Journal of Modern Power Systems and Clean Energy

ISSN 2196-5625 CN 32-1884/TK

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Stability Analysis of DC Microgrid Clusters Based on Step-by-step System Matrix Building Algorithm  PDF

  • K. Jithin (Graduate Student Member, IEEE)
  • N. Mayadevi
  • R. Hari Kumar (Senior Member, IEEE)
  • V. P. Mini
Department of Electrical Engineering, College of Engineering Trivandrum, APJ Abdul Kalam Technological University (APJAKTU), Thiruvananthapuram, India

Updated:2024-05-20

DOI:10.35833/MPCE.2023.000054

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Abstract

DC microgrids (DCMGs) are made up of a network of sources and loads that are connected by a number of power electronic converters (PECs). The increase in the number of these PECs instigates major concerns in system stability. While interconnecting the microgrids to form a cluster, the system stability must be ensured. This paper proposes a novel step-by-step system matrix building (SMB) algorithm to update the system matrix of an existing DCMG cluster when a new microgrid is added to the cluster through a distribution line. The stability of the individual DCMGs and the DCMG cluster is analyzed using the eigenvalue method. Further, the particle swarm optimization (PSO) algorithm is used to retune the controller gains if the newly formed cluster is not stable. The simulation of the DCMG cluster is carried out in MATLAB/Simulink environment to test the proposed algorithm. The results are also validated using the OP4510 real-time simulator (RTS).

I. Introduction

DUE to the growing market for DC loads and sources as well as their numerous advantages, DC microgrids (DCMGs) have attained a prominent role nowadays [

1], [2]. The power electronic converters (PECs) interface the majority of the sources and loads of a DCMG to a shared DC bus. Therefore, it is possible to characterize power flow, stability, power quality, etc, as a function of converter behavior and their control loops [3], [4].

Various energy storage devices like batteries and flywheels are introduced in the DCMG to tackle the inertia challenges that arise due to the high penetration of renewable energy sources (RESs). However, the capacity limitations and charging and discharging complications raise concerns [

5], [6]. As a result, the clustering of grids becomes a more effective and dependable way to get around these problems. The development of clusters of various DCMGs can increase system resilience by diminishing the need for energy storage and enabling seamless power sharing within the grid. However, for clustered DCMGs as autonomous grids, clustered DCMGs are faced with challenges in many aspects including stability, voltage regulation, variable consumer needs, and power quality pose challenges [7]-[9]. Numerous researchers have recently examined DCMG stability problems. One of the widely used techniques for assessing the dynamic stability of the DCMG is small-signal (SS) stability analysis [10]. An investigation of the SS stability of the DCMG cluster based on impedance is performed in [11]. The SS model based on the impedance of the cluster is generated, and the system stability is examined using the eigenvalues. The stability of the low-frequency modes of the DCMG is examined using mesh analysis employing the control method in [12]. A sensitivity analysis is also performed to study how the system parameters affect the stability. A grid-connected DCMG with a hybrid photovoltaic (PV) and wind is assessed for its stability in [13]. Reference [14] discusses several restrictions connected to the traditional stability criteria such as right-half-plane (RHP) poles or subsystem zeros. Additionally, impedance-based criteria for stability are proposed for multi-bus DCMGs in the light of generalized Bode plots.

Using eigenvalue observation, a state space model of the DCMG is created in [

15] to examine the effects of the suggested optimized secondary controller on DCMG stability. The admittance matrix is generated and factorized using a separate methodology and the attainment of stability requirements for droop-controlled DCMGs with different loads is facilitated and ensured by the admittance matrix modification technique in [16]. The design and stability study of a DCMG with a hybrid energy storage system with operating voltages of different supercapacitor is covered in [17]. Reference [18] discusses large-signal stability of DCMG using a system-level method. Reference [19] suggests a Takagi-Sugeno multi-modeling method based large-signal stability study of DCMG clusters. The analysis is simplified by reducing the large-signal Lyapunov stability of DCMG cluster to the computation of a number of linear matrix inequalities. Using a boost converter and a bidirectional DC-DC converter with the Lyapunov function, the DCMG stability is investigated in [20]. The relationship between system parameters and stability is presented and discussed in [21].

