Virtual Inertia and Mechanical Power-Based Control Strategy to Provide Stable Grid Operation under High Renewables Penetration

This paper presents a virtual inertia and mechanical power-based control strategy to provide a stable operation of the power grid under high penetration of renewable energy sources (RESs). The proposed control technique is based on a new active and reactive power-based dynamic model with the permanent magnet synchronous generator (PMSG) swing equation, in which all PMSG features i.e., inertia and mechanical power are embedded within the controller as the main contribution of this paper. To present an accurate analysis of the virtual PMSG-based parameters, the desired zero dynamics of the grid angular frequency are considered to evaluate the effects of virtual mechanical power (VMP) on the active and reactive power sharing, as well as the investigation of virtual inertia variations for the grid angular frequency responses. Moreover, by considering various active power errors and virtual inertia, the impacts of active power error on reactive power in the proposed control technique, are precisely assessed. Simulation results are employed in Matlab/Simulink software to verify the stabilizing abilities of the proposed control technique.


Introduction
Emulating the behaviours of a synchronous generator (SG) in the structure of novel control techniques for power electronics converters has been increased in recent years in order to guarantee the stability of the power grid under high penetration of renewable energy sources [1][2][3][4][5][6][7].
Several studies have been reported in the literature of the modelling behaviour of SG in power electronics-based generators [8][9][10][11][12][13][14][15][16][17]. Reference [8] utilizes the concept of power hardware-in-the-loop for emulating a synchronous generator in aircraft power systems applications. A virtual synchronous generator-based control technique is presented in [12], in which the differences between inverters and real synchronous generators are considered by emulating and analysing swing and power equations [12].

Introduction of the Proposed Control Technique
The general structure of the proposed model for high penetration of renewable energy sources can be seen in Figure 1. A load is connected to the power grid and a large-scale of RESs-based converters are ready for integration into power grid. The proposed controller-based converter has a duty to provide the stability for the power grid in the period of large scale integration of RESs. In this section, a new dynamic model will be extracted based on the proposed model; then, a new controller will be proposed based on the obtained dynamic model.

The Dynamic Model Analysis
To design an appropriate control technique for the control of interfaced converters, the dynamic model of the proposed model is needed. By considering the interfaced converter in Figure 1, the initial mathematical model is obtained as: As it is known that P = idvd and Q = −iqvd. Considering the aforementioned relationships, (1) can be rewritten as: As a result, by the use of (4), the grid angular frequency variation is achieved as:

The Dynamic Model Analysis
To design an appropriate control technique for the control of interfaced converters, the dynamic model of the proposed model is needed. By considering the interfaced converter in Figure 1, the initial mathematical model is obtained as: As it is known that P = i d v d and Q = −i q v d . Considering the aforementioned relationships, (1) can be rewritten as: L R dP dt + P + ωL R Q − u d P c1 + P d = 0 L R dQ dt + Q − ωL R P + u q P c1 − P dq = 0 RC dP c1 dt + u d P − u q Q + P c2 = 0 (2) where, P c1 = v dc v d /R, P c2 = i dc v d , P pd = v d 2 /R, and P pdq = v d v q /R. The PMSG swing equation is equal to: Small signal linearization is applied to (3) which leads to, As a result, by the use of (4), the grid angular frequency variation is achieved as: The relationship (5) can be rewritten as the following, Based on the active power control, the active power error relationship is driven through dividing the Laplace operator "s" by (6) as: It is assumed that the interfaced converter in this paper has a nominal apparent power to generate total limited currents. It means that S = constant. Thus, by considering the relations P 2 + Q 2 = S 2 = constant, the relation between the small signal variables of active and reactive power can be written as, Using Equations (7) and (8), the reactive power control-based term can be achieved as: Equations (5), (7) and (9) are employed to design the proposed controller as: In fact, the Equation (10) is a new description of the interfaced converter that now has the PMSG characteristics included the virtual inertia, VMPE (virtual mechanical power error), and angular frequency error. According to (10), both d and q components of the proposed control technique can be drawn as Figure 2a,b. Noticing Equation (10) and Figure 2, the compensators coefficients can be stated as:

