Optimum management of power and energy in low voltage microgrids using evolutionary algorithms and energy storage

Electrical Power and Energy Systems 119 (2020) 105886

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Optimum management of power and energy in low voltage microgrids using evolutionary algorithms and energy storage

T

Mirosław Parola, , Tomasz Wójtowicza, Krzysztof Księżykb, Christoph Wengec, ⁎ Stephan Balischewskic, Bartlomiej Arendarskic, ⁎

a

Faculty of Electrical Engineering, Warsaw University of Technology, Warsaw, Poland Plans Ltd., Warsaw, Poland c Fraunhofer Institute for Factory Operation and Automation IFF, Magdeburg, Germany b

ARTICLE INFO

ABSTRACT

Keywords: Battery energy storage unit Distributed generation Evolutionary algorithm Microgrid optimization Power and energy management

Microgrids are subsystems in which some loads and distributed energy resources are controlled in a coordinated manner. In recent years, microgrids have been proposed as a solution to enhance critical infrastructures’ resilience and the integration of distributed energy resources. There are many solutions on microgrid planning, as well as some practical experience on microgrids’ implementation. However, choosing microgrid optimal control strategy is strongly related to the individual structure, components and configuration of microgrid. Among others, the advantages of microgrids include improved energy efficiencies, minimized operating costs and improved environmental impacts. Achieving these targets necessitates optimal control of all energy components in the microgrid. Main contribution of this paper are two control strategies of power and energy management for synchronous microgrid operation, which have been analyzed for a specific low voltage microgrid configuration. The first strategy reduces power and energy losses, thus improving the entire microgrid system’s efficiency. The second minimizes operating costs. An evolutionary algorithm was developed to control the components of the microgrid, including e.g. micro-sources and energy storage. The method of technical and economic energy storage system sizing for microgrid optimal operation is also proposed.

1. Introduction Many definitions of microgrids have been proposed. Cigré Working Group C6.22 defines microgrids thus: “Microgrids are electricity distribution systems containing loads and distributed energy resources, (such as distributed generators, storage devices, or controllable loads) that can be operated in a controlled, coordinated way either while connected to the main power network or while islanded” [1,2]. Low voltage (LV) microgrids are subsystems in which power and electricity are generated, stored and consumed [3–5]. Microsources, energy storage units and controllable loads are connected to microgrids by local controllers (microsource controllers, energy storage unit controllers and load controllers). A microgrid can operate in synchronous mode with a distribution system operator’s (DSO) grid or in island mode. In both cases, the operating points of the microsources, energy storage units and controllable loads and their modes of operation based on the management strategy implemented must be ascertained. A control strategy of microgrid in interconnected operation, which

optimizes production of the local distributed generation (DG) and power exchanges with the main distribution grid is proposed in [6]. An overview of microgrid structure, components control, power and energy management strategies, microgrid supervisory centralized and decentralized control architectures and information flows are given in [7]. The effect of renewable resources penetration and surplus power generation issues have been investigated in [8], where advantages and defects of three main strategies which include storing the surplus power by electrical storage, converting the extra power to the hydrogen as a powerful energy carrier and transferring the surplus power to the main grid are discussed. Real and reactive power management strategies of electronically interfaced DG units in the context of a multiple-DG microgrid system are addressed in [9]. The operation of a multi-agent system (MAS) for the control of a microgrid, based on the symmetrical assignment problem for the optimal energy exchange between the microgrid production units and the local loads, as well the main grid is proposed in [10]. The agent based framework of a microgrid management system and its application to the effective management of

Corresponding authors. E-mail addresses: [email protected] (M. Parol), [email protected] (T. Wójtowicz), [email protected] (K. Księżyk), [email protected] (C. Wenge), [email protected] (S. Balischewski), [email protected] (B. Arendarski). ⁎

https://doi.org/10.1016/j.ijepes.2020.105886 Received 26 April 2019; Received in revised form 24 December 2019; Accepted 27 January 2020 0142-0615/ © 2020 Elsevier Ltd. All rights reserved.

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generation and storage devices is described and tested on laboratory facilities in [11]. In [12], a microgrid management system is developed in a stochastic framework as a constraint-based system that employs forecasts and stochastic techniques to manage microgrid operations. The performance of the on-line optimization-based control strategy is analyzed within case study and the effectiveness of the proposed approach is tested in an experimental microgrid. Distributed robust operation method for coordinating multiple microgrids in a distribution system is presented in [13] and [14]. A comprehensive overview and guidelines for practical implementations and operation of microgrids based on selected major microgrid projects realized worldwide is given in [15]. It summarizes microgrid’s operating modes, transitions, architectures and topologies, protection, communication, management and control features. Furthermore, it compiles the more relevant studies on unit commitment for microgrids, on economic dispatch of a microgrid, voltage control strategies, candidates for non-bumpless islanding transition, recommended bumpless islanding procedures to increase microgrid resilience, preventive control strategies to restore both grid-connected and isolated microgrids to the normal state, emergency control strategies for isolated microgrids, re-synchronization procedures, and off-grid black start procedures. In [16], a comparative study of differential evolution algorithm, symbiotic organisms search algorithm, grey wolf optimization implemented to minimize the microgrid operating cost abiding by various equality and inequality constraints and considering load uncertainties and market bids is made. A stochastic mixed-integer nonlinear programming (MINLP) model for the optimal operation of islanded microgrids in the presence of stochastic demands and renewable resources is given in [17]. Proposed model considers the uncertainty while reducing the average operational costs and load curtailments, when compared with a deterministic model. In [18], a multiple chance-constrained scheduling model is developed for optimal scheduling of microgrid considering the most important factors that affect the power balance of the microgrid such as the uncertainty of demand and renewable resources, sudden outage of DGs, and unwanted islanding. Steady state analysis of microgrid operation is presented in [19]. Optimization of microsources’ operating points is discussed and solved to minimize microgrid operating costs in [20]. Three separate problems of exchanging electricity between a microgrid and a DSO’s grid, of optimally siting and sizing energy storage units to minimize microgrid operating costs, and of defining optimal energy storage unit charge and discharge time intervals to minimize microgrid operation costs are formulated in [20] as well. The problem of exchanging electricity between a microgrid and a DSO’s grid is addressed and solved in [21]. The microgrid operation in electric power system was broadly presented and discussed in [22]. Numerous other problems in microgrids have been examined and solved in recent years. The issue of voltage stability in low voltage microgrids in aspects of active and reactive power demand is presented in [23]. A variety of issues are presented and discussed in [24]. A design for microgrid cluster structures and an autonomous coordination control strategy are proposed in [25]. An iterative consistency algorithm for the optimization of voltage and frequency references in AC microgrids as well as the use of a multi-agent communication topology to exchange information between different distributed generators and energy storage systems is described in [26]. In [27], an algorithm for energy management system (EMS) based on multi-layer ant colony optimization (EMS-MACO) is presented to find energy scheduling in microgrid. Comparison of MACO, which is able to analyze the technical and economic time dependent constraints, with modified conventional EMS and particle swarm optimization (PSO) based EMS shows that the system performance is improved using MACO algorithm. An Improved Artificial Bee Colony algorithm for modeling and managing microgrid connected system is given in [28]. Proposed method achieves the required multi-objective function to

