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Adaptive H event-triggered load frequency control in islanded microgirds with limited spinning reserve constraints

Abstract

Using an islanded microgrid (MG) with large-scale integration of renewable energy is the most popular way of solving the reliable power supply problem for remote areas and critical electrical users. However, compared with traditional power systems, the limited spinning reserves and network communication bandwidth may cause weak frequency stability in the presence of stochastic renewable active outputs and load demand fluctuations. In this paper, an adaptive event-triggered control (ETC) strategy for a load frequency control (LFC) system in an islanded MG is proposed. First, a bounded adaptive event-triggered communication scheme is designed. This not only saves on network resources, but also ensures that the control center has a sensitive monitoring ability for the MG operating status when the frequency deviations have been effectively damped. Secondly, by fully considering the spinning reserve constraints and uncertain communication delays, the LFC system is described as a nonlinear model with saturation terms. Design criteria for ETC parameters are strictly deduced based on Lyapunov stability theory. Finally, an ETC parameter optimization algorithm based on random direction search is developed to reconcile the bandwidth occupancy and control performance. The effectiveness of the proposed method is verified in an MG test system.

1 Introduction

Increased attention on carbon emissions has promoted the development of distributed generation technologies with sun and wind as the energy sources [1, 2]. Correspondingly, a microgrid (MG) which integrates large-scale distributed generators has become an important approach towards providing reliable power supply service for remote areas or sensitive loads [3,4,5]. However, because of the uncertainties in distributed generation and load demand fluctuations, the power supply and demand in an MG are often unmatched, which can cause an adverse effect on frequency stability [6,7,8]. Therefore, an MG control center is usually equipped with a load frequency control (LFC) system to regulate the active power outputs of generators according to the frequency deviation so that the power supply–demand balance can be maintained [9, 10].

It should be mentioned that compared to traditional power systems with large-capacity synchronous generators, the frequency stability in an MG is more vulnerable to the double challenges from physical and cyber layers. An MG is generally equipped with small-capability and low-inertia generators [11, 12], so that under the same level of power supply–demand imbalance, the frequency deviation in an MG is larger and the corresponding recovery time is longer than in a conventional power system [13, 14]. Hence, the use of energy storage systems (ESSs) to provide auxiliary frequency regulation services has attracted attention. For example, an MG frequency regulation scheme considering electric vehicle clusters is proposed in [15], where the delay-independent control design criteria are strictly deduced based on Lyapunov stability theory. A delay-dependent sliding LFC strategy for an islanded MG containing ESSs and diesel generators is designed [16]. In the above works, the inverters in ESSs mainly adopt a constant-power control scheme with no inertia support for the MG. Therefore, some have proposed the virtual synchronous generator (VSG) control method for the inverters. By designing the algorithm to simulate the characteristics of synchronous generators, the inverters can provide extra inertia support for the MG [17]. A frequency regulation framework with VSG controlled ESSs is proposed to provide ancillary services in [18].

On the other hand, the MG control center must rely on the communication network to transmit control instructions and operating status with the equipment participating in the LFC. This will inevitably bring in uncertain time delay-induced problems [19]. Under the weakening inertia trend, the uncertain time delays will further enlarge the frequency deviations [20]. While the periodic triggered control (PTC) scheme is adopted in the above literature, the main drawback of PTC is that the operating status and control instructions are required to be transmitted over the network at each sampling period. This will cause a heavy communication burden in the MG. Therefore, some have proposed event triggered control (ETC) schemes, where the communication interaction is only triggered if the control performance has degenerated to under a preset threshold. The main objective of ETC is to reduce the communication frequency along with sacrificing partial dynamic performance [21,22,23]. With consideration of uncertain time delays, reference [24] deduces the ETC parameter criteria with a fixed triggered threshold by constructing a Lyapunov function. To further reduce the bandwidth occupancy, an adaptive event-triggered communication scheme is proposed where the triggered threshold is adaptively increased after frequency deviations have been well damped [25]. However, in existing adaptive ETC research, the exponential increase mechanism for adjusting the triggered threshold is usually adopted, while the upper bound of triggered threshold is restricted. That means that the triggered difficulty will rise rapidly when the frequency deviations start to be effectively damped. However, higher triggered difficulty will cause the MG control center to lose the monitoring sensitivity for operating status of the LFC system, resulting in slow recovery speed of frequency deviations. For a practical MG, continuous frequency deviation is deleterious to its safe and economic operation [26].

Most existing event-triggered LFC strategies have not taken the spinning reserve constraints of the MG into consideration. Spinning reserve is the maximum adjustable releasing or absorbing capability of the MG to maintain frequency stability. This is generally 10–20% of the total rated capacity of the MG. When the power supply–demand imbalance exceeds the spinning reserve limitation, the generation devices can only participate in LFC with their maximum adjustable capabilities. In other words, the power supply–demand imbalance can only be finitely reduced and the frequency deviations cannot be damped to zero. In this case, the MG control center should quickly adjust the control instruction to the maximum spinning reserve to dampen the frequency deviations as much as possible while avoiding the frequency deviation exceeding the allowable range. The current ETC schemes without considering the spinning reserve constraints may exhibit frequency over-limitation risk or a slow recovery problem [27].