Evaluation the Impacts of Virtual Inertia and VMPE on the Proposed Dynamic Model
By use of the achieved active and reactive power-based dynamic model (2) along with the proposed controller-based time domain dynamic Equation (10), an evaluation related to the virtual inertia and VMPE effects is executed in this section. All power and angular frequency variations are firstly defined as: According to Figure 2, performance of the proposed controller is highly influenced by the embedded characteristics of PMSG that can consequently improve the controller operation with accurately regulating the virtual parameters. Also, the coefficients of α 1 , α 2 , β 1 and β 2 are the control factors of the proposed controller.

Evaluation the Impacts of Virtual Inertia and VMPE on the Proposed Dynamic Model
By use of the achieved active and reactive power-based dynamic model (2) along with the proposed controller-based time domain dynamic Equation (10), an evaluation related to the virtual inertia and VMPE effects is executed in this section. All power and angular frequency variations are firstly defined as: The time-based dynamic model of (10) can be achieved as: As can be understood from (13) and (14), the zero dynamic of angular frequency i.e., dω/dt is dependent on the active power, reactive power, virtual inertia, VMP, the angular frequency, and also their reference values. Considering the desired value for dω/dt in (10) and (11) that is dω/dt = 0, the relationships (13) and (14) can be rewritten based on linear relationships for both active and reactive power as, The diagrams related to (15) and (16) are drawn in Figure 3a,b, respectively. As can be seen from Figure 3a, a constant ramp is achieved for the active power diagram. The reference value increment and decrement of the VMPE can change the initial value of the active power diagram according to Figure 3a. On the other hand, as demonstrated in Figure 3b, a changeable ramp generated by varying the reactive power reference value can be observed for the reactive power diagram. Moreover, the initial value of the reactive power diagram can be altered by P * m and ∆ω. These diagrams illustrate that how improved control of P m and ω can help both active and reactive power approach to desired values.   In order to reach an accurate zero dynamic-based analysis associated with the proposed controller-based model, the second order derivative of the angular frequency should be driven by use of (13) and (14) as: In order to reach an accurate zero dynamic-based analysis associated with the proposed controller-based model, the second order derivative of the angular frequency should be driven by use of (13) and (14) as: The zero dynamic of the VMP is found in (17) and (18). Moreover, it can be realized from (17) and (18) that the zero dynamics of the active and reactive power can highly impact on the angular frequency variations. By obtaining dP/dt and dQ/dt from the first and second terms of (2) respectively, and substituting the results into (17) and (18), two accurate zero dynamic models for the grid angular frequency with acceptable approximations can be achieved as: Except for the state variables involved with the converter operating conditions, both virtual inertia and mechanical power can highly impact on performance of the grid angular frequency as demonstrated in (19) and (20). By solving the differential Equations of (19) and (20), the closed-loop frequencies can be achieved as: Considering the right-side terms of (19) and (20) and (21) and (22), the grid angular frequency can be achieved as: Various states of (23) are drawn in Figure 4. As can be seen from this figure, changing the virtual inertia can be utilized to reach a desired value for the angular frequency of grid. For positive values of s 1 , s 2 , s 3 , and s 4 , two unstable responses in Figure 4 are achieved. But in the cases of stable responses, the magnitude of fluctuations of the response is noticeably decreased as shown in Figure 4 as a blue colour. On the other hand, very low and high values of the virtual inertia can lead to the non-compensable transient reaction of the grid angular frequency. The low-high values of the virtual inertia can provide appropriate response instead. The low-high values for virtual inertia in this paper mean that the virtual inertia locates in the interval [1 × 10 −1 s, 1 × 10 3 s]. In another word, the virtual inertia, which is smaller than 1 × 10 −1 s, or bigger than 1 × 10 3 s, will be called very low, and high values of virtual inertia, respectively. The proposed control technique in the interval [1 × 10 −1 s, 1 × 10 3 s] will work very well with high stability. To more understand the effects of these values, both Sections 4 and 5 are presented.

