Mathematical Model and Stability Analysis of Inverter-Based Distributed Generator

Abstract

This paper presents a mathematical (small-signal) model of an electronically interfaced distributed generator (DG) by considering the effect of voltage and frequency variations of the prime source. Dynamic equations are found by linearization about an operating point. In this study, the dynamic of DC part of the interface is included in the model. The stability analysis shows with proper selection of system parameters; the system is stable during steady-state and dynamic situations, and oscillatory modes are well damped. The proposed model is useful to study stability analysis of a standalone DG or a Microgrid.

1. Introduction

Distributed generation (DG) systems have been expected to be an important electric power supply system for next generation. DGs are able to be installed near the loads, so they can increase the power quality and reliability of electricity delivered to sensitive loads. Some of the DG technologies require a power electronics interface in order to convert the energy into the grid compatible AC power. These interface devices make the sources more flexible in their operation and control compared to the conventional electrical machines. However, due to their negligible physical inertia, they also make the system potentially susceptible to oscillation resulting from network disturbances [1].

The coordinated operation and control of DGs together with loads and storage devices are central to the concept of microgrid [2]. The analysis of the dynamic stability of conventional power systems is well established, but for microgrid there is a need to investigate how circuit and control features give rise to particular oscillatory modes, and which of these have poor damping. Finding an exact mathematical model by considering DGs and their control is needed to investigate dynamic stability of the microgrid under transient events such as islanding from main grid and small-signal deviation like slow changing in load.

Reference [3] presented a small-signal model for inverter in stand-alone AC supply system by using only the droop controller's variables as state variables of the DG model. An averaged current source model has been suggested for the converter in [4]. In this investigation, the high frequency converter current dynamics have been neglected in order to focus exclusively on the dynamics and control of the islanded microgrid. Regardless of the type of the DG, an equivalent RLC circuit including output filter and transformers impedances has been modeled as a power circuit of DG in most of the literatures [1, 5–8]. In these cases, output currents and voltages are considered as state variables. References [1, 6, 7] by adding controller equations to the DG system made an accurate model for small signal stability analysis of a microgrid.

In most articles, DC part of the DG (DC filter’s voltage and current) is not considered in dynamic studies, or it is assumed to be a constant value [6], so its effect is neglected in stability analysis. In the reference [8], the input DC voltage variation of the inverter has been represented as the external perturbation in the open-loop model of the DG, but it has been set as a constant during control loop design and frequency-domain analysis. The DC voltage has been related to the input and output equations of electronically interface of the DG by using switching function of rectifier and inverter in [5]. Moreover, two types of models have been presented in this paper for the prime source of the DG. While the proposed models have been used in steady-state and load flow analysis, but their linear forms can be used also in dynamic studies.

The objective of this paper is to find a comprehensive dynamic model of DG, including the prime source, power electronically interfaces, output filter, and controller. The proposed model represents all components of the DG in a dq0 reference frame, thus it ensures any application such as steady-state and dynamic analysis that meets requirements and constraints of both AC and DC parts of the system.

The dynamic stability of the DG is investigated by the small signal and step response analysis. In this paper, the DG is connected only to the static load, but it can be extended into the network or typical microgrid for more applications.

2. System Studied

2.1. Power Circuit

A simplified single-line diagram of the studied DG is illustrated in Figure 1. DG is included of a three-phase source, a three-phase rectifier, an inverter, and a three-phase static load on the output bus. The DG source can be representatives of wind turbine or microturbine generator for medium and low voltage levels, respectively. Regardless of the power measurement and control loops, the system shown in Figure 1 can be studied as a power circuit of DG in microgrid stability analysis.

Figure 1 

Single line equivalent of DG.

2.2. Control System

As shown in Figure 1, the control of DG included power, voltage, and current controllers. The power control has been achieved by applying droop control for both active and reactive powers. The droop control for an electronically interfaced DG has the role of the governor for a synchronously generator. Where, by an increase in load, the reference frequency is decreased. In the similar way, the reactive power is controlled by a droop characteristic in the voltage magnitude.