The stability criterion evaluates and guarantees the system stability under significant disturbance in the design phase rather than repeatedly computing and simulating. In [

22], various stability analysis and control techniques with interconnected MGs are studied and compared. The MG is described as a medium- and high-dimensional negative feedback loop in [23]. Also, a stability criterion based on the extended Gershgorin theorem with a wide eigenvalue range is developed. In [24], stability analysis considering constant delay and time-varying delay utilizing linear matrix inequality is performed. In order to satisfy the requirements for power balance and grid voltage stability, the non-linear MG model is introduced, and a thorough stability study is performed in [25].

Reference [

26] proposes a partitioning strategy with stability constraints based on SS stability. Moreover, a novel index for the marginal stability of the MG is defined using sensitivity analysis of droop gain. A buck converter-based DCMG is developed in [27] and its robustness and stability are assessed using the Kharitonov theorem and root locus, respectively. It is proven that an increase in constant power loads causes a decrease in system stability.

Even though, various techniques are available for the stability analysis of DCMG, and the effect of the interconnection of MGs to form a cluster on stability is not considered yet. This paper examines the stability of a DCMG cluster using an eigenvalue-based method. A step-by-step system matrix building (SMB) algorithm that avoids the need to completely rebuild a system matrix of cluster whenever a new DCMG is interconnected to it when a distribution line is developed and analyzed. To ensure the reliable functioning of the cluster, the optimization of controller gains utilizing particle swarm optimization (PSO) is also carried out. Through time response analysis, simulation, and real-time experimentation, the proposed algorithm has been validated.

The rest of the paper is structured as follows. Section II presents the transfer function of converter-based DCMGs. The stability analysis of individual DCMG is presented in Section III. The development of the system matrix of a clustered network from the system matrix of the individual networks, when interconnected through a distribution line, is presented in Section IV. Section V presents the stability analysis of the DCMG clusters. Section VI presents the simulation results. Section VII presents a real-time analysis of the DCMG cluster and Section VIII concludes this paper.

II. Transfer Function of Converter-based DCMGs

A DCMG consists of distributed energy resources (DERs), loads, converters, and controllers. All the sources and loads in a DCMG are interfaced with the main bus using DC-DC converters and hence they have a significant role in the stable and reliable operation of the MG. The transfer function of DCMG is developed using the SS model.

The transfer functions are shown in Table I, where D'=1-D and den(s)=s2LCR+sL+RD'2, and D is the duty cycle.The duty cycle to the output voltage and duty cycle to inductor current transfer functions Gvd and Gid of the converters are listed in Table I [

28], where L and C are the converter reactive elements; Vin is the input to the converter; Vo is the output of the converter; and R is the load in the system. The simplified representations of DCMGs with non-dispatchable and dispatchable sources are given in Figs. 1 and 2, respectively, where Vbus is the bus voltage; ILn is the inductor current; and d is the duty ratio. PWM stands for paulse width modulation. For systems with non-dispatchable sources, the reference input to the controller is Vref, and for systems with dispatchable sources, it is Iref.

TABLE I  Transfer Functions of Converters
ConverterGvdGid
Buck RVin/(s2 LCR+sL+R) (l+sCR)Vin/(s2LCR+sL+R)
Boost (RD'2-sL)Vin/den(s)D'2 (2+sCR)Vin/den(s)D'

Fig. 1  Block diagram of DCMG with non-dispatchable sources.

Fig. 2  Block diagram of DCMG with dispatchable sources.

The central controller, which employs an algorithm for the optimal dispatch, provides the value of Iref [

29]. The controller gains are obtained using Ziegler-Nichols method [30], [31].