determine optimal microgrid’s configuration based on load demand by reducing the fuel cost, emission factors, operating and maintenance cost. In [29], different strategies for the synthesis of a Fuzzy Inference System (FIS) based EMS by means of a hierarchical Genetic Algorithm (GA) with the aim to maximize the profit generated by the energy exchange with the grid, assuming a Time Of Use (TOU) energy price policy are investigated. Types and potential applications of energy storage systems (ESS) and control methods for ESS in microgrids are reviewed in [30]. Battery energy storage system (BESS) optimal sizing process has been formulated as a linear programming based optimization problem in [31]. Fuzzy-Logic is proposed to control the charging/discharging time and quantity for batteries. Authors compare various optimization techniques, such as Particle Swarm Optimization, Genetic Algorithm and Flower Pollination Algorithm to perform the economic dispatch calculation. In [32], a mix-mode energy management strategy (MM-EMS) is developed by combining three operating strategies, such as continuous run mode, power sharing mode and ON/OFF mode for a 24 h time period. Furthermore, an appropriate battery sizing method using PSO technique for operating the microgrid at the lowest possible operating cost is proposed. An approach to sizing energy storage systems and inventory reserves as flexibility options that maximize the integration of power generated by volatile sources in manufacturing processes is presented in [33]. A method for collaboratively controlling battery energy storage (BES) units in both charge and discharge mode to maintain the microgrid’s (MG) power balance and a virtual impedance strategy are introduced in [34] to enhance the precision of reactive power sharing among BES units and renewable energy sources (RES) in the MG. An energy management scheme based on conditional value at risk (CVaR) that optimizes resilience and operating costs in microgrids of commercial buildings and PV and battery energy storage systems is analyzed in [35]. The number of publications on the topic of optimization of control of low voltage AC microgrids is quite significant. However, in the existing papers on this topic there is quite often a lack of any detailed information on exact formulations of the optimization problems defined in these papers (forms of the objective functions, sets of optimization constraints), on methods used to finding solutions to these problems and on detailed descriptions of the algorithms based on these methods. It is also very common that the details on the microgrids being optimized as part of case studies described in these papers are missing. We mean here both presenting such details in a direct form (in the paper) and in a more indirect one (as references to other articles). As a result, in our opinion there is still some place in the research community for presenting different, more detailed papers on the topic of microgrid’s control optimization (for different microgrid’s structures and input data). This paper focuses on the formulation and solution of two select power and energy management strategies for microgrid operation in synchronous mode. Control of active power generated in microsources and of energy storage units’ mode of operation and active power is implemented in the first strategy. Control of power generated in microsources and active power received by controllable loads is implemented in the second strategy. Both strategies assume that a twolevel hierarchical control system with local controllers and a supervisory controller is in operation. Furthermore, an approach of technical and economic battery energy storage sizing relevant for optimal operation in the microgrid is proposed. The main contribution of the paper is treating the considered topic in a more comprehensive way, that includes: giving detailed descriptions on the formulated optimization problems, presenting in detail the methods and algorithms applied to solve the problems, describing a test (model) microgrid for which the computations have been made (including specific technical data), presenting the results obtained from the optimization calculations and last but not least discussing the 2

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results. It should be emphasized, that the description of the first considered optimization problem and the method of dealing with it is based on [22]. What is important is that the current paper formulates the optimization problem related to the strategy of the management of the energy storage’s operation more precisely. In turn, the description of the second considered optimization problem and the method of dealing with it is based on the publication [24]. What is more, in case of both considered problems the detailed results of the optimization computations, which were not presented in such form in the publication [24], have been given and precisely commented. This paper examines the efficacy of two different strategies of power and energy management for microgrid operation in synchronous mode and presents the results of optimization calculations obtained for a model low voltage microgrid with evolutionary algorithms. The greater complexity and accuracy in microgrid modeling enables the design of more advanced control strategies that can be implemented to manage real microgrids and their components.

=

Pg , i, t

2.1. Problem formulation

M

Et = t=1

M

dt · Pt = t=1

0 U s ·IDs max · = Q Us· d t

i, t

t

t=1

dQD = IDs max · dt ·

Pi, t i=1

if Qt = 0 if Qt > 0 if Qt > 0

Qt > dQD Qt dQD

(3)

(4)

i, t

In charge mode, the active power Pl,i,t charged by the i-th energy storage unit in the t-th time interval can be formulated as follows:

N

dt ·

(2)

where Us is the operating voltage of the energy storage unit, IDsmax the maximum permitted current discharged by the energy storage unit, δi,t a random number from the range (0,1 > , Qt the amount of electric charge stored in the energy storage unit in the t-th time interval, and dQD the amount of electric charge that can be received from the energy storage unit in discharge mode in the t-th time interval. The latter value can be formulated as follows:

