Based on the practical engineering needs, the FEC strategy considering the participation of ESCs should meet the following three requirements: (i) Since the designed control strategy mainly deals with emergency faults, ESCs should only work in the case of emergency faults and not participate in the conventional frequency control (e.g., primary frequency regulation). (ii) In order to enable ESCs to respond quickly and reliably to emergency faults, the FEC strategy should be designed as a decentralized strategy that does not require communication among control units. (iii) In order to reasonably distribute the unbalanced power among multiple control units during control process, the control parameters should be optimally selected to meet certain control objectives (e.g., the two control objectives selected in Section 3.2).

According to the above three requirements, this paper adopts the decentralized droop control idea to design the frequency emergency control strategy. We assume that ESCs and the generators can provide enough power support to deal with the unbalanced power, and so, the load-shedding strategies are not considered. The overall scheme of the optimal-droop-based FEC strategy is shown in Figure 2, which mainly comprises three parts. Since ESCs do not participate in conventional primary frequency regulation, the ESC droop control and the primary frequency control are independent of each other, and the control scheme shown in Figure 2 should be designed to make the ESC droop control to work as a supplementary support for primary frequency control in emergency situations.

In the FEC process, in order to reasonably distribute the unbalanced power among multiple ESCs and multiple synchronous generators, the optimal droop coefficients are obtained by formulating the optimal frequency emergency control problem. The optimization objective of the optimal frequency emergency control problem is to minimize the defined control cost. In the following, the two typical control objectives were selected to illustrate the general formulation method of the optimal frequency emergency control (OFC) problem.

Since the OFC problem shown in Eq. 9 is strictly convex, there exists at most one global optimal solution and no suboptimal solution. We assume that the OFC problem shown in Eq. 9 is feasible; then, the optimal droop coefficients of the generators and ESCs are analytically expressed as follows: r i g = 1 a i , i ∈ N g , r i e = 1 2 λ 1 i b i M i e 2 + λ 2 i c i S i e 2 , i ∈ N e . (10)

Furthermore, the design idea of the optimal droop coefficients is illustrated; i.e., when the optimal droop coefficients shown in Eq. 10 are adopted, the closed-loop system dynamics are equivalent to the partial primal-dual algorithm which solves the OFC problem shown in Eq. 9. Let d i = λ 1 i b i M i e 2 + λ 2 i c i S i e 2 . (11)

From Eq. 9, the objective function of the dual OFC problem is (Boyd, Boyd, and Vandenberghe, 2004) Ψ ν = inf u i g , u i e ∑ i ∈ N g 1 2 a i u i g 2 + ν P i g + ν u i g + ∑ i ∈ N e d i u i e 2 + ν P i e + ν u i e + ∑ i ∈ N ν P i , (12) where ν represents the dual variable. Then, solving the infimum problem of Eq. 12, we have Ψ ν = ∑ i ∈ N g − 1 2 a i ν 2 + ν P i g + ∑ i ∈ N e d i u i e ν 2 + ν P i e + ν u i e ν + ∑ i ∈ N ν P i , (13) u i g ν = − 1 a i ν , u i e ν = − 1 2 d i ν = − 1 2 λ 1 i b i M i e 2 + λ 2 i c i S i e 2 ν . (14)

According to Eqs 13 and 14, the dual variable ν represents the common variable of all buses; thus, the communication among buses is required when solving the OFC problem. In order to make the control process decentralized, let ν = ν i , i ∈ N , and we have ν i = ν j , i , j ∈ E at the optimal solution of the OFC problem.

The dual OFC problem (DOFC) is expressed as follows: max ν   Ψ ν , s . t .   ν i = ν j , i , j ∈ E . (15)

The Lagrangian function of the DOFC problem is L D ν , η = Ψ ν + ∑ i , j ∈ E η i j ν i − ν j , (16) where η = η i j , i , j ∈ E represent the Lagrangian multiplier. Applying the partial primal-dual algorithm to solve the DOFC problem, we have ν ˙ i = γ i ∂ L D ν , η ∂ ν i = γ i P i + P i g + ∑ j ∈ N η i j − 1 a i ν i , i ∈ N g , (17) 0 = ∂ L D ν , η ∂ ν i = P i + P i e + ∑ j ∈ N η i j + u i e ν i , i ∈ N e , (18) 0 = ∂ L D ν , η ∂ ν i = P i + ∑ j ∈ N η i j , i ∈ N p , (19) η ˙ i j = − ρ i j ∂ L D ν , η ∂ η i j = − ρ i j ν i − ν j , i , j ∈ E , (20) where γ i and ρ i j represent the iteration step sizes. If we equivalently replace ν i and η i j by ω i and − P i j , respectively, and set the step size γ i = 1 / T i and ρ i j = 1 / B i j , then Eqs 17–20 are identical to the closed-loop system dynamics in Eqs 1–4. Therefore, under the setting of the optimal droop coefficients, the closed-loop system dynamics is equivalent to a partial primal-dual algorithm which solves the OFC problem shown in Eq. 9. Furthermore, according to Eq. 14, when ν i is equivalent to ω i , the optimal droop coefficients are shown as Eq. 10.

