# Grid-Tied Hybrid AC-DC Microgrid: Finding Optimal Number of Parallel-Connected AC-DC Bidirectional Interfacing Converters

Authors: Mohsen Rezaie Estabragh; Ali Dastfan; Morteza Rahimiyan
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

Linking AC and DC microgrids via a bidirectional AC-DC interfacing converter has emerged the hybrid AC-DC microgrid. The DC bus is connected to the grid by a series converter to mitigate the grid voltage’s power quality problems. To enhance the reliability and to increase transferring active and nonactive powers between DC and AC main buses, the AC-DC bidirectional interfacing converters are connected in parallel. In this paper, the optimal number (n) of parallel-connected bidirectional interfacing converters is obtained to minimize the annual cost of investment and the reliability cost. A two-step deciding algorithm is proposed to find n. First, active and nonactive powers between DC and AC buses are obtained with power quality and power flow considerations. Based on two types of powers, two reliability indices are calculated, including the expected energy shortage (EES) based on active power and the expected Volt-Amps shortage (EVAS) based on nonactive power. The sum of two reliability index costs is considered as the reliability cost. Next, a decision-making strategy is executed to determine the optimal n by making a trade-off between minimization of two objectives: the annual cost of investment and the reliability cost. The simulations of a grid-tied hybrid AC-DC microgrid are done with the experimental measurement data of DC and AC loads and distributed generations. Simulation results verify the performance of the represented approach.


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## Body

## 1. Introduction

The microgrid (MG) is a controllable set of sources and loads that operates in grid-tied or standalone mode [1]. The AC and DC MGs are combined to form the hybrid AC-DC microgrid (HMG) through an interfacing converter (IC). Due to the specifications of the HMG, it seems this type of MGs has formed the future of distribution systems [2].Microgrids are experiencing significant power quality problems because of the intermission and randomness specifications of renewable power generations, nonlinear and unbalanced loads, and power electronic connections [3]. Many studies were carried out on the HMG topics like the construction of HMG, grid-tied mode and standalone mode of operation, control of DC and AC voltages, power quality, energy management, etc. [2, 4]. In [4–9], several control methods were done to enhance the power quality of MGs. The grid voltage may have power quality problems. In this situation, the series active power filter (SeAPF) as a proper solution is used to compensate them [10, 11]. The SeAPF is added to the traditional grid-tied HMG structure between the DC bus and the grid to reduce power quality problems of the grid (the AC bus). The economic and reliability considerations have received less attention in HMG studies.The use of an interfacing converter limits the amount of transmitting power. Many papers, including [12, 13], claimed that power transmission is enhanced by paralleling the interfacing converters. The reliability of MG is also increased [14, 15]. The parallel-connected converters have a higher availability [15]. Based on [16–18], the parallel-connected converters enhance the range of the load power as well as the reliability of the MG. In [19], several advantages are listed about paralleling the inverters over a single module:(i)
Improved heat dissipation (each module generates a fraction of the total heat which can be dissipated more easily)(ii)
Improve current handling (each module switches a portion of the total current; power transistors are easier to find, and circuit design is much simpler)(iii)
Higher reliability (more reliable than a single inverter for two reasons: (1) each module handles a lower current and dissipates less heat which improves module reliability, and (2) a fault detection circuit in each module would isolate the module when the module fails)(iv)
Lower cost (component and manufacturing costs drop due to increased volume)(v)
Easier shipping and installation (shipping and installation become easier and more cost-effective as module sizes decrease)(vi)
Field expandable (a modular system can be easily expanded to increase output power or degree of redundancy by adding more modules; customers can easily expand the system in the field as their requirements change).Paralleling the interfacing converters has created some vital challenges in terms of control methods. In [20–23], some control methods are proposed to remove or mitigate the challenges. The control for parallel converters is not discussed in this paper.The economic and reliability aspects are the shortcomings in the researches on the subject of parallel converters. By using a big AC-DC converter as the interfacing converter, HMG’s efficient and reliable operation is not obtained. Hence, the economic and reliability analyses in the construction of the HMG are attractive issues for researchers.In this paper, the optimal number of parallel-connected bidirectional interfacing converters in the grid-tied HMG is obtained with minimization of two objectives: the reliability cost and annual cost of investment. The process is done with considering the power quality problem mitigation and satisfaction of the power flow. So, the problem’s objectives for obtaining the optimal number of interfacing converters are power quality improvement and reliability enhancement at the lowest annual cost of investment. The MG planner uses a two-stage decision-making strategy to ascertain the optimal number (n) of the parallel-connected bidirectional interfacing converters. In the first stage, with the purpose of improving the power quality and satisfying the power flow of the HMG, the values of the transferring apparent power (SIC, kVA), active power (PIC, kW), and nonactive power (PfIC, kVA) [24] between DC and AC main buses are computed. Then, the reliability index, the expected energy shortage (EES), is calculated that uses PIC. In this paper, the interfacing converter also supplies nonactive power of load, PfIC. So, the value of PfIC is not negligible. Due to the importance of PfIC, another reliability index, the expected Volt-Amps shortage (EVAS), is computed that uses the PfIC [25]. In this paper, two separate reliability indices are used to analyze the HMG’s reliability because of showing the importance of nonactive power on determining the optimal number of interfacing converters. The reliability index costs are obtained with multiplying the kWh cost ($/kWh) by the EES (kWh) and multiplying the kVAh cost ($/kVAh) by the EVAS (kVAh) [26]. The sum of two reliability index costs should be minimized as a single objective. As a result, the HMG’s reliability is enhanced. The minimization of the annual cost of investment is the second objective. The capital recovery factor (CRF) is multiplied by the buying cost of n-interfacing converters to calculate the annual cost of investment. The simulations are performed on the grid-tied HMG test system with the experimental measurement of loads and DGs. The experimental measurements of the DC and the AC, DGs, and loads for four arbitrary days (one day of every season) are measured that represented four seasons.The main contributions of this paper could be summarized as follows:(i)
Two reliability indices have been calculated, namely, the expected energy shortage (EES) based on the transferring active powerPIC and the expected Volt-Amps shortage (EVAS) based on the transferring nonactive power PfIC (kVA). These two indices are used to calculate the cost of reliability and incorporated into the objective function.(ii)
To find the optimal number of parallel-connected bidirectional interfacing converters, a multi-objective optimization problem is applied by minimizing the cost of reliability and investment. A two-step deciding algorithm is used to solve the problem. In first step, two objectives are calculated based on the number of parallel interfacing converters with consideration of power quality and power flow. In second step, the optimal number of parallel converters is obtained while minimizing the total cost of reliability and investment.(iii)
To report results from a realistic case study that shows the effectiveness of the proposed decision-making algorithm for the microgrid planner.