The object of interest is a low voltage microgrid powered by a single MV/LV transformer. This LV microgrid contains loads specified by daily characteristics and active and reactive peak powers. The microgrid also contains microsources, e.g. wind turbines, gas microturbines, photovoltaic panels and fuel cells. Since battery energy storage systems are often used as energy storage units in microgrids, one such microgrid is the object of optimization problems. The main goal in the problem of optimizing the microgrid interacting with a DSO’s grid is to minimize the active power and energy losses [22]. M

..., 1, M , 2,1, ..., 2, M , G,1, ..., G, M , ..., G + 1, M , G + S,1, ..., G + S, M

where G is the number of microsources in the microgrid, S the number of electrical energy storage units in the microgrid, δi,t the level of active power generation in i-th microsource in the t-th time interval or mode of operation of i-th electrical energy storage unit in the t-th time interval, and M the number of specified time intervals during optimization period T. It was assumed that if δi,t > 0, i.e. its value is in the range of (0, 1 > , then the energy storage unit operates in discharge mode and if δi,t < 0, i.e. its value is equal to –1, then energy storage unit operates in charge mode. If δi,t = 0, then the energy storage unit is in stop mode. In discharge mode, the value of δi,t specifies the active power currently generated by the energy storage unit. Assuming constant voltage and constant active power generation (discharge) by the energy storage unit, the generation level Pg,i,t, in the t-th time interval can be determined as follows:

2. First management strategy

ET =

1,1,

G + 1,1,

(1)

Pl, i, t

where ΔET is the microgrid’s total active energy losses during optimization period T, M the number of specified time intervals t during the optimization period T, ΔEt the active energy losses in the t-th time interval, dt the length of t-th time interval (in the analyzed strategy dt = 15 min = 0.25 h), ΔPt – the total active power losses in the t-th time interval, ΔPi,t the active power losses in i-th network element (line or transformer) in the t-th time interval, and N the number of network elements. 15-minute-long optimization time periods, which were used in the optimization calculations have been chosen in an arbitrary way and should not be treated as any limitation of the method and algorithm presented further in the paper. Some shorter time intervals can be chosen as well, without any obstacles. The problem analyzed is to identify the optimal level of power generation in microsources and to determine the optimal mode of operation of the electrical energy storage unit.

0 U s · ILs · = Q Us· N d

if Qt = QN if Qt < QN Qt · i, t if Qt < QN t i, t

Qt + dQL < QN Qt + dQL QN

(5)

where ILs is the energy storage unit’s charge current, QN the energy storage unit’s rated capacity, and dQL the amount of electric charge supplied to the energy storage unit in charge mode in the t-th time interval. The latter value can be formulated as follows:

dQL =

ILs ·dt wLs

(6)

where wLs is the energy storage unit’s loading factor specified by manufacturer. The objective function in the problem considered can be formulated as follows [22]:

min { ET ( ) = FCT ( )}

(7)

The search for the optimal solution is accompanied by a set of constraints [22]. Current flows in specific microgrid components (LV lines and MV/LV transformers) may not exceed their current-carrying capacity. Voltage levels in all of the microgrid’s nodes should remain within permissible values (between the lower and upper voltage limit). The constraints can be presented as inequalities:

2.2. Mathematical model An appropriate mathematical model that solves the problem was formulated. A vector δ representing a set of possible solutions was defined. This vector is composed of ninety-six parts or subvectors (if M = 96). Each subvector represents the operating characteristic for a microsource or for an electrical energy storage system. It was assumed that the characteristic corresponding to active power generation needs to be defined for each microsource and each electricity storage system. Reactive power generation is yielded by the established value of the power factor. Vector δ can be formulated as follows [22]:

|Ik|

Iz, k for lines

(8)

|Ik|

IN , k for transformers

(9)

Uimin

Ui

Uimax

(10)

where |Ik| is the current modulus in k-th network branch, Iz,k the currentcarrying capacity of k-th line, IN,k the nominal current of k-th transformer, Ui the voltage in i-th node, Uimin the minimum permissible 3

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voltage level in i-th node, and Uimax the maximum permissible voltage level in i-th node. Active power generated in each microsource should remain within permissible values (between the lower and upper power limit). The levels of reactive power generated should remain within permissible values derived from the constraint of the maximum permissible apparent power generated in the microsource. The constraints on the power generated in microsources can be described by the following relationships:

Pimin

Pi

Pi2 + Qi2

2.3. Problem solution using evolutionary algorithms In the event the constraints (8)–(10) and (13)–(15) are violated, appropriate penalty functions are formulated: - ΨI,k for constraints (8) and (9)

Si2,max

(12)

- ΨU for constraint (10)

where Pi is the active power generated in i-th microsource, the minimum permissible active power value generated in i-th microsource, Pimax the maximum permissible active power value generated in i-th microsource, Qi the reactive power generated in i-th microsource, and Si, max the maximum permissible apparent power value generated in i-th microsource. Constraints on the magnitude of the power generated will be verified when the optimization problem is solved and are not directly reflected in the problem model. Constraints on excessive depth of discharge by energy storage units can be formulated as follows:

Pimin

Dt = 1

Qt ·100% QN

Dmax

1

Qt = 0

Qt = QN

D1| = 0

(13)

DM

DD

2

if voltage

constraints are violated

(20)

=

1

( )

2 Dt Dmax

if Dt Dmax if Dt > Dmax

(21)

= kD·(1 + ND )

(22)

where kD is the penalty factor for excessive depth of discharge in the energy storage unit, ND is the number of time intervals during which attempts are made to continue discharging the already fully discharged energy storage unit before the current time interval t. If ND = 0, then DD = 1

(14)

(15)

- ΨDL for constraint (15) DL

(16)

= (1 + NLN )2

(23)

where NLN is the number of time intervals during which attempts are made to continue charging the already fully charged energy storage unit before the current time interval t. An equivalent penalty function for excessive energy storage unit discharge or charge can be represented as follows

=

D

DM · DD· DL

(24)

The penalty function ΨR for the constraint (16) can be formulated as follows

(17)

NCN

Uj U max j

- ΨDD for constraint (14)

where NTI is the number of time intervals t in which the operating mode is switched from charge to discharge mode or from discharge to charge mode. Stop mode is disregarded here. The higher the number of the energy storage unit’s charge and discharge cycles is, the shorter its service life is assumed to be. Any reduction in the number of charge and discharge cycles should therefore be taken into account in the optimization problem considered. It can be presented as follows:

Y ·365·NC

U n max j=1

- ΨDM for constraint (13)

where DM is the energy storage unit’s depth of discharge in the time interval M and D1 is the energy storage unit’s depth of discharge in the first time interval. The constraint on switching the operating mode from charge to discharge mode or vice versa one or more times can be formulated as follows:

NTI > 0

Ui

2

where is the number of nodes in which a “voltage” constraint on U is the number of the minimum permissible voltage level is violated, n max nodes in which a “voltage” constraint on the maximum permissible voltage level is violated.