A. Solar PV System with Boost Converter

The solar PV system with boost converter can be represented by the block diagram as shown in Fig. 3, where Vpv is the output voltage of PV; Vref,pv(s) is the reference voltage of PV; IL,pv(s) is the inductor current of PV converter; and Vo,pv(s) is the output voltage of PV converter. The overall transfer function of the system is derived by reducing the control loops and substituting the parameter values from Table II.

Fig. 3  Block diagram of solar PV system with boost converter.

TABLE II  Design Parameters for Solar PV System and Wind Generation System
ParameterSolar PV systemWind generation system
Converter inductance L1=45 mH L2=45 mH
Converter capacitance C1=25 mF C2=5 mF
Load R1=30 Ω R2=27 Ω
Output voltage Vpv=80 V VW=300 V
Proportional gain of voltage controller K1=0.09 K5=0.1
Integral gain of voltage controller K2=1 K6=0.55
Proportional gain of current controller K3=0.1 K7=0.1
Integral gain of current controller K4=5 K8=0.1
PWM gain Vm=4 Vm=4
Duty cycle D1=0.22 D2=0.33

The values of these gains are obtained by tuning.

The overall transfer function of solar PV system with boost converter Vo,pv(s)/Vref,pv(s) is obtained as (1).

Vo,pv(s)Vref,pv(s)=-0.0024×10-4s4+0.82s3+109s2+2796s+960000.00065s5+0.06s4+8.72s3+332s2+4256s+19200 (1)

B. Wind Generation System with Buck Converter

The block diagram of a wind generation system with buck converter is shown in Fig. 4, where the subscript WG denotes the wind generation.

Fig. 4  Block diagram of a wind generation system with buck converter.

Rectifier dynamics are neglected in this paper. The values of the gains (K5-K8) are obtained by tuning the controllers. The overall transfer function of the wind generation system Vo,WG(s)/Vref,WG(s) is derived by reducing the block diagram and substituting the parameter values from Table II and is obtained as (2).

Vo,WG(s)Vref,WG(s)=10.93s3+152.07s2+586.64s+4460.0033s5+0.06s4+34.3s3+298s2+204s+445.5 (2)

C. Fuel Cell with Boost Converter

Block diagram of a fuel cell with the boost converter is shown in Fig. 5, where the subscript FC denotes the fuel cell.

Fig. 5  Block of a fuel cell with boost converter.

The overall transfer function of the system, derived by reducing the block diagram in Fig. 5 and substituting the parameter values from Table III, is obtained as (3). The values of the proportional and integral gains of the current controller, K9 and K10, respectively, are obtained by tuning the controller.

Vo,FC(s)Iref,FC(s)=-0.00275s2+1.114s+332.80.00025s3+0.029s2+8.8s+121 (3)
TABLE III  Design Parameters for Fuel Cell and Battery System
ParameterFuel cellBattery system
Converter inductance L3=0.5 mH L4=1 mH
Converter capacitance C3=7.5 mF C4=1 mF
Load R3=10 Ω R4=10 Ω
Output voltage VFC=55 V Vbat=48 V
Proportional gain of current controller K9=0.01 K11=K12=0.01
Integral gain of current controller K10=2 K13=K14=8
PWM gain vm=4 vm=4
Duty cycle D3=0.45 D4=0.6

D. Battery Energy Storage System (BESS) with Bidirectional Converter

The BESS works in either a buck or boost mode based on the charging/discharging operation. For deriving the transfer function, each mode is considered separately.

1) BESS Operating in Buck Mode

Figure 6 shows the block diagram of BESS operating in buck mode, where the subscript B1 denotes the BESS in buck mode. K11 and K12 are the proportional and integral gains of the current controller. The overall transfer function of BESS with converter operating in buck mode, after substituting the values from Table III, is obtained as (4).

Fig. 6  Block diagram of BESS operating in buck mode.