## 2. Parallel-Connected Bidirectional Interfacing Converters in the HMG

Determination of the optimal number of the parallel-connected bidirectional AC-DC interfacing converters is represented as a two-objective deciding problem. The problem is modeled with the technical and economic objectives containing the reliability cost and the annual cost of investment. Two reliability indices are calculated based on transferring active and nonactive powers between DC and AC main buses. The sum of two reliability indices is considered as the reliability cost. Furthermore, the HMG’s power quality is enhanced and the HMG’s power flow is satisfied. The two problem objectives are obtained as follows.


### 2.1. Grid-Tied HMG Configuration

Figure1 shows the grid-tied HMG as a single-line diagram layout (DC and AC MGs, series converter, interfacing converter, and DC and AC loads). The interfacing converter connects DC and AC main buses. In this paper, along with bidirectional transmitting active power of DGs and loads, the interfacing converter provides the reactive and harmonic power of the AC load to mitigate the grid current’s power quality problems. As a result, the grid current will be a unity power factor sinusoidal current. The interfacing converter acts similar to a shunt active power filter.Figure 1
Parallel-connected AC-DC interfacing converters of the grid-tied HMG.In the HMG, the DC bus is linked to the grid through a series converter and a series transformer. The series converter deals with the mitigation of the grid voltage’s power quality problems (sag/swell/harmonics). As a result, the HMG’s AC bus voltage will be sinusoidal by nominal amplitude and frequency. The series converter acts similar to a SeAPF. Consequently, the MG’s and the grid’s power quality problems are not passed down from one side to the next. The quantity of transferring power between DC and AC main buses must be computed to determine the optimal number of parallel-connected bidirectional interfacing converters.


