A sufficient stochastic framework for optimal operation of micro-grids considering high penetration of renewable energy sources and electric vehicles

Abstract

This paper investigates the optimal operation and management of micro-grids (MGs) considering the high penetration of different types of renewable energy sources (RESs) and plug-in electric vehicles (PEVs). Optimal operation of MG is a precious and economic strategy for improving the situation of MG from different aspects such as reducing power losses and total costs, improving the reliability and enhancing the voltage level of buses. In order to achieve these goals, a new formulation is proposed to model the stochastic and volatile behaviors of RESs as well as PEVs. Also, the idea of vehicle-2-grid (V2G) is employed to provide a bi-directional power exchange (either charging/discharging) between PEV and the main grid during the parking hours. In order to model the uncertainties of the problem, a new sufficient stochastic optimization framework based on unscented transform (UT) and krill herd algorithm (KH) is proposed too. The feasibility and performance of the proposed method are examined on a typical grid-connected MG considering different types of RESs including Photovoltaics (PVs), Fuel Cell (FC), Micro Turbine (MT) and Wind Turbine (WT).

Keywords

Micro grid (RMG)renewable energy sources (RESs)plug-in electric vehicle (PEV) and Vehicle-2-Grid (V2G)

Nomenclature

a, b

Wöhler curve parameters.

Cost _sub

Cost of power supply by the

up-steam network.

Cost _PEV

Cost of aggregated PEVs.

Cost _Rel

Cost of reliability.

Cost _DG

Cost of power generation

by DGs.

Cost

Total network cost.

C_{sub}^{t}

Hourly price of energy

purchased from main grid

energy/loss/V2G.

C_{PEV}^{t}

Hourly price of V2G.

C _i

Cost of interruption of the ith

bus in ($/kW).

C _Bat

Battery investment cost ($).

C_{DG, k}^{t}

Hourly price of energy

purchased from DGs.

Dimension of the problem

investigated.

f_i/f_b/f_w

Objective value of ith/best/

worst individual.

f_{i}^{b}

Objective value of best

individual in ith movement.

Iter

Iteration counter.

DoD_i/DoD_f

Initial/final DoD in a

discharge cycle.

E _bat

Usable energy of the battery

(kWh).

E_{D, v}^{t}

Energy for PEVs in fleet v to

drive at time t.

E_{v}^{t}

Available energy in batteries of

fleet v at time t.

E_{v}^{ini} / E_{v}^{fin}

Initial/final energy in PEV

fleet v.

E_{v}^{min} / E_{v}^{max}

Min/max energy in batteries of

PEV fleet v.

f _s

Optimal function value for sth

scenario.

L _a(i)

Average load connected to

ith bus.

Number of uncertain variables.

N_L/N_br/N_bus

Number of loops/branches/buses

of network.

N _v

Total number of PEVs.

N _dis

Number of discharge cycles.

N _c

Number of life cycles.

N _tie

Number of tie switches of grid.

N _Sw

Number of sectionalizing

switches of grid.

N _U

Number of uncertain variables

of the problem.

N _s

Number of surrounding

neighbors.

N _DG

Number of DG units including

MTs and FCs.

N _Cus

Total number of customers

served.

N _p

Population size.

P_{DG, k}^{t}

Output power of kth DG at

hour t.

P_{sub}^{t} / P_{sub}^{max}

Hourly/max imported power

from upstream grid.

P_{c, v}^{t} / P_{d, v}^{t}

Charging/discharging capacity

of PEV fleet v.

P_{c, v}^{min} / P_{c, v}^{max}

Min/max charging capacity of

PEV fleet v.

P_{d, v}^{min} / P_{d, v}^{max}

Min/max discharging capacity

of PEV fleet v.

P_{DGi, min}^{t} / P_{DGi, max}^{t}

Min/max power capacity of ith

DG at time t.

P_{v}^{t}

Charge/discharge power rate of

PEV fleet v at time t.

P_{i}^{t} / Q_{i}^{t}

Hourly injected active/reactive

power at bus i.

Number of uncertain

parameters.

P_xx/P_yy

Covariance of input variable X/

output variable Y.

S_{ij}^{t} / S_{ij}^{max}

Hourly/max apparent power

flow between bus i and j.

Planning horizon.

t′

Time in which SOC is set to a

specific value.

u_j/l_j

Upper/lower bound of jth

control variable.

U_{v}^{t}

Status of grid connection of

fleet v at time t.

U_{c, v}^{t} / U_{d, v}^{t} / U_{i, v}^{t}

Indicator of fleet v in charge/

discharge/idle mode.