Vo,B1(s)Iref,B1(s)=4.8s+38400.00004s3+0.048s2+50.3s+384 (4)

2) BESS Operating in Boost Mode

Figure 7 shows the block diagram of BESS operating in boost mode, where the subscript B2 denotes the BESS in boost mode. The overall transfer function of the BESS operating in boost mode, after substituting the parameter values from Table III, is obtained as (5).

Fig. 7  Block diagram of BESS operating in boost mode.

Vo,B2(s)Iref,B2(s)=-0.00048s2+0.384s+614.40.0000025s3+0.0027s2+2.9s+307.2 (5)

III. Stability Analysis of Individual DCMGs

The stability analysis of individual DCMGs is carried out by computing the eigenvalues using the state space model obtained from the transfer function of sources.

A. MG1

The transfer function of the solar PV system in (1) is used to model MG1. The transfer function is converted to the state space model and the system matrix AMG1 of MG1 is obtained as (6).

AMG1=-91.81-105.1-62.53-25.05-14.13128000006400000320000080 (6)

B. MG2

The transfer function of the wind generation system in (2) is used to model MG2, and the system matrix AMG2 is obtained as (7).

AMG2=-181.5-81.78-22.19-3.798-2.07212800000320000040000040 (7)

C. MG3

The system matrix of MG3 AMG3 is obtained using the transfer function of the fuel cell in (3) and is given in (8).

AMG3=-115.2-137.8-59.19256000320 (8)

D. MG4

In this analysis, the BESS operating in boost mode is considered. The transfer function in (5) is used for the model and the system matrix is obtained as (9).

AMG4=-1000-1123-457.810240002560 (9)

The eigenvalues of individual DCMGs shown in Fig. 8 is generated using the system matrices of the individual DCMGs. As long as the systems are stable, all their eigenvalues are located in the left half of the s-plane.

Fig. 8  Eigenvalues of individual DCMGs.

IV. System Matrix of a Clustered Network: Theoretical Evaluation

To analyze the impact of interconnection during the formation of clusters, the RLC network shown in Fig. 9 with V1 as input and V2 as output is considered.

Fig. 9  RLC network.

VL is the voltage across inductor; iC is the current through capacitor; iRL is the current through the load; RL is the load resistance; and iL is the state variable.

The state space model is given in (10). In this paper, three cases analyzed are ① adding a new network to an existing network via a distribution line, ② adding a new network in between a distribution line, and ③ adding a new network at one end of a distribution in a cluster.

diLdtdVcdt=-RL-1L1C-1RLCiLVc+1L0V1 (10)

where Vc is the voltage across capacitor.

A. Case 1: Adding a New Network to an Existing Network

This subsection analyses the modifications that need to be made to the system matrix when a new network is interconnected through the distribution line to an existing network. Figure 10 shows new RLC network added to an existing network through a distribution line. RTL and LTL are the resistance and the inductance of the distribution line, respectively; VR1 and VR2 are the voltages across the line resistances R1 and R2, respectively; and iLTL is the current through the distribution line.

Fig. 10  New RLC network added to an existing network through a distribution line.

The system matrix of the interconnected network is represented by (11).

(11)

It can be observed from a comparison of the system matrices in (10) and (11) that the system matrices of the individual networks are added diagonally. In addition, an additional row and column are introduced to accommodate the coupling due to the distribution line.

The value of the diagonal element corresponding to the additional row and column is equal to the negative of the resistance to inductance ratio of the distribution line. The magnitude of the off-diagonal elements is equal to the inverse of the inductance and capacitance of the distribution line at both ends. The paper reveals that it is possible to create a step-by-step algorithm for updating the system matrix.

Consider the graphical representation of the interconnected networks shown in Fig. 11. We can let: ① AC be the system matrix of the single network with dimension of m×m; ② AS be the system matrix of the incoming network with dimension of n×n; ③ ith row of AC correspond to the state equation of the state variable associated with the capacitor Ci connected to node i; and ④ jth row of AS correspond to the state equation of the state variable associated with the capacitor Cj connected to node j.