The constraint on the energy storage’s level of charge at the end of the optimization period T can be formulated as follows:

|DM

Uimin

U n min i=1

U n min

The constraint on further charging a fully charged energy storage unit can be formulated as follows:

Pl, t = 0 if

if no voltage constraints are violated

=

U

where Dt is the energy storage unit’s level of discharge in the t-th time interval and Dmax is the energy storage unit’s maximum depth of discharge. The constraint on further discharging a fully discharged energy storage unit can be formulated as follows:

Pg , t = 0 if

(19)

where kI is the penalty factor for “current” violations and LI,k is the overcurrent rate of k-th network branch.

(11)

Pimax

0 if "current" constraint is not violated kI · LI2, k if "current" constraint is violated

=

I ,k

R

= 1 + |DM

D1 |2

(25)

In the event the constraints (17)-(18) are violated, the appropriate penalty functions are formulated: - ΨC0 for constraint (17) C0

(18)

=

{10 1

kNTI0

if NTI > 0 if NTI = 0

(26)

where kNTI0 is the fixed exponent of 10 for the penalty function, if NTI = 0

where Y is the number of years of scheduled energy storage unit operation, NC the number of charge and discharge cycles in the energy storage unit’s operating characteristic, NCN the nominal number of charge and discharge cycles specified by the manufacturer as the guaranteed number of cycles for which the energy storage unit will retain its capability to store (supply) energy.

- ΨCN for constraint (18) CN

4

=

kC · Y ·365·NC NCN

(27)

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where kC is the penalty factor for an excessive number of energy storage unit charge and discharge cycles. If this is not the case, then CN = 1. An equivalent function for the number of energy storage unit charge and discharge cycles can be formulated as follows: C

=

(28)

C 0· CN

In the event “current” constraints are violated, the penalty function ΨI,k, is formulated as a function of branch overloading as demonstrated. This function increases the amount of losses in the branches affected by an overcurrent and is present in components in formula (29). In the event “voltage” constraints are violated, the penalty function ΨU is formulated as presented above. Finally, the objective function FCT(δ) can be written as [22]

min

FCT ( )= C · R·(

M d( U t=1 t

D(

Pt ( ) + Pk, li ( ) + Pk . ti ( ))))

(29)

where ΔPk,li(δ) is the penalty component derived from overcurrents in microgrid lines and ΔPk,ti(δ) is the penalty component derived from overcurrents in microgrid transformers. An evolutionary algorithm (EA) was employed to solve the optimization problem considered. The evolutionary algorithm strives to maximize the utility function [36]. Since the problem presented above is based on the search for the function minimum, the objective function must be converted into the evaluation function eval(δ). The fitness (evaluation) function can be formulated as follows [22]:

max{eval

E

( ) = Cmax,

E

FCT ( )}

Fig. 1. Energy bid scheme for the second management strategy (CL – controllable load, G – generating unit, UCL – uncontrollable load).

consumers connected to the microgrid. The cost of delivering power (energy) to the microgrid in t-th elementary time interval Ct is defined as in Eqs. (31)–(35) [24]:

Ct = CP, t

(30)

(31)

IS, t

CP , t = 0, 25

where Cmax,ΔE is the evaluation constant equal to the maximum prospective active energy losses during the optimization period T. Three genetic operations are performed on the population of individuals: selection, crossover and mutation [36]. Selection is performed with the roulette wheel method. An elitist strategy is also used. A simple crossover method is employed to exchange information between two randomly chosen individuals in a population. Uniform mutation is used to make a random change in one individual. Linear scaling is employed to prevent premature convergence in the evolutionary algorithm.

(cp, i, t Pg , i, t ) + cp, t PBIL, t

(32)

i MS

IS, t = 0, 25

cs, t + cp, t 2

Plh, i, t +

Pl, t = i L1

L = L1

Plh, i, t + i L1

Pll, i, t + i L1

L2

Pl, i, t + P , t i L2

cpl , i, t Pll, i, t

Pl, i, t + i L2

i L1

(33) (34) (35)

where CP,t is the cost of the purchase of electricity in t-th time interval, IS,t the proceeds from sales of electricity in t-th time interval, Pg,i,t the 15-minute average of the active power generated by i-th microsource in t-th time interval, cp,i,t the energy purchase per unit price from i-th microsource in t-th time interval, cp,t the purchase of energy per unit price on the market (from the DSO’s grid) in t-th time interval, PBIL,t the 15-minute average of active power purchased from the DSO’s grid in tth time interval, cs,t energy sales per unit price by the microgrid in t-th time interval, cpl , i, t the purchase of low-priority energy per unit price in t-th time interval sent by i-th controllable load to the MGCC in the form of a bid, Pl,t the 15-minute average of the power received in the microgrid (including power losses) in t-th time interval, Plh, i, t the 15-minute average of the high-priority active power received by i-th controllable load in t-th time interval, Pll, i, t the 15-minute average of the low-priority active power received by i-th controllable load in t-th time interval, Pl,i,t the 15-minute average of the active power received by uncontrollable ith load in t-th time interval, PΔ,t the power losses in the microgrid in t-th time interval, MS the set (number) of microsources, L the set (number) of loads in the microgrid, L1 the set (number) of controllable loads, and L2 the set (number) of uncontrollable loads. The problem does not consider the power exported to the DSO’s grid nor the mode of operation (charge or discharge) of the energy storage units installed in the microgrid.