Evaluation of the Impacts of Active Power Error on Reactive Power in the Proposed Controller
Reactive power sharing of the proposed controller is affected by the active power error. In this section, the reactive power operation is evaluated through the step and ramp changes of active power error, and also different values of the virtual inertia. Based on Figure 2 and Equation (2), the transfer function of reactive power and active power error can be achieved as: where, AP and Δ are given in the Appendix A. Figure 5 shows the Bode and Phase diagrams for Q/∆P, as well as the step and ramp changes of active power error around the operating angular frequency. It can be seen from the Phase diagram of Figure 5, that the Q/∆P and ramp active power error generate a phase difference of 90 degrees for the reactive power of the proposed controller. But, by carefully analysing the magnitude diagram in Figure 5, it can be achieved that the magnitude of the reactive power generated by the active power error is extremely low. To complete the aforementioned analysis, the Root Locus diagrams of reactive power for step and ramp active power error are shown in Figure 6. It can be seen from Figure 6, that a step change of active power error leads to fluctuating reactive power responses and in another case of ramp change, the reactive power becomes completely unstable as shown in Figure 6b. Changing the virtual inertia is considered in Figure 7. As it can be seen, decreasing the virtual inertia makes a little difference in the magnitude of Q/∆P related to larger value and totally for all values of the virtual inertia and the same results are approximately achieved.

Evaluation of the Impacts of Active Power Error on Reactive Power in the Proposed Controller
Reactive power sharing of the proposed controller is affected by the active power error. In this section, the reactive power operation is evaluated through the step and ramp changes of active power error, and also different values of the virtual inertia. Based on Figure 2 and Equation (2), the transfer function of reactive power and active power error can be achieved as: where, A P and ∆ are given in the Appendix A. Figure 5 shows the Bode and Phase diagrams for Q/∆P, as well as the step and ramp changes of active power error around the operating angular frequency. It can be seen from the Phase diagram of Figure 5, that the Q/∆P and ramp active power error generate a phase difference of 90 degrees for the reactive power of the proposed controller. But, by carefully analysing the magnitude diagram in Figure 5, it can be achieved that the magnitude of the reactive power generated by the active power error is extremely low. To complete the aforementioned analysis, the Root Locus diagrams of reactive power for step and ramp active power error are shown in Figure 6. It can be seen from Figure 6, that a step change of active power error leads to fluctuating reactive power responses and in another case of ramp change, the reactive power becomes completely unstable as shown in Figure 6b. Changing the virtual inertia is considered in Figure 7. As it can be seen, decreasing the virtual inertia makes a little difference in the magnitude of Q/∆P related to larger value and totally for all values of the virtual inertia and the same results are approximately achieved.

Results and Discussions
Performance of the proposed controller in the dynamic operating condition is surveyed in this section. Also, effects of the different virtual mechanical factors including the virtual inertia and the mechanical power error are assessed. The values of simulation parameters are specified in Table 1. The general structure of the proposed controller is given in Figure 8.

Active and Reactive Power Sharing Assessment
The active and reactive power sharing ability of the proposed controller-based converter can be influenced by variation of the virtual inertia and mechanical power. Firstly, the proposed controller-based converter is supplying a 3 kW + j2 kVar grid connected load and then an abrupt disconnection of the PV sources with 4 Kw + j3 kVar occurs at the PCC at t = 0.1 s which will be supplied by the proposed controller at t = 0.25s. Figure 9 shows the active power variations under different values for virtual inertia and mechanical power.

Results and Discussions
Performance of the proposed controller in the dynamic operating condition is surveyed in this section. Also, effects of the different virtual mechanical factors including the virtual inertia and the mechanical power error are assessed. The values of simulation parameters are specified in Table 1. The general structure of the proposed controller is given in Figure 8.