Fig. 11  Graphical representation of adding a new network to an existing network.

The system matrix can be updated as follows.

1) Create matrix AU1 by adding one row and one column to the system matrix AC to accommodate the elements corresponding to the distribution line.

2) Obtain AC1 by adding AS to the system matrix AU1 diagonally.

3) Update the elements corresponding to the distribution line in AC1 as follows: AC1(m+1,m+1)=-RTL/LTL; AC1(i,m+1)=-1/Ci; AC1(m+1+j,m+1)=1/Cj; AC1(m+1,i)=1/LTL; and AC1(m+1,m+1+j)=-1/LTL.

The system matrix of the interconnected system will be a (m+1+n)×(m+1+n) matrix and is given by (12).

(12)

The proposed algorithm has the advantage that, the addition of a new system to the existing cluster does not require a complete rebuilding of the system matrix.

B. Case 2: Adding a New Network in Between a Distribution Line

Figure 12 shows a new RLC network in between the distribution line, which has system 1 and system 2 connected via a distribution line TL and system 3 connected through a distribution line TL3 in between the former distribution line. V0 is the voltage assumed at the point of interconnection. The system matrix of the complete network is given by (13).The current cluster consists of two systems, system 1 and system 2, which are coupled by a distribution line running between nodes i and j. The incoming network, system 3, is connected to a point (node p) on the distribution line that links nodes i and j through a distribution line as shown in Fig. 13. It is assumed that the current of new distribution line flows from node l to node p.

Fig. 12  A new network in between a distribution line.

  (13)

  

Fig. 13  Graphical representation of adding a new network in between a distribution line.

We can let: ① AC be the system matrix of cluster with dimension of m × m; ② AS be the system matrix of the incoming network with dimension of n × n; ③ ith row of AC correspond to the state equation of the state variable associated with the capacitor Ci connected to node i; ④ jth row of AC correspond to the state equation of the state variable associated with the capacitor Cj connected to node j; ⑤ kth row of AC correspond to the state equation of the state variable associated with the inductance of line between node i and node j; and ⑥ first row of AS correspond to the state equation of the state variable associated with the capacitor C connected to node l.

The existing distribution line with resistance RTL and inductance LTL is split into two sections: ① the first section has resistance RTL1=aRTL and inductance LTL1=aLTL; and ② the second section has resistance RTL2=bRTL and inductance LTL2=bLTL, where a+b=1. The system matrix of the original cluster must be augmented with an additional row and column in order to account for this splitting. The system matrix of the interconnected system can be obtained as follows.

1) Create matrix AU1 by adding one row and one column to the system matrix AC to accommodate the elements corresponding to the distribution line that is split into two sections due to interconnection.

2) Create matrix AU2 by adding one row and one column to the system matrix AC to accommodate the elements corresponding to the new distribution line with RTL3 and LTL3.

3) Obtain AC2 by adding AS to the system matrix AU2 diagonally.

4) Update the values corresponding to the distribution line in AC2 as follows: AC2(k,k)=-RTL1/LTL1=-aRTL/(aLTL)=-RTL/LTL; AC2(k,i)=1/LTL1=1/(aLTL); AC2(k,j)=0; AC2(j,k)= 0; AC2(m+1,m+1)=-RTL2/LTL2=-bRTL/(bLTL)=-RTL/LTL; AC2(j,m+1)=-1/C2; AC2(m+1,j)=-1/LTL2=-LTL/b; AC2(m+2,m+2)=-RTL3/LTL3; AC2(m+2,m+2+l)=-1/LTL3; and AC2(m+2+l,m+2)=-1/Cl.

The dimension of the system matrix of interconnected system AC2 will be (m+2+n)(m+2+n).

C. Case 3: Adding a New Network at One End of a Distribution Line in a Cluster

The new network interconnected to one end of a distribution line in a cluster is shown in Fig. 14.

Fig. 14  New network interconnected to one end of a distribution line in a cluster.