Based on the above assessments, a bounded adaptive event-triggered LFC strategy considering spinning reserve constraints in islanded MGs is proposed in this paper, and the main contributions are as follows:

  1. 1.

    By fully taking the spinning reserve constraints into consideration, the networked LFC system with auxiliary VSG controlled ESSs is described as a nonlinear saturation model. Compared with current modelling methods without considering the spinning reserve constraints, the established model can exactly reflect the frequency dynamics of the actual MG.

  2. 2.

    A bounded adaptive ETC strategy is developed. Compared with the existing unbounded adaptive ETC methods, the triggered threshold in the proposed method is limited within a certain range to guarantee a sensitive monitoring ability of the MG operating status. By constructing a Lyapunov function considering time delays and spinning reserve limitations, the ETC parameter constraints with H damping performance to external power uncertainties are deduced. In addition, an ETC parameter optimization algorithm considering both cyber and control performances is developed.

The rest of the paper is organized as follows. Section 2 presents the LFC framework of islanded MG with limited spinning reserves. The design criteria and optimization algorithm are given in Section 3. Case studies are conducted in Section 4 to illustrate the effectiveness of the proposed method. Section 5 draws the conclusions.

2 LFC model in islanded MG with limited spinning reserves

2.1 LFC system with auxiliary VSG-controlled ESSs

As shown in Fig. 1, the power sources in an islanded MG contain wind turbines (WTs), photovoltaic plants (PVs), diesel generators (DGs) and ESSs. Considering weather unpredictability, the WTs and PVs generally do not participate in frequency regulation while injecting active power into the MG by adopting maximum power point tracking (MPPT) technology. The frequency regulation task is usually undertaken by the DGs and ESSs because of their controllable characteristics.

Fig. 1
figure 1

Networked LFC system

The detailed LFC mechanism in an islanded MG is as follows. The fluctuations of renewable generations or load demands will cause imbalance between the mechanical torque (produced by the prime motor and imposed on the rotating shaft of the synchronous generator) and the electromagnetic torque generated by the synchronous generator. Correspondingly, the rotational angular velocity of the synchronous generator (i.e., the frequency of the MG) will deviate from the rated value. In this case, the DG will adjust the mechanical power output by the primary motor to realize power supply–demand re-balance at the rated frequency point.

The above dynamic process can be described as a first-order inertial model. For a given power supply–demand imbalance, the frequency deviation and restoration speed are positively correlated to the damping and inertia of the synchronous generator in the DGs, respectively [19]. However, it should be noted that a high proportion of renewable generation integrated into the MG will reduce the installed capacities of the DGs equipped with synchronous generators. In this case, the frequency deviation, rate of change of frequency (RoCoF), and restoration speed will be enlarged. Therefore, in this paper, to provide auxiliary inertia and damping support for the MG, a VSG control strategy, which simulates the rotating characteristics of the synchronous generator, is adopted in the ESSs [28].

Assume that the amounts of DG and ESS in the islanded MG are N and M, respectively. Let H and D be the equivalent rotational inertia and damping coefficient of the MG. The relationships between (H, D) and each generation device are given by:

$$H = \sum\limits_{n = 1}^{N} {H_{{{\text{DG}}n}} \times \frac{{S_{{{\text{DG}}n}} }}{S}} + \sum\limits_{n = 1}^{M} {H_{{{\text{ESS}}n}} \times \frac{{S_{{{\text{ESS}}n}} }}{S}}$$
(1)
$$D = \sum\limits_{n = 1}^{N} {D_{{{\text{DG}}n}} \times \frac{{S_{{{\text{DG}}n}} }}{S}} + \sum\limits_{n = 1}^{M} {D_{{{\text{ESS}}n}} \times \frac{{S_{{{\text{ESS}}n}} }}{S}}$$
(2)

where HDGn and HESSm are the inertia of the nth DG and mth VSG-controlled ESS, respectively, and DDGn and DESSm are the corresponding damping coefficients. SDGn, SESSm, and S are the rated capabilities of the nth DG, mth VSG-ESS and the total MG, respectively. The dynamics of the LFC system in an islanded MG with VSG-controlled ESSs can be described as [29–31]:

$$\left\{ \begin{gathered} \dot{x}(t) = Ax(t) + Bu(t) + Ew(t) \hfill \\ y(t) = Cx(t) \hfill \\ \end{gathered} \right.$$
(3)

