Optimal operational management of a micro-grid,including high penetration level of renewable energy resources

Abstract

This paper presents a proper scenario-based method for hourly operation management of a Micro-Grid (MG) in a stochastic environment. The proposed method can model the uncertainty of the power produced by wind and solar resources, load demand and electricity market price simultaneously. In order to reduce the computational time of the problem, a Linear Programming (LP) is proposed to reduce and rearrange the number of scenarios and their probabilities respectively. The optimal objective value of the original multi-scenario problem is closely approximated by the optimal objective value of the reduced problem. In fact, the original stochastic model is transformed into a deterministic equivalent using the reduced set of scenarios. Each new obtained scenario is considered as the input for a deterministic problem with a specific probability. In this regard, each problem is formulated as a multi-objective optimization problem to minimize the total operating cost of the MG considering reliability issues. Then, the Shuffled Frog Leaping Algorithm (SFLA) is employed to solve the optimization problem. The SFLA is also compared with conventional heuristic algorithms (i.e., Simulated Annealing (SA), Particle Swarm Optimization (PSO) and Genetic Algorithm (GA)) in terms of capability and superiority. The simulations are conducted on a typical low-voltage grid-connected MG which includes Micro-Turbine (MT), high penetration of Wind Turbine (WT) and Photovoltaic (PV) generation and Energy Storage System (ESS).

Keywords

Optimal operation management micro-grid shuffled frog leaping linear programming

Nomenclature

|S|

Set of initial scenarios

|S′|

Subset of initial scenarios

p _s

Probability of scenario s

p _s′

Probability of scenario s′

N _g

Number of conventional power units

N _WT

Number of wind power units

N _PV

Number of photo-voltaic power units

N _ES

Number of energy storage units

N _s

greaterthan Appropriate for gestational age

N _V

Number of problem variables

Number of hour

P_{G_{k}}^{s} (t)

Power generated by conventional unit k at hour t in scenario s

P_{{WT}_{i}}^{s} (t)

Power generated by WT unit i at hour t in scenario s

P_{{PV}_{j}}^{s} (t)

Power generated by PV unit j at t in scenario s

P_{{ES}_{n}}^{s} (t)

Power generated (consumed) by ESS n at hour t in scenario s

P_{MG}^{s} (t)

Power imported (exported) from (to) external grid at hour t in scenario s

P_{LD}^{s} (t)

Load demand at hour t in scenario s

λ_{MG}^{s} (t)

Bid of utility at hour t in scenario s

λ _{MT
_k}

Bid of conventional unit k

I_{k}^{s} (t)

Commitment state of unit k at hour t in scenario s

SU_k/SD_k

Start-up/shut down cost of conventional unit k

P_{MG}^{Min} / P_{MG}^{Max}

Minimum/ maximum power imported (exported) from (to) external grid

P_{G_{k}}^{Min} / P_{G_{k}}^{Max}

Minimum/ maximum power generated by conventional unit k

LSH^s (t)

Load shedding at hour t in scenario s

SCDF

Sector customer damage functions

P_{CH
_n}/P_{DCH
_n}

Permitted rate of charge/discharge of ESS n at hour t

E_{{DCH}_{n}}^{Min} / E_{{DCH}_{n}}^{Max}

Minimum/maximum capacity of ESS n

P_{{CH}_{n}}^{Max} / P_{{DCH}_{n}}^{Max}

Maximum rate of charge/discharge of ESS n

η_{CH
_n}/η_{DCH
_n}

Charging/leakage loss of ESS n

E_{ES
_n} (t)

Available energy in ESS n at hour t

\begin{matrix} β_{WT}^{l_{WT}^{t}} / β_{PV}^{l_{PV}^{t}} \end{matrix}

Probability corresponding to

\begin{matrix} β_{LD}^{l_{LD}^{t}} / β_{MG}^{l_{MG}^{t}} \end{matrix}

error interval $l_{LD}^{t} / l_{MG}^{t}$

\begin{matrix} L_{{WT}_{i}} / L_{{PV}_{j}} \end{matrix}

Number of probability level

\begin{matrix} L_{{LD}_{f}} / L_{MG} \end{matrix}

of discrete distribution function related to WTi PVj/LDf/λMG

Standard deviation of uncertain variables

\cap_{X^{t = 1}}^{T} (l_{W T}^{t}, l_{P V}^{t}, l_{E D}^{t}, l_{M G}^{t})

