Recently, distributed power generation system (DPGS) with renewable energy sources, such as wind turbines and photovoltaic, has been attracting more and more attentions for solving energy crisis and environmental issues. However, when the penetration level of renewable energy is high, the grid equivalent rotational inertia becomes low (Blaabjerg et al., 2017). DPGS with low inertia, which is also known as weak grid, may result in poor voltage and frequency response during large disturbances (Xiao et al., 2021a).

In general, there are two ways to solve this problem. One is to install large number of storage batteries in the DPGS, which is not practical considering the high construction cost. The other solution is to develop new control method that can increase the grid inertia or make the interfacing voltage source converter (VSC) participate in voltage and frequency regulation of the grid (Ashabani and Mohamed, 2014a). This paper will look for a solution from the latter.

In conventional VSC control system, the dq-axis decoupling method is used. That is, the active power is controlled by regulating the d-axis current component, while the reactive power is controlled by regulating the q-axis current component. In this way, the active power and reactive power can be controlled independently. The exiting studies found that the widely used current vector controller for VSC based high voltage direct current (HVDC) systems is not applicable when the AC system is very weak (Zhang et al., 2011a; Zhou and Gole, 2012; Hu et al., 2019). When the AC system is weak (the short circuit ratio is small), the d-axis and q-axis of the current is no longer decoupled. The active power and reactive power cannot be regulated independently. Therefore, in weak AC system, the dq-axis decoupling characteristic of the current vector controller is destroyed, which may lead to the instability of the system (Zhang et al., 2011b).

In order to overcome the above limitation, the conventional droop control (CDC) method is usually adopted for the VSC station (Chandorkar et al., 1993; Coelho et al., 1999; Xiao et al., 2022). With the CDC method, the frequency and the amplitude of the output voltage are directly proportional to the real-time transmission active and reactive power. But there are some serious drawbacks for this method. Important among them are permanent offset of frequency due to the change of power, inaccurate power sharing and no contribution for the system inertia (Mohamed and El-Saadany, 2008), (Ashabani et al., 2015).

Another control method for the VSC connected to weak grid or islanded system is the virtual synchronous generator (VSG) control, which operates the VSC in the similar way as the synchronous generator (SG) (Beck and Hesse, 20072007). So far there have been many VSG models (Zhang et al., 2010; Zhong and Weiss, 2011; Zhong et al., 2014; Ashabani and Mohamed, 2014b; Guan et al., 2015). One of them is presented in (Zhong and Weiss, 2011), with which the characteristics of SG are well mimicked. But the additional control loop makes the control scheme complicated, and also this model is lack of current limiting capability. Another VSG model which is well known as power-synchronization control is proposed in (Zhang et al., 2010). This model is successfully used in powering a weak grid and achieves good performance. But the secondary frequency regulation is not realized by this model, which will cause the frequency offset. A simple and practical VSG model is realized in (Guan et al., 2015). The secondary frequency regulation is also achieved. But the exciter of the SG is not emulated, which means the reactive power cannot be controlled precisely.

In order to precisely control the power and optimize power sharing among distributed generators (DGs), higher level controllers (secondary or tertiary controllers) are usually activated. In general, the controller can be divided into two categories: centralized and decentralized (Xin et al., 2015; Olivares et al., 2014; Xiao et al., 2021b). In a decentralized system each DG is controlled by its local controller and the new DGs can be easily plugged into the grid (Xin et al., 2013). However, it is difficult for the decentralized controller to handle the operation of DPGS requiring high levels of coordination (Olivares et al., 2014). Therefore, centralized controllers containing communication system are needed in a DPGS with strong coupling between DGs. A power sharing method with communication is presented in (Liang et al., 2013) to precisely control the reactive power. But the frequency offset is not eliminated and also the system stability is very sensitive to the communication delays (Kahrobaeian and Ibrahim Mohamed, 2015).