### 2.2. Calculate Active and Nonactive Powers between DC and AC Main Buses

The procedure for calculating the apparent power (SIC, kVA), the active power (PIC, kW), and the nonactive power (PfIC, kVA) between DC and AC main buses is illustrated to enhance the HMG's power quality and to meet the HMG’s power flow. The power flow equations of real, reactive, and distorted powers in the AC bus are as follows:(1)PGrid+PSeC+PAC_Gen+PIC=PAC_Load,QIC=QAC_Load,DIC=DAC_Load,SIC=PIC+j⟶QIC+k⟶DIC,where PGrid, PSeC, PAC_Gen, PIC, PAC_Load, PAC_Load, QIC, QAC_Load, DIC, DAC_Load, and SIC are the grid’s real power, the real power of series converter, the AC DG real power, the interfacing converter real power, the AC load real power, the interfacing converter reactive power, the reactive power of the AC load, the interfacing converter distorted power, the load distorted power, and the interfacing converter apparent power, respectively.The DC bus active power flow equation is as follows:(2)PDC_Gen=PIC+PDC_Load+PBatt+PV_DC_Bus,where PDC_Gen, PDC_Load, PBatt, and PV_DC_Bus are the DC DG real power, the DC load real power, the battery real power, and the real power for voltage regulation of the DC bus, respectively.The series converter could regulate the AC load phase voltage (AC bus) at the rated grid voltage.VLp=Vsp, for improving the power quality, where VLp and Vsp are the load phase voltage and the rated grid voltage, respectively.Another aim of this paper is improving the grid current's power quality problems. The interfacing converter performs this role. For this purpose, the load current is divided into two parts: (1) the interfacing converter provides the load current’s reactive and harmonic power component. (2) The grid provides the load current’s active power component. So, the power factor is corrected and load harmonic currents are reduced. The value of therms load current and the fundamental component of load current are computed as follows:(3)IAC_L1a=PAC_Load3VLp,IAC_L1=IAC_L1acosφ,IAC_L=IAC_L11+THD2,where PAC_Load, cosφ, and THD are the AC load real power, the AC load power factor, and the AC load current total harmonic distortion, respectively.The interfacing converter nominal voltage is identical to the AC load’s phase voltage since the AC side of the converter is linked to the AC bus,VIC=VLp. The interfacing converter supplies the load’s reactive and harmonic power component in addition to transferring the DGs and load powers. The nominal current of the interfacing converter is obtained as follows.The load current’s reactive and harmonic power component is calculated as follows:(4)Inon−activeIC=IAC_L2−IAC_L1a2.The grid current under voltage sag is obtained as follows:(5)Is′=Ps3Vsp′,Ps=PGrid+PSeC,Ps=PAC_Load−PAC_Gen−PDC_Gen+PDC_Load,Vsp′=Vsp1−X,where Ps, X, and Vsp′ are the supply power, the maximum p.u. variations of grid voltage, and the voltage of AC bus under the condition of voltage sag (sag/swell), respectively.Figure2 shows the AC bus power flow.Figure 2
The AC bus real power flow.When the sag condition occurs, the grid voltage’s fundamental component is lower than the nominal voltage of the load. So, the real power is absorbed by the interfacing converter during voltage sag condition and the power injected by the series converter during voltage sag condition.The real power component of the interfacing converter current is obtained as follows:(6)IactiveIC=Is′+IAC_Gen+IDC_Gen_g−IAC_L1a−IDC_Load_g,IAC_Gen_g=PAC_Gen3Vsp,IDC_Gen_g=PDC_Gen3Vsp,IDC_Load_g=PDC_Load3Vsp,where IAC_Gen, IDC_Load_g, and IDC_Gen_g are the grid side equivalent current of AC DG, DC load, and DC DG, respectively.The interfacing converter’s current, active power, apparent power, and nonactive power are calculated as follows:(7)IIC=IactiveIC2+Inon−activeIC2,PIC=3VICIactiveIC,SIC=3VICIIC=PIC2+QIC2+DIC2=PIC2+PfIC2,PfIC=QIC2+DIC2.


### 2.3. Reliability Analysis

Figure3 depicts a AC-DC three-phase bidirectional converter with its components. The components’ failure rates (λ) and the AC-DC converter’s repair rate (μ) are reported and are calculated in Table 1 [15]. The AC-DC converter’s repair time is 24 days [15].Figure 3
: Three-phase AC-DC converter with details.Table 1
The AC-DC converter components: the failure rates and repair time [15].
ComponentCapacitorIGBTDiodeSnubberFilter inductorCurrent sensorVoltage sensorGate driveCooling systemλtotalμtotalFORλone3.030.90.80.0010.0380.50.5611.3623.89 (Failures/106 hours) = 0.2093/year365/24 = 15.2/yearFOR=λλ+μ = 0.0136No.166633361λNoλc = 3.03λIGBT = 5.4λD = 4.8λS = 0.006λLf = 0.114λI = 1.5λV = 1.68λGD = 6λCS = 1.36Supposen parallel-connected bidirectional interfacing converters are chosen. In state i, i-1 converters are failed. The probability of state i is computed as follows:(8)pi=nk×pk×qn−k=n!k!n−k!×pk×qn−k,Success:p=FOR,Failure:q=1−FOR,FOR=λλ+μ,k=i−1.The number of states (SN) (i = 1, 2, …, SN = n+1) for failed converters (0 to n) when success is the failure converter.The active power shortage (PSi) between the DC bus and the AC bus is obtained as follows:(9)Pbase=Sbase×cosφkW,cosφ=0.8,ni=n−i−1,PSi=PIC−niPbasekW.where Sbase, Pbase, and ni are the apparent power nominal size of one converter, the active power nominal size of one converter, and the number of remaining converters in state i, respectively. The converters are considered similar.TheEES is calculated as follows:(10)EES=∑i=1SNPSi×pi×8760kWh.In this paper, the interfacing converter deals with compensation of the nonactive component of the load power (kVA) containing reactive and distortion power in addition to bidirectional active power flow. The measurements show that the transmission of nonactive power is not negligible. Active and reactive powers are almost always elaborated, but distortion power is rarely measured and paid for. Inevitably, nonactive power must also be generated, transmitted, and distributed. The nonactive power shortage (PFSi) between the DC bus and the AC bus is obtained as follows:(11)Pbasef=Sbase2−Pbase2kVA,PFSi=PfIC−niPbasefkVA,where Pbasef is the nonactive power’s nominal size of one converter.Hence, in this paper, based on the nonactive power a reliability index is proposed as expected Volt-Amps shortage (EVAS):(12)EVAS=∑i=1SNPFSi×pi×8760kVAh.