Probability assigned to a new scenario in the reduced set of scenarios

Number of virtual frogs

Number of memeplexes

Number of virtual frogs in each memeplex

X _r

Location of virtual frog r

X_b/X_w

The locations of frog with the best/worst fitness value in each memeplex

X _g

The location of a frog with the best global fitness value

Search acceleration factor

D_{i}^{min} / D_{i}^{max}

Minimum/maximum leap frog

Index for conventional unit

Index for hour

Index for scenario

Index for WT unit

Index for PV unit

Index for ESS unit

Index for virtual frog

Index for iteration

Index for memeplex

1 Introduction

Conventional energy resources are decreasing and public concern for adverse environmental impacts are rapidly increasing; hence, renewable energy is known as a proper alternative in the last few decades. Renewable energy resources, particularly wind and solar energy have acquired more significance in the power system generation, and more and more Wind Turbine (WT) generator and Photovoltaic (PV) cells are built and connected to electric power systems. However, due to the variable and relatively unpredictable nature of renewable energies, they have had and will continue to have serious impacts on power system operation and reliability. Previously, this problem could be disregarded due to the low penetration of renewable energy resources. In other words, the intermittency and variability of renewable energy resources did not have a remarkable impact on power system from operation, management and stability points of view. Nowadays, with the increasing penetration of renewable energy resources, the major concern, especially for the new power grids is to derive an optimal operation management that ensures a high reliability level, energy cost minimization and better utilization of renewable energy resources.

Accordingly, a Micro-Grid (MG) was introduced as a small intelligent power network which takes into consideration these important factors. It includes controllable loads, distributed generations (mainly renewable energy resources such as WT and PV), Energy Storage Systems (ESSs) and control devices that are normally connected to or isolated from the utility grid [1 –3].

Storage systems have a significant role in the operation of the MG. The utilization of these types of energy sources can add the facility to the operation of the MG and provide reliability and economic benefits. ESSs are used to smooth out the volatility and variation of renewable power and shift it for a period of time when the demand or energy requirement is high. Moreover, ESSs can store energy during low-price periods and discharge it back to the MG during high-price periods, thus ensuring reliability improvement and economic benefits for the MG[4, 5].

In order to maximize economic and reliability benefits, there is a need for accurate operation strategies to efficiently optimize the use of ESS and renewable energy resources. However, the simultaneous use of renewable energy resources and energy storage devices could add more complexity to network operation. Considering the importance of this issue, a large number of studies have examined approaches to the optimization of the operation and management of the MG from different perspectives.

Chakraborty et al. [6] and Farag et al. [7] proposed a Linear Programming (LP) model to solve the MG operation cost and economic dispatch problems. Recently, evolutionary programming methods have been applied to solve such kinds of problems due to their simplicity, robustness and powerful search capability. Moghaddam et al. [8], for example, employed Fuzzy Self Adaptive Particle Swarm Optimization (FSAPSO) to find an optimal schedule for the operation of the MG taking into account minimum levels of operating cost and emission simultaneously. Moghaddam et al. [9], also used the PSO algorithm to minimize the cost of an MG considering controllable load and ESS. A methodology for the operation of the MG based on matrix real-coded was presented in [10] to maximize profits. The reliability benefits of power system containing wind power and energy storage under different operating strategies were investigated in [11] from another perspective. As another example, [12] carried out the adequacy assessment of power system, including wind and hydro power and ESS to improve power fluctuation. All the above-mentioned studies were focused on either reliability analysis or economic operation of renewable energy resources and ESS integration. However, these two aspects are closely interrelated. Therefore, it is necessary to determine an optimal operation strategy considering reliability and economical operation simultaneously. Additionally, the main flaw of the previous studies is that the operation of the MG is carried out in a deterministic framework. Deterministic scheduling techniques are based on the accuracy of input information. However, due to the variability and randomness of the natural phenomena, the input information will be always associated with uncertainty. If the uncertainty of the load forecast error, energy prices in the electricity market and power produced by WT and PV are ignored, the MG operators and consumers will be faced with many technical and economic problems.

The main goal of this study is to present an efficient approach to examine the impact of uncertainty on the optimal operation management of the MG. In this regard, a scenario-based stochastic method is proposed to take into account the uncertainties of electricity market price, wind and solar generations and load forecast error. The proposed stochastic method is conducted in two steps. In the first step, the scenarios are generated to translate the continuous description of the stochastic variable to a set of discrete realizations of that stochastic variable. The computational time of solving the optimal operation management problems with a large number of scenarios is extremely high. Therefore, a new scenario reduction approach is used to reduce the number of scenarios needed in the problem formulation. This approach introduces an objective function with LP formulation. Its main goal is to select a minimum subset of scenarios along with their probabilities that would help in approximating the optimal objective value of original multi-scenario problem. In the second step, the optimization problem is separately solved for each new obtained scenario using the SFLA. Finally, the expected operation cost and generation points are calculated by taking into account the probability of each scenario. Application examples in deterministic and stochastic environment framework are presented to demonstrate the effectiveness of the proposed procedure.