Active and Reactive Power Sharing Assessment
The active and reactive power sharing ability of the proposed controller-based converter can be influenced by variation of the virtual inertia and mechanical power. Firstly, the proposed controllerbased converter is supplying a 3 kW + j2 kVar grid connected load and then an abrupt disconnection of the PV sources with 4 Kw + j3 kVar occurs at the PCC at t = 0.1 s which will be supplied by the proposed controller at t = 0.25s. Figure 9 shows the active power variations under different values for virtual inertia and mechanical power.   As it can be seen from Figure 9a,b, the best response is due to the low-high values of virtual inertia as well as low VMP error. In contrast, the worst response can be achieved by very high values of both virtual inertia and mechanical power. Moreover, the active power with high VMP error is trying to follow its desired value that verifies the importance of choosing appropriately the VMP in the proposed controller. According to Figure 9a, for very low inertia, the active power reaches its reference value ultimately, however the transient takes very long time along with very high value of the transient magnitude. The reactive power of the proposed controller-based converter is illustrated in Figure 10. The same scenario associated with the best response happens for the reactive power. Also, the worst case of the reactive power response is belonging to very high value of the virtual inertia and mechanical power as shown in Figure 9a,b. In the cases of very low inertia and high VMP error, the reactive power is trying to make minimum divergence in its entire responses.
As it can be seen from Figure 9a,b, the best response is due to the low-high values of virtual inertia as well as low VMP error. In contrast, the worst response can be achieved by very high values of both virtual inertia and mechanical power. Moreover, the active power with high VMP error is trying to follow its desired value that verifies the importance of choosing appropriately the VMP in the proposed controller. According to Figure 9a, for very low inertia, the active power reaches its reference value ultimately, however the transient takes very long time along with very high value of the transient magnitude. The reactive power of the proposed controller-based converter is illustrated in Figure 10. The same scenario associated with the best response happens for the reactive power. Also, the worst case of the reactive power response is belonging to very high value of the virtual inertia and mechanical power as shown in Figure 9a,b. In the cases of very low inertia and high VMP error, the reactive power is trying to make minimum divergence in its entire responses. As it can be seen from Figure 9a,b, the best response is due to the low-high values of virtual inertia as well as low VMP error. In contrast, the worst response can be achieved by very high values of both virtual inertia and mechanical power. Moreover, the active power with high VMP error is trying to follow its desired value that verifies the importance of choosing appropriately the VMP in the proposed controller. According to Figure 9a, for very low inertia, the active power reaches its reference value ultimately, however the transient takes very long time along with very high value of the transient magnitude. The reactive power of the proposed controller-based converter is illustrated in Figure 10. The same scenario associated with the best response happens for the reactive power. Also, the worst case of the reactive power response is belonging to very high value of the virtual inertia and mechanical power as shown in Figure 9a,b. In the cases of very low inertia and high VMP error, the reactive power is trying to make minimum divergence in its entire responses.

Evaluation of the Grid Frequency and the Voltage Magnitude
The same load variations are considered in this sub-section. Employing various values, for the virtual inertia and VMPE in the proposed controller, can impact on both steady stated and transient responses of the grid frequency and voltage magnitude under high renewable energy sources penetration.
The grid frequency responses, along with the virtual inertia and VMPE changes, are illustrated in Figure 11. Figure 11a shows the best response of the proposed controller for the grid frequency. According to Figure 11b, both very high and low values of the virtual inertia will lead to the worst response for the grid frequency. The grid frequency in the case of the high VMPE is making effort to remain at desired value as depicted in Figure 11c, which is in contrast to the very high values of the VMPE that leads to unstable reaction for the grid frequency. The responses of grid voltage magnitude are shown in Figure 12. Based on Figure 12a, for the appropriate values of virtual inertia and VMPE, the proposed controller is able to force the grid voltage following its reference value, with a very good transient response. The grid voltage magnitude is collapsed for both very high and low values of virtual inertia. The same scenario happens for very high VMPE and inversely the high value of VMPE can somewhat provide stable response for the grid voltage magnitude as illustrated in Figure 12c.