### 2.4. The Objectives of Problem

The parallel-connected bidirectional interfacing converters’ optimal number (n) is determined by solving a two-objective problem containing the annual cost of investment and combination of two reliability index costs.


#### 2.4.1. First Objective: The Reliability Index Costs

The values of the indices (EES (kWh) and EVAS (kVAh)) are converted to cost ($) as follows:(13)CostEES=CostkWh×EES$,(14)CostEVAS=CostkVAh×EVAS$,where CostkWh ($/kWh) and CostkVAh ($/kVAh) are the cost of EES and the cost of EVAS that are considered as 0.15 $/kWh and 0.15 $/kVAh, respectively [25].The first objective is considered as follows:(15)OF1:CostRI=CostEES+CostEVAS$.The reliability index costs are added together with equal coefficients because of the same significance of the transferring active and nonactive powers.


#### 2.4.2. Second Objective: The Annual Cost of Investment

The active power filters’ buying costs are reported in Table2.where n and CostIC are the number of interfacing converters and the buying cost of one interfacing converter, respectively.Table 2
Buying costs of active power filters.
Current (A)S (VA)Cost ($)5033000180075495002400100660002800150990003900The total investment cost of the interfacing converters is calculated as follows:(16)Costn−IC=n×CostIC$.The capital recovery factor (CRF) is multiplied by the total investment cost, and the annual cost of investment (AC) is calculated as follows [27]:(17)OF2:AC=CRF×Costn−IC$,CRF=IR1+IRLT1+IRLT−1,where LT and IR are an interfacing converter’s lifetime and the interest rate.


## 2.1. Grid-Tied HMG Configuration

Figure1 shows the grid-tied HMG as a single-line diagram layout (DC and AC MGs, series converter, interfacing converter, and DC and AC loads). The interfacing converter connects DC and AC main buses. In this paper, along with bidirectional transmitting active power of DGs and loads, the interfacing converter provides the reactive and harmonic power of the AC load to mitigate the grid current’s power quality problems. As a result, the grid current will be a unity power factor sinusoidal current. The interfacing converter acts similar to a shunt active power filter.Figure 1
Parallel-connected AC-DC interfacing converters of the grid-tied HMG.In the HMG, the DC bus is linked to the grid through a series converter and a series transformer. The series converter deals with the mitigation of the grid voltage’s power quality problems (sag/swell/harmonics). As a result, the HMG’s AC bus voltage will be sinusoidal by nominal amplitude and frequency. The series converter acts similar to a SeAPF. Consequently, the MG’s and the grid’s power quality problems are not passed down from one side to the next. The quantity of transferring power between DC and AC main buses must be computed to determine the optimal number of parallel-connected bidirectional interfacing converters.