2 Reliability and economy evaluation framework

The basic purpose of an MG is to supply its customers with electrical energy as economically as possible and with a reasonable degree of continuity and quality [13]. In the new electric power systems, reliable electric supply is acquiring greater significance to increase customer satisfaction. It is important to consider both outages and power quality problems because from a customer’s perspective which can affect the price of the electrical energy. The customer interruption costs are the customer unsupported energy costs due to electricity supply interruptions [13]. The customer interruption cost is a proper index, which can be used in the reliability evaluation of an electric system. The customer interruption costs depend on customer type and outage duration. Customer Damage Functions (CDF) can be used to describe the cost due to a power supply failure as a function of the outage duration [14]. The CDF can be divided into seven customer sectors including large users, agricultural, residential, commercial, industrial, government and institutions and office and building. The CDF of these seven customer types is known as Sector Customer Damage Functions (SCDF) [14]. Table 1 shows the SCDF for the IEEE-RTS [15].

The integration of renewable energy resources and ESS can reduce energy purchasing cost and may lead to increased profit from integrating more renewable energy and from improved MG reliability. However, different operating strategies, can affect the MG from economic and reliability perspective. It is worth mentioning that during different operation strategies the ESS is considered sometimes as a load (in charging periods) and sometimes as a generator (in discharging periods).

3 Problem formulation

The optimal power dispatch and the operation management problem in the MG can be defined as a multi-objective optimization problem to determine the optimal generation set points besides ON/OFF states of DGs taking into account economic indices. Daily energy purchasing cost from external grid and DGs in the MG and customer interruption cost can be used as the economic indices. The energy purchasing cost and customer interruption cost are obtained by actual operation in each hour of a day. Therefore, the following assumptions are employed in this formulation.

3.1 Objective function

Minimizing the total cost of the MG is chosen as an objective function, which is expressed in Equation (1). It consists of three main parts. The first part calculates the cost for power generation of the units as well as the cost of startup/shutdown which is formulated in Equation (2). The second one calculates the power exchanging cost between the MG and external grid, which is shown in Equation (3). The third and last part calculates the cost of unserved power which is expressed in Equation (4). The proposed optimization problem is constrained with mixed integer variables and it may be solved with different evolutionaryalgorithms. $Min f (x) = \sum_{s = 1}^{N_{s}} p_{s} \sum_{t = 1}^{T} {{OF}_{1} + {OF}_{2} + {OF}_{3}}$ (1) ${OF}_{1} = \sum_{k = 1}^{N_{g}} {\begin{matrix} λ_{{MT}_{k}} P_{G_{k}}^{s} (t) I_{k}^{s} (t) \\ + {SU}_{k} \times max (0, I_{k}^{s} (t) - I_{k}^{s} (t - 1)) \\ + {SD}_{k} \times max (0, I_{k}^{s} (t - 1) - I_{k}^{s} (t)) \end{matrix}}$ (2) ${OF}_{2} = λ_{MG}^{s} (t) P_{MG}^{s} (t)$ (3) ${OF}_{3} = {LSH}^{s} (t) SCDF$ (4)

3.2 Micro-grid and unit constraints

The objective function is minimized subject to different operating constraints to satisfy the electrical requirements for the MG, constraints on DG and ESS operation, as well as constraints on power exchanging between the MG and external grid. These constraints are described as follows.

3.2.1 Power balance constraint

For each hour, the following equation ensures that the power generated from DG units in the MG and power imported (exported) from (to) the external grid satisfy the hourly MG load. It should be noted that a load shedding variable is considered in the Equation (5), if the total generation cannot satisfy the load. $\begin{matrix} \sum_{i = 1}^{N_{WT}} P_{{WT}_{i}}^{s} (t) + \sum_{j = 1}^{N_{PV}} P_{{PV}_{j}}^{s} (t) + P_{MG}^{s} (t) + \sum_{k = 1}^{N_{g}} P_{G_{k}}^{s} (t) I_{k}^{s} (t) \\ + \sum_{n = 1}^{N_{ES}} P_{{ES}_{n}}^{s} (t) + {LSH}^{s} (t) = P_{LD}^{s} (t) \end{matrix}$ (5)

The ESS power is positive (negative) when the storage is in discharging (charging) periods, and zero in idle periods. Additionally, the external grid power is positive (negative) when power is imported (exported) to (from) the MG, and zero when the MG operates in island mode.