In this study, the sin-triangle modulation strategy is used to prepare applied voltage in the converter. The next parts of the control system are the outer voltage and inner current control loops, which are designed to reject high-frequency disturbances and provide sufficient damping for the output filter by using the proportional integral (PI) compensators.

3. System Equations

3.1. Power Circuit

The dynamic model of AC side of DG in three-phase abc frame is obtained from 𝑣_{𝑖𝑎𝑏𝑐}=𝑅_𝐺2𝑖_𝑖^𝐺2_𝑎𝑏𝑐+𝐿_𝐺2𝜌𝑖_𝑖^𝐺2_𝑎𝑏𝑐+𝑣^𝐺2_𝑎𝑏𝑐, (1) where 𝑖_𝑖^𝐺2_𝑎𝑏𝑐, 𝑣^𝐺2_𝑎𝑏𝑐, and 𝑣_{𝑖𝑎𝑏𝑐} are vectors of the instantaneous amounts of the output currents, bus voltages, and the converter output voltages, respectively. Then (1) is transformed to the 𝑑-𝑞 reference frame of DG that rotates at frequency 𝜔_𝐺2 related to the angular velocity of the voltage space vector of the bus. The transferred equations are 𝐸₁^𝐺2𝜌𝑖^𝐺2_𝑞𝑑𝑖=𝐴₁^𝐺2𝑖^𝐺2_𝑞𝑑𝑖+𝑣^𝐺2_𝑞𝑑𝑖−𝑣^𝐺2_𝑞𝑑, (2) where 𝜌 is the operator 𝑑/𝑑𝑡.

3.2. DC Interface Equations

The average voltage and current of capacitor and inductance of dc filter will change with time during transient. For the rectifier under operation without phase delay and with commutating inductance, we have [9] 𝑉_𝑑^𝐺2=3√3𝜋𝐸_𝑞^𝐺2−3𝜋𝑋_{𝐿𝐶𝐺2}𝜔_𝑏𝜔_𝐸𝐼_𝑑^𝐺2−2𝜔_𝑏𝑋_{𝐿𝑐𝐺2}𝜌𝐼_𝑑^𝐺2. (3) We also have 𝑋_{𝐿𝑑𝑐𝐺2}𝜌𝐼_𝑑^𝐺2=𝑉_𝑑^𝐺2−𝑉^𝐺2_𝑑𝑐. (4) From (3) and (4), 𝑉^𝐺2_𝑑𝑐=3√3𝜋𝐸_𝑞^𝐺2−3𝜋𝑋_{𝐿𝑐𝐺2}𝜔_𝑏𝜔_𝐸𝐼_𝑑^𝐺2−1𝜔_𝑏𝑋_{𝐿𝑑𝑐𝐺2}+2𝑋_{𝐿𝑐𝐺2}𝜌𝐼_𝑑^𝐺2. (5) Equation (5) expresses the relationship between dynamic parts of the dc filter and DG source voltage. The 𝜔_𝐸 is the frequency of the source, and 𝐸_𝑞^𝐺2 is the network side voltage’s 𝑞 component. The voltage is wrote in the form of 𝜌𝑉^𝐺2_𝑑𝑐=𝜔_𝑏𝑋_{𝑐𝑑𝑐𝐺2}𝐼_𝑑^𝐺2−𝐼^𝐺2_𝑑𝑐. (6) Apart from the harmonics in the voltage waveform and considering sin-triangle modulation for inverter, voltages become 𝑣_𝑞^𝐺2_𝑖=𝑑_𝐺22𝑉^𝐺2_𝑑𝑐,𝑣_𝑑^𝐺2_𝑖=0, (7) where 𝑑_𝐺2 is duty cycle.

