-FCPPs is profitable. Under this circumstance, the extra electrical energy of the PEM-FCPP can be sold to the grid while the PEM-FCPP unused capacity is possible to be used for producing the hydrogen. Taking the best advantage of this strategy, the efficiency of the PEM-FCPPs would be satisfyingly increased to a much desired quantity.

The main concept of the hydrogen production in the suggested economical version is to use the PEM-FCPP extra electrical capacity when its power production is less than its maximum power production capacity. Therefore, P_H,j is considered as the equivalent electrical power energy to model the amount of produced hydrogen. The equivalent electrical power is supposed to be placed at the fuel cell stack output such that it will be able to determine the equivalent hydrogen (kg/s) in terms of electrical power (P_H,j). Notwithstanding, in the present paper, the amount of produced hydrogen is assumed to change between zero and the difference between PEM-FCPP maximum capacity and its current generated electrical power. The following equation [10] is used to measure the amount of the equivalent hydrogen: $(H_{2}) amount = F \frac{{\tilde{P}}_{H}^{t}}{v_{cell}}$ (13)

The mentioned strategy is considered to be highly profitable since it makes the production of hydrogen possible. Noticeably, selling the produced hydrogen can compensate some costs of the system.

The thermal energy production is one significant aspects of PEM-FCPP. The amount of thermal energy has a one to one corresponding relationship with the amount of electrical energy produced by the PEM-FCPP. Along with the normal operation, the PEM-FCPP efficiency is estimated 36% whereas this value can vary based on the supplied electrical energy. On the other hand, the efficiency of the PEM-FCPP gets increased at full load such that the total produced electrical power is equal to the produced thermal energy. The relationship between the PLR and the efficiency is mathematically represented as follows [24]: $\begin{matrix} For {PLR}_{t} ≺ 0.05 \\ η_{t} = 0.2716, r_{TE, t} = 0.6801 \\ For {PLR}_{t} 0.05 \\ η_{t} = 0.9033 {PLR}_{t}^{5} - 2.9996 {PLR}_{t}^{4} + 3.6503 {PLR}_{t}^{3} \\ - 2.0704 {PLR}_{t}^{2} + 0.4623 {PLR}_{t} + 0.3747 \\ r_{TE, t} = 1.0758 {PLR}_{t}^{4} - 1.9739 {PLR}_{t}^{3} + 1.5005 {PLR}_{t}^{2} \\ - 0.2817 {PLR}_{t} + 0.6838 \end{matrix}$ (14)

Figure 1 represents the relationship between the Part Load Ratio (PLR) and the PEM-FCPP efficiency. In the followings, the PEM-FCPP produced thermal power is measured as: $P_{th, j}^{t} = r_{TE} (P_{H, j}^{t} + P_{FC}^{t} + P_{a}^{t})$ (15)

In the following equation the overall efficiency of PEM-FCPP is assessed considering the hydrogen, thermal and electrical energy production $η_{new, t} = \frac{P_{FC}^{t} + P_{a}^{t} + P_{H}^{t} + min (L_{th, t}^{t}, P_{th, t}^{t})}{(P_{FC}^{t} + P_{a}^{t}) / η_{t} + P_{H}^{t}}$ (16)

4 The stochastic load flow based on two-point estimate method

In this part the Point Estimate Method (PEM) is completely described. The moments of the random quantity that is a function of one or several random variables are measured by two-point estimate method [25 –29]. Mathematically, deterministic power flow can mathematically be expressed as: $S = F (z)$ (17)

Here, the network data, the load consumption, the generation dispatch, etc. get together and form the uncertain variable z. Therefore, the vector z uncertainty is moved to the output variable S or better to put it is transferred to the bus voltage or line flows. In this situation, if z_l is supposed as a random variable with the probability density function f_zl, by matching the first three moments of f_zl, the two-point estimate technique can use two probability concentrations to replace f_zl [20]. According to 2m PEM, Equation 17 is solved 2m times. Figure 2 is shown the two-dimension representation of the scheme in detail.