3.2.2 Power exchanging between the MGand external grid constraint

For each hour, the following inequality constraint limits power exchanging between the MG and external grid. $P_{MG}^{Min} (t) \leq P_{MG} (t) \leq P_{MG}^{Max} (t)$ (6)

3.2.3 Micro-grid power generation constraint

The conventional power generations in the MG are limited as follows: $P_{G_{k}}^{Min} \leq P_{G_{k}} (t) \leq P_{G_{k}}^{Max}$ (7)

3.2.4 Energy storage constraints

The available energy in the ESS at time t can be described as:

$\begin{matrix} E_{{ES}_{n}} (t) & = & E_{{ES}_{n}} (t - 1) \\ + η_{{CH}_{n}} P_{{CH}_{n}} (t) Δ t - \frac{1}{η_{{DCH}_{n}}} P_{{DCH}_{n}} (t) Δ t \end{matrix}$ (8)

Also, available energy in the ESS should be within the maximal and minimal capacities. $E_{E S_{n}}^{M i n} \leq E_{E S_{n}} (t) \leq E_{E S_{n}}^{M a x}$ (9)

The charge (discharge) rate of ESS is considered to be less than or equal to a specific value. $P_{{CH}_{n}} (t) \leq P_{{CH}_{n}}^{Max}$ (10)

$P_{{DCH}_{n}} (t) \leq P_{{DCH}_{n}}^{Max}$ (11)

3.2.5 Load shedding constraint

The maximum load curtailment is limited by the following inequality constraint. $0 \leq LSH (t) \leq P_{LD} (t)$ (12)

4 Stochastic programming

A stochastic programming problem is a math programming problem, so that the values of the variables are replaced by statistical distributions. In this section, the problem of reducing the number of scenarios and rearranging their probabilities in multi-scenario optimization problems is addressed. The validity of deterministic scheduling techniques is based on the accuracy of input information. However, due to the nature of some variables, most of them will be always associated with uncertainty. Uncertainty refers to the random and variable nature of the input variables such as wind and solar energy, the price in power markets and load demand in the new deregulated power systems. In other words, they do not follow a fixed pattern and vary with several factors. Usually stochastic programming is known as a proper approach for solving the problems associated with uncertainty. Therefore, there is an urgent need to analyze power systems in the stochastic framework instead of deterministic framework, to lessen sensitivity of output solutions to uncertain input variables. The combination of the values of the uncertain variables results in a specific scenario. In this way, the problems associated with uncertainties are transformed into an equivalent deterministic set of problems. Each of them is known as a scenario or realization. However, in any system, the continuity of the distribution function of uncertain variables can create an infinite number of different scenarios. Moreover, since the probability of any point on a continuous distribution function is zero, no specific probability can be considered for any of the obtained scenarios. Using the discrete distribution function instead of continuous distribution function can be helpful. Normal probability distribution function is reasonably able to show the forecast error of load demand, wind and solar power generation and electricity prices in the market [16]. The continuous and discrete probability distribution function with seven probability levels is shown in Fig. 1; where δ is the difference between each of the two probability levels and is equal to the standard deviation of each uncertain variable, β₁, β₂, β₇ are the probabilities corresponding to each error interval. Scenarios are created by considering all the input variables together. Each possible system state is represented by one scenario. The length of each scenario equals to the number of input variables.

$\begin{matrix} S c e n a r i o S = {P_{W T}^{l_{_{W T}}^{t}}, P_{P V}^{l_{_{P V}}^{t}}, P_{L D}^{l_{_{L D}}^{t}}, P_{M G}^{l_{_{M G}}^{t}}} \\ l_{WT}^{t} / l_{PV}^{t} / l_{LD}^{t} / l_{MG}^{t} = 1, 2, ..., L_{W T} / L_{P V} / L_{L D} / L_{M G} \\ t = 1, 2, ..., T \end{matrix}$ (13)

The probability of each scenario is obtained by multiplying the probability of each of the input variables shown as follows: $p_{S} = \prod_{t = 1}^{T} {β_{W T}^{l_{_{W T}}^{t}} . β_{P V}^{l_{_{P V}}^{t}} . β_{L D}^{l_{_{L D}}^{t}} . β_{M G}^{l_{_{M G}}^{t}}}$ (14)

4.1 Scenario reduction

Based on the above description, combining all the uncertain variables will lead to a quite large number of scenarios result in an explosion in the number of scenarios. With the increased number of scenarios, the size of the optimization problem becomes larger so that it will be very hard to solve.