We will see that the 𝑞 component of converter voltage is controlled by controller and 𝑣^𝐺2_𝑑𝑖 is considered to be zero. The instantaneous power balance between two sides of the inverter is established by 𝑉^𝐺2_𝑑𝑐𝐼^𝐺2_𝑑𝑐=32𝑣^𝐺2_𝑞𝑖𝑖^𝐺2_𝑞𝑖. (8) According to (7), we have 𝐼^𝐺2_𝑑𝑐=3𝑑_𝐺24𝑖^𝐺2_𝑞𝑖. (9) Rewriting (6) with regard to (9), 𝜌𝑉^𝐺2_𝑑𝑐=𝜔_𝑏𝑋_{𝐶𝑑𝑐𝐺2}𝐼_𝑑^𝐺2−3𝑑_𝐺24𝑖^𝐺2_𝑞𝑖. (10) The dynamic equations of DC part are represented by (5) and (10).

3.3. Control System Equations

Figure 2 shows inverter power controller, where 𝑃 and 𝑄 are measured values of inverter active and reactive output powers. Moreover, 𝑃 and 𝑄 are filtered values of power after passing through the low-pass filter. As shown in Figure 3, the inverter output voltage 𝑣_𝑖^𝐺2 and inverter output frequency 𝜔_𝐺2 are adjusted by the droop controller characteristics described by (11) and (12), respectively, as follows: 𝜔_𝐺2=𝜔_𝑛−𝑘_𝑝𝑣𝑃,_{𝐺2∗𝑞𝑖}=𝑣_𝑛−𝑘_𝑣𝑄. (11)(12) Here, 𝜔_𝑛 and 𝑣_𝑛 represent the nominal frequency and 𝑞-axis output voltage set points, respectively. 𝑣^𝐺2∗_𝑞𝑖 is the duty cycle (𝑑_𝐺2).

Figure 2 

Inverter power controller.

Figure 3 

Common reference frame.

The power measurement outputs are given by (13) and (14), where 𝜔_𝑓 is the cutoff frequency of the low-pass filter and 𝑠 is the Laplace operator 𝜔𝑃=^𝑓𝑠+𝜔_𝑓𝜔𝑃,𝑄=^𝑓𝑠+𝜔_𝑓𝑄. (13)(14)

3.4. Common Reference Frame Theory

All equations of DG have been written in its reference frame. In power system analysis, it is necessary to relate all components’ reference frames to a common frame. This common reference frame would be represented as constant voltages in the synchronous reference frame. In microgrid studies, DG with the biggest nominal power is considered as a common reference frame.

Figure 3 shows the common reference frame of a microgrid by 𝑞 and 𝑑 axis rotating at the synchronous angular velocity of 𝜔_𝑒. The 𝑞 axis is in orientation of the voltage vector of the reference DG, 𝑣^𝐺_𝑞. By assuming 𝑣_𝑞^𝐺2 on the 𝑞 axis of the DG2 reference frame, 𝑞^𝐺2, 𝛿_𝐺2 is the angle between the DG2 reference frame and common reference frame. In order to transfer variables in the DG2 reference frame to common reference frame, the transformation matrix is defined as 𝑓^𝐺=𝑇_𝐺2𝑓^𝐺2,𝑓^𝐺2=𝑇⁻¹_𝐺2𝑓^𝐺,𝑇_𝐺2=cos𝛿⁰_𝐺2sin𝛿⁰_𝐺2−sin𝛿⁰_𝐺2cos𝛿⁰_𝐺2𝑇_−1𝐺2,=cos𝛿⁰_𝐺2−sin𝛿⁰_𝐺2sin𝛿⁰_𝐺2cos𝛿⁰_𝐺2. (15)(16)(17) To connect an inverter to the whole system the output variables that have the connection with the network need to be converted to the common reference frame. In this case the output variables are the output currents 𝑖^𝐺2_𝑞𝑑𝑖 and the network side voltages 𝑣^𝐺2_𝑞𝑑. Because the dynamic of one DG is studied in this paper, the 𝛿_𝐺2 is zero, and transformation matrix is the unit.

