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).
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 [
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 [
Using eigenvalue observation, a state space model of the DCMG is created in [
The stability criterion evaluates and guarantees the system stability under significant disturbance in the design phase rather than repeatedly computing and simulating. In [
Reference [
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.
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
Converter | ||
---|---|---|
Buck | ||
Boost |

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 [
The solar PV system with boost converter can be represented by the block diagram as shown in

Fig. 3 Block diagram of solar PV system with boost converter.
Parameter | Solar PV system | Wind generation system |
---|---|---|
Converter inductance | mH | mH |
Converter capacitance | mF | mF |
Load | Ω | Ω |
Output voltage | V | V |
Proportional gain of voltage controller | ||
Integral gain of voltage controller | ||
Proportional gain of current controller | ||
Integral gain of current controller | ||
PWM gain | ||
Duty cycle |
The values of these gains are obtained by tuning.
The overall transfer function of solar PV system with boost converter is obtained as (1).
(1) |
The block diagram of a wind generation system with buck converter is shown in

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 is derived by reducing the block diagram and substituting the parameter values from
(2) |
Block diagram of a fuel cell with the boost converter is shown in

Fig. 5 Block of a fuel cell with boost converter.
The overall transfer function of the system, derived by reducing the block diagram in
(3) |
Parameter | Fuel cell | Battery system |
---|---|---|
Converter inductance | mH | mH |
Converter capacitance | mF | mF |
Load | Ω | Ω |
Output voltage | V | V |
Proportional gain of current controller | ||
Integral gain of current controller | ||
PWM gain | ||
Duty cycle |
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.

Fig. 6 Block diagram of BESS operating in buck mode.
(4) |

Fig. 7 Block diagram of BESS operating in boost mode.
(5) |
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.
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).
(6) |
The transfer function of the wind generation system in (2) is used to model MG2, and the system matrix is obtained as (7).
(7) |
The system matrix of MG3 is obtained using the transfer function of the fuel cell in (3) and is given in (8).
(8) |
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).
(9) |
The eigenvalues of individual DCMGs shown in

Fig. 8 Eigenvalues of individual DCMGs.
To analyze the impact of interconnection during the formation of clusters, the RLC network shown in

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.
(10) |
where Vc is the voltage across capacitor.
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.

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 Graphical representation of adding a new network to an existing network.
The system matrix can be updated as follows.
1) Create matrix by adding one row and one column to the system matrix to accommodate the elements corresponding to the distribution line.
2) Obtain by adding to the system matrix diagonally.
3) Update the elements corresponding to the distribution line in as follows: ; ; ; ; and .
The system matrix of the interconnected system will be a 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.

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
The existing distribution line with resistance RTL and inductance LTL is split into two sections: ① the first section has resistance and inductance ; and ② the second section has resistance and inductance , where . 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 by adding one row and one column to the system matrix to accommodate the elements corresponding to the distribution line that is split into two sections due to interconnection.
2) Create matrix by adding one row and one column to the system matrix to accommodate the elements corresponding to the new distribution line with and .
3) Obtain by adding to the system matrix diagonally.
4) Update the values corresponding to the distribution line in as follows: ; ; ; ; ; ; ; ; ; and .
The dimension of the system matrix of interconnected system will be .
The new network interconnected to one end of a distribution line in a cluster is shown in

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 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 by adding one row and one column to the system matrix to accommodate the elements corresponding to the distribution line with and .
2) Obtain by adding to the system matrix diagonally.
3) Update the values corresponding to the distribution line in as follows: ; ; ; ; and .
The dimension of will be . 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) |
By interconnecting DCMGs to form a DCMG cluster as shown in

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 DCMG cluster interconnecting two MGs with a distribution line is shown in

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 [
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 [
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 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 [
(17) |
s.t.
(18) |
where real() is the real part of the eigenvalue; is the number of eigenvalues in the system; is the minimum value of the controller gain; and is the maximum value of the 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 ; the maximum iteration is 20; the maximum inertia weight ; and the minimum inertia weight .
The controller gains for case 1 are given in
Controller parameter | Initial value | Optimal 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
Controller parameter | Initial value | Optimum 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.
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 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 Eigenvalues for DCMG cluster in scenario 3 with actual and optimized gain values of controller.
Controller parameter | Initial value | Optimal value | Controller parameter | Initial value | Optimal value |
---|---|---|---|---|---|
0.0900 | 0.0298 | 1.2669 | 0.2033 | ||
1.0000 | 0.0927 | 0.4823 | 1.1945 | ||
0.1000 | 0.1516 | 0.1004 | 0.0664 | ||
5.0000 | 7.6143 | 0.0299 | 0.0892 | ||
0.3494 | 0.2736 | 0.6696 | 0.1241 |
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.
![]() | (21) |
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.

Fig. 23 Variation of eigenvalues for different controller gains.
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 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 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 Step response of cluster with initial gain values.

Fig. 27 Step response of cluster with optimal gain values.
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 Experimental setup for real-time analysis.
The step response of the individual DCMGs with the initial gain values is shown in

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.
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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