Approximate methods can be used as a suitable solution to overcome this challenge by reducing the number of scenarios. The total number of scenarios can be obtained as follows: $| S | = (L_{WT} . L_{PV} . L_{LD} . L_{MG})^{T}$ (15)

The first objective of this paper is to minimize the number of scenarios. An LP model is proposed in order to reduce the number of scenarios. By means of this model, it is possible to select a minimum subset of scenarios |S^′| from the initial scenarios |S| with new probability corresponding to each of the |S^′| scenarios. In fact, we try to rearrange the probability of new scenarios with respect to the following objective function and constraints. $\begin{matrix} minf = \sum_{S = 1}^{N_{S}} {\frac{1}{p_{S}}} p_{S^{'}} = \\ \underset{t = 1}{\underset{︸}{\sum_{l_{WT}^{1} = 1}^{L_{WT}} \sum_{l_{PV}^{1} = 1}^{L_{PV}} \sum_{l_{LD}^{1} = 1}^{L_{LD}} \sum_{l_{MG}^{1} = 1}^{L_{MG}}}} \dots \underset{t = T}{\underset{︸}{\sum_{l_{WT}^{T} = 1}^{L_{WT}} \sum_{l_{PV}^{T} = 1}^{L_{PV}} \sum_{l_{LD}^{T} = 1}^{L_{LD}} \sum_{l_{MG}^{T} = 1}^{L_{MG}}}} \\ {\frac{1}{\prod_{t = 1}^{T} β_{WT}^{l_{WT}^{t}} β_{PV}^{l_{PV}^{t}} β_{LD}^{l_{LD}^{t}} β_{MG}^{l_{MG}^{t}}}} . χ^{⋂_{t = 1}^{T} (l_{WT}^{t}, l_{PV}^{t}, l_{LD}^{t}, l_{MG}^{t})} \end{matrix}$ (16)

$\begin{matrix} \underset{t = 1}{\underset{︸}{\begin{matrix} S . t . \\ \sum_{l_{WT}^{1} = 1}^{L_{WT}} \sum_{l_{PV}^{1} = 1}^{L_{PV}} \sum_{l_{LD}^{1} = 1}^{L_{LD}} \sum_{l_{MG}^{1} = 1}^{L_{MG}} \end{matrix}}} \dots \\ \underset{t = T}{\underset{︸}{\sum_{l_{WT}^{T} = 1}^{L_{WT}} \sum_{l_{PV}^{T} = 1}^{L_{PV}} \sum_{l_{LD}^{T} = 1}^{L_{LD}} \sum_{l_{MG}^{T} = 1}^{L_{MG}}}} χ^{⋂_{t = 1}^{T} (l_{WT}^{t}, l_{PV}^{t}, l_{LD}^{t}, l_{MG}^{t})} = 1 \end{matrix}$ (17) $0 \leq χ^{⋂_{t = 1}^{T} (l_{WT}^{t}, l_{PV}^{t}, l_{LD}^{t}, l_{MG}^{t})} \leq 1$ (18)

$\begin{matrix} \underset{t = 1}{\underset{︸}{\sum_{l_{PV}^{1} = 1}^{L_{PV}} \sum_{l_{LD}^{1} = 1}^{L_{LD}} \sum_{l_{MG}^{1} = 1}^{L_{MG}}}} \dots \\ \underset{t = T}{\underset{︸}{\sum_{l_{PV}^{T} = 1}^{L_{PV}} \sum_{l_{LD}^{T} = 1}^{L_{LD}} \sum_{l_{MG}^{T} = 1}^{L_{MG}}}} χ^{⋂_{t = 1}^{T} (l_{WT}^{t}, l_{PV}^{t}, l_{LD}^{t}, l_{MG}^{t})} = β_{WT}^{l_{WT}^{t}} \\ l_{WT}^{t} = 1, 2, \dots, L_{WT} \end{matrix}$ (19)

$\begin{matrix} \underset{t = 1}{\underset{︸}{\sum_{l_{WT}^{1} = 1}^{L_{WT}} \sum_{l_{LD}^{1} = 1}^{L_{LD}} \sum_{l_{MG}^{1} = 1}^{L_{MG}}}} . . . \\ \underset{t = T}{\underset{︸}{\sum_{l_{WT}^{T} = 1}^{L_{WT}} \sum_{l_{LD}^{T} = 1}^{L_{LD}} \sum_{l_{MG}^{T} = 1}^{L_{MG}}}} χ^{⋂_{t = 1}^{T} (l_{WT}^{t}, l_{PV}^{t}, l_{LD}^{t}, l_{MG}^{t})} = β_{PV}^{l_{PV}^{t}} \\ l_{PV}^{t} = 1, 2, \dots, L_{PV} \end{matrix}$ (20)