where \(A = \left[ {\begin{array}{*{20}c} { - \frac{D}{H}} & {A_{12} } & 0 & {A_{14} } & 0 \\ 0 & {A_{22} } & {A_{23} } & 0 & 0 \\ {A_{31} } & 0 & {A_{33} } & 0 & 0 \\ 0 & 0 & 0 & {A_{44} } & 0 \\ 1 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\), \(B = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ {B_{3} } \\ {B_{4} } \\ 0 \\ \end{array} } \right]\), \(E = \left[ {\begin{array}{*{20}c} { - \frac{1}{H}} \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{array} } \right]\),\(A_{12} = \left[ \frac{1}{H} \right]_{1 \times N}\), \(A_{14} = \left[ \frac{1}{H} \right]_{1 \times M}\), \(A_{22} = - A_{23} = {\text{diag}}\{ - \frac{1}{{T_{{{\text{t}}n}} }}\}\), \(A_{31} = \left[ { - \frac{1}{{T_{{{\text{g}}n}} R_{{{\text{DG}}n}} }}} \right]_{N \times 1}\), \(A_{33} = {\text{diag}}\{ - \frac{1}{{T_{{{\text{g}}n}} }}\}\), \(A_{44} = {\text{diag}}\{ - \frac{1}{{T_{{{\text{ESS}}m}} }}\}\), \(B_{3} = \left[ {\frac{{\alpha_{n} }}{{T_{{{\text{g}}n}} }}} \right]_{N \times 1}\), \(B_{4} = \left[ {\frac{{\beta_{m} }}{{T_{{{\text{ESS}}m}} }}} \right]_{M \times 1}\), C = [1, 0, 0, 0, 0], and x(t) = [Δf, ΔPDG1, …, ΔPDGN, ΔPv1, …, ΔPvN, ΔPref ESS1, …, ΔPref ESSM, ΔPESS1, …, ΔPESSM, ∫Δf]T. Δf is the frequency deviation, ΔPDGn (n = 1, 2, …, N) and ΔPESSm (m = 1, 2, …, M) are the active power output increments of the nth DG and mth ESS, respectively. ΔPvn is the valve opening variation of the governor in the nth DG, and ΔPref ESSm is the active power output instruction of the mth ESS. u is the instruction of the spinning reserve injected into the MG, w(t) = ΔPWT + ΔPPVPD represent the external power uncertainties, ΔPWT and ΔPPV are the outputs of WTs and PVs, respectively, and ΔPD are the load demands. RDGn is the droop coefficient of the DG, TESSn is the time constant of the inverter in the mth ESS, whereas Tgn and Ttn are the time constants of the governor and diesel engine, respectively. αn and βm are the participation factors of the DG and ESS satisfying Σαn + Σβm = 1.

2.2 Bounded adaptive ETC scheme

To maintain power supply and demand balance at the rated frequency point, the MG control center usually adjusts the control instructions (i.e., the spinning reserve injected into the MG) according to the current operating status of the LFC systems. Clearly, the transmission of control instructions and operating status must rely on the communication network, so the inevitable transmission delays may adversely affect the stability of the LFC system or even cause an instability accident. To reduce the dependence on the network and guarantee a sensitive monitoring ability to the LFC system simultaneously, the following bounded adaptive event-triggered communication scheme is proposed:

$$t_{h + 1} T_{{\text{s}}} = t_{h} T_{{\text{s}}} + \min \{ l|e^{{\text{T}}} (i_{l} T_{{\text{s}}} )\Phi e(i_{l} T_{{\text{s}}} ) \ge \delta (t_{h} T_{{\text{s}}} )x^{{\text{T}}} (t_{h} T_{{\text{s}}} )\Phi x(t_{h} T_{{\text{s}}} )\}$$
(4)

where thTs is the triggered moment, ilTs = thTs + lTs, e(ilTs) = x(ilTs)-x(thTs), Φ is the performance weighting matrix, and Ts is sampling period. δ(thTs) = min{δM, max{δm, νδ(th−1Ts)} is the adaptive triggered threshold. The coefficient ν is given by:

$$\nu = \left\{ {\begin{array}{*{20}l} {0{, }||y(t_{h + 1} T_{{\text{s}}} )|| \ge ||y(t_{h} T_{{\text{s}}} )||} \hfill \\ {1 - \frac{2\mu }{\pi }{\text{atan}} \left( {\frac{{||y(t_{h + 1} T_{{\text{s}}} )|| - ||y(t_{h} T_{{\text{s}}} )||}}{{||y(t_{h} T_{{\text{s}}} )||}}} \right){\text{, others}}} \hfill \\ \end{array} } \right.$$
(5)

where μ > 0 and δm, δM (0, 1), δm ≤ δM. According to (5), the triggered threshold is limited within [δm, δM].

Remark 1

Equation (4) shows that only when the weighting value of the difference between the sampled system status and the last triggered one is larger than the pre-set value then, and only then, can the current system status be transmitted to the control center through the network. If δm = δM, the bounded adaptive event-triggered communication scheme (4) is equivalent to a fixed-triggered threshold scheme. In particular, if δm = δM = 0, the proposed bounded adaptive ETC scheme is equivalent to a periodical triggered one. In addition, when the system status of the LFC starts to be damped, the increasing rate of triggered threshold is characterized by the coefficient ν.