V_{i}^{t} / δ_{i}^{t}

Voltage magnitude/phase of

bus i at hour t.

V_{ind, i}^{m} / V_{frg, i}^{m} / V_{dif, i}^{m}

Induced/foraging/diffusion

velocity of ith krill at the mth

movement.

V_{ind}^{max}

Maximum induced velocity.

V_{r, i}^{m}

Resultant velocity of ith krill

at mth movement.

V_{i}^{min} / V_{i}^{max}

Minimum/maximum voltage

at bus i.

W ^k

kth weighting factor of the

sample point.

X_i/X^best/X_r

Position of ith/best/randomly

selected individual.

X_{i}^{b}

Position of best individual in

ith movement.

Y_ij/θ_ij

Magnitude/phase of impedance

between bus i and j.

ω_ind/_frg/_dif

Inertia of induction/foraging/

diffusion motion.

Small positive number.

ρ _i

ith random number between

[0, 1].

α_ind,I

Attractive/repulsive tendency

factor.

η_c/η_d

Charge/discharge efficiency.

θ, χ

Empirical constant factors.

distance between the two krill

in the water.

β₀

initial attractiveness at r = 0

(while krill fireflies are

completely near each other).

λ _i

Failure rate of the ith

component.

μ_i

Mean value of ith input random

variable.

1 Introduction

In recent years, the increasing concerns over the high amount of pollutions caused by the fossil fuel based devices have motivated the electrical engineers and companies to make use of renewable energy sources (RESs). The use of RESs in the network can provide many benefits including more reliable and secure electrical services, higher power quality, less interruptions, less power losses and lower amount of pollution [1 –4]. In the new competitive power market, the use of different types of distributed generations (DGs) such as wind turbine (WT), fuel cell (FC), photovoltaics (PVs) and micro turbine (MT) can help the network to have effective role in the market. Along with these advantages, the high penetration of DGs especially in the form of RESs can create some unforeseen and unwanted challenges for the distribution system. Parts of these challenges are addressed in the issues of Micro-Grid (MG). By definition, MG is a group of interconnected loads and distributed generations (DGs) which can either connect or disconnect to/from the grid to enable its operation in either grid-connected or islanded mode, respectively [5]. In the view of the main grid, MG is either a controllable unit inside the network which can be regarded as a single load to be supplied or as a power source to help supply a portion of the electrical loads [6]. According to the attractiveness of RESs and MGs, wide researches have been implemented in recent years.

In [7], Chedid et al. made use of a linear programming technique to lessen the average power production charges in a solar and wind MG considering the environmental factors. In [8], Niknam et al. proposed a heuristic approach to operate a FC-based MG considering thermal and hydrogen generation ability of FC. In [9], Chakraborty et al. suggested a linear programming method for handling and scheduling the battery charge/discharge rate when minimizing the total cost. In [10], Tsikalakis et al. assessed the bi-directional effect of MG and the up-stream grid on each other while optimizing the total power production. In [11], Hernandez-Aramburo et al. proposed a new formulation for minimizing the total electrical/thermal fuel cost considering minimum reserve power. In [12], Dukpa et al. suggested a price-base participation solution to schedule the WT and storage device for the 24-hour program in the unit commitment problem. A practical three-phase approach based on matrix real-coded genetic evolutionary algorithm was suggested by Chen et al. in [13]. The proposed formulation is constructed based on the prediction, storage and management modules for optimizing the MG costs. While each of these works have considered special aspects of MG, none of them has considered the penetration of electric vehicles (EVs). EV technology is a newly introduced device than has the same purposes as RESs for reducing the air pollutions. Therefore, it has attracted the attention of most of researchers in recent years [14]. According to the statistical reports, EVs would be occupy notable portion of the future transportation system [15, 16]. In this regard, a high number of researches are done on the EVs and their effects on the smart grids [17 –20]. In addition, the high amount of uncertainty that is injected to MG should be modeled using an appropriate stochastic method.