Evaluation of the Grid Frequency and the Voltage Magnitude
The same load variations are considered in this sub-section. Employing various values, for the virtual inertia and VMPE in the proposed controller, can impact on both steady stated and transient responses of the grid frequency and voltage magnitude under high renewable energy sources penetration.
The grid frequency responses, along with the virtual inertia and VMPE changes, are illustrated in Figure 11. Figure 11a shows the best response of the proposed controller for the grid frequency. According to Figure 11b, both very high and low values of the virtual inertia will lead to the worst response for the grid frequency. The grid frequency in the case of the high VMPE is making effort to remain at desired value as depicted in Figure 11c, which is in contrast to the very high values of the VMPE that leads to unstable reaction for the grid frequency. The responses of grid voltage magnitude are shown in Figure 12. Based on Figure 12a, for the appropriate values of virtual inertia and VMPE, the proposed controller is able to force the grid voltage following its reference value, with a very good transient response. The grid voltage magnitude is collapsed for both very high and low values of virtual inertia. The same scenario happens for very high VMPE and inversely the high value of VMPE can somewhat provide stable response for the grid voltage magnitude as illustrated in Figure  12c.

Conclusions
A control technique based on the characteristics of PMSG has been proposed in this paper to deal with the power grid stability issues under high penetration of renewable energy sources. Initially, virtual inertia and mechanical power (VMP) have been embedded in an active and reactive power-based dynamic model of the interfaced converter as the main contribution of this paper, that could provide a stable operation for the power grid, with properly varying the virtual parameters. By considering the first order zero dynamic of the grid angular frequency, the active and reactive power variations, based on VMP, have been assessed. Based on this assessment, the accurate active and reactive power sharing have been improved by appropriately controlling VMP and the error of angular frequency. Moreover, the second order zero dynamic of the grid angular frequency has been used to investigate precisely the virtual inertia effects. This investigation has shown that the stable responses of the angular frequency have driven through the so-called low-high virtual inertia. From another side, the effects of the active power error on the reactive power, under the changing virtual inertia, have been comprehensively analysed. Within the future work, authors suggest that more detailed dynamic model of PMSG can be embedded into the active and reactive power-based dynamic model, presented in this paper to design more detailed controller. Simulation results confirmed that the proposed control technique can guarantee the stability of the power grid under high penetration of renewable energy sources. The main findings of this paper are that, for a wide range of virtual inertial with VMP, stable responses with very short transient time, have been achieved for the grid frequency and voltage magnitude, as well as the active and reactive power under high penetration condition.

Conclusions
A control technique based on the characteristics of PMSG has been proposed in this paper to deal with the power grid stability issues under high penetration of renewable energy sources. Initially, virtual inertia and mechanical power (VMP) have been embedded in an active and reactive power-based dynamic model of the interfaced converter as the main contribution of this paper, that could provide a stable operation for the power grid, with properly varying the virtual parameters. By considering the first order zero dynamic of the grid angular frequency, the active and reactive power variations, based on VMP, have been assessed. Based on this assessment, the accurate active and reactive power sharing have been improved by appropriately controlling VMP and the error of angular frequency. Moreover, the second order zero dynamic of the grid angular frequency has been used to investigate precisely the virtual inertia effects. This investigation has shown that the stable responses of the angular frequency have driven through the so-called low-high virtual inertia. From another side, the effects of the active power error on the reactive power, under the changing virtual inertia, have been comprehensively analysed. Within the future work, authors suggest that more detailed dynamic model of PMSG can be embedded into the active and reactive power-based dynamic model, presented in this paper to design more detailed controller. Simulation results confirmed that the proposed control technique can guarantee the stability of the power grid under high penetration of renewable energy sources. The main findings of this paper are that, for a wide range of virtual inertial with VMP, stable responses with very short transient time, have been achieved for the grid frequency and voltage magnitude, as well as the active and reactive power under high penetration condition.

Conflicts of Interest:
The authors declare no conflict of interest.