This section proves the optimality of the droop coefficients in Eq. 10 rigorously. Since the OFC problem (Eq. 9) is a strictly convex optimization problem and the constraints are all linear, the Karush–Kuhn–Tucker (KKT) conditions can describe the optimal solution of the optimization problem (Shen et al., 2017; Yang et al., 2021b). We choose the phase angle of bus 1 as the reference phase angle, and the relative phase angle is defined as φ i = θ i − θ 1 ; then, we have φ ˙ i = ω i − ω 1 . (21)

Let φ = φ i , i ∈ N , ω g = ω i , i ∈ N g , u g = u i g , i ∈ N g , and u e = u i e , i ∈ N e . The Lagrangian function of the OFC problem (Eq. 9) is L u g , u e , ν = ∑ i ∈ N g 1 2 a i u i g 2 + ν P i g + ν u i g + ∑ i ∈ N e d i u i e 2 + ν P i e + ν u i e + ∑ i ∈ N ν P i . (22)

Then, we have the following theorem about the optimality.

Theorem 1

When the droop coefficients (Eq. 10) are adopted, if φ * , ω g * is the equilibrium point of the closed-loop system (Eqs 1–4), then the u g * , u e * corresponding to the equilibrium is the optimal solution of the OFC problem (Eq. 9).

Proof: To prove Theorem 1, we only need to show that the KKT conditions of the OFC problem are satisfied at the equilibrium point of the closed-loop system (Eqs 1–4). When at the equilibrium point, we have φ ˙ i = 0 , i ∈ N and ω ˙ i = 0 , i ∈ N g ; thus, we observe ω i * = ω j * = ω s y n , (23) where ω s y n represents the synchronous frequency deviation at the equilibrium point. As mentioned before, ω i is equivalent to ν i ; then, we let ω s y n = ν * . According to the optimal droop coefficients (Eq. 10), we have ∂ L ∂ u i g | u g * , u e * , ν * = a i u i g * ν * = 0 , i ∈ N g , ∂ L ∂ u i e | u g * , u e * , ν * = 2 d i u i e * + ν * = 0 , i ∈ N e . (24)

Therefore, the stationary conditions of KKT conditions are satisfied (M. Li, 2019). Furthermore, by substituting ω ˙ i = 0 , i ∈ N g into Eqs 2–4, and adding up Eqs 2–4, under the concept of the relative phase angle, we have 0 = ∑ i ∈ N P i + ∑ i ∈ N g P i g + ∑ i ∈ N e P i e + ∑ i ∈ N g u i g * ∑ i ∈ N e * u i e . (25)

Thus, the feasibility conditions of KKT conditions are also satisfied. In summary, KKT conditions are satisfied at the equilibrium point of the closed-loop system, and the corresponding u g * , u e * is the optimal solution of the OFC problem (Eq. 9), which completes the proof.

The implementation scheme of the optimal-droop-based FEC strategy in the ESC system is shown in Figure 3.

At the droop control layer, the frequency signal at the grid-connected point is first measured by the phase-locked loop (PLL), and then, the frequency signal is input into the droop controller through the filter. The droop controller outputs the power control command for ESC. At the ESC control layer, the power control of ESC is mainly realized by the power conversion system (PCS), which is usually comprised of the DC–AC bidirectional converter and the control unit. The PCS is connected between the battery system and the power grid and realizes the bidirectional power conversion, which makes the ESC system have a constant power control mode, constant power factor control mode, and constant current control mode. Therefore, when the constant power control mode is adopted, ESC can realize rapid power regulation, according to the power control command from the droop controller, and provide emergency power support.