According to the above discussion, the main purpose of this paper is to investigate the optimal operation of MGs considering the high penetration of different types of DGs including MT, FC, WT and PV as well as EVs. In this work, plug-in EVs (PEVs) are considered in the MG due to the high popularity and success in the EV market. A PEV is any motor vehicle that can be recharged from an external source of electricity, such as wall sockets [21, 22]. The most significant issue with the appearance of PEVs in the MG is the stochastic behavior of EV owners which can affect all the pre-determined schedules in the MG central control (MGCC). In fact, the high penetration of DGs, RESs and PEVs can inject much uncertainty in the MG which should be handled, appropriately. In this way, a new stochastic framework based on unscented transform (UT) is proposed to model the uncertainties of the problem. UT is a newly introduced superposition approach which has revealed great performance in modeling nonlinear transformations and state estimators [23]. The structure of UT is such that it can model either dependent or correlated uncertainty of parameters. Also, in order to make the existence of PEVs as an opportunity for the MG, the idea of veicle-2-grid (V2G) technology is used to make the control of charge/discharge of energy between the MG and main grid possible. According to the high complexity and nonlinearity of the scheduling RESs and PEVs, a powerful optimization algorithm is required to solve the problem optimally. In this regard, a new optimization method called modified krill herd (MKH) algorithm is proposed too [24]. KH is a novel evolutionary algorithm that mimics the foraging behavior of krill animals in the ocean. Also, a new modification method is proposed for this algorithm to increase its search ability and reduce the possibility of trapping in local optima. The cost function is defined as the total cost of the microgrid including the cost of power generation by DGs and RESs, cost of power supply for PEVs, cost of V2G (battery aging cost), cost of power purchased from the main grid and cost of energy not supplied (ENS) as a reliability index. The optimization problem tries to minimize this cost by optimal scheduling of DGs, RESs and PEVs. This cost is the total microgrid cost which is on the owner of the microgrid. Though the minimization of this cost, the microgrid owner is profited. This is a significant target in the operation of the distribution systems in the electricity market. The feasibility and satisfying performance of the proposed stochastic optimization method is examined on a grid-connected MG.

The rest of this paper is organized as follows: Section 2 describes the PEV technology and the problem formulation. Section 3 describes the proposed UT method. The optimization algorithm based on MKH is explained in Section 4. In Section 5, the application procedure is explained in some steps. The simulation results are discussed in Section 6. Finally, the main conclusions and concepts are given in Section 7.

2 PEV technology and MG problem

This section first describes the PEV technology and then explains the problem formulation including the cost function and the practical limitations of the problem.

2.1 PEV technology

PEV, or Plug-in Electric Vehicle, refers to all vehicles with a battery that can be charged by plugging into an electrical outlet. One of the main benefits of PEVs over other substitute vehicles such as gas-based cars is that the required infrastructure already exists; it means the electricity network. In addition, some of the main benefits of PEVs in comparison with other internal combustion engines are lower operation and maintenance costs and generation of less air pollution. While there are many benefits for PEVs, they can be yet problematic especially for the electricity grid. In the view of the electricity network, high number of PEVs (generally in the form of PEV fleet) are mobile loads that should be managed to avoid any electricity mismatch between the generation and supply. In other words, without a smart and optimal scheduling program, PEVs charging demand can coincide the peak-load hours and thus cause interruptions in the network. In order to deal with this issue, the behavior of PEVs should be first modeled. Each PEV fleet can be known by some characteristics such as number of PEVs in each fleet, departure time, arrival time and the departure locations and destinations. By the use of these characteristics, the operation features of PEVs such as energy consumption, min/max capacity and state of charge (SOC) can be determined directly. In the technical terms, SOC is defined as the ratio of available energy to maximum storable energy in the battery [21]. In accordance with the statistical information, each driver experiences two main trips per day with few short trips up to 10 minutes [25]. Short trips can be neglected in the hourly scheduling. Therefore, each PEV fleet will experience two main trips; one trip from a starting point (from home to work location) in the morning and the other trip is coming back to the same point at the end of the day or at evening (from work to the home). According to the statistics, each PEV fleet experiences an average driving distance of 12,000 miles per year which means 32.88 miles per day [26]. By the average energy consumption of 9 kWh per day, each PEV needs 3.65 kWh per mile [27]. During the hours that PEVs are in the park; either at home or at work; the idea of V2G can be employed to control the charge and discharge processes in PEVs for reducing their costs. This bi-directional power exchange between PEV and MG can help to reduce the costs and support the high penetration of PEVs. Figure 1 shows the V2G idea in the smart grids. During the charge/discharge of PEV, battery depth of discharge (DOD) is considered 20% to avoid high aging phenomenon in the battery [21].

2.2 Problem formulation

This section explains the problem formulation including the objective function and limitations. The problem is formulated as a single-objective minimization problem with several equality and inequality constraints which are described in the rest.

2.2.1 Minimization of the cost function

The cost function consists of several parts: 1) cost of power generation by DGs and RESs, 2) cost of PEVs including the cost of charging and cost of V2G (battery aging cost), 3) cost of power purchased from the main grid and 4) cost of energy not supplied (ENS) as a reliability index.