The state feedback control scheme is adopted in this paper. Since only the sampled system status satisfying the triggered condition (4) can be transmitted to the control center through the network, the control instruction update intervals are prolonged compared to the traditional periodic triggered control methods. In other words, the control instructions remain unchanged during two neighboring triggered moments. Hence, the ideal control instruction of spinning reserve injected into the MG satisfying u(t) = Kx(thTs), t [thTs + τ th, th+1Ts + τ th+1], where K and \(\tau_{{t_{h} }} \in [0, \, \overline{\tau }]\) are the controller gain and transmission delay, respectively. Then the interval of two neighboring triggered moments can be divided into a series of subsets as \(\Omega_{h} = \cup_{l = 0}^{{t_{h + 1} - t_{h} - 1}} \Omega_{h}^{l}\), where \(\Omega_{h}^{l} = [i_{h} T_{s} + \tau_{{i_{h} }} ,(i_{h} + 1)T_{s} + \tau_{{i_{h} + 1}} )\) and ih = tkh + lh. Defining τ(t) = t-ihTs satisfying \(0 \le \tau (t) \le \overline{\tau } + T_{{\text{s}}} \triangleq d\), the ideal control instruction of spinning reserve injected into the MG is equivalent to u(t) = K[x(t-τ(t))-e(ihTs)], t\(\Omega_{h}^{l}\).

However, because of the generation rate constraints of DGs and capacity limitations of inverters, the adjustable spinning reserves of the MG are always limited. If the power supply–demand imbalance exceeds the maximum spinning reserve constraints, the imbalance can only be finitely reduced. In this case, each generation device participated in the LFC can only provide the maximum regulation capacity (i.e., inject external power into or absorb redundant power from the MG). Therefore, the actual spinning reserve provided for the LFC system can be described as the following nonlinear saturation model:

$$\begin{aligned} u(t) & = \left\{ {\begin{array}{*{20}l} {\Delta P_{r\max } , \, Kx(t - \tau ) \ge \Delta P_{r\max } } \hfill \\ {Kx(i_{h} T_{s} ), \, - \Delta P_{r\max } \le Kx(t - \tau ) \le \Delta P_{r\max } } \hfill \\ { - \Delta P_{r\max } , \, Kx(t - \tau ) \le - \Delta P_{r\max } } \hfill \\ \end{array} } \right. \\ & = K[x(t - \tau (t)) - e(i_{h} T_{s} )] + \varphi (Kx(i_{h} T_{s} )) \\ \end{aligned}$$
(6)

where ΔPrmax is the maximum allowable spinning reserve in the MG, and φ(·) is the function describing nonlinear saturation characteristics of spinning reserve constraints. Finally, the close-loop dynamics of the LFC system in an islanded MG adopting the proposed unbounded adaptive event-triggered communication scheme (4) can be described as:

$$\dot{x}(t) = Ax(t) + BK[x(t - \tau (t)) - e(i_{h} T_{s} )] + Ew(t) + B\varphi (Kx(i_{h} T_{s} ))$$
(7)

Remark 2

The nonlinear saturation item φ(Kx(ihTs)) can be explained from Fig. 2. The abscissas of the intersection points of φ(Kx(ihTs)) and Kx(ihTs) are the maximum allowable spinning reserves of the MG. The dashed line represents the function g(Kx(ihTs)) = -λKx(ihTs) with λ [0, 1]. Let − ΔPλ rmax, ΔPλ rmax be the abscissas of the intersection points of g(Kx(ihTs)) and φ(Kx(ihTs)), satisfying ΔPλ rmax = ΔPrmax/(1 − λ). From Fig. 2, it can be seen that [− ΔPrmax, ΔPrmax]  [− ΔPλ rmax, ΔPλ rmax]. Clearly, there are:

$$\left\{ \begin{gathered} - \lambda Kx(i_{h} T_{s} ) \le \varphi (Kx(i_{h} T_{s} )) \le 0, \, 0 \le Kx(i_{h} T_{s} ) \le \Delta P_{r\max } \hfill \\ 0 \le \varphi (Kx(i_{h} T_{s} )) \le - \lambda Kx(i_{h} T_{s} ), \, - \Delta P_{r\max } \le Kx(i_{h} T_{s} ) \le 0 \hfill \\ \end{gathered} \right.$$
(8)
Fig. 2
figure 2

Saturation model of spinning reserve constraints

Hence, for an arbitrary positive defined matrix Ξ, the following inequality always holds:

$$\varphi^{{\text{T}}} (Kx(i_{h} T_{s} ))\Xi (\varphi (Kx(i_{h} T_{s} )) + \lambda Kx(i_{h} T_{s} )) \le 0$$
(9)

3 Optimal design for ETC parameters

For the LFC system in the MG, the ETC parameters should satisfy the following three requirements:

  1. 1.

    The closed-loop LFC system described in (7) should be asymptotically stable if the external power uncertainties w(t) = 0.

  2. 2.