3.5. Small-Signal Model

By liberalization of nonlinear differential equations around the operating steady-state point, the small-signal model of the system has been created [9].

For inverter equations, first output variables must be transferred to the common reference frame and then be linearized. By using (15) inverter output currents and voltages in the common reference frame can be achieved as 𝑖^𝐺2_𝑞𝑑𝑖=𝑇⁻¹_𝐺2𝑖^𝐺_𝑞𝑑𝑖,𝑣^𝐺2_𝑞𝑑=𝑇⁻¹_𝐺2𝑣^𝐺_𝑞𝑑. (18) Then with linearization technique, we have Δ𝑖^𝐺2_𝑞𝑑𝑖=𝑇⁻¹_𝐺2Δ𝑖^𝐺_𝑞𝑑𝑖+𝑖⁰_𝛿𝐺2Δ𝛿_𝐺2,Δ𝑣^𝐺2_𝑞𝑑=𝑇⁻¹_𝐺2Δ𝑣^𝐺_𝑞𝑑+𝑣⁰_𝛿𝐺2Δ𝛿_𝐺2, (19) where 𝑖⁰_𝛿𝐺2=−𝑖^𝐺2_𝑑𝑖0𝑖^𝐺2_𝑞𝑖0^𝑇,𝑣⁰_𝛿𝐺2=−𝑣^𝐺2_𝑑0𝑣^𝐺2_𝑞0^𝑇. (20) The subscript 0 denotes steady-state quantities. The Δ𝛿_𝐺2 could be rewritten as a function of output voltages in common reference frame as follows 𝛿_𝐺2=tan⁻¹𝑣^𝐺_𝑑𝑣^𝐺_𝑞. (21) With linearization of (21), Δ𝛿_𝐺2=𝑚_{𝑞𝛿𝐺2}Δ𝑣^𝐺_𝑞+𝑚_{𝑑𝛿𝐺2}Δ𝑣^𝐺_𝑑, (22) where 𝑚_{𝑞𝛿𝐺2}𝑣=−^𝐺𝑑0𝑣^𝐺2_𝑞0+𝑣^𝐺2_𝑑0,𝑚_{𝑑𝛿𝐺2}=𝑣^𝐺_𝑞0𝑣^𝐺2_𝑞0+𝑣^𝐺2_𝑑0. (23) For more information, see [3, 7]. Furthermore, the linear equation of the inverter output voltage could be written as Δ𝑣_𝑞^𝐺2_𝑖=𝑑_𝐺202Δ𝑉^𝐺2_𝑑𝑐+𝑉^𝐺2_𝑑𝑐02Δ𝑑_𝐺2. (24) By replacing (7) and (18) into the inverter equations and linearization with considering (19), (22), and (24) the linear equations of the DG have been arranged in the basic style of the linear differential equations as 𝜌Δ𝑋^DG=𝐴^DGΔ𝑋^DG+𝐵^DGΔ𝑈^DG,Δ𝑋^DG=Δ𝑋^𝐺2Δ𝑋_𝐶^𝐺2,Δ𝑈^DG=Δ𝑈^𝐺2Δ𝑈_𝐶^𝐺2, (25)(26) where Δ𝑈^DG and Δ𝑋^DG are control and state variables respectively. The state matrix of the system is expressed by 𝐴^DG that is used in dynamic analysis of the DG.

The output vector of the system is defined as a linear combination of control and state variables in the form of Δ𝑌^DG=𝐶Δ𝑋^DG+𝐷Δ𝑈^DG. (27) In the power system small signal stability analysis, the current injection into the network from the device is considered as the output signal [10]. For the DG, the output current is one of the state variables, so we have Δ𝑖^𝐺_𝑞𝑑𝑖=𝐶^DGΔ𝑋^DG. (28) All matrices are available in Appendix A in detail.