3. Second management strategy 3.1. Problem formulation The optimization problem presented here determines the values of active and reactive power generated by microsources and the values of active power supplied to controllable loads during the optimization period, i.e. every fifteen minute interval (elementary time interval) during twenty-four hours, in order to minimize the cost of delivering power (energy) to the microgrid. Similarly as in the first considered problem, 15-minute-long optimization time periods, which were used in the optimization calculations have been chosen in an arbitrary way and should not be treated as any limitation of the method and algorithm presented further in the paper. Some shorter time intervals can be chosen as well, without any obstacles. Both controllable and uncontrollable loads are assumed to be present in the microgrid. Two components are defined for each controllable load, namely power requiring and power not requiring high service reliability, i.e. high and low priority power, respectively. Each controllable load only sends bids for low-priority power to the microgrid central controller (MGCC). The MGCC prices high-priority energy for controllable loads (and uncontrollable loads) based on the price of energy generated by microsources and the price of energy offered on the market (by the DSO) (see Fig. 1). The “good citizen” concept was adopted in the problem. The main problem for the microsources is to meet the energy demand of

3.2. Mathematical model The optimization problem, i.e. minimizing the cost of delivering power (energy) to the microgrid by controlling the power generated by microsources and controlling the active power of controllable loads, can 5

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be defined as a search for a vector of matrix g + l , that minimizes the value of the criterion function while fulfilling conditions from the set of existing constraints. The vector g + l has a number of columns equal to 2MS1 + L1, where MS1 is equal to the number of microsources based on nonrenewable energy carriers. The objective function for the problem of minimizing the cost of delivering power (energy) to the microgrid is formulated in formula (36):

min

g+ l

{Ct (

g + l, t )}

necessitates converting the original objective function into a fitness (evaluation) function. The following form of fitness function is proposed

(Plh, i, t + Pll, i, t ),

Pgmin ,i

Pg , i, t

Qgmin ,i

Q g , i, t

|Sj (

g + l, t )|

Uimin

Ui (

Pgmax ,i , Qgmax ,i , cj, g + l, t )

Pg, i, t i MS

i

Pl, i, t

i

Uimax,

P ,t (

i g + l, t )

(MS

L)

0

i L

=

p,2 (vr , t )

=

(41) (42)

p,3 (vr , t )

Cmax

nI i=1

( )

IB, i 2 Iz, i

if current constraints are violated

if no voltage constraints are violated U n min i=1

Uimin Ui

2

U n max j=1

Uj U max j

2

if voltage

constraints are violated

(46)

=

if energy export constraint is not violated kpenalty if energy export constraint is violated

(47)

where nI is the number of microgrid branches exceeding current-carrying capacity, k I the proportionality factor in the penalty function, IB, i the calculated (operational) current in the microgrid’s i-th branch, Iz , i the current-carrying capacity of i-th branch, and kpenalty the penalty factor in the event the “energy export” constraint is violated. The value of the equivalent penalty function p ( g + l, t ) in the event the aforementioned constraints are violated can be determined with the equation p(

3.3. Problem solution using an evolutionary algorithm

g + l, t )

=

p,1 (vr , t )· p,2 (vr , t )· p,3 (vr , t )

(48)

Finally, the optimization problem can be defined as follows:

An evolutionary algorithm was used to solve the optimization problem considered. In the optimization problem, each r-th individual vr , t in the current population in t-th time interval (representing vector g + l, t ) is denoted as follows

= Pg ,1, t . ..Pg , MS1, t Qg,1, t . ..Qg , MS1, t Pl,1, t . ..Pl, L1, t

g + l, t )

1

where is the minimum permissible value of the active power generated by i-th microsource, Pgmax , i the maximum permissible value of the active power generated by i-th microsource, Qg,i,t the reactive power generated by i-th microsource in t-th time interval, Qgmin , i the minimum permissible value of the reactive power generated by i-th microsource, Qgmax , i the maximum permissible value of the reactive power generated by i-th microsource, |Sj| the modulus of the maximum apparent power flow in j – th microgrid branch, cj the power-carrying capacity of j – th microgrid branch, m the number of microgrid branches, Ui the voltage in i-th node, Uimin the minimum permissible voltage level in i-th node, and Uimax the maximum permissible voltage level in i-th node.

g + l, t

p(

> Cmax

p,3 (vr , t ) – penalty function in the event the “energy export” constraint in the microgrid is violated for r-th individual in t-th time interval

Pgmin ,i

vr , t =

g + l, t )

if no current constraints are violated

kI

1

(40)

j = 1, 2, ...,m

if Ct (vr , t ) p(

p,2 (vr , t ) – penalty function in the event “voltage” constraints in the microgrid are violated by r-th individual in t-th time interval

(39)

MS

g + l , t );

(45)

(38)

MS i

1 p,1 (vr , t )

(37)

L1

p(

0; if Ct (vr , t )

where Ct(vr,t) is the value of the objective function, i.e. the cost of delivering energy in t-th time interval for r-th individual determined from Eqs. (31)–(33), p ( g + l, t ) the value of the equivalent penalty function if constraints are violated, and Cmax the constant estimated in initial generation as Cmax = max{Ct (vr , t ) p ( g + l, t )} . The problem includes equality and inequality constraints, some of which are incorporated in the evolutionary algorithm and methods of calculation that determine the fitness function. Inequality constraints (5), (6) and (7) (see Eqs. (40), (41), (42)) must be treated separately. The introduction of respective penalty functions in the event the aforementioned constraints are violated is proposed. Three penalty functions were formulated in the problem [21,24]: p,1 (vr , t ) – penalty function in the event “current” constraints are violated by r-th individual in t-th time interval

(36)

(1) The active power balance and the reactive power balance should be ensured in the microgrid. (2) The supply of active and reactive power to particular nodes of microgrids and the generation of power by microsources should be constant in 15-minute time intervals. (3) The active power of controllable loads should have appropriate ranges of values (between minimum power equal to high-priority active power and maximum power totaling high- and low-priority active power) (see inequality (37)). (4) The active and reactive power generated by specific microsources should remain within permissible values (see inequalities (38) and (39)). (5) The power flow in each component of the microgrid may not exceed each component’s power-carrying capacity (see inequality (40)). (6) The voltage level in each node of the microgrid should remain within permissible limits (see inequality (41)). (7) Energy may not be exported from the microgrid to the DSO’s grid (see inequality (42)).