Motivated by the aforementioned reasons, an enhanced control and a novel power sharing for DPGS are proposed in this paper. In this control scheme, each DG adopts a novel control method called synchronous generator imitation control (SGIC). The novel SGIC consists of three loops, including active power and frequency loop (Pf loop), reactive power and voltage loop (QU loop) as well as inner current loop. The novelty and contribution of this paper is summarized as follows:

1) An enhanced control method called synchronous generator imitation control is proposed in this paper. With the proposed control, the VSC could imitate the behavior of a synchronous generator so that the multiple VSCs could operate stably in the weak AC grid or islanded system as well as provide frequency and voltage support for the power systems.

This paper is organized as follows. Synchronous Generator Imitation Control of VSC Section presents the principle of SGIC method. The dynamic power sharing for DGPS is introduced in Dynamic Power Sharing for DPGS Section. The small-signal model of a DGPS with multiple DGs is built and the stability performance is analysis in Small-Signal Analysis for a DPGS With the Proposed Control Scheme Section. Case Study Section shows the simulation results of the system in different operation conditions. Finally, Conclusion Section concludes the paper.

Due to the high penetration level of renewable energy in the power system, the equivalent rotational inertia of the grid is dramatically reduced. It will be of great benefit if the VSC stations can increase the inertia of the grid just like a SG. In such a manner, the VSC stations can also provide frequency and voltage support for the local grid. In this section, the SGIC method, including inner current loop, Pf loop and QU loop, is presented.

When the CDC method or the proposed SGIC method is used in a DPGS, there exist three problems for the whole system. 1) The output power of the VSC stations will deviate from its original set value due to the change of the load demand, which in turn causes the frequency offsets. 2) The reactive power cannot be shared accurately among VSC stations. 3) System stability is highly sensitive to the droop coefficients and the communication delays, which results in low stability margin.

In this section, a novel method named dynamic power sharing is proposed to solve the above three problems.

When the SGIC method is used, each VSC station can be regarded as a SG. In this paper, the real SG and the VSC stations with SGIC method are collectively called generation units (GUs).

In order to eliminate the frequency offset, the power references are adjusted dynamically. The references of active power and reactive power are given by P G U i = λ i P t o t a l (13) Q G U i = γ i Q t o t a l (14) where, P GUi is the active power reference of GU _i and Q GUi is the reactive power reference. P _total = ∑P GUi and Q _total = ∑Q GUi are the total active and reactive powers of the local grid. λ _i is the active power allocation factor of GU _i . γ _i is the reactive power allocation factor of GU _i . The active and reactive power allocation factors satisfy the following equations ∑ i = 1 n λ i = 1 (15) ∑ i = 1 n γ i = 1 (16)

Figure 3 is the schematic diagram of the dynamic power sharing. The active and reactive powers generated by each GU are transmitted to the energy management system (EMS) so that the total power demand can be calculated. Then the desired active and reactive powers λ _i P _total , γ _i Q _total are sent back to the GUs as their references.

Because the power references are dynamically adjusted according to the real-time power demand, the frequency offset can be eliminated.

Inserting Eqs 13, 14 to Eqs 5, 7, the following equations can be obtained ω i ∗ = ω 0 + 1 J i s + D i ( λ i P t o t a l − P i ) (17) U i ∗ = U 0 + ( k G i + k T i s ) ( γ i Q t o t a l − Q i ) (18) One of the issues that need to be considered in a centralized based EMS is the handling for a failure of the communication system. When the communication system fails, the power reference for each GU becomes zero (that is λ _i = γ _i = 0). In order to stabilize the voltage, the integral term of the QU loop should be set to zero (that is k _Ti = 0). In this way, the control equations can be rewritten as in Eqs 19, 20 ω i ∗ = ω 0 − 1 J i s + D i P i (19) U i ∗ = U 0 − k G i Q i (20)

In order to analyze the dynamic performance of the system and optimize the controller parameters, it is necessary to carry out small-signal modeling and analysis on the system with the proposed control scheme. This section will perform a small-signal analysis of a DPGS containing n distributed power supplies.