$\begin{matrix} \underset{t = 1}{\underset{︸}{\sum_{l_{WT}^{1} = 1}^{L_{WT}} \sum_{l_{PV}^{1} = 1}^{L_{PV}} \sum_{l_{MG}^{1} = 1}^{L_{MG}}}} \dots \\ \underset{t = T}{\underset{︸}{\sum_{l_{WT}^{T} = 1}^{L_{WT}} \sum_{l_{PV}^{T} = 1}^{L_{PV}} \sum_{l_{MG}^{T} = 1}^{L_{MG}}}} χ^{⋂_{t = 1}^{T} (l_{WT}^{t}, l_{PV}^{t}, l_{LD}^{t}, l_{MG}^{t})} = β_{LD}^{l_{LD}^{t}} \\ l_{LD}^{t} = 1, 2, \dots, L_{LD} \end{matrix}$ (21)

$\begin{matrix} \underset{t = 1}{\underset{︸}{\sum_{l_{WT}^{1} = 1}^{L_{WT}} \sum_{l_{PV}^{1} = 1}^{L_{PV}} \sum_{l_{LD}^{1} = 1}^{L_{LD}}}} \dots \\ \underset{t = T}{\underset{︸}{\sum_{l_{WT}^{T} = 1}^{L_{WT}} \sum_{l_{PV}^{T} = 1}^{L_{PV}} \sum_{l_{LD}^{T} = 1}^{L_{LD}}}} χ^{⋂_{t = 1}^{T} (l_{WT}^{t}, l_{PV}^{t}, l_{LD}^{t}, l_{MG}^{t})} = β_{MG}^{l_{MG}^{t}} \\ l_{MG}^{t} = 1, 2, \dots, L_{MG} \end{matrix}$ (22)

It is worth mentioning that the weighting coefficients {1/p_s} in the objective function lead to re-selection of the initial scenarios with a higher probability and assign new probabilities to them. Equation (17) indicates that the sum of the probability of the new scenarios should be equal to 1. The probability of each new scenario is truly limited in Equation (18). The Equations (19–22) indicate that the sum of the probabilities of the new scenarios which include the uncertain variables with $l_{WT}^{t} / l_{PV}^{t} / l_{LD}^{t} / l_{MG}^{t}$ error interval is equal to $β_{WT}^{l_{WT}^{t}} / β_{PV}^{l_{PV}^{t}} / β_{LD}^{l_{LD}^{t}} / β_{MG}^{l_{MG}^{t}}$ respectively.

5 Shuffled frog-leaping algorithm

The first proposal for the SFLA was introduced and used for optimal design of a water distribution network by Eusuff and Lansey in [17]. SFLA is a population-based meta-heuristic optimization technique which is based on the memetic evolution of a group of frogs when searching for the location with maximum amount of food [18]. This algorithm has been recently tested for several engineering optimization problems such as AC-DC optimal power flow dispatch [19] and parameter identification of Jiles-Atherton model [20]. The main advantage of this algorithm is its fast convergence with accuracy in finding global solutions [21]. In this method, an initial population of Z virtual frogs is created randomly in a multidimensional search space. The location of each frog (e. g. rth frog) can be represented as X_r = (x_r1, x_r2, …, x_rNv), where N_v is the number of problem variables. Next, the fitness functions of the frogs are calculated and the frogs stored in descending order considering their fitness values. Then, the whole population is portioned into m separate groups (memeplexes), where each memeplex include n frogs (i.e., Z = m × n). In this algorithm, the first frog goes to the first memeplex, the second frog goes to the second memeplex, the mth frog goes to the mth memeplex, and the (m + 1) th frog goes to the first memeplex, and so on [17]. The locations of the frogs with the best and worst fitness values in each memeplex are identified as X_b, X_w respectively. Moreover, the location of a frog with the best global fitness value (considering all of the memeplexes) is identified as X_g. In each memeplex, the frog with worst fitness value (not all frogs) leaps towards the best frog in order to improve its location as follows: $D_{i} = rand () \times (X_{b} - X_{w})$ (23) $X_{w}^{new} = X_{w}^{current} + D_{i} (D_{i}^{min} < D_{i} < D_{i}^{max})$ (24)

In order to promote the ability of the algorithm in local and global search, the search acceleration factor, C, is considered in Equation (24) as follows: $X_{w}^{new} = X_{w}^{current} + C \times D_{i} (D_{i}^{min} < D_{i} < D_{i}^{max})$ (25)