$\begin{matrix} Min Cost \\ = {Cost}_{DG} + {Cost}_{PEV} + {Cost}_{sub} + {Cost}_{Rel} \end{matrix}$ (1)

Cost of Power Supplied by DGs:

The cost of power generation by different types of DGs is calculated as follows: ${Cost}_{DG} = \sum_{t = 1}^{T} C_{{DG, k}^{t} P_{DG, k}^{t}}$ (2)

Cost of PEV:

The cost of PEV includes both the cost of charge/discharge (Cost_Op) and the cost of battery aging due to the V2G technology (Cost_Age). The cost of charge and discharge is calculated as follows: ${Cost}_{Op} = \sum_{t = 1}^{T} \sum_{n = 1}^{N_{v}} U_{v}^{t} C_{v}^{t} P_{v}^{t}$ (3)

The sign of $P_{v}^{t}$ determines either the PEV is charging (positive sign) or discharging (negative sign). The cost of battery aging Cost_Age is calculated according to the Wöhler curve [21]. Figure 2 shows Wöhler curve for a typical battery. Wöhler curve shows the number of cycles that the battery experiences before a complete failure. Mathematically, Wöhler curve is formulated as below [21]: $N_{c} (DoD) = a . {DoD}^{b}$ (4) where, parameters a and b are determined based on the battery type.

By the use of the number of cycles calculated from Wöhler curve, the degradation cost of the battery from fully charged state (DoD = 0) to a particular charge level (DoD = DoD_s) is computed as follows [28]: $C_{d} (0, {DoD}_{s}) = \frac{C_{Bat} \times {DoD}_{s} \times E_{bat}}{N_{c} ({DoD}_{s})}$ (5)

By the use of Equation 5, the degradation cost of battery from DoD_i to DoD_f is calculated as below [28]: $C_{d} ({DoD}_{i}, {DoD}_{f}) = C_{d} (0, {DoD}_{f}) - C_{d} (0, {DoD}_{i})$ (6)

Therefore, the total degradation cost in a calculated as the sum of degradation costs over the number of discharges as follows [28]: ${Cost}_{deg} = \sum_{k = 1}^{N_{dis}} C_{d}^{k} ({DoD}_{i}, {DoD}_{f})$ (7)

Main Grid Cost:

This term is the cost of energy purchased by MG from the main grid as follows: ${Cost}_{sub} = \sum_{t = 1}^{T} C_{sub}^{t} P_{sub}^{t}$ (8)

Cost of ENS:

ENS is a significant index in the reliability of power systems which calculates the amount of expected energy not supplied by the MG as follows:

$\begin{matrix} {Cost}_{sub} & = & \sum_{t = 1}^{T} C_{ENS}^{t} {ENS}^{t} \\ {ENS}^{t} & = & \sum_{i = 1}^{N_{Cus}} {La}_{i}^{t} U_{i}^{t} \end{matrix}$ (9)

2.2.2 Limitations and constraints

The problem constrains are some operation and security limitation that should be met during the optimization as follows:

DG Capacity:

P_{DGi, min}^{t} \leq P_{DGi}^{t} \leq P_{DGi, max}^{t}

(10)

Power flow equality constraints:

{\begin{matrix} P_{i}^{t} = \sum_{j} | V_{i}^{t} | | V_{j}^{t} | | Y_{ij} | cos (θ_{ij} + δ_{i}^{t} - δ_{j}^{t}) \\ Q_{i}^{t} = \sum_{j} | V_{i}^{t} | | V_{j}^{t} | | Y_{ij} | sin (θ_{ij} + δ_{i}^{t} - δ_{j}^{t}) \end{matrix}

(11)

Bus voltage limit:

V_{i}^{min} \leq V_{i}^{t} \leq V_{i}^{max}

(12)

Maximum power flow between the MG and the main grid:

| P_{sub}^{t} | \leq P_{sub}^{max}

(13)

Maximum power flow in each branch:

S_{ij}^{t} \leq S_{ij}^{max}

(14)

Hourly charge/discharge/idle states of fleets: As long as PEVs are in the parking, they can be charged, discharged or be in idle mode depending on the optimization cost function:

U_{c, v}^{t} + U_{d, v}^{t} + U_{i, v}^{t} = U_{v}^{t}

(15)

There the parameter $U_{v}^{t}$ guaranties the attendance of PEV fleet v in the charging station at time t. When $U_{v}^{t} = 1$ it means that PEV fleet is connected to the grid; either charging or discharging. On the opposite point, $U_{v}^{t} = 0$ shows that the PEV fleets is in idle mode.