The load angle derivative is given by Δ δ ˙ i = Δ ω i (21) Linearization of Eq. 17 yields Δ ω ˙ i = − D i J i Δ ω i − 1 J i Δ P i + λ i J i ∑ j = 1 n Δ P i (22) The average active and reactive powers are obtained through low-pass filter as following equations P i = ω c s + ω c p i (23) Q i = ω c s + ω c q i (24) where ω _c is the bandwidth of the low-pass filter. p _i and q _i are the instantaneous active and reactive power. The small-signal equations of Eqs 23, 24 are given by Δ P ˙ i = − ω c Δ P i + ω c p i (25) Δ Q ˙ i = − ω c Δ Q i + ω c q i (26) The linearized form of Eq. 18 can be expressed as Δ U ˙ i = ( k T i − ω c k G i ) γ i ∑ j = 1 n Δ Q j − ( k T i − ω c k G i ) Δ Q i + ω c k G i γ i ∑ j = 1 n Δ Q j − ω c Δ Q i (27) For a DPGS with n power-generating nodes and m load nodes, the network equations can be written as [ I n ⋯ 0 ] = [ Y n n ⋮ Y n m ⋯ ⋯ ⋯ Y m n ⋮ Y m m ] [ U n ⋯ U m ] (28) In order to simplify the analysis, Kron reduction (Anderson and Foad, 2003)- (Dorfler and Bullo, 2013) is used on Eq. 28. Then we can get the system equation containing only n power-generating nodes as I n = Y e q U n Y e q = Y n n − Y n m Y m m − 1 Y m n } (29) The active and reactive power injected by each generation unit can be expressed as p i = U i ⁡ cos ⁡ δ i ∑ j = 1 n U j ( G i j ⁡ cos ⁡ δ j − B i j ⁡ sin ⁡ δ j ) + U i ⁡ sin ⁡ δ i ∑ j = 1 n U j ( G i j ⁡ sin ⁡ δ j + B i j ⁡ cos ⁡ δ j ) (30) q i = U i ⁡ sin ⁡ δ i ∑ j = 1 n U j ( G i j ⁡ cos ⁡ δ j − B i j ⁡ sin ⁡ δ j ) − U i ⁡ cos ⁡ δ i ∑ j = 1 n U j ( G i j ⁡ sin ⁡ δ j + B i j ⁡ cos ⁡ δ j ) (31) where G _ij and B _ij are the real and imaginary parts of the element Y _ij in the equivalent admittance matrix Y _eq (ie, Y _ij = G _ij + jB _ij ).

The linearized form of Eqs 30, 31 can be presented as Δ p i = ∑ j = 1 n ∂ p i ∂ δ j Δ δ j + ∑ j = 1 n ∂ p i ∂ U j Δ U j (32) Δ q i = ∑ j = 1 n ∂ q i ∂ δ j Δ δ j + ∑ j = 1 n ∂ q i ∂ U j Δ U j (33)

The small-signal model of a DPGS with n distributed power supplies is described in Eqs 21, 22; Eqs 25–27; Eqs 30–33. The overall system small-signal state-space model can be derived as Δ x ˙ = A Δ x (34) A = [ 0 n × n I n × n 0 n × n 0 n × n 0 n × n 0 n × n ( − D 1 J 1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ − D n J n ) T 3 0 n × n 0 n × n ω c ( ∂ p ∂ δ ) n × n 0 n × n − ω c I n × n 0 n × n ω c ( ∂ p ∂ U ) n × n ω c ( ∂ q ∂ δ ) n × n 0 n × n 0 n × n − ω c I n × n ω c ( ∂ q ∂ U ) n × n T 2 × ( ∂ q ∂ δ ) n × n 0 n × n 0 n × n T 1 T 2 × ( ∂ q ∂ U ) n × n ] (35) where state variable set is Δ x = [ Δ δ i Δ ω i Δ P i Δ Q i Δ U i ] T (for i = 1,2…n). T ₁, T ₂ and T ₃ are given by T 1 = [ ( k T 1 − ω c k G 1 ) ( γ 1 − 1 ) ( k T 1 − ω c k G 1 ) γ 1 ⋯ ( k T 1 − ω c k G 1 ) γ 1 ( k T 2 − ω c k G 2 ) γ 2 ( k T 2 − ω c k G 2 ) ( γ 2 − 1 ) ⋯ ( k T 2 − ω c k G 2 ) γ 2 ⋮ ⋮ ⋱ ⋮ ( k T n − ω c k G n ) γ n ( k T n − ω c k G n ) γ n ⋯ ( k T n − ω c k G n ) ( γ n − 1 ) ] T 2 = [ ω c k G 1 ( γ 1 − 1 ) ω c k G 1 γ 1 ⋯ ω c k G 1 γ 1 ω c k G 2 γ 2 ω c k G 2 ( γ 2 − 1 ) ⋯ ω c k G 2 γ 2 ⋮ ⋮ ⋱ ⋮ ω c k G n γ n ω c k G n γ n ⋯ ω c k G n ( γ n − 1 ) ] T 3 = [ ( λ 1 − 1 ) / J 1 λ 1 / J 1 ⋯ λ 1 / J 1 λ 2 / J 2 ( λ 2 − 1 ) / J 2 ⋯ λ 2 / J 2 ⋮ ⋮ ⋱ ⋮ λ n / J n λ n / J n ⋯ ( λ n − 1 ) / J n ]