A graphical representation of this interconnected network in Fig. 15 is used to create a step-by-step procedure for updating the system matrix whenever a new network is interconnected with an existing network at one end of a distribution line linking two networks in the cluster.

Fig. 15  Graphical representation of adding a new network at one end of distribution line.

The new network is added to the cluster formed in the first case and the assumptions are also the same as in the previous case. The system matrix can be updated as:

  

(14)

1) Create matrix AU1 by adding one row and one column to the system matrix AC to accommodate the elements corresponding to the distribution line with RTL2 and LTL2.

2) Obtain AC3 by adding AS to the system matrix AU1 diagonally.

3) Update the values corresponding to the distribution line in AC3 as follows: AC3(m+1,m+1)=-RTL1/LTL1; AC3(m+1,i)=-1/LTL1; AC3(m+1,l)=1/LTL1; AC3(i,m+1)=1/Ci; and AC3(l,m+1)=-1/Cl.

The dimension of AC3 will be (m+1+n)(m+1+n). The aforementioned analysis shows that, rather than rebuilding the entire system matrix, the system matrix of a cluster, when a new network is interconnected through a distribution line, can be obtained by updating the system matrix of the existing cluster through a simple and direct step-by-step procedure. This method can be applied to develop the system matrix of interconnected DCMGs in order to assess its stability:

(15)

V. Stability Analysis of DCMG Clusters

By interconnecting DCMGs to form a DCMG cluster as shown in Fig. 16, the reliability can be enhanced as it overcomes the issues due to storage and intermittent behavior of RESs. However, the stability of DCMGs with high penetration of converter interfaced sources and loads might be a big hurdle, particularly when interconnecting with the existing DCMG clusters. Therefore, it is crucial to investigate how the interconnection affects system stability.

Fig. 16  DCMG cluster.

The proposed SMB algorithm developed in Section IV is used to developing the system matrix of the DCMG cluster. Three different scenarios considered are: ① interconnecting two DCMGs through a distribution line; ② adding a DCMG between two buses in a distribution line; and ③ adding a new DCMG at one end of a distribution line in the DCMG cluster.

A. Scenario 1: Formation of a DCMG Cluster Interconnecting Two MGs Using a Distribution Line

A DCMG cluster interconnecting two MGs with a distribution line is shown in Fig. 17, one with a wind generation system as a source and the other with a BESS connected using a distribution line with 0.2 Ω resistance and 50 μH inductance [

19]. During the stability analysis, the reliability of the system and dynamic thermal rating (DTR) of the system that determines the optimum rating of power components based on thermal limit can be considered to obtain much more accurate results.

Fig. 17  DCMG cluster interconnecting two MGs with a distribution line.

Rather than choosing random values for cables or overhead lines with some static ratings, the selection of parameters can be done by considering the DTR. Standards like IEEE 738 describe how parameters for overhead lines are selected based on DTR. In this paper, the static rating of the cable is considered for the analysis [

32], [33].

In order to improve the precision of a DCMG analysis, all system uncertainties need to be taken into account. Climate, temperature, etc., are very significant when analyzing RESs integrating MG, as intermittent sources like wind and PV are often used in these systems [

34]-[36]. These dynamic factors are neglected as the analysis is performed from the perspective of control system.

The system matrix of the DCMG cluster, using the system matrix of the MG2 given in (7) and the system matrix of the MG4 in (9), is obtained using the algorithm developed in Section IV-A and is given in (16).

  

(16)

Using (16), the eigenvalues for DCMG cluster in scenario 1 with actual gain and optimized gain values of the controller are shown in Fig. 18. It can be observed that the eigenvalues of the cluster after interconnection migrate to the right side of the s-plane, indicating an unstable state even though MG2 and MG4 are stable independently. Therefore, it can be inferred that the presence of a distribution line has affected the interconnected system stability. Hence, the controller gains need to be modified in order to bring the interconnected system to its stable mode of operation.