## 2.2. Calculate Active and Nonactive Powers between DC and AC Main Buses

The procedure for calculating the apparent power (SIC, kVA), the active power (PIC, kW), and the nonactive power (PfIC, kVA) between DC and AC main buses is illustrated to enhance the HMG's power quality and to meet the HMG’s power flow. The power flow equations of real, reactive, and distorted powers in the AC bus are as follows:(1)PGrid+PSeC+PAC_Gen+PIC=PAC_Load,QIC=QAC_Load,DIC=DAC_Load,SIC=PIC+j⟶QIC+k⟶DIC,where PGrid, PSeC, PAC_Gen, PIC, PAC_Load, PAC_Load, QIC, QAC_Load, DIC, DAC_Load, and SIC are the grid’s real power, the real power of series converter, the AC DG real power, the interfacing converter real power, the AC load real power, the interfacing converter reactive power, the reactive power of the AC load, the interfacing converter distorted power, the load distorted power, and the interfacing converter apparent power, respectively.The DC bus active power flow equation is as follows:(2)PDC_Gen=PIC+PDC_Load+PBatt+PV_DC_Bus,where PDC_Gen, PDC_Load, PBatt, and PV_DC_Bus are the DC DG real power, the DC load real power, the battery real power, and the real power for voltage regulation of the DC bus, respectively.The series converter could regulate the AC load phase voltage (AC bus) at the rated grid voltage.VLp=Vsp, for improving the power quality, where VLp and Vsp are the load phase voltage and the rated grid voltage, respectively.Another aim of this paper is improving the grid current's power quality problems. The interfacing converter performs this role. For this purpose, the load current is divided into two parts: (1) the interfacing converter provides the load current’s reactive and harmonic power component. (2) The grid provides the load current’s active power component. So, the power factor is corrected and load harmonic currents are reduced. The value of therms load current and the fundamental component of load current are computed as follows:(3)IAC_L1a=PAC_Load3VLp,IAC_L1=IAC_L1acosφ,IAC_L=IAC_L11+THD2,where PAC_Load, cosφ, and THD are the AC load real power, the AC load power factor, and the AC load current total harmonic distortion, respectively.The interfacing converter nominal voltage is identical to the AC load’s phase voltage since the AC side of the converter is linked to the AC bus,VIC=VLp. The interfacing converter supplies the load’s reactive and harmonic power component in addition to transferring the DGs and load powers. The nominal current of the interfacing converter is obtained as follows.The load current’s reactive and harmonic power component is calculated as follows:(4)Inon−activeIC=IAC_L2−IAC_L1a2.The grid current under voltage sag is obtained as follows:(5)Is′=Ps3Vsp′,Ps=PGrid+PSeC,Ps=PAC_Load−PAC_Gen−PDC_Gen+PDC_Load,Vsp′=Vsp1−X,where Ps, X, and Vsp′ are the supply power, the maximum p.u. variations of grid voltage, and the voltage of AC bus under the condition of voltage sag (sag/swell), respectively.Figure2 shows the AC bus power flow.Figure 2
The AC bus real power flow.When the sag condition occurs, the grid voltage’s fundamental component is lower than the nominal voltage of the load. So, the real power is absorbed by the interfacing converter during voltage sag condition and the power injected by the series converter during voltage sag condition.The real power component of the interfacing converter current is obtained as follows:(6)IactiveIC=Is′+IAC_Gen+IDC_Gen_g−IAC_L1a−IDC_Load_g,IAC_Gen_g=PAC_Gen3Vsp,IDC_Gen_g=PDC_Gen3Vsp,IDC_Load_g=PDC_Load3Vsp,where IAC_Gen, IDC_Load_g, and IDC_Gen_g are the grid side equivalent current of AC DG, DC load, and DC DG, respectively.The interfacing converter’s current, active power, apparent power, and nonactive power are calculated as follows:(7)IIC=IactiveIC2+Inon−activeIC2,PIC=3VICIactiveIC,SIC=3VICIIC=PIC2+QIC2+DIC2=PIC2+PfIC2,PfIC=QIC2+DIC2.