If the worst frog can obtain better fitness value, it will be replaced, otherwise, the Equations (23) and (24) are recalculated with the replacement of X_b by X_g. By the way, if there is no improvement in fitness value of worst frog again, a new X_w will be generated randomly within the feasible space. The calculation continues for a specific number of iterations [17]. After a predefined number of memetic evolutionary steps within each memeplex, the solution of evolved memeplexes comes together to share their ideas in shuffling process [17]. The shuffling process and local search continue until the convergence criteria are met. It should be noted that in the SFLA, there are some parameters such as number of memeplexes and population and search acceleration factor which can affect the performance of this algorithm. These parameters should be appropriately tuned in each optimization problem. Figure 2 shows the flowchart of the SFLA. As mentioned earlier, the optimal operation management of the MG cannot be easily solved by conventional optimization method. In the present study, the SFLA is used to solve the optimal operation management problem in a scenario-based structure.

6 Case study

The system used for the case study, as shown in Fig. 3, is a modified low-voltage grid-connected MG consists of different energy sources such as MT, WT generators, PV cells and ESS (Ni-MH-Battery) [9]. The maximum and minimum power limits of DGs are presented in Table 2. In the same table, the start-up and shut down costs of each generation unit are given. A step-down transformer (20 kV/400V) also connects the macro grid and the MG. In this study, it is assumed that the transformer has no fault and the MG operates in a grid-connected mode in the examined period of time. The MG, as shown in Fig. 3, consists of three types of customers (i.e., commercial, industrial and residential customers). The percentage of each feeder from total load of the MG is presented in Table 3. The SCDF of these three types of customers is also given in this table. The forecasted load demand at each hour of the day, the forecasted electricity market price and the forecasted power generation of renewable energy sources (WT and PV) are shown in Fig. 4.

7 Simulation results and analysis

The problem of optimal operation management involves a combination of both continuous and discrete variables. In this study, the SFLA is employed to obtain an optimal operational planning of the modified MG considering cost minimization. In order to make the study more applicable, both deterministic and stochastic analysis are implemented. In each analysis, two cases are investigated separately. In the first case, the battery is supposed to be partially charged (about 50%) while in the second case, the initial charge of the battery is zero. It is evident that the battery should be charged in some hours in order to be discharged in the other hours. It means that the amount of energy released from the battery is limited to the energy charged during the sameday only.

The following assumptions are considered:

All of the power units in the MG work at unity power factor and there is no reactive power trading.

No heat load demand is considered and all of the loads are supposed to be electrically typed.

All of the power generated by PV and WT is bought by the MG Central Controller (MGCC).

A scheduling horizon of one day is considered.

In this study, the parameters of each optimization algorithm are tuned so that the algorithms achieve acceptable fitness tolerance (10^–2) in the reasonable number of iterations. The experiences show that:

The GA algorithm has a good performance with 40 individuals and 100 generations. Also, the mutation and crossover probabilities are set to 5% and 90%, respectively.

The perturbation parameter and the number of iterations of the SA algorithm are tuned to 0.977 and 2000, respectively.

In the PSO algorithm, the population size of 200 and 20 generations can provide appropriate results. The social and cognitive attractions are tuned to 2 and 1.75, respectively.

The SFLA can find appropriate results when the population size, the number of iterations, the number of memeplexes and weight factor are tuned as 150, 20, 10 and 2, respectively.

All simulations are conducted on a laptop computer with 2.5 GHz Dual core processor and 1 GB of RAM.

7.1 Analysis in deterministic environment

In the deterministic framework, all variables are considered as definite. This means that the predicted values for the uncertain variables are correct and will definitely happen. Therefore, in this section, the predicted values of the power output of PV and WT, the electricity price in the market and the load demand in each hour of a day are utilized as input variables for solving the optimal operation management problem.

At first, in order to show the higher capability of the SFLA compared to the other heuristic algorithm (i.e., GA, PSO and SA), the simulations are implemented for the two mentioned cases. The performance of each algorithm can be evaluated by the fitness value (the value of the objective function) and computational time of the simulation. Due to the random nature of the heuristic algorithms, their performances are compared in several trials with different initialization. The simulation results for case 1 and 2 for 30 independent trials are presented in Tables 4 and 5 respectively. Moreover, the best and worst fitness values with mean and variance of the fitness values for each algorithm are shown in the same tables. It can be clearly seen from Tables 4 and 5 that the SFLA not only gives the lowest fitness value compared to the other implemented algorithms but also finds the optimal solution in the shortest time. Furthermore, a comparison of the average and standard deviation values reveals the superiority and reliability of the SFLA. It is also notable that due to the high penetration of renewable energy resources into the MG and considering their high cost of operation, the total operating cost of the MG does not vary greatly in case 1 and 2. However, the difference observed in the results is due to the initial status of the battery.