Maximum/Minimum charging rate:

U_{c, v}^{t} P_{c, v}^{min} \leq P_{c, v}^{t} \leq U_{c, v}^{t} P_{c, v}^{max}

(16)

Maximum/Minimum discharging rate:

U_{d, v}^{t} P_{d, v}^{min} \leq P_{d, v}^{t} \leq U_{d, v}^{t} P_{d, v}^{max}

(17)

Available energy balance in PEV batteries:

$\begin{matrix} E_{v}^{t} & = & E_{v}^{ini} + \sum_{m = 1}^{t} (U_{c, v}^{m} P_{c, v}^{m} η_{c, v} - U_{d, v}^{m} P_{d, v}^{m} η_{d, v}) \\ - \sum_{m = 1}^{t} (1 - U_{v}^{m}) E_{D, v}^{m} \end{matrix}$ (18)

Energy capacity limit of each fleet:

P_{v}^{t} = E_{v}^{t} - E_{v}^{t - 1}

(19)

E_{v}^{min} \leq E_{v}^{t} \leq E_{v}^{max}

(20)

E_{v}^{fin} = E_{v}^{ini}

(21)

SOC of PEV battery in the beginning of the first trip: In the morning and at the beginning of the first trip, all PEVs are supposed to start the first trip being fully charged. This is a practical constraint since most of EV owners tend to charge their cars during night hours:

E_{v}^{t^{'}} = E_{v}^{max}

(22)

In the proposed problem, the control variables are 1) optimal output power of DGs including FC and MT shown by $P_{DG, k}^{t}$ ; 2) charge/discharge/idle states of PEV fleets $U_{c, v}^{t} / U_{d, v}^{t} / U_{i, v}^{t}$ , respectively; 3) amount of exchanged power between PEV fleet and MG at each hour $(P_{v}^{t})$ as follows: $X = [\bar{X_{DG}}, \bar{X_{P_{v}}}, \bar{X_{U_{c}}}, \bar{X_{U_{d}}}, \bar{X_{U_{i}}}]$ (23) $\bar{X_{DG}} = [\bar{P_{DG, 1}}, \bar{P_{DG, 2}}, \dots, \bar{P_{DG,_{N_{DG}}}}]$ (24) $X_{P_{v}} = [{\bar{P}}_{1}, {\bar{P}}_{2}, \dots, {\bar{P}}_{N_{v}}]$ (25) $X_{U_{c}} = [{\bar{U}}_{c, 1}, {\bar{U}}_{c, 2}, \dots, {\bar{U}}_{c, N_{v}}]$ (26) $X_{U_{d}} = [{\bar{U}}_{d, 1}, {\bar{U}}_{d, 2}, \dots, {\bar{U}}_{d, N_{v}}]$ (27) $X_{U_{i}} = [{\bar{U}}_{i, 1}, {\bar{U}}_{i, 2}, \dots, {\bar{U}}_{i, N_{v}}]$ (28) ${\bar{P}}_{k} = [P_{k}^{1}, P_{k}^{2}, \dots, P_{k}^{T}] k = 1 : N_{v}$ (29) ${\bar{U}}_{c, k} = [U_{c, k}^{1}, U_{c, k}^{2}, \dots, U_{c, k}^{T}] k = 1 : N_{v}$ (30) ${\bar{U}}_{d, k} = [U_{d, k}^{1}, U_{d, k}^{2}, \dots, U_{d, k}^{T}] k = 1 : N_{v}$ (31) ${\bar{U}}_{i, k} = [U_{i, k}^{1}, U_{i, k}^{2}, \dots, U_{i, k}^{T}] k = 1 : N_{v}$ (32)

3 Stochastic method based on UT

As mentioned before, the proposed problem includes a high number of uncertain parameters including the charge/discharge values of PEVs, active and reactive loads, hourly market price, WT/PV output power and number of PEVs in each fleet. In order to model the uncertainty effects of these parameters, UT method is proposed in this section. UT method is one of the most popular and successful methods among the approximate techniques. This method was first introduced for modeling uncertainty in the nonlinear correlated transformations [23]. The existence of some special characteristics such as simplicity, high uncertainty modeling ability, low computational burden and ability of usage in correlated environment make UT method an appropriate choice amongst the other well-known stochastic methods. The main idea of UT is constructed on the fact that it is easier to approximate a probability distribution function (PDF) than an arbitrary nonlinear function [23]. The superiority of UT over some other well-known stochastic methods is demonstrated in the literature [29]. In order to explain this technique, suppose the nonlinear function y = f (X); in which y is the output vector, f is the nonlinear function and X is the input uncertain variables. Considering m uncertain parameters, X would be a vector with length m with the mean value μ and covariance P_xx. In the independent environment, P_xx is a symmetrical matrix. In a correlated environment, P_xx would become non-symmetrical. The UT method converts a stochastic problem with m uncertain parameters to 2m + 1 deterministic problems that each is solved individually. The result is the mean μ_y and covariance matrix P_yy of the output y. Therefore. the below steps are required to solve the problem [29]: $x^{0} = μ$ (33) $x^{k} = μ + {(\sqrt{\frac{n}{1 - W^{0}} P_{xx}})}_{k}; k = 1, 2, \dots, m$ (34) $x^{k} = μ - {(\sqrt{\frac{n}{1 - W^{0}} P_{xx}})}_{k}; k = 1, 2, \dots, m$ (35)