## 2.3. Reliability Analysis

Figure3 depicts a AC-DC three-phase bidirectional converter with its components. The components’ failure rates (λ) and the AC-DC converter’s repair rate (μ) are reported and are calculated in Table 1 [15]. The AC-DC converter’s repair time is 24 days [15].Figure 3
: Three-phase AC-DC converter with details.Table 1
The AC-DC converter components: the failure rates and repair time [15].
ComponentCapacitorIGBTDiodeSnubberFilter inductorCurrent sensorVoltage sensorGate driveCooling systemλtotalμtotalFORλone3.030.90.80.0010.0380.50.5611.3623.89 (Failures/106 hours) = 0.2093/year365/24 = 15.2/yearFOR=λλ+μ = 0.0136No.166633361λNoλc = 3.03λIGBT = 5.4λD = 4.8λS = 0.006λLf = 0.114λI = 1.5λV = 1.68λGD = 6λCS = 1.36Supposen parallel-connected bidirectional interfacing converters are chosen. In state i, i-1 converters are failed. The probability of state i is computed as follows:(8)pi=nk×pk×qn−k=n!k!n−k!×pk×qn−k,Success:p=FOR,Failure:q=1−FOR,FOR=λλ+μ,k=i−1.The number of states (SN) (i = 1, 2, …, SN = n+1) for failed converters (0 to n) when success is the failure converter.The active power shortage (PSi) between the DC bus and the AC bus is obtained as follows:(9)Pbase=Sbase×cosφkW,cosφ=0.8,ni=n−i−1,PSi=PIC−niPbasekW.where Sbase, Pbase, and ni are the apparent power nominal size of one converter, the active power nominal size of one converter, and the number of remaining converters in state i, respectively. The converters are considered similar.TheEES is calculated as follows:(10)EES=∑i=1SNPSi×pi×8760kWh.In this paper, the interfacing converter deals with compensation of the nonactive component of the load power (kVA) containing reactive and distortion power in addition to bidirectional active power flow. The measurements show that the transmission of nonactive power is not negligible. Active and reactive powers are almost always elaborated, but distortion power is rarely measured and paid for. Inevitably, nonactive power must also be generated, transmitted, and distributed. The nonactive power shortage (PFSi) between the DC bus and the AC bus is obtained as follows:(11)Pbasef=Sbase2−Pbase2kVA,PFSi=PfIC−niPbasefkVA,where Pbasef is the nonactive power’s nominal size of one converter.Hence, in this paper, based on the nonactive power a reliability index is proposed as expected Volt-Amps shortage (EVAS):(12)EVAS=∑i=1SNPFSi×pi×8760kVAh.


## 2.4. The Objectives of Problem

The parallel-connected bidirectional interfacing converters’ optimal number (n) is determined by solving a two-objective problem containing the annual cost of investment and combination of two reliability index costs.


### 2.4.1. First Objective: The Reliability Index Costs

The values of the indices (EES (kWh) and EVAS (kVAh)) are converted to cost ($) as follows:(13)CostEES=CostkWh×EES$,(14)CostEVAS=CostkVAh×EVAS$,where CostkWh ($/kWh) and CostkVAh ($/kVAh) are the cost of EES and the cost of EVAS that are considered as 0.15 $/kWh and 0.15 $/kVAh, respectively [25].The first objective is considered as follows:(15)OF1:CostRI=CostEES+CostEVAS$.The reliability index costs are added together with equal coefficients because of the same significance of the transferring active and nonactive powers.


### 2.4.2. Second Objective: The Annual Cost of Investment

The active power filters’ buying costs are reported in Table2.where n and CostIC are the number of interfacing converters and the buying cost of one interfacing converter, respectively.Table 2
Buying costs of active power filters.
Current (A)S (VA)Cost ($)5033000180075495002400100660002800150990003900The total investment cost of the interfacing converters is calculated as follows:(16)Costn−IC=n×CostIC$.The capital recovery factor (CRF) is multiplied by the total investment cost, and the annual cost of investment (AC) is calculated as follows [27]:(17)OF2:AC=CRF×Costn−IC$,CRF=IR1+IRLT1+IRLT−1,where LT and IR are an interfacing converter’s lifetime and the interest rate.


## 2.4.1. First Objective: The Reliability Index Costs

The values of the indices (EES (kWh) and EVAS (kVAh)) are converted to cost ($) as follows:(13)CostEES=CostkWh×EES$,(14)CostEVAS=CostkVAh×EVAS$,where CostkWh ($/kWh) and CostkVAh ($/kVAh) are the cost of EES and the cost of EVAS that are considered as 0.15 $/kWh and 0.15 $/kVAh, respectively [25].The first objective is considered as follows:(15)OF1:CostRI=CostEES+CostEVAS$.The reliability index costs are added together with equal coefficients because of the same significance of the transferring active and nonactive powers.


## 2.4.2. Second Objective: The Annual Cost of Investment

The active power filters’ buying costs are reported in Table2.where n and CostIC are the number of interfacing converters and the buying cost of one interfacing converter, respectively.Table 2
Buying costs of active power filters.
Current (A)S (VA)Cost ($)5033000180075495002400100660002800150990003900The total investment cost of the interfacing converters is calculated as follows:(16)Costn−IC=n×CostIC$.The capital recovery factor (CRF) is multiplied by the total investment cost, and the annual cost of investment (AC) is calculated as follows [27]:(17)OF2:AC=CRF×Costn−IC$,CRF=IR1+IRLT1+IRLT−1,where LT and IR are an interfacing converter’s lifetime and the interest rate.