Finally, the transfer function is defined by Δ𝑌^DGΔ𝑈^DG=𝐶^DG𝑆𝐼−𝐴^DG−1𝐵^DG+𝐷^DG. (29)

3.6. Eigenvalue Analysis

In power systems small-signal stability analysis is aimed to determine the properties of operation parameter variations that are independent from disturbance intensity. Eigenvalue analysis is used to show the information of different stability modes for power system small-signal stability problems. A system is stable when all of its modes are stable. Furthermore, it is required that all oscillations are well and quickly damped. This technique is declared in [11] in details.

In this paper, the participation factors and eigenvalues of the state matrix are computed and analyzed. Eigenvalues with negative real parts show that the system is stable. the state variable with the highest normalized participation factor is the best choice for feedback signal if it may be a measurable physical variable.

4. Case Study and Results

To study small-signal stability of the model, a 10 kVA DG connected to a 9 kW static load is considered as the case study. All DG information is taken from [1]. The output impedance of DG is 0.1+𝑗0.4239Ω. The cutoff frequency 𝜔_𝑓 is considered 31.41 Rad/s.

The rectifier’s input inductance 𝑋_𝐿𝑐 is used to amend the input current. The inductor with low value may not restrict the peak current, and its high amount may reduce the input voltage [10]. Hence, a 1.06 (mH) inductor is chosen as an optimized value.

The DC filter is involved in a small series inductance, 𝑋_𝐿𝑑𝑐, (here 0.03 mH) and a shunt capacitance, 𝑋_𝑐𝑑𝑐, to smooth the diode rectifier output. With regard to [10], a 165 μF/kW capacitance is selected in this study. For DG, we have Power=10kW. (30) Hence 𝐶=165∗100=1650𝜇F. (31) The steady-state values of the system are calculated by load flow program.

In order to realize the sensitivity and dynamic behaviour, eigenvalues trajectory plots are drawn as a function of some system parameters. Figure 4 shows the locations of eigenvalues as a function of variation of 𝑘_𝑝 from 0.00000004 to 0.000094 with the rate of 0.000001. In this case, 𝑘_𝑣 is equal to 0.0013.

Figure 4 

Loci of eigenvalues according to kp changing.

Figure 5 shows the loci of eigenvalues corresponding to variation of 𝑘_𝑣 from 0.00003 to 0.1 with the rate of 0.001. 𝑘_𝑝 is considered to be 0.000015.

Figure 5 

Loci of eigenvalues according to kv changing.

Trajectory plots show instability in modes of 1, 2, 3, and 4 if the 𝑘_𝑝 and 𝑘_𝑣 increase to big values.

With regard to 𝑘_𝑝 and 𝑘_𝑣 as constant values of 0.000015 and 0.0013, Figure 6 shows the root locus plot for the model as a function of the filter cutoff frequency 𝜔_𝑓 from 37 to 25 with the rate of −0.5. It can be seen that eigenvalues 5 and 6 move to right side of the plot.

Figure 6 

Loci of eigenvalues according to wf changing.

Results are used to adjust the gains of the power controller. With optimal values of controller gains and appropriate selection of the DG parameters, all oscillatory modes are well damped, and the proposed model is stable.

By using QR-decomposition method in the MATLAB software environment, the eigenvectors and eigenvalues of the state matrix have been computed in operating point. The DG system along with its controller is described by six eigenvalues: 𝜆_1,2=−44.90±698.37𝑖, 𝜆_3,4=−31.14±244.66𝑖,𝜆₅=−156.58, and 𝜆₆=−32.62.

The normalized participation factors of the dynamic model are represented in Table 1. It is a useful indication of the role of a state on the mode. As seen in oscillatory modes of 1, 2, 3, and 4, the state variables of DC part have the main role.

Table 1 

Participation factors.