where the term (A) _k in the above equations is the kth row or column of matrix A. Also, W⁰ is the initial weight of the mean value μ.

Step 2: Compute the weighting factor of concentration points as follows: $W^{0} = W^{0}$ (36) $W^{k} = \frac{1 - W^{0}}{2 m}; k = 1, 2, \dots, m$ (37) $W^{k + n} = \frac{1 - W^{0}}{2 m}; k + m = m + 1, \dots, 2 m$ (38)

It should be noted that the summation of weighting factors should become one unit: $\sum_{k = 0}^{2 m} W^{k} = 1$ (39)

Step 3: Input 2m + 1 sample points to the nonlinear function to find the output samples: $y^{k} = f (X^{k})$ (40)

Step 4: Compute μ_y and P_yy for the output variable Y as below [29]: $μ_{y} = \sum_{k = 0}^{2 m} W^{k} Y^{k}$ (41) $P_{yy} = \sum_{k = 0}^{2 m} W^{k} (Y^{k} - μ_{y}) (Y^{k} - μ_{y})^{T}$ (42)

4 Optimization algorithm: Modified krill herd (MKH) algorithm

The proposed problem is a kind of nonlinear, complex optimization problem with several equality and inequality constraints. In order to solve the problem optimally, krill herd (KH) algorithm with a new modification method are explained in this section.

4.1 Original KH algorithm

KH algorithm was first introduced by Gandomi et al. [30] in 2012 to model the foraging behavior of krill in the oceans. In comparison to other evolutionary based optimization algorithms, KH is equipped with some special mechanisms such as crossover and mutation from the genetic algorithm and appropriate balance between the local and global search which can increase the search ability of the algorithm. Similar to other evolutionary algorithms, KH starts with the generation of an initial random population. After calculating the objective function for each krill, the best krill X^b is determined. Then the positions of other krill are updated in the population using the below equation [30]: $X_{i}^{Iter + 1} = X_{i}^{Iter} + V_{r, i}^{Iter} χ \sum_{j = 1}^{N_{v}} (u_{j} - l_{j})$ (43)

As it can be seen from the above equation, the updating process is limited by the lower and upper bounds of control variables and a position constant factor χ. The variable $V_{r, i}^{Iter}$ as the velocity is controlled by three different effective parameters: 1) Induction speed, 2) Foraging speed, and 3) Diffusion speed as follows [30]: $V_{r, i}^{Iter} = V_{ind, i}^{Iter} + V_{frgi, i}^{Iter} + V_{dif, i}^{Iter}$ (44)

Induction Speed $V_{ind, i}^{Iter}$ : The movement of each krill is affected by surrounding krill using the below equation: $V_{ind, i}^{Iter} = α_{ind, i} V_{ind, i}^{max} + ω_{ind} V_{ind, i}^{Iter - 1}$ (45)

The coefficient α_ind models the attractive/repulsive effect of surrounding krill as well as the best krill as follows:

$\begin{matrix} α_{ind, i} & = & \sum_{j = 1}^{N_{s}} [\frac{f_{i} - f_{j}}{f_{w} - f_{b}} \times \frac{X_{i} - X_{j}}{| X_{i} - X_{j} | + ɛ}] \\ + 2 [ρ_{1} + \frac{i}{N_{p}}] f_{i}^{b} X_{i}^{b} \end{matrix}$ (46)

Each krill is affected by surrounding krill in the neighboring circle region with radius of: $R_{Vicinity} = \frac{1}{5 N_{p}} \sum_{j = 1}^{N_{p}} | X_{i} - X_{j} |$ (47)

Foraging Speed $V_{frg, i}^{Iter}$ : Each krill will update its position using the current and previous position of the food as follows:

$\begin{matrix} V_{frg, i}^{Iter} & = & 0.02 [2 (1 - \frac{i}{N_{p}}) f_{i} \frac{\sum_{i = 1}^{N_{s}} \frac{X_{i}}{f_{i}}}{\sum_{i = 1}^{N_{s}} \frac{1}{f_{i}}} \\ + f_{i}^{b} X_{i}^{b}] + ω_{frg} V_{frg, i}^{Iter - 1} \end{matrix}$ (48)

Diffusion Speed $V_{dif, i}^{Iter}$ : Random diffusion pledges the variety of the population using a normal random operator as follows: $V_{dif, i}^{Iter} = υ \times ω_{dif}$ (49)

By the use of the above process, krill herd population is updated. This process is repeated until the termination criterion is satisfied and the best krill position is improved.