A observed in Fig. 2, the data of z_l,1 and z_l,2 are transformed to the output variables S_l,1 and S_l,2 by the help of the function S = F (z). Both estimates of variant S that is S_l,1, S_l,2 are scaled by two variables ω_l,1 and ω_l,2. Each concentration point holds two pairs (z_l,k, ω_l,k), k = 1, 2; in which z_l,k and ω_l,k are the location and the weighting factor, respectively [20]. The location of each concentration is computed as:

$z_{l, k} = μ_{z_{l}} + ξ_{l, k} . σ_{z_{l}}; k = 1, 2$ (18) where μ_zl and σ_zl are the mean and standard deviation of f_zl., respectively. Besides, ζ_l,k is computed as below: $ξ_{l, k} = \frac{λ_{l, 3}}{2} + (- 1)^{3 - k} \sqrt{n - (λ_{l, 3}^{2} / 2)^{2}}, k = 1, 2$ (19)

The variables λ_l,3 which is mathematically expressed as below is known as the coefficient of skew-ness [20] and is assumed as the third central moments of z_l: $λ_{l, 3} = \frac{E [{(z_{l} - μ_{z_{l}})}^{3}]}{{(σ_{z_{l}})}^{3}}$ (20) $σ = \sqrt{var (S_{i})} = \sqrt{E (S_{i}^{2}) - {[E (S_{i})]}^{2}}$ (21)

5 Solution methodology

5.1 The Fuzzy modeling of objective functions

Since objective functions have different features and due to their different optimal values corresponding to each of them, the following fuzzy membership is utilized to formulate all objective functions in the same basis. $μ_{fi} (X) = {\begin{matrix} 1; f_{i} (X) \leq f_{i}^{min} \\ 0; f_{i} (X) \geq f_{i}^{max} \\ \frac{f_{i}^{max} - f_{i} (X)}{f_{i}^{max} - f_{i}^{min}}; f_{i}^{min} \leq f_{i} (X) \leq f_{i}^{max} \end{matrix}$ (22)

It is appropriate to consider that $f_{i}^{min}$ and $f_{i}^{max}$ are evaluated by single optimization of the Ith objective function.

5.2 Improved PSO (IPSO)

The IPSO is explained completely in this section.

5.2.1 The original PSO descriptions

One of the inspired evolutionary based optimization techniques is known as PSO algorithm which was first introduced in by Kennedy and Eberhart [22]. The central concept composing this algorithm is to imitate the searching process of particles for the food to long distances during the immigration. Each particle X position represents the optimal solution of the investigated problem which here is considered as the output power of the units in the system. The initial population is generated randomly Just like other evolutionary algorithms. Rationally, each particle ought to move to update its position. To update the movement of each particle, three basic ideas are used: the inertia (W), the global best position (gbest) and the particle best position (pbest_j). $\begin{matrix} V_{i}^{k + 1} & = ρ V_{i}^{k} + c_{1} {rand}_{1} (P_{best, i}^{k} - X_{i}^{k}) \\ + c_{2} {rand}_{2} (G_{best}^{k} - X_{i}^{k}); i = 1, 2, \dots, N_{SW} \end{matrix}$ (23) $X_{i}^{k + 1} = X_{i}^{k} + V_{i}^{k + 1}; i = 1, 2, \dots, N_{SW}$ (24)

Here, the speed of each particle is between a specific pre-determined bound as: $V_{i, min} \leq V_{i}^{k + 1} \leq V_{i, max}$ . A high rate for V_i,max can misinform the particles to soar beyond the optimal solution when a low value of V_i,max can diminish the capability of the algorithm to escape from local optima.

5.2.2 The improved version of PSO (IPSO)

Since the modification is regarded as the process of an algorithm performance improvement in both local and global search, the present paper is devoted to represent a new modification process to decrease the probability of being trapped in local optima as well as being dependable on the algorithm initial parameters. In this paper, simulation results depict the efficiency of present modification in improving the ability of the PSO. Indeed, the present study suggests a sufficient modification method to enhance the search ability of the PSO algorithm effectively. This proposed two-phase approach modification method is explained completely as follows: In the first phase, the idea of Levy flight known as an algorithm powerful tool to provide a more powerful local search is utilized. ${Le}^{'} vy (ρ) \sim τ = k^{- ρ}; (1 < ρ \leq 3)$ (25)

In this regard, a novel candidate solution is generated as: $X_{i}^{k} = X_{i}^{k - 1} + φ_{1} \oplus {Le}^{'} vy (ρ)$ (26)

In the second modification phase, the population is moved toward the best current particle within each iteration. Here, first the mean value of the population is measured (M_D); then, the distance between the best particle and M_D is computed and added to the whole population as below: $X_{i}^{k} = X_{i}^{k - 1} + round (1 + rand) (X_{Gbest} - M_{D})$ (27)

The above movement will force the whole population to change its position toward the best solution, rapidly. At last, the diversity of the population can be increased, effectively. Applying these two simple modifications makes the search ability and the PSO convergence improved.