Figures 7 and 8 show step responses of the output current according to the source voltage and frequency. The small steady-state error in 𝑖_𝑞𝑖 is caused by the effects of them on DC voltage and DC current. The variation in 𝑖_𝑑𝑖 is due to coupling between 𝑞 and 𝑑 components of current. By control of rectifier and making a constant DC voltage this effects will be removed. Error specifications also could be adjusted by adding current and voltage control loops.

Figure 7 

Step response of current to the source voltage.

Figure 8 

Step response of current to the source angular velocity.

5. Conclusion

The mathematical model of the inverter-based DG has been proposed in this paper. The proposed model has included power and control circuits of the DG. The dynamics of the DC filter and the effects of the prime source have been also considered in the DG model. By using eigenvalue analysis, the stability of proposed model has been studied in detail. This study presented that all oscillatory modes are well damped with suitable selection of system parameters. Since the proposed model is stable, it can be used in the dynamic and steady-state study of a microgrid.

Appendices

Linear equations of the DG.

A. Power Circuit

A.1. AC Part

By substituting (18) into the (2) and linearization 𝐸₁^𝐺2Δ𝑖̇^𝐺_𝑞𝑑𝑖=𝐴₁^𝐺2Δ𝑖^𝐺2_𝑞𝑑𝑖+𝐵₁^𝐺2Δ𝑣^𝐺2_𝑞𝑑𝑖+𝐵₂^𝐺2Δ𝑣^𝐺_𝑞𝑑+𝐵₃^𝐺2Δ𝜔_𝐺2, (A.1) where 𝐸₁^𝐺2=𝑋_𝑙𝐺2𝜔_𝑏00𝑋_𝑙𝐺2𝜔_𝑏,𝐴₁^𝐺2=−𝑅_𝐺2−𝜔_𝐺20𝑋_𝑙𝐺2𝜔_𝑏𝜔_𝐺20𝑋_𝑙𝐺2𝜔_𝑏−𝑅_𝐺2,𝐵₁^𝐺2=𝑇_𝐺210,𝐵₂^𝐺2=𝑇_𝐺2𝐴₁^𝐺2𝑖⁰_𝛿𝐺2−𝑇_𝐺2𝑣⁰_𝛿𝐺2𝑀^𝐺2_𝑞𝑑,𝐵−𝐼_3𝐺2=−𝑇_𝐺2𝐸₁^𝐺2𝑖⁰_𝛿𝐺2,𝑀^𝐺2_𝑞𝑑=𝑚_{𝑞𝛿𝐺2}𝑚_{𝑑𝛿𝐺2},Δ𝜔_𝐺2=Δ𝛿̇_𝐺2. (A.2) By (23) and Δ𝑣_𝑑^𝐺2_𝑖=0𝐸₁^𝐺2Δ𝑖̇^𝐺_𝑞𝑑𝑖=𝐴₁^𝐺2Δ𝑖^𝐺_𝑞𝑑𝑖+𝐵₁^𝐺2𝑑_𝐺202Δ𝑉^𝐺2_𝑑𝑐+𝐵₁^𝐺2𝑉^𝐺2_𝑑𝑐02Δ𝑑_𝐺2+𝐵₂^𝐺2Δ𝑣^𝐺_𝑞𝑑+𝐵₃^𝐺2Δ𝜔_𝐺2,𝐸₁^𝐺2Δ𝑖̇^𝐺_𝑞𝑑𝑖=𝐴₁^𝐺2Δ𝑖^𝐺_𝑞𝑑𝑖+𝐹₁^𝐺2Δ𝑉^𝐺2_𝑑𝑐Δ𝐼_𝑑^𝐺2+𝐵₇^𝐺2Δ𝑑_𝐺2+𝐵₂^𝐺2Δ𝑣^𝐺_𝑞𝑑+𝐵₃^𝐺2Δ𝜔_𝐺2,𝐹₁^𝐺2=𝐵₁^𝐺2𝑑_𝐺20210,𝐵₇^𝐺2=𝐵₁^𝐺2𝑉^𝐺2_𝑑𝑐02. (A.3)