4.2 Modification method based on KH algorithm

KH algorithm is a powerful optimization algorithm with special aspects that makes it suitable for optimization of both continuous and discrete optimization problems. Nevertheless, a new modification method is proposed in this section to increase the diversity of the krill population. As a result, the convergence ability of the algorithm is increased and the possibility of trapping in local optima is decreased. In the proposed modification method, a less successful krill (X_j) (krill with worse position in the population) is attracted to a more successful krill (X_i) to improve its position. In this way, a monotonically decreasing function is defined as follows: $β (r) = β_{0} \times exp (- γ r^{m}); m \geq 1$ (50)

The distance between the two krill can be calculated in different frameworks such as the Cartesian (or Euclidean) distance, Manhattan distance or Mahalanobis distance. The distance between the krill (X_i) and krill (X_j) in the Cartesian (or Euclidean) framework is computed as follows: $r_{ij} = ∥ X_{i} - X_{j} ∥ = \sqrt{\sum_{k = 1}^{d} (x_{i, k} - x_{j, k})^{2}}$ (51) where X_i,k is the kth element of the special vector X_i associated with the ith krill.

By the use of the above equations, the position of Krill X_i is improved toward the position of krill X_j using the below equation:

$\begin{matrix} X_{j} & = & X_{j} + β_{0} \times exp (- γ r^{m}) \times (X_{i} - X_{j}) + u_{j} \\ u_{j} & = & α (rand - \frac{1}{2}) \end{matrix}$ (52)

There the first term is the current position of the jth krill; the second term is the attractiveness of krill X_i for krill X_j and the third term is a random movement when no more attractive (more successful) krill is seen in the neighborhood.

5 Solution procedure

The below steps are required to apply the proposed stochastic optimization method on the problem:

Step 1: Collect the input data.

Step 2: Convert the constrained problem into an unconstraint optimization problem using the penalty factors.

Step 3: Generate the initial population of krill. In this problem, each krill contains the optimal output power of the DGs as well as the optimal charging/discharging values of PEVs during the operation time horizon.

Step 4: Run the probabilistic load flow based on UT method.

Step 5: Evaluate the objective function value for the krill population as described in (1–9).

Step 6: Choose the best solution (krill) with lowest cost function value as X^b.

Step 7: Update the krill population as described in Section 5 part A.

Step 8: Apply the proposed modification method on krill population as described in Section 5 part B.

Step 9: Update the population.

Step 10: Repeat steps 6 to 10 until the termination criterion is satisfied.

6 Simulation results

This section investigates the performance of the proposed method on a standard test system. The test system is a typical MG with 32 buses which is connected to main grid through a circuit breaker [31]. The nominal voltage of the MG is 12.66 kV [31]. The single line diagram of the test system is shown in Fig. 3. As it can be seen from this figure, MG contains two WTs, two MTs, one FC and two PEV fleets. The location of each DG is shown in Fig. 3. It is assumed that there is one circuit breaker in the main feeder and there is a sectionalizer at the beginning of each feeder. The proposed stochastic framework models the uncertainties of charge/discharge pattern in PEVs, active and reactive loads of the MG, hourly market price, WT/PV output power, number of PEVs in each fleet and the arrival or departure time of PEV fleets.

Table 1 shows the number of customers served in each busbar. Also, the complete information of DGs including the capacity, position, start up/shut down costs and cost of energy generation are shown in Table 2. For the sake of simplicity, similar power curves are considered for WTs. The main distinction between the WTs is that the output power of WT-2 is supposed to be 1.2 times of WT-1. The normalized forecast output power of WT and PV sources are depicted in Fig. 4A and B. The hourly total load demand as well as the market price are shown in Fig. 4C and D, respectively.

In terms of PEVs, two fleets with different characteristics are considered in the MG. One of the PEV fleets starts its first trip from inside the MG while the other one comes to MG from the outside area. The number of PEVs in each fleet along with the capacity and parking location are given in Table 4. It is worth noting that in this work lithium-ion battery due to the high popularity is considered for PEVs. The parameters of Wöhler curve for this type of battery are a = 1331 and b = –1.825.