6 Simulation results

This section makes use of the IEEE standard test system incorporating 69 buses to check the performance of the proposed method. The single line diagram of the test system is illustrated in Fig. 3. The networks works on the voltage level of 12.66 and the complete data are given in [23]. Initially, a sensitivity analysis is performed to reduce the number of candidates for DG placement. Also, each DG is supposed to be operated to maximum of 250 kW.

Since the proposed IPSO algorithm is solving the proposed problem for the first time, at the beginning a list of optimal solutions in the area is given for comparison. For better comparison, the uncertainty effects are neglected initially.

Table 1 shows the results of genetic algorithm (GA), particle swarm optimization (PSO) algorithm and the proposed IPSO comparatively. According to these results, two main conclusions can be made. First, the proposed IPSO has shown superior performance than the GA and original PSO by minimizing all the targets more optimally. This can be deduced from power losses, voltage deviation and total cost function. Second, the use of DG in the system has enhanced all targets greatly. It is worth noting that the initial value of power loss, voltage deviation and total cost has been 2.227912 kW, 0.09055946 pu and 165.988 $, correspondingly. In other words, the use of DG has enhanced all objective functions satisfactorily.

So as to get the consequence of considering uncertainty in the problem, the single-objective optimization process is repeated in the stochastic structure. Here, by the use of 2m-PEM, the problem with m number of uncertain parameters is solved 2 m times to model the uncertainty. This is a useful number of scenarios since it can preserve the computational burden. Table 2 displays the outcomes of simulation in the stochastic context. T better understanding, the deterministic outcomes are also given in Table 2. In keeping with these results, seeing uncertainty has caused incremental progress in all objectives. This rise in the optimal values of the targets is the charge of modeling uncertainty in the problem. Nonetheless, the new values are more reliable than the deterministic outcomes. Based on the standard deviation values, it is realized that incorporating DGs in the system can raise the trustworthiness of the operating status of the system from the sight of all objects.

By comparing the results in Tables 1 and 2, it is seen that optimizing a specific objective function has moved the optimal generation of DGs toward a different value different from the other objectives. This event shows that the objective functions have different tendencies in setting and placement of DGs in the network. Therefore, the idea of multi-objective optimization should be used to satisfy all the objectives simultaneously. In order to provide a set of solutions for the operator, we make use of the idea of non-dominated solutions. By definition, the solution X₁ will dominate the solution X₂ if the below two criteria are satisfied: $\begin{matrix} 1) \forall j \in {1, 2, \dots, n}, f_{j} (X_{1}) \leq f_{j} (X_{2}) \\ 2) \exists k \in {1, 2, \dots, n}, f_{k} (X_{1}) < f_{k} (X_{2}) \end{matrix}$ (28) where n is the number of objectives wherein here is n = 3. Each non-dominated solution can be considered as an optimal situation for the network. This can give the operator the choice of preferences. Table 3 shows a set of non-dominated solutions for the problem. As it can be seen, none of these solutions can be supposed as a more superior point. The optimal power productions of DGs are also shown simultaneously.

The influence of optimal placement and operation of the DGs on the voltage level of the buses in the system is depicted in Fig. 4. This figure reveals the voltage profile of the system before and after the use of DG in the system. From this figure, the voltage level of all buses is improved sufficiently.

Finally, the thermal recovery and the hydrogen production are both considered in the DG models. The simulation results depicted completely in Table 4 manifest the effective improvements of all targets. Noticeably, the efficiency values are improved, too. Statistically, the standard deviation of each objective function is evaluated to highlight the influence of considering uncertainties of the load forecast error and price factors in the evaluations.

7 Conclusion

The present paper investigates optimal placement and operation of DGs in the radial distribution systems. Scientifically, uncertainties of both load forecast error and the cost error were considered through the probabilistic PEM load flow to study the problem in a more realistic environment. Moreover, the optimal operation management of a specific DG, FCPP was minutely studied after applying placement in the system. Rationally, because the suggested problem is regarded as a complex, discrete nonlinear multi-objective optimization problem, a new algorithm based on IPSO is needed to be utilized to search the exploration space, globally. Based on simulation results, the efficiency quantities of the proposed method are improved. Moreover, it was represented that the standard deviation of the objective functions was reduced with respect to the effect of load and cost uncertainties along with the evaluations. Consequently, it was approved that the IPSO performs more appropriately in comparison to other renowned algorithms within this field (such as GA and original PSO).

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