Pl, i, t

Ct (vr , t )

(44)

In this optimization problem the following constraints should be fulfilled in every specified time interval:

Plh, i, t

Cmax

eval p (vr , t ) =

max eval p (

g + l, t )

(49)

Selection and recombination operations (crossover and mutation) for two problems are performed on the population of individuals [36]. Just as in the first optimization problem, selection may be performed in this problem with the roulette wheel method. Linear scaling may be also be applied to the evaluation function to prevent premature convergence in the EA.

(43)

The objective function in the optimization problem considered was determined in formula (36). The form of the optimization problem 6

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Fig. 2. Model microgrid structure (WT – wind turbine, MT – gas microturbine, FC – fuel cell, BS – battery storage; based on [19–21].

Fig. 4. Total active power generation in microsources and energy storage unit day and night.

4. Model optimization calculations

storage is medium (Fig. 4), factoring in the minimum and maximum amount of power potentially generated. The amount of average power generated in the microsources is around 97 kW. This represents about 42% of the usable range of generation installed in both microsources and energy storage unit. The active energy losses after optimization are 96.4 kWh. Losses were thus almost 55% lower than in the grid without microsources. The active power loss curve after optimization is presented in Fig. 5.

4.1. Description of the test microgrid Model optimization calculations were performed in the test microgrid presented in Fig. 2, which is a modified version of the microgrid presented in [37]. It is a low voltage grid powered by a single MV/LV transformer and contains fourteen loads described by daily (residential, commercial and industrial) characteristics and active and reactive peak powers. The microgrid also contains microsources: wind turbines (WT), gas microturbines (MT) and fuel cells (FC). Only one energy storage unit, a standard chemical cell battery (consisting of conventional liquid electrolyte OPzS cells), located on the node N34 is assumed to be installed in the test microgrid. Detailed data on the microgrid analyzed can be found in [19–22,24] and are presented in Appendix.

4.3. Results of the second optimization problem calculations The values of the following economic data have to be determined to perform the optimization calculations: (1) the coefficients of cost functions for specific kinds of microsources, (2) the per unit prices of energy purchases from the DSO’s grid and per unit prices of energy sales by the microgrid in every time interval day and night, (3) the status of each (controllable or uncontrollable) load, and (4) the high-priority power, total power and per unit prices of purchases of energy sent by controllable loads to MGCC as bids for lowpriority power in every time interval.

4.2. Results of the first optimization problem calculations The active energy losses in the test microgrid without microsources during the day analyzed equal 211.9 kWh. The results of the calculations to optimize the energy storage unit’s mode of operation are presented in Figs. 3 and 4. One discharge period and three charge periods can be distinguished in the electrical energy storage unit’s characteristics (see Fig. 3). The discharge period corresponds to the evening load peak. The energy storage unit is mainly charged during the night (from 0:00 to 5:00 and from 22:15 to 23:00). One charge period is from 15:30 to 16:45 in the afternoon. The total microgrid load varies within the range of 100 kW to 300 kW during the day. The level of active power generation in microsources and energy

Some of the economic data used in the optimization problem considered are prices of energy purchasable from the electricity utility network and prices of energy sellable by the microgrid in every time interval a of twenty-four-hour period (see Table 1) [21]. There assumed to be two controllable loads, load L20-IND and load L31-COM, in the model optimization calculations performed. The

Fig. 3. Optimum mode of energy storage unit operation.

Fig. 5. Active power loss curve before and after optimization. 7

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Table 1 Electrical energy trade data during specific time intervals of a twenty-four-hour period [21]. Specific hour in a 24-hour period –

Energy Sale price

Energy purchase price

USc/kWh

USc/kWh

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

11.78 12.09 12.09 12.79 12.86 12.44 11.24 10.05 9.50 9.02 8.63 8.30 8.16 8.13 8.21 8.50 9.06 9.85 10.56 10.74 10.87 10.82 10.82 11.00

11.20 10.13 9.85 9.85 9.97 10.59 12.88 13.00 14.09 14.60 14.83 15.21 15.17 15.06 14.34 13.76 13.46 13.64 13.76 13.92 13.94 13.36 14.13 12.69

Fig. 8. Active power curve of a model gas microturbine (located in node N33) during a twenty-four-hour period.

Fig. 9. Reactive power curve of a model gas microturbine (located in node N33) during a twenty-four-hour period.

Fig. 6. Value of the objective function as a function of the time intervals of a twenty-four-hour period (PLN stands for złoty; US$ is around 3.80 PLN). Fig. 10. Active power curve of industrial load L20-IND (located in node N20) during a twenty-four-hour period.

Fig. 7. Active power curve of a model fuel cell (located in node N09) during a twenty-four-hour period. Fig. 11. Active power curve of commercial load L31-COM (located in node N31) during a twenty-four-hour period.

8

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results of the model optimization calculations are presented in Figs. 6–11. The values of the objective function during specific time intervals of one day and night are presented in Fig. 6. The value of the objective function (the cost of delivering power to the microgrid) is less than zero for the greater part of one day and night. The values of the objective function are smallest early in the morning and late in the evening. The power curves of a model fuel cell and a model gas microturbine are presented in Figs. 7–9. The active power generated by these microsources (Figs. 7 and 8) vary substantially during one day and night. The reactive power generated by the gas microturbine varies far more than the active power (Fig. 9). The active power curves of the controllable loads are presented in Figs. 10 and 11. The high active power for the industrial load (Fig. 10) is received during the load’s main hours of operation of 7:30 to 16:00. The high power for the commercial load (Fig. 11) is received from 10:00 to 20:00.