Figure 5 shows the optimal generation points of DG units in the MG and of the external grid for case 1 obtained using the SFLA. In this case, as mentioned earlier, the Ni-MH-Battery is assumed to be partially charged (about 50%). To charge the remaining capacity, it is beneficial to charge the battery in the first hours of the day when the MG has a light load level and bid of the utility is lower in comparison with the other DGs. However, in the midday, the load demand and electricity price are increased simultaneously. This leads the process of discharging to start which can reduce the amount of energy imported from the external grid and therefore the total operating cost of the MG. The amount of energy released from the battery is determined by and limited to the amount of energy which can be recharged in the next hours (i.e., at 3 and 4 PM). If the recharging process does not happen, the total operation cost increases due to the increase in customer interruption cost, which seems to be logical. It should be noted that, during these hours the MT produces its maximum power for further reducing the total operation cost by exporting or selling energy to the external grid. In the next hours (from 5 PM to 10 PM), the MG experiences a high load level; therefore, the optimization problem forces the MT to produce their maximum power in order to reduce load shedding.

In case 2, as mentioned before, it is supposed that the Ni-MH-Battery does not have any initial electrical charge. In this case, the charging (discharging) process of the battery during the day is slightly different than the previous one. Charging of the battery, as shown in Fig. 6, is not limited to the first hours of the day and continues until the midday. Although, at first glance, the sale of energy to the external grid seems to be beneficial in the midday, the MG operator prefers to purchase energy from it. This is due to the fact that the customer interruption cost and therefore, total cost increases in the next hours. In order to get a better insight into the charging (discharging) process, the state of charge (SOC) of the battery in case 1 and 2 is shown in Fig. 7. It can be seen that the SOC is basically higher when the load level is low and vice versa. It is worth noting that the SOC is less dependent on energy prices offered by the external grid and DGs at different times of the day.

7.2 Analysis in stochastic environment

As mentioned earlier, wind speed and solar radiation are random in nature and different techniques cannot predict them accurately. Similarly, load demand and energy price in the market can be considered as uncertain variables. In this way, a normal probability distribution function with seven probability levels for each uncertain variable is used and all of the scenarios are generated. It should be noted that the standard deviation of the uncertain variables is assumed to be 0.03 (i.e., δ= 0.03). Next, the original stochastic model should be transformed into a deterministic equivalent using the reduced set of scenarios. By means of the proposed LP model, a subset of the original scenarios is selected. The minimum number of new scenarios is reduced to 34. The probability of the new scenarios is rearranged considering the objective function and the constraints previously proposed in Section 4.1. In fact, a weight is designated for each new scenario that can indicate the probability of occurrence of the scenario. Afterwards, the optimization problem of the MG is solved for each of the new scenarios obtained. At last, the results obtained in the previous step are aggregated based on their probabilities. The simulation results for 30 independent trials are presented in Table 6. In addition, the expected power generation for DGs for case 1 and 2 in stochastic environment is presented in Tables 7 and 8, respectively. A comparison of the results in deterministic and stochastic environments reveals that the total operation cost has increased due to the uncertainty of the optimization problem variables. In this analysis, unlike the deterministic analysis, the available generations are not able to supply the load at all times of the day; Therefore, the load shedding occurs in some hours of the day, which increases the total operating cost. However, the analysis of the information obtained from simulation results in the stochastic environment may provide the MG operator with a more comprehensive attitude to manage the MG with higher reliability and lower cost of operation.

8 Conclusion

In this paper, a probabilistic approach was presented to model the uncertainty of WT and PV power generation, electricity market price and load demand in the optimal operation management of the MG. The problem of large number of scenarios, which is due to the uncertainty of the input variables, was solved by utilizing an efficient LP model. The simulation results showed that the proposed model can effectively select a minimum subset of scenarios (i.e., 34 scenario with new probabilities) when normal probability distribution function with seven probability levels is considered for each of the input variables. Considering the nature of the operation management problem of the MG, the SFLA was utilized and compared with other heuristic algorithms. Thorough examination of the results showed that the SFLA not only can help to achieve the optimal solution, but also finds the optimal solution in the shortest time. However, the presented framework can help the MGCC manage the generation of the DG units in the MG and external grid, especially in a stochastic environment. However, The methodology adopted permits the MG operator to decide the best compromise between the cost of purchasing energy from external grid and DGs in the MG and the cost of energy notsupplied.

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