A.2. DC Part

By rearranging (5) and (10) in the frame linear differential equation 𝐸₂^𝐺2𝜌Δ𝑉^𝐺2_𝑑𝑐Δ𝐼_𝑑^𝐺2=𝐴^𝐺2₂Δ𝑉^𝐺2_𝑑𝑐Δ𝐼_𝑑^𝐺2+𝐴₃^𝐺2Δ𝑖^𝐺2_𝑞𝑑𝑖+𝐵₄^𝐺2Δ𝑑_𝐺2+𝐵₅^𝐺2Δ𝐸^𝐺2Δ𝜔_𝐸,𝐸₂^𝐺2=,𝐴1001_𝐺22=3𝑑_𝐺20𝑖^𝐺2_𝑞𝑖0𝑋_{𝐶𝑑𝑐𝐺2}𝜔_𝑏4𝑉^𝐺2_𝑑𝑐0𝑋_{𝐶𝑑𝑐𝐺2}𝜔_𝑏−1𝑍₂−3𝑋_{𝐿𝑐𝐺2}𝜔_𝐸0𝜋𝑍₂𝜔_𝑏,𝐴^𝐺2₃=−3𝑑_𝐺20𝑋_{𝐶𝑑𝑐𝐺2}𝜔_𝑏40,𝐵00_𝐺24=−3𝑖^𝐺2_𝑞𝑖0𝑋_{𝐶𝑑𝑐𝐺2}𝜔_𝑏2𝑉^𝐺2_𝑑𝑐00,𝐵^𝐺2₅=3√003𝜋𝑍₂−3𝐼_𝑑^𝐺2₀𝑋_{𝐿𝑐𝐺2}𝜋𝑍₂𝜔_𝑏,𝑍₂=1𝜔_𝑏𝑋_𝐿𝑑𝑐+2𝑋_𝐿𝑐. (A.4) By replacing (19) and (22) into the (A.4), 𝐸₂^𝐺2𝜌Δ𝑉^𝐺2_𝑑𝑐Δ𝐼_𝑑^𝐺2=𝐴₂^𝐺2Δ𝑉^𝐺2_𝑑𝑐Δ𝐼_𝑑^𝐺2+𝐴₄^𝐺2Δ𝑖^𝐺_𝑞𝑑𝑖+𝐵₄^𝐺2Δ𝑑_𝐺2+𝐵₅^𝐺2Δ𝐸^𝐺2Δ𝜔_𝐸+𝐵₆^𝐺2Δ𝑣^𝐺_𝑞𝑑,𝐴^𝐺2₄=𝐴^𝐺2₃𝑇_𝐺2⁻¹,𝐵^𝐺2₆=𝐴^𝐺2₃𝑖⁰_𝛿𝐺2𝑀^𝐺2_𝑞𝑑. (A.5)

A.3. Differential Equation of Power Circuit

𝐸^𝐺2Δ𝑋̇^𝐺2=𝐴^𝐺2Δ𝑋^𝐺2+𝐵^𝐺2Δ𝑈^𝐺2+𝐶^𝐺2Δ𝑋_𝐶^𝐺2,Δ𝑋^𝐺2=Δ𝑖^𝐺2_𝑞𝑖Δ𝑖^𝐺2_𝑑𝑖Δ𝑣^𝐺2_𝑑𝑐Δ𝐼_𝑑^𝐺2,𝐸^𝐺2=𝐸₁^𝐺200𝐸₂^𝐺2,𝐵^𝐺2=𝐵₂^𝐺20𝐵₆^𝐺2𝐵₅^𝐺2,𝐶^𝐺2=𝐵₃^𝐺2𝐵₇^𝐺20𝐵₄^𝐺2,𝐴^𝐺2=𝐴₁^𝐺2𝐹₁^𝐺2𝐴₄^𝐺2𝐴₂^𝐺2. (A.6)