Fig. 13. Energy exchange in the energy storage system.

4.4. Optimal design of a battery energy storage system for distribution grids

caused by technology related behavior and specifics of the batteries such as approved operation limits e.g. SOC (state-of-charge). Operating at low or high SOC, i.e. below 10% or above 90% SOC, affects the battery performance as well as maximum power reduction, battery ageing and battery life. The demand for energy required to operate the system must be factored in, too. The operating strategy proposed here affects battery life and overall system efficiency. The power required for a real battery system’s self-consumption can generally be estimated between 1.5% and 3% of nominal battery power, depending on the system’s configuration and size. The heating, ventilation and air conditioning (HVAC) system is one of the main consumers. Every component such as the battery, power converter and transformer has losses that contribute to heating the system. The permissible operating temperature of the battery cells is also a limiting factor for the overall system. Environmental conditions and the entire system’s housing influence the cooling system. The energy storage system’s main components and the energy flow between them are presented in Fig. 13. The storage system is run by the primary instrumentation and control system. The control unit receives input data from monitoring sensors that continuously measure predefined operating parameters. The storage system’s ICT interface is used for communication with the higher-level system. It enables data exchange with the MGCC, thus making it possible to integrate the storage system as an active element of the microgrid. The energy storage system’s design primarily incorporates the technical and economic factors (see Fig. 14). The analysis of generation and loads in the local energy system as the starting point for optimally sizing the energy storage system for microgrids is presented in Fig. 12. Local synergies of energy processes and infrastructure such as consumption, generation and storage potential in electrical and thermal energy form have to be identified and analyzed. This specific analysis helps optimize energy use and cost effectiveness in the operating strategy for a local microgrid, specifically the second management strategy proposed in Section 3. The results of the BES sizing relevant for optimized 1-day-operation in the modeled microgrid are shown in Fig. 15.

Balancing power and stabilizing power generation and demand are the main challenges presented by battery energy storage systems in microgrids or complex power infrastructures. Advanced power electronics enable battery energy storage systems to provide additional ancillary services. The required storage power Pst (50), storage capacity Cst (51) and power gradient αst (52) in Fig. 12 ascertained in the first step of a general evaluation of a storage system following [38]. Calculations, system simulations or historical data on characteristics of system components such as load and generation are used to determine the system parameters. The required storage power Pst can be calculated with the following equation: (50)

PSt = max |Pres| where, Pres is the analyzed residual load profile. The storage capacity Cst is represented by the equation: t ,max

Cst = max(Est ) = max t=1

ti + 1(P = 0) ti (P = 0)

Pres dti

(51)

where, Est is the maximal required amount of energy needed to harmonize the analyzed load profile. The power gradient αst can be calculated using the equation: k,max

St

= max k=1

Pk tk

(52)

where, ΔPk is the change of power during the analyzed time period Δtk. Defining the optimal parameters for storage system configuration is much more complex than determining the maximum and minimum system configuration using the system’s load and generation profiles. Several factors have to be taken into account because of the differences in real technical battery systems’ performance. One example is the difference between usable and nominal battery capacity. The system reserve is typically calculated 10% to 20% higher in order to satisfy requirements. This reserve is

Fig. 12. Energy storage system sizing to balance residual loads [38,39].

Fig. 14. Energy storage system sizing algorithm. 9

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optimization calculations using evolutionary algorithms are presented and discussed. The results obtained demonstrate that the optimization problems considered are important economically and technically. Research has proven that evolutionary algorithms are an efficient optimization tool since they yield optimal results for a wide variety of cases. Solutions obtained from the optimization problems for the model microgrid support the thesis that optimization can be a source of additional savings. Optimization of the microgrid in the first problem delivered tangible benefits in terms of the magnitude of active power losses. Active energy losses during the one-day optimization period analyzed were reduced by over 55%. This outcome is the result of a combination of two facts. First, installing microsources close to the load reduces losses. Second, optimization reveals how to control these sources to obtain the optimal result. The magnitude of microsource generation for the optimal solution was medium and more than 40% of the total active power installed in microsources. The mode of energy storage system operation reflects the daily change characteristics of active power losses in the microgrid. The energy storage system operates in discharge mode during the periods when the losses are greatest (evening hours), thus helping reduce total energy losses. The energy storage system operates in charge mode during nighttime hours when the active power losses are smallest. The availability of a variety of telecommunications technologies, the possibility of obtaining measurement data in the microgrid, and the short time required to obtain results from optimization calculations for the microsources’ operating points make it possible to implement these algorithms in real microgrid controllers. Only a few seconds were needed in the model microgrid analyzed to obtain a solution for a single time interval. The issues presented in this paper require further research. Future work on the analysis of and search for optimal solutions for microgrid operation ought to focus on calculations for other kinds of microsources or energy storage systems. Work on the first optimization problem ought to analyze the use of various types of microsources or energy storage systems and the adoption of other evaluation criteria such as minimized microgrid operating costs. Work on the second optimization problem ought to explore different (charge or discharge) modes of operation of energy storage systems installed in microgrids.

Fig. 15. Power and capacity of battery energy storage usage in modeled microgrid.

Based on generation and load profiles and after optimizing the operation of the considered microgrid, the maximum values for storage power Pst and storage capacity Cst are 8.9 kW and 30.7 kWh, respectively. Due to optimization strategy the needed BES could be reduced to 17.5% of nominal energy capacity while battery storage power is reduced to 31.3% in this example. It is necessary to underline that such reductions are a result and consequence of applied criterion function and defined set of constraints. Taking specific power and energy costs like in [39] into account the costs of BES are set to 24.000 € including 20% for technical overheads. The minimization of storage system parameters shows significant benefits of optimal microgrid operation. However, it is difficult to assume appropriate values of these parameters in the microgrid design stage as used criterion function and limiting conditions have significant impact on the results of optimization calculations and BES sizing, respectively. 5. Summary and conclusion The main section of this paper deals with the formulation of mathematical models and algorithmic solutions for two optimization problems. The control of active power generated by microsources and the energy storage unit’s mode of operation and active power are optimized in the first problem. The control of power generated by microsources and the active power received by controllable loads are optimized in the second problem. The “good citizen” concept was adopted in the second problem. The methodology for designing battery energy storage systems for microgrids and distribution grids as well as an example of its use in practice are also presented. The results obtained for each optimization problem from model

Declaration of Competing Interest The authors declare no conflicts of interes in relation to this work. Acknowledgments This research project has been funded in part with research funds from the Polish Ministry of Science and Higher Education (grant: N N511 0573 33) and in part with the funds from Faculty of Electrical Engineering of Warsaw University of Technology (project: 504/G/1041/0160).