    The LFC system should have an H damping performance to the external power uncertainties, i.e., the output of the LFC system should satisfy ||y(t)||2 ≤ γ||w(t)||2, where w(t) L2[0, ∞) and γ > 0 is the damping coefficient.

  3. 3.

    To guarantee safe operation of the MG system, the frequency deviation under uncertain time delays and external power fluctuations must be limited within the allowable ranges.

In this section, the constraints of ETC parameters are first deduced by adopting Lyapunov stability theory. Then an optimization algorithm for the ETC parameters is developed to improve the dynamic performance under uncertain communication delays and external power fluctuations.

3.1 Design criteria of ETC parameters

Theorem 1

(Closed-loop stability criteria under spinning reserve constraints) For given scalars ρ, γ, δM, λ, d > 0, if there exist matrices X, Ξ and positive definite matrices P, Q, R, Φ satisfying (10) and (11), then the closed-loop LFC system (7) with the proposed bounded adaptive event-triggered communication scheme (4) is H asymptotically stable under uncertain communication delays and external power fluctuations.

$$\left[ {\begin{array}{*{20}c} R & X \\ {X^{{\text{T}}} } & R \\ \end{array} } \right] \ge 0$$
(10)
$$\left[ {\begin{array}{*{20}l} {\Omega_{11} } \hfill & {\Omega_{12} } \hfill & {\Omega_{13} } \hfill & {\Omega_{14} } \hfill & {\Omega_{15} } \hfill & {\Omega_{16} } \hfill & {\Omega_{17} } \hfill & {\Omega_{18} } \hfill \\ * \hfill & {\Omega_{22} } \hfill & {\Omega_{23} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ * \hfill & * \hfill & {\Omega_{33} } \hfill & {\Omega_{34} } \hfill & 0 \hfill & {\Omega_{36} } \hfill & {\Omega_{37} } \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & {\Omega_{44} } \hfill & 0 \hfill & {\Omega_{46} } \hfill & {\Omega_{47} } \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & {\Omega_{55} } \hfill & 0 \hfill & {\Omega_{57} } \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & {\Omega_{66} } \hfill & {\Omega_{67} } \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & {\Omega_{77} } \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & {\Omega_{88} } \hfill \\ \end{array} } \right] \le 0$$
(11)

where Ω11 = Q − R + PA + ATP, Ω12 = XT, Ω13 = PBK − XT + R, Ω14 = − PBK, Ω15 = PE, Ω16 = PB, Ω17 = ATP, Ω18 = CT, Ω22 = − Q − R, Ω23 = R − X, Ω33 = − 2R + XT + X + δMΦ, Ω34 = − δMΦ, Ω36 = − λKTΞ, Ω37 = KTBTP, Ω44 = − Φ + δMΦ, Ω46 = λKTΞ, Ω47 = − KTBTP, Ω55 = − γ2I, Ω57 = ETP, Ω66 = − 2Ξ, Ω67 = BTP, Ω77 = ρ2d2R-2ρP, Ω88 = − I.

Proof

Choose the following Lyapunov function:

$$V(t) = x^{{\text{T}}} (t)Px(t) + \int_{t - d}^{t} {x^{{\text{T}}} (s)Qx(s){\text{d}}s} + d\int_{ - d}^{0} {\int_{t + r}^{t} {\dot{x}^{{\text{T}}} (s)R\dot{x}(s){\text{d}}s{\text{d}}r} }$$
(12)

According to event-triggered communication scheme (4), eT(ilTse(ilTs) ≤ δMxT(thTsx(thTs) holds during two neighboring triggered moments. Applying (9), Jensen's inequality [32] and reciprocally convex theorem [33], if there exists a matrix X with suitable dimensions, the time derivative of V(t) satisfies:

$$\begin{aligned} \dot{V}(t) & \le \dot{x}^{{\text{T}}} (t)Px(t) + x^{{\text{T}}} (t)P\dot{x}(t) + x^{{\text{T}}} (t)Qx(t) - x^{{\text{T}}} (t - d)Qx(t - d) \\ & \quad + d^{2} \dot{x}^{{\text{T}}} (t)R\dot{x}(t) - \left[ {\begin{array}{*{20}c} {x(t - \tau (t)) - x(t - d)} \\ {x(t) - x(t - \tau (t))} \\ \end{array} } \right]^{{\text{T}}} \left[ {\begin{array}{*{20}c} R & X \\ {X^{{\text{T}}} } & R \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {x(t - \tau (t)) - x(t - d)} \\ {x(t) - x(t - \tau (t))} \\ \end{array} } \right] \\ & \quad + y^{{\text{T}}} (t)y(t) - \gamma^{2} w^{{\text{T}}} (t)w(t) + \delta_{M} x^{{\text{T}}} (t - \tau (t))\Phi x(t - \tau (t)) \\ & \quad - e^{{\text{T}}} (i_{l} T_{{\text{s}}} )\Phi e(i_{l} T_{{\text{s}}} ) - 2\varphi^{T} \left( {Kx\left( {i_{h} T_{{\text{s}}} } \right)} \right)\Xi (\varphi \left( {Kx\left( {i_{h} T_{{\text{s}}} } \right)} \right) + \lambda Kx\left( {i_{h} T_{{\text{s}}} } \right)) \\ & \quad - y^{{\text{T}}} (t)y(t) + \gamma^{2} w^{{\text{T}}} (t)w(t) \\ \end{aligned}$$
(13)