Fig. 18  Eigenvalues for DCMG cluster in scenario 1 with actual gain and optimized gain values of controller.

PSO is one of the most popular algorithms used for optimal tuning the controller gains [

37], [38]. The objective function used for optimization is given in (17).

J=real(λi)<0i=1,2,,n (17)

s.t.

KjminKjKjmax (18)

where real(λi) is the real part of the ith eigenvalue; n is the number of eigenvalues in the system; Kjmin is the minimum value of the jth controller gain; and Kjmax is the maximum value of the jth controller gain.

The algorithm is implemented in MATLAB and the parameters used in the optimization algorithm are as follows: population size is 100; acceleration constants C1=C2=2; the maximum iteration is 20; the maximum inertia weight wmax=0.9; and the minimum inertia weight wmin=0.4.

The controller gains for case 1 are given in Table IV. The system matrix is given in (19) and the corresponding eigenvalue plot is given in Fig. 18. From Fig. 18, it can be observed that the eigenvalues of the tuned DCMG cluster are on the left half of the s-plane and thus the system will operate in a stable mode.

TABLE IV  Controller Gains for Case 1
Controller parameterInitial valueOptimal value
K5 0.10 0.3494
K 0.55 1.2669
K 0.10 0.4823
K8 0.10 0.1004
K13 0.01 0.0299
K14 8.00 0.6696

B.  Scenario 2: Adding a DCMG Between Two Buses in a Distribution Line

In scenario 2, the cluster is formed by integrating MG3 in the middle of the distribution line connecting two DCMGs in a cluster, as shown in Fig. 19.

  

Fig. 19  DCMG cluster interconnecting a new DCMG in between a distribution line in existing cluster.

It can be observed that the system is unstable as it has eigenvalues on the right half of s-plane.

(20)

The controller gains for scenario 2 are shown in Table V. It can be inferred from Fig. 20 that the system is now stable as all eigenvalues are on the left half of the s-plane.

TABLE V  Controller Gains for Scenario 2
Controller parameterInitial valueOptimum value
K5 0.3494 0.9716
K6 1.2669 1.7434
K7 0.4823 1.4560
K8 0.1004 0.0207
K9 0.0100 0.0641
K10 2.0000 1.3629
K13 0.0299 0.0094
K14 0.6696 1.2990

Fig. 20  Eigenvalues for DCMG cluster in scenario 2 with actual and optimized gain values of controller.

C. Scenario 3: Adding a New DCMG at One End of a Distribution Line in DCMG Cluster

The cluster in scenario 3 is obtained by interconnecting MG1 at one end of a distribution line in a DCMG cluster, as shown in Fig. 21.

Fig. 21  DCMG cluster interconnecting a new DCMG at one end of distribution line.

The system matrix of the DCMG cluster after interconnection is obtained as (21). The eigenvalue plot of the system is presented in Fig. 22. As in previous cases, here also, the system is unstable due to the presence of eigenvalues in the right half of s-plane. The controller gains for scenario 3 are given in Table VI.

Fig. 22  Eigenvalues for DCMG cluster in scenario 3 with actual and optimized gain values of controller.

Table VI  Controller Gains for Scenario 3
Controller parameterInitial valueOptimal valueController parameterInitial valueOptimal value
K1 0.0900 0.0298 K6 1.2669 0.2033
K2 1.0000 0.0927 K7 0.4823 1.1945
K3 0.1000 0.1516 K8 0.1004 0.0664
K4 5.0000 7.6143 K13 0.0299 0.0892
K5 0.3494 0.2736 K14 0.6696 0.1241

Figure 22 shows that the system became stable after tuning the controller gains.

The stability of a DCMG cluster during interconnection, considering the impact of the distribution line, is examined in this investigation. Additionally, it is demonstrated how beneficial is the controller gain adjustment in maintaining the stability of DCMG cluster.