## 3. Decision-Making Algorithm

The hierarchy decision-making algorithm is represented to determine the optimaln as Figure 4.Figure 4
Algorithm for finding the optimal number of interfacing converters.Figure5 shows the schematic diagram of the determination process of the optimal number of parallel-connected bidirectional interfacing converters.Figure 5
The solving method's schematic diagram.The decision-making function is composed of (14) and (16) as follows:(18)OF:minOF1+OF2.s.t.(19)1≤n≤SmaxICSbase+x,(20)nPbase≥PIC,(21)nPbasef≥PfIC,where SmaxIC, Sbase, and x are the maximum apparent power between the main AC and DC buses, the converter capacity, and arbitrary positive integer value, respectively.The planner determines the best solution by considering the reliability and the economic aspects. There are many optimization algorithms to solve the multi-objective optimization problems, but based on the above discussions a two-step algorithm is proposed to find the optimal solution. In first step, the problem objectives are calculated in consideration with the number of parallel interfacing converters (1, 2, …, n). In second step, the optimal number of parallel converters is chosen among several scenarios by considering the objective function minimization. Hence, based on the proposed two-step algorithm, the microgrid planner can easily find the optimal number of parallel interfacing converters.


## 4. Simulation and Results

The HMG setup is depicted in Figure1 as a single-line diagram [2]. The typical HMG includes DC load, AC nonlinear load, DC DG (PV), and AC DG (wind turbine). Measurements are made, and the experimental data are recorded, analyzed, and reported. The hourly experimental measurements of DC load, wind turbine, and PV of the four days that represent four seasons are shown in Figures 6–8 (daily curves of spring, summer, autumn, and winter, one day of each season), and the data of three-phase nonlinear load of the four days that represent four seasons are displayed in Figure 9 (daily curves of active power, THD, and power factor in spring, summer, autumn, and winter, one day of each season). The power quality analyzer unipower 25003928 is used for measuring the experimental data of nonlinear load. The data are reported every 10 mins.Figure 6
Seasonal (autumn, winter, spring, summer) daily DC load real power.Figure 7
Seasonal (autumn, winter, spring, summer) daily wind output real power.Figure 8
Seasonal (autumn, winter, spring, summer) daily PV output real power.Figure 9
Nonlinear load, (a) real power data (kW), (b) THD data, (c) power factor data. (a) Seasonal (autumn, winter, spring, summer) daily nonlinear load real power. (b) Seasonal (autumn, winter, spring, summer) daily nonlinear load THD. (c) Seasonal (autumn, winter, spring, summer) daily nonlinear load power factor.
(a)(b)(c)The required simulation parameters are given in Table3.Table 3
Simulation parameters.
ParameterValueParameterValueIR1.5% [28]Pbasef29700 VALT12 years [29]Vs (AC bus voltage)380 VSbase49500 VAX (maximum p.u. variations of grid voltage)20%Pbase39600 WVDC (DC bus voltage)650 VIn order to calculateSIC, PIC, and PfIC, a MATLAB toolbox is written based on equations (1)–(7). The values are displayed in Figures 10–12 that are calculated in every 10 mins. The maximum of (SIC, PIC, and PfIC) peak values are 383.27 kVA, 183.75 kW, and 380.0862 kVA, respectively.Figure 10
Seasonal (autumn, winter, spring, summer) dailySIC (apparent power, kVA).Figure 11
Seasonal (autumn, winter, spring, summer) dailyPIC (active power, kW).Figure 12
Seasonal (autumn, winter, spring, summer) dailyPfIC (nonactive power, kVA).Using the illustrated decision-making strategy (Figure5), the optimal number of interfacing converters is selected. The interfacing converters have the same size (49500 VA, Table 2).The decision-making strategy is run and based on different values ofn; the objectives are computed and are expressed in Tables 4 and 5. It could be seen by increasing n, the reliability index costs are decreased, and the annual costs are increased. Hence, the reliability and the cost of the HMG are enhanced. The planner of the MG chooses the optimal number of the interfacing converters by observing the objective values. When the load power supply is important, an enormous number (n) is chosen. When the load is not sensitive, a smaller number (n) is determined, and a piece or whole of the load should be shed.Table 4
The values of reliability and cost analysis.
nEES (kWh)EES cost =  EEScost=CostkWh×EES$ (CostkWh=0.15)EVAS (kVAh)EVAScost=CostkVAh×EVAS$ (CostkWh=0.15)Total investment costTIC=n×Costbase$11545600231840759350011390002400270643010597062394009359204800318899028348504760075713072004305684585.14011900601780960051744.7261.7087305890045883012000666.712710.006921927003289101440072.0460.306914212002131801680080.05450.00827821801173301920090.00131.9712e − 043439905159821600102.9463e − 054.4194e − 061052901579324000116.2429e − 079.3644e − 08262973944.526400121.2647e − 081.8971e − 095050.5757.57928800132.4701e − 103.7051e − 11580.920387.13831200144.6795e − 127.0193e − 1346.37836.956733600158.6404e − 141.2961e − 142.86840.430336000161.5607e − 152.3410e − 160.14710.022138400Table 5