Appendix Table A.1 presents data on parameters of MV/LV transformer in the grid. In turn, data on parameters of MV and LV power lines in the test microgrid are presented in Table A.2. Table A.3 includes technical data of loads in the test microgrid. Daily profiles of different loads (see Fig. A.1) are taken from [37]. Total active power load of the test microgrid is presented in Fig. A.2. Table A1 Parameters of MV/LV transformer in the test microgrid [22]. Transformer

Sn [kVA]

Un_HV [kV]

Un_LV [kV]

ukr [%]

Pkr [kW]

ΔPFe [kW]

Transformer 15/0.4

400

15.75

0.4

4.5

4.10

0.72

where: Sn is a nominal apparent power, Un_HV is a high voltage side nominal voltage, Un_LV is a low voltage side nominal voltage, ukr is a short-circuit voltage, Pkr is a load active power losses, ΔPFe is a no-load active power losses. 10

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Table A2 Parameters of MV and LV power lines in the test microgrid microgrid [19,22,24]. Line

Start node

End node

Rl [mΩ]

Xl [mΩ]

Iz [A]

L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L20 L30 L31 L32 L33 L34 L35 L36 L37 L38 L39 L40 L41 MV line

LVBB N01 N02 N03 N04 N05 N06 N07 N08 N03 N10 N11 LVBB LVBB N30 N31 N32 N33 N34 N35 N36 N33 N38 N31 N40 MVBB

N01 N02 N03 N04 N05 N06 N07 N08 N09 N10 N11 N12 N20 N30 N31 N32 N33 N34 N35 N36 N37 N38 N39 N40 N41 BIL

9.94 9.94 9.94 9.94 9.94 9.94 9.94 9.94 9.94 17.395 17.395 17.395 52.8 11.91 11.91 11.91 11.91 17.22 17.22 17.22 17.22 36.54 36.54 17.22 17.22 126.5

2.905 2.905 2.905 2.905 2.905 2.905 2.905 2.905 2.905 3.5 3.5 3.5 14.2 8.37 8.37 8.37 8.37 8.82 8.82 8.82 8.82 9.54 9.54 8.82 8.82 44.0

280 280 280 280 280 280 280 280 280 195 195 195 270 235 235 235 235 190 190 190 190 100 100 190 190 245

where: Rl is a line resistance, Xl is a line reactance, Iz is a long term current-carrying capacity. Table A3 Technical data of loads in the test microgrid [19,22,24]. Load

Connection node

Load type

Pmax [kW]

Qmax [kvar]

N01_R N05_R N07_R N09_R N12_R N20_I N31_C N35_C N37_C N38_C N39_C N40_C N41_C1 N41_C2

N01 N05 N07 N09 N12 N20 N31 N35 N37 N38 N39 N40 N41 N41

Residential Load Residential Load Residential Load Residential Load Residential Load Industrial Load Commercial Load Commercial Load Commercial Load Commercial Load Commercial Load Commercial Load Commercial Load Commercial Load

15.0 55.0 15.0 37.0 35.0 70.0 30.0 26.0 20.0 16.0 8.0 26.0 8.0 20.0

4.93 18.08 4.93 12.16 11.50 17.54 6.09 5.28 4.06 3.25 1.62 5.28 1.62 4.06

where: Pmax is a maximum active power, Qmax is a maximum reactive power.

Fig A1. Daily profiles for different loads in the test microgrid [19,22,24,37].

Table A.4 contains parameters of microsources installed in the test microgrid. It is assumed for wind turbine-generator sets that wind velocity changes from 4 m/s to 12 m/s. The cut-in velocity for the sets is equal to 4 m/s. Calculated daily generation profile for wind turbine-generator set is presented in Fig. A.3. 11

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Fig. A2. Total active power load of the test microgrid during 24 h [22,24]. Table A4 Parameters of microsources installed in the test microgrid [20–22,24]. Device type

Technology

Pn [kW]

N03-FC N05-WT N07-MT N09-FC N20-MT N32-WT N33-MT N34-FC N40-WT

Fuel cell Wind turbine-generator set Gas microturbine Fuel cell Gas microturbine Wind turbine-generator set Gas microturbine Fuel cell Wind turbine-generator set

10 50 30 10 60 10 30 10 10

where Pn is a nominal active power.

Fig. A3. Daily generation profile for wind turbine-generator set [22]. Table A5 Data on battery energy storage [22]. Energy storage type

No. of cycles

Operation time [years]

Capacity [kWh]

Maximum active power [kW]

110x8OPzS800

1500

15

176

28.424

Table A.5 presents data on battery energy storage installed in the test microgrid. “No. of cycles” informs on the permissible number of cycles of full discharging and charging of battery energy storage (BES), when it preserves its properties of energy storing. “Operation time” describes guaranteed time of BES operation during proper exploitation assuming not increased number of cycles with deep discharging. “Maximum active power” is a power possible to be achieved during the continuous operation in discharging mode within the time of 5 h. It is a power for which during the discharging mode of operation the state of charge of BES does not become smaller than 20%. It is assumed, that BES can work in three modes. These are: charging mode, stoppage mode and discharging mode. In discharging mode BES operates with constant power and acts as the source of active power. In turn, in charging mode BES acts as a load for microgrid to which is connected. The power of this load is determined by the charging current of BES. 12

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