The proposed model is applied to a DPGS shown in Figure 4. In this system, two VSC stations supply power to a common load through respective connected lines. In order to simplify the analysis, it is assumed that the parameters of the two VSC stations are the same. The active power allocation factor and reactive power allocation factor for the two VSC are 0.5.

Figure 5 shows the system eigenvalue spectrum with different control parameters.

Figure 5A shows the system root locus when k _G = 0, k _T = 0, D = 1,000 and J changes from 0.1 to 0.5. It can be seen from the figure that with the increase of J, eigen2 and eigen3 move from the left half plane to the imaginary axis. In particular, when J > 0.35, the two conjugate complex roots go into the right half plane, indicating that the system is unstable.

Figure 5B shows the location of eigenvalue with k _G = 0, k _T = 0.03, J = 0.1, D from 1e5 to 5e5. As can be seen from the figure, eigen2 and eigen3 are very sensitive to parameter D while eigen5 and eigen6 are almost constant. As D increases, eigen2 and eigen3 change from a pair of conjugate complex roots to two real roots. The absolute value of a real root becomes smaller gradually, which indicates that the stability margin of the system decreases.

Figure 5C shows the system eigenvalue spectrum when k _T = 0.03, D = 2e5, J = 0.1 and k _G changes from 2e-5 to 2e-3. It can be seen that eigen5 and eigen6 are greatly influenced by k _G . When k _G is less than 1e-3, with the increase of k _G , the speed of the system response to the disturbance is accelerated.

Figure 5D shows the system root locus when k _G = 5e-4, D = 2e5, J = 0.1 and k _T changes from 0.005 to 0.05. When k _T < 0.015, eigen5 and eigen6 are negative real roots; when k _T > 0.015, a pair of conjugate complex roots is generated and the oscillation frequency of the system increases. But other dominant roots are almost constant, which means satisfactory performance and fast response can be achieved without the loss of stability.

In order to verify the accuracy and validity of the proposed control strategy, a DPGS with three distributed generators shown in Figure 6 is built in PSCAD/EMTDC simulation software. The control parameters of the VSC stations can be seen in Table 1.

This paper presents a control and power sharing method for the distributed power generation system. In this scheme, each of the VSC stations adopts the proposed SGIC method. In order to eliminate the frequency offset, control the power precisely and improve the system stability when there are communication delays, a novel control called dynamic power sharing is proposed. Small-signal model of the whole system has been built and analyzed. A DPGS with three DGs is built in PSCAD/EMTDC software. Various operation conditions including load changes, power allocation factors adjustment, station fault and communication delay are simulated. According to this paper, the important advantages for the proposed scheme can be concluded as follows:

1) The Pf loop is designed through simply adding a first-order inertia element to the CDC method, by which the VSC has the similar inertia as the SG. The QU loop is used to imitate the exciter of the SG. With this loop the VSC station is able to keep the output voltage stable and control the reactive power at the same time.