The first and second objectives and objective function.
nOF1:CostRI=CostEES+CostEVAS$OF2:Annual costAC=TIC∗CRF$OF=OF1+OF2$11370840220.0321371060.03221041890440.0641042330.0643785478660.0959786138.09594606365.1880.1279607245.22795459091.70871100.2460191.90876328920.00691320.2330240.20697213180.30691540.2214720.50698117330.00821760.3119090.30829515981980.353578.310157932200.317993.3113944.52420.46364.912757.5792640.43397.9791387.1382860.42947.538146.95673080.43087.3567150.43033300.53300.9303160.02213520.53520.5221In Table4, the cost of the EES becomes near zero in the state of n = 7 with the annual cost of 1540.2 $, while the cost of the EVAS becomes near zero in n = 15 with the annual cost of 3300.5 $. The grid current’s power quality problems are considered in the cost of the EVAS.In Table5, the problem objectives and total objective function are reported. By increasing the values of n, the costs of the reliability indices are decreased, and consequently, the reliability is increased. But, the annual costs are increased. The objective function values are constituted from the first objective (sum of the reliability index costs) plus the second objective (annual cost).Based on Tables4 and 5, the planner selects the optimal value of n. Four plans are discussed for determining n in the following.(i)
Plan 1: ifn = 1, the annual cost is 220.032 $ (the lowest cost). The EES cost is 231840 $, and the EVAS cost is 1139000 $. As a result, the reliability cost is 1370840 $ that shows the maximum reliability cost and the least reliability of the HMG. Hence, a big or some piece of the load should be shed. It could be seen that the plan is not proper for interfacing converters. Based on the EVAS cost value, the importance of nonactive power transferring between the main buses in the decision-making process is clear. Also, the value of the objective function is 1371120.032 $.(ii)
Plan 2: ifn = 7, the annual cost is 1540.2 $ that is bigger than the previous plan. The EES cost is 0.3069 $, which is almost near zero. The EVAS cost is 213180 $. As a result, the reliability cost is 213180.3069 $ which is lower than previous plan and still shows the nonproper reliability for the HMG. In this plan, if n is chosen based on the EES cost, the transferring nonactive power is neglected, and the full load is not supplied. It can be seen the plan is not suitable for the interfacing converters. Also, the value of the objective function is 214720.5069 $.(iii)
Plan 3: ifn = 13, the annual cost is 2860.4 $. The EES cost is 3.7051e-11 $, which is zero. The EVAS cost is 87.138 $. As a result, the reliability cost is 87.138 $, which almost showed the HMG's good reliability. In the plan, approximately the full load is supplied. Hence, it can be seen; the plan is suitable for the interfacing converters compared to others. Also, the value of the objective function is 2947.538 $.(iv)
Plan 4: ifn = 15, the annual cost is 3300.5 $. The EES cost is 1.2961e-14 $, that is, zero. The EVAS cost is 0.4303 $, which is almost zero. As a result, the reliability cost is 0.4303 $, near zero that showed a good reliability condition of the HMG. In the plan, near full load is supplied. Based on the value of the annual cost, it is possible that the plan is not acceptable for the interfacing converters compared to others. Hence, for selecting the optimal n, it is necessary to make a trade-off between the economic and technical aspects of the problem.Based on the more importance of the annual cost or the reliability cost (economic or technical), the planner selects among the feasible plans.


## 5. Conclusions

In this paper, by taking into account both reliability and economic aspects, a two-step decision-making problem is represented with two objectives to select the optimum number of parallel-connected bidirectional interfacing converters of the HMG. In the first step, the values of transferring apparent power, active power, and nonactive power (SIC, PIC and PfIC) between DC and AC main buses are calculated by taking into account of improving the power quality and satisfying the power flow. The reliability indices, expected energy shortage (EES) and expected VA shortage (EVAS), are obtained based on the PIC and the PfIC, respectively. The summation of two reliability index costs is used to calculate the reliability cost as the first objective. The annual cost of parallel-connected bidirectional interfacing converters is calculated as the second objective. Then, in the second step, the objective function is obtained by summation of two objectives based on feasible values of parallel AC-DC interfacing converters’ number. The HMG planner runs the decision-making strategy to select the optimal number of parallel-connected bidirectional interfacing converters based on the values of objectives. When n is enhanced, the reliability cost is reduced and the annual cost is raised. With the bigger n, the SIC, PIC, and PfIC power transferring shortages are decreased, the reliability cost is reduced, and the reliability is enhanced while the annual cost is also increased.



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*Source: 1932818-2022-06-02.xml*