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D. Variation of Eigenvalues with Different Controller Gains

Every time, tuning the controller gains brings the unstable cluster to the stable operation mode. It implies that the system eigenvalues move from the right half of s-plane to the left during optimization. Figure 23 shows the variation of eigenvalues for different controller gains. It can be observed that the eigenvalues of the interconnected system move from the right half of s-plane to the left half of s-plane, making the interconnected system stable.

Fig. 23  Variation of eigenvalues for different controller gains.

VI. Simulation Results

The step response of the system is assessed to verify the proposed SMB algorithm. To determine the system stability, the step response of the individual DCMGs is first assessed separately. The step response for the cluster is then assessed after the stable DCMGs are interconnected through a distribution line.

The step response of the two independent DCMGs (MG2 and MG4) is given in Section III, and the response indicates that the individual DCMGs are stable and are shown in Fig. 24.

Fig. 24  Step response of individual DCMGs (MG2 and MG4).

The DCMG cluster is formed by interconnecting MG2 and MG4, which are stable while operating independently. The general block diagram representation of the DCMG cluster with the MGs interconnected using a distribution line is is given in Fig. 25.

Fig. 25  Block diagram representation of DCMG cluster with MGs interconnected using a distribution line.

The block diagram includes a transfer function model for both MGs and a distribution line connecting the two systems. The step response of the cluster with initial gain values is shown in Fig. 26. It can be observed that the response deviates significantly and the cluster becomes unstable due to the presence of a distribution line in the cluster. PSO algorithm is employed to fine-tune the controller gains in order to stabilize the system. As a result, as illustrated in Fig. 27, the unstable cluster is stabilized with the modified optimal gains and the response settles without any steady state error. The results obtained during the simulation studies are validated using real-time analysis.

Fig. 26  Step response of cluster with initial gain values.

Fig. 27  Step response of cluster with optimal gain values.

VII. Experimental Study: Real-time Analysis

The experimental set-up using OP4510 real-time simulator is used to experimentally validate the proposed SMB algorithm in real time and is shown in Fig. 28. For the validation, the DCMG cluster of case 1 shown in Fig. 25 is considered. The system parameters used in the paper are listed in Table IV.

Fig. 28  Experimental setup for real-time analysis.

The step response of the individual DCMGs with the initial gain values is shown in Fig. 29 and it can be observed that both the MGs are operating stably. The response of the DCMG cluster after being interconnected through a distribution line is shown in Fig. 30. It is evident that the resulting DCMG cluster is unstable. The best method for stabilizing the resulting DCMG cluster is to re-tune the controller gains in the new cluster. The response of the DCMG cluster with the controller gains tuned using PSO is shown in Fig. 31. It is clear that the response of the DGMG cluster with optimally tuned controller gains is stable. These outcomes vouch for the viability of the proposed SMB algorithm for stability analysis of DCMG cluster.

Fig. 29  Step response of individual DCMGs (MG2 and MG4) with initial gain values.

Fig. 30  Step response of DCMG cluster with initial gain values.

Fig. 31  Step response of DCMG cluster with controller gains tuned using PSO.

VIII. Conclusion

In this paper, modeling and stability problems in DCMG clusters are examined. Three possible cases are considered in this paper: ① adding a new network to an existing network, ② adding a new network in between a distribution line, and ③ adding a new network at one end of a distribution line in a cluster. A step-by-step programmable technique has been developed to update the system matrix of the DCMG cluster after interconnection.

With the developed method, adding a new network to an existing network has the benefit of not necessitating a total rebuilding of the system matrix.

Even though the extant DCMG network and the incoming DCMG are both stable, it has been found that, while interconnecting through a distribution line, the system stability is affected. To stabilize the new cluster, PSO is applied to modify the controller gains. The response of the interconnected DCMG with optimally tuned controller gains indicates that stable operation is ensured. In addition, the novel step-by-step SMB algorithm can be adapted to build the system matrix of the interconnected system to perform the stability analysis. The simulation and real-time results prove the significance of the proposed SMB algorithm.

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