Defining ξ(t) = [xT(t), xT(t-d), xT(t-τ(t)), eT(ihTs), wT(t), φT(Kx(ihTs))]T, there are:

$$\begin{aligned} & x(t) = [I \, \;0 \, \;0 \, \;0 \, \;0\; \, 0]\xi (t) = e_{1} \xi (t) \\ & \dot{x}(t) = [A\; \, 0 \, \;BK\; \, - BK \, \;H\; \, B]\xi (t) = e_{2} \xi (t) \\ & x(t - d) = [0 \, \;I \, \;0 \, \;0\; \, 0 \, \;0]\xi (t) = e_{3} \xi (t) \\ & x(t - \tau (t)) = [0 \, \;0\; \, I \, \,0 \, \;0 \, \;0]\xi (t) = e_{4} \xi (t) \\ & e(i_{l} T_{s} ) = [0\; \, 0\; \, 0 \, \;I\; \, 0\; \, 0]\xi (t) = e_{5} \xi (t) \\ & w(t) = [0 \, \;0\; \, 0\; \, 0 \, I \, \;0]\xi (t) = e_{6} \xi (t) \\ & \varphi (Kx(i_{h} T_{s} )) = [0\; \, 0\; \, 0\; \, 0\; \, 0\; \, I]\xi (t) = e_{7} \xi (t) \\ & y(t) = [C\; \, 0\; \, 0\; \, 0\; \, 0\; \, 0]\xi (t) = e_{8} \xi (t) \\ \end{aligned}$$

If the inequality (14) holds, there is \(\dot{V}(t) \le - y^{{\text{T}}} (t)y(t) + \gamma^{2} w^{{\text{T}}} (t)w(t)\).

$$\begin{aligned} & e_{2}^{{\text{T}}} Pe_{1} + e_{1}^{{\text{T}}} Pe_{2} + e_{1}^{{\text{T}}} Qe_{1} - e_{3}^{{\text{T}}} Qe_{3} + d^{2} e_{2}^{{\text{T}}} Re_{2} \\ & \quad - \left[ {\begin{array}{*{20}c} {e_{4} - e_{3} } \\ {e_{1} - e_{4} } \\ \end{array} } \right]^{{\text{T}}} \left[ {\begin{array}{*{20}c} R & X \\ {X^{{\text{T}}} } & R \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {e_{4} - e_{3} } \\ {e_{1} - e_{4} } \\ \end{array} } \right] + e_{8}^{{\text{T}}} e_{8} - \gamma^{2} e_{6}^{{\text{T}}} e_{6} \\ & \quad + \delta_{M} e_{4}^{{\text{T}}} \Phi e_{4} - e_{5}^{{\text{T}}} \Phi e_{5} - 2e_{7}^{{\text{T}}} \Xi (e_{7} + \lambda K(e_{4} - e_{5} )) \\ & \le 0 \\ \end{aligned}$$
(14)

Clearly, \(\dot{V}(t) \le 0\) always holds when w(t) = 0. In addition, under the zero initial condition, \(\int_{0}^{ + \infty } {y^{{\text{T}}} (t)y(t){\text{d}}t} \le \gamma^{2} \int_{0}^{ + \infty } {w^{{\text{T}}} (t)w(t){\text{d}}t}\) holds, i.e., ||y(t)||2 ≤ γ||w(t)||2. This means that the closed-loop LFC system with spinning reserve constraints is H asymptotically stable. Moreover, inequalities (11) and (14) are equivalent. Thus this completes the proof.

Because of the existence of nonlinear items (e.g., Ω13, Ω14, Ω36), inequality (10) is in a bilinear matrix inequality (BLMI) form which is difficult to solve. Therefore, the congruent transformation approach is adopted in this paper. Define Λ = P−1, Ψ = Ω−1, KΛ = Γ, \(\tilde{Q} = \Lambda Q\Lambda\), \(\tilde{R} = \Lambda R\Lambda\), \(\tilde{\Phi } = \Lambda \Phi \Lambda\), \(\tilde{X} = \Lambda X\Lambda\). By pre- and post-multiplying both sides of (10) and (11) with diag{Λ, Λ} and diag{Λ, Λ, Λ, Λ, I, Ξ−1, Λ, I} respectively, the following corollary is obtained.