This section verifies the effectiveness and optimality of the proposed FEC strategy. A test system with ESCs is built on the electromagnetic transient (EMT) simulation platform, and the topology of the test system is shown in Figure 4. In this test system, the AC transmission network adopts the IEEE 39-bus system, which contains seven generators and is connected to three ESCs (ESC1, ESC2, and ESC3). The proposed optimal-droop-based FEC strategy is implemented in ESCs, and the generators adopt the primary droop control.

Let the active power base-value of the test system be P b a s e = 100 MW and the cost coefficients of the generators be a 1 = a 5 = 0.1 p.u., a 2 = a 3 = a 4 = a 6 = a 7 = 0.2 p.u. The relevant parameters of ESCs are shown in Table 1.

Based on the above parameter settings, the optimal droop coefficients can be obtained, according to Eq. 10. It should be noted that in order to reflect the control effect of the FEC strategy under two control objectives, the two control objectives are verified separately in the subsequent case analysis; i.e., when adopting control objective I, λ 1 i = 1 and λ 2 i = 0 , and when adopting control objective II, λ 1 i = 0 and λ 2 i = 1 . In order to verify the optimality of the droop coefficients, we define the average droop coefficient as the average value of the optimal droop coefficients. The optimal droop coefficients of the generators under both control objectives are r 1 g = r 5 g = 10 p.u., r 2 g = r 3 g = r 4 g = r 6 g = r 7 g = 5 p.u. The average droop coefficient of the generators is 6.43 p.u. The optimal/average droop coefficients of ESCs under both control objectives are shown in Table 2.

We set a − 300 -MW unbalanced power as the emergency fault in the transmission network and conduct EMT simulation. The simulation results and analysis are shown below.

First, the effectiveness of the FEC strategy considering the participation of ESCs is verified. We set two groups of simulations: (i) ESCs have no droop control. (ii) ESCs are implemented with droop control, and the droop coefficients are the optimal droop coefficients under control objective I. The system frequencies in the two groups of simulations are shown in Figure 5, and the active power of ESCs is shown in Figure 6.

According to Figure 5, it can be seen that when ESCs have no droop control, after the emergency fault occurs, the steady-state frequency of the system is lower than 49.8 Hz, and the frequency overshoot is large, which would have caused serious frequency instability in practical engineering. When ESCs are implemented with the designed optimal droop control, after the emergency fault occurs, the steady-state frequency of the system is higher than 49.8 Hz, and the frequency overshoot is small. Therefore, the FEC strategy, considering the participation of ESCs, can effectively enhance the frequency stability of the power system and can improve the steady-state and transient characteristics of the system frequency response.

Furthermore, as shown in Figure 6A, since ESCs have no droop control, after the emergency fault occurs, the steady-state active power of ESCs remains unchanged. Figure 6B shows that the power of ESCs is rapidly increasing under the effect of the droop control, which effectively supports the system frequency after the emergency fault. The above results verify the effectiveness of the FEC strategy, considering the participation of the ESCs.

This section verifies the optimality of the FEC control strategy. For the two control objectives, let the ESCs adopt the optimal droop coefficients and the average droop coefficients, respectively. Under the same emergency fault setting, the active power of ESCs is output to verify the control objectives. The active power of ESCs under control objective I is shown in Figure 7, and the active power of ESCs under control objective II is shown in Figure 8.

According to Figure 7, since the active power upper bounds of the three ESCs are set to be the same, the power regulation margins of the three ESCs satisfy M 2 e > M 3 e > M 1 e . Figure 7A shows that when ESCs adopt the optimal droop coefficients, the power regulation satisfies u 2 e > u 3 e > u 1 e . Figure 7B shows that when ESCs adopt the average droop coefficients, the power regulation of each ESC is the same. Therefore, the optimal droop coefficients satisfy control objective I; i.e., the ESC with a larger power regulation margin provides more power support.

As shown in Figure 8, when considering control objective II, the SoC of the three ESCs satisfy S 2 e > S 1 e > S 3 e . Figure 8A shows that when ESCs adopt the optimal droop coefficients under control objective II, the power regulation satisfies u 2 e > u 1 e > u 3 e . Figure 8B shows that when the average droop coefficients are adopted for ESCs, the power regulation of each ESC is identical. As a result, the optimal droop coefficients satisfy control objective II; i.e., the ESC with larger SoC provides more power support. In summary, the optimality of the FEC strategy is verified, and the design method of the optimal droop coefficients proposed in this paper can be applied to different control objectives, which illustrates the generality of the design method.