B. Control Circuit

B.1. Active Power Droop Controller

Linear form of (11) and (13), Δ𝜔_𝐺2=−𝑘_𝑝ΔΔ𝜔𝑃,𝑃=^𝑓𝑠+𝜔_𝑓Δ𝑃. (B.1)(B.2) By combining (B.1) and (B.2), 𝑠Δ𝜔_𝐺2=−𝜔_𝑓Δ𝜔_𝐺2−𝑘_𝑝𝜔_𝑓Δ𝑃. (B.3)

B.2. Reactive Power Droop Controller

In the same way of the active power control, from (12) and (14) we have 𝑠Δ𝑑_𝐺2=−𝜔_𝑓Δ𝑑_𝐺2−𝑘_𝑣𝜔_𝑓Δ𝑄. (B.4)

B.3. Differential Equation of Control Circuit

Δ𝑋̇_𝐶^𝐺2=𝐴_𝐶^𝐺2Δ𝑋_𝐶^𝐺2+𝐵^𝐺2_𝐶0Δ𝑈_𝐶^𝐺2,Δ𝑋_𝐶^𝐺2=Δ𝜔_𝐺2Δ𝑑_𝐺2,Δ𝑈_𝐶^𝐺2=Δ𝑃Δ𝑄,𝐴_𝐶^𝐺2=−𝜔_𝑓00−𝜔_𝑓. (B.5) With writing the active and reactive power in terms of voltage and current Δ𝑋̇_𝐶^𝐺2=𝐴_𝐶^𝐺2Δ𝑋_𝐶^𝐺2+𝐶_𝐶^𝐺2Δ𝑋^𝐺2+𝐵^𝐺2_𝐶6Δ𝑣^𝐺_𝑞𝑑,𝐶_𝐶^𝐺2=𝐵^𝐺2_𝐶50,𝐵^𝐺2_𝐶5=𝐵^𝐺2_𝐶1𝐵^𝐺2_𝐶2𝑇⁻¹_𝐺2,𝐵^𝐺2_𝐶6=𝐵^𝐺2_𝐶1𝐵^𝐺2_𝐶3𝑇⁻¹_𝐺2+𝐵^𝐺2_𝐶4𝑀^𝐺2_𝑞𝑑,𝐵^𝐺2_𝐶4=𝐵^𝐺2_𝐶1𝐵^𝐺2_𝐶2𝑖⁰_𝛿𝐺2+𝐵^𝐺2_𝐶1𝐵^𝐺2_𝐶3𝑣⁰_𝛿𝐺2,𝐵^𝐺2_𝐶1=−32𝑘_𝑝𝜔_𝑓030−2𝑘_𝑣𝜔_𝑓,𝐵^𝐺2_𝐶2=−𝑣^𝐺2_𝑑0𝑣^𝐺2_𝑞0𝑣^𝐺2_𝑞0𝑣^𝐺2_𝑑0,𝐵^𝐺2_𝐶3=𝑖^𝐺2_𝑑𝑖0−𝑖^𝐺2_𝑞𝑖0𝑖^𝐺2_𝑞𝑖0𝑖^𝐺2_𝑑𝑖0. (B.6)

C. Linear Differential Equations of DG (25)

𝜌Δ𝑋^DG=𝐴^DGΔ𝑋^DG+𝐵^DGΔ𝑈^DG,𝐴^DG=𝐸^𝐺2−1𝐴^𝐺2𝐸^𝐺2−1𝐶^𝐺2𝐶_𝐶^𝐺2𝐴_𝐶^𝐺2,𝐵^DG=𝐸^𝐺2−1𝐵^𝐺2𝐵^𝐺2_𝐶7,𝐵^𝐺2_𝐶7=𝐵^𝐺2_𝐶60. (C.1)