Corollary 1

For given scalars ρ, γ, δM, λ, d > 0, if there exist matrices X and positive definite matrices Λ, Ψ, \(\tilde{Q}\), \(\tilde{R}\) and \(\tilde{\Phi }\) satisfying (15) and (16), then the closed-loop LFC system (7) with the proposed bounded event-triggered communication scheme (4) is H-asymptotical. The controller gain is given by K = ΓΛ−1 and the performance weighting matrix is given by \(\Phi = \Lambda^{ - 1} \tilde{\Phi }\Lambda^{ - 1}\).

$$\left[ {\begin{array}{*{20}c} {\tilde{R}} & {\tilde{X}} \\ {\tilde{X}^{{\text{T}}} } & {\tilde{R}} \\ \end{array} } \right] \ge 0$$
(15)
$$\left[ {\begin{array}{*{20}l} {\tilde{\Omega }_{11} } \hfill & {\tilde{\Omega }_{12} } \hfill & {\tilde{\Omega }_{13} } \hfill & {\tilde{\Omega }_{14} } \hfill & {\tilde{\Omega }_{15} } \hfill & {\tilde{\Omega }_{16} } \hfill & {\tilde{\Omega }_{17} } \hfill & {\tilde{\Omega }_{18} } \hfill \\ * \hfill & {\tilde{\Omega }_{22} } \hfill & {\tilde{\Omega }_{23} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ * \hfill & * \hfill & {\tilde{\Omega }_{33} } \hfill & 0 \hfill & 0 \hfill & {\tilde{\Omega }_{36} } \hfill & {\tilde{\Omega }_{37} } \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & {\tilde{\Omega }_{44} } \hfill & 0 \hfill & {\tilde{\Omega }_{46} } \hfill & {\tilde{\Omega }_{47} } \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & {\tilde{\Omega }_{55} } \hfill & 0 \hfill & {\tilde{\Omega }_{57} } \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & {\tilde{\Omega }_{66} } \hfill & {\tilde{\Omega }_{67} } \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & {\tilde{\Omega }_{77} } \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & {\tilde{\Omega }_{88} } \hfill \\ \end{array} } \right] \le 0$$
(16)

where \(\tilde{\Omega }_{11} = \tilde{Q} - \tilde{R} + AP + PA^{{\text{T}}}\), \(\tilde{\Omega }_{12} = \tilde{X}^{{\text{T}}}\), \(\tilde{\Omega }_{14} = - \tilde{X}^{{\text{T}}} + \tilde{R} + B\Gamma\), \(\tilde{\Omega }_{15} = - B\Gamma\), \(\tilde{\Omega }_{16} = E\), \(\tilde{\Omega }_{17} = B\Psi\), \(\tilde{\Omega }_{18} = \Lambda C^{{\text{T}}}\), \(\tilde{\Omega }_{22} = - \tilde{R} - \tilde{Q}\), \(\tilde{\Omega }_{33} = - 2\tilde{R} + \tilde{X}^{{\text{T}}} + \tilde{X} + \delta_{M} \tilde{\Phi }\), \(\tilde{\Omega }_{34} = - \delta_{M} \tilde{\Phi }\), \(\tilde{\Omega }_{36} = - \lambda \Gamma^{{\text{T}}}\), \(\tilde{\Omega }_{37} = \Gamma^{{\text{T}}} B^{{\text{T}}}\), \(\tilde{\Omega }_{44} = - \tilde{\Phi } + \delta_{M} \tilde{\Phi }\), \(\tilde{\Omega }_{46} = \lambda \Gamma^{{\text{T}}}\), \(\tilde{\Omega }_{47} = - \Gamma^{{\text{T}}} B^{{\text{T}}}\), \(\tilde{\Omega }_{55} = - \gamma^{2} I\), \(\tilde{\Omega }_{57} = - H^{{\text{T}}}\), \(\tilde{\Omega }_{66} = - 2\Psi\), \(\tilde{\Omega }_{67} = \Psi B^{{\text{T}}}\), \(\tilde{\Omega }_{77} = \rho^{2} d^{2} \tilde{R} - 2\rho \Lambda\), \(\tilde{\Omega }_{88} = - I\).

3.2 Optimization algorithm

The design criteria of controller gain K and performance weighting matrix Φ under given H damping coefficient and upper triggered threshold bound δM have been discussed in Sect. 2.1. However, the dynamic performance and triggered frequency are contradictory designing requirements in event-triggered LFC systems. Specifically, more satisfied dynamic performance usually requires more frequent communications [34], which will increase the burden of the communication network in the MG. Therefore, an optimization algorithm is designed in this subsection to reconcile the conflict between bandwidth occupancy and dynamic control performance. The following objective function is chosen:

$$J(K,\delta_{m} ,\delta_{M} ,\Phi )|_{{|\Delta f| \le 0.1\% |\Delta f|_{\max } }} = \sigma_{1} \frac{{C_{{{\text{triggered}}}} }}{{C_{\Sigma } }} + \sigma_{2} \frac{{|\Delta f|_{\max } }}{{|\Delta f|_{{\max \_{\text{rated}}}} }}$$
(17)

where σ1 and σ2 are the weighting coefficients satisfying σ1 + σ2 = 1. Ctriggered is the triggered times during the frequency regulation process, and CΣ is the total number of sampling periods. |Δf|max is the maximum frequency deviation, and |Δf|max_rated is the allowable maximum frequency deviation.