The operation problem is scheduled in 24 hours. It is supposed that all DGs work at unit power factor and thus just produce active power. In addition, the MG can decide to exchange power with the main grid at any hour of the scheduling time based on the benefits. All loads are considered to be electric type and thermal loads are not considered here. According to Table 2, PV and WT have the highest cost of power generation among the DGs. This event roots in the high initial capital cost of these DGs. In order to support the idea of RESs, the MG will buy all the power produced by these DGs at any hour of the day. For better understanding of problem, two different status of operation (SOP) are defined. In SOP1, PEVs are neglected in the MG and it is assumed that all DGs would be ON. In SOP2, the situation is similar to SOP1 just DGs are allowed to turn ON or OFF depending on the benefits in the cost function. Finally in SOP3, DGs can switch between ON/OFF status and PEVs are considered in the MG. All scenarios are solved in both stochastic and deterministic environment. Table 5 shows the optimization of the cost function in SOP1. For better comparison, the simulation results of some other well-known algorithms such as genetic algorithm (GA), particle swarm optimization (PSO) and original krill herd (KH) are given simultaneously. For each algorithm, the problem is solved 20 trails and the results of the best solution, worst solution, average value and standard deviation value are provided. As it can be seen from Table 5, the proposed MKH algorithm could reach more optimal solution than the other algorithms. Moreover, comparing the results of worst solution, average solution and standard deviation of different algorithms, the high robustness of the proposed MKH algorithm is deduced. Table 6 shows the optimal output power of each unit. As it can been from these results, the MG tendency is to keep the output power of dispatchable units at minimum during the hours that the market price is low. In the opposite point, during the mid-day hours that market price is high, most of DGs are forced to work at higher power capacity.

Tables 7 and 8 show the simulation results for SOP2. In this scenario, the power units are allowed to switch between ON and OFF status. In comparison to SOP1, the total cost of the MG is reduced in this scenario. From the optimization point of view, the proposed MKH algorithm shows superior performance in both search and stability. According to Table 8, the lower cost of the MG is SOP2 roots in the authority given to DGs to shut down. As it can be seen from Table 8, FC and MT as dispatchable units are OFF at low cost hours and therefore the required energy is purchased from the main grid. This is an economical policy to reduce the total cost of MG during the day.

Until now, the MG was operated neglecting PEVs effects. In SOP3, the high penetration of PEVs in the form of two fleets is considered. It is assumed that PEV drivers prefer to have their cars fully charged at the beginning of the first trip. Also, it is supposed that the SOC of each PEV battery would be 50% when entering or leaving the MG. According to the simulation results in Table 9, considering PEV fleets has resulted in proper reduction in the total MG costs. In contrast to the first expectation that the high penetration of PEVs in the MG can be supposed as additional load and will increase the MG costs, but the existence of these vehicles has reduced the cost effectively. In fact, the use of V2G technology has made PEVs as movable storages that can be scheduled by the MGCC during the parking hours. As it can be seen from Table 10, PEVs are charged at light load hours (for fleet 1 at early morning) to be able to discharge at heavy load hours.

Finally, Fig. 5 shows the comparative values of the objective function for different SOPs in both deterministic and stochastic frameworks. The stochastic problem considers the uncertainties of charge/discharge pattern in PEVs, active and reactive loads of the MG, hourly market price, WT/PV output power, number of PEVs in each fleet and the arrival or departure time of PEV fleets. According to Fig. 5, considering uncertainty has increased the optimal value of cost function in all SOPs. Nevertheless, it should considered that this amount of additional cost is considered to operate the MG in a more reliable and realistic level.

7 Conclusion

This paper proposed a new stochastic optimization framework based on UT method to solve the optimal operation and management of MGs with high penetration of RESs and PEVs. The proposed method makes use of KH algorithm with a new modification method to search the problem space globally. The simulation results on a standard MG shows that optimal scheduling of power units as well as the charge and discharge of PEVs can reduce the total cost of the MG, effectively. In addition, it is more economical to turn off DGs at light-load hours for avoiding high generation costs. Regarding the PEVs, the use of V2G technology can help to support the idea of PEVs in the MGs. From the optimization point of view, the proposed MKH algorithm showed superior performance than some other famous algorithms such as GA, PSO and original KH algorithm. Last but not least, the high search ability and stability of MKH algorithm were deduced too.

Footnotes

Acknowledgments

The authors would like to thank Alireza Heidari for his help to improve the technical quality of the paper.

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