From Corollary 1, the controller K and weighting matrix Φ can be directly obtained with given triggered threshold bounds (δm, δM). It means that the actual decision variables in objective function (17) are (δm, δM). For such a two-dimensional optimization problem, a random direction search algorithm is developed taking advantage of its fast convergence speed [35]. The proposed optimization algorithm includes the following steps:

Step 1 Input system matrices (A, B, C, E), allowable maximum spinning reserve ΔPrmax, upper delay bound d, maximum allowable frequency deviation |Δf|max_rated, H performance index γ, weighting coefficients (σ1, σ2), initial triggered threshold boundaries [δm0, δM0], (δm0 ≤ δM0), step size θ > 0, scaling factor {εmin, εmax}, (0 < εmin < 1, εmax > 1), termination threshold of iteration χ.

Step 2 Calculate the controller gain K0 and performance weighting matrix Φ0 under given δM0 by using Corollary 1. Then calculate the corresponding value (denoted as J0) of objective function (16).

Step 3 Randomly generate H unit vectors satisfying \(\{ (g_{mh} ,g_{Mh} )|\sqrt {g_{mh}^{2} + g_{Mh}^{2} } = 1, \, h = 1, \, 2, \ldots , \, H\}\). Then update the triggered threshold boundaries as follows:

$$\delta_{mh} = \delta_{m0} + \theta g_{mh}$$
(18)
$$\delta_{Mh} = \delta_{M0} + \theta g_{Mh}$$
(19)

Step 4 For each [δmh, δMh] (h′  {1, 2, …, H′}  {1, 2, …, H}) satisfying 0 ≤ δmh ≤ δMh ≤ 1, calculate the controller gain Kh and weighting performance matrix Φh by applying Corollary 1. Then calculate the corresponding objective function value (denoted as {Jh}), and obtain the optimal feasible solution [δml, δMl], l {1, 2, …, H′} with the minimal objective function value in current iteration.

Step 5 If Jl < J0, then δml → δm0, δMl → δM0 and θ = εmaxθ. Otherwise, keep δm0 and δM0 unchanged and let θ = εminθ.

Step 6 If |Jl-J0|< χ, then output [δm0, δM0], K0, and Φ0. Otherwise, repeat Steps 2–5.

4 Case studies and discussions

In this section, the effectiveness of the proposed unbounded adaptive ETC method is verified in an islanded MG test system shown in Fig. 3. The simulated MG system contains one DG and one ESS, and the simulation parameters are listed in Table 1. The DG and VSG-controlled ESS are responsible for damping the frequency deviation caused by the uncertain electrical load demand and renewable generation fluctuations to zero via the proposed LFC scheme.

Fig. 3
figure 3

Topology of the MG test system

Table 1 Simulation parameters

4.1 Control performance with different control schemes

The ETC parameters are optimized under the external step power disturbance of w(t) = − 0.2 p.u., as: [δm, δM] = [0.005, 0.01], K = [− 1.1459, − 0.2503, − 0.3366, − 0.7073, − 0.4465], and

\(\Phi = \left[ {\begin{array}{*{20}l} {76.7023} \hfill & {{18}{\text{.9487}}} \hfill & {{20}{\text{.4452}}} \hfill & {{30}{\text{.4929}}} \hfill & {{19}{\text{.5897}}} \hfill \\ {{18}{\text{.9487}}} \hfill & {9.3862} \hfill & {{3}{\text{.5811}}} \hfill & {{8}{\text{.5611}}} \hfill & {5.5355} \hfill \\ {{20}{\text{.4452}}} \hfill & {{3}{\text{.5811}}} \hfill & {10.7711} \hfill & {{8}{\text{.8567}}} \hfill & {{5}{\text{.5355}}} \hfill \\ {{30}{\text{.4929}}} \hfill & {{8}{\text{.5611}}} \hfill & {{8}{\text{.8567}}} \hfill & {19.4274} \hfill & {{8}{\text{.7429}}} \hfill \\ {{19}{\text{.5897}}} \hfill & {5.5355} \hfill & {{5}{\text{.5355}}} \hfill & {{8}{\text{.7429}}} \hfill & {7.1263} \hfill \\ \end{array} } \right]\).

The following schemes are chosen for comparison. It should be mentioned that all the compared schemes have not taken the comprehensive performance improvement into consideration.

  1. 1

    Periodic triggered control (PTC) without considering saturation [26];

  2. 2

    Periodic triggered control (PTC) considering saturation [27];

  3. 3

    Fixed-threshold ETC (FTETC) without considering saturation [23];

  4. 4

    Fixed-threshold ETC (FTETC) considering saturation designed based on Corollary 1;

  5. 5

    Unbounded adaptive ETC (UAETC) without considering saturation [25];

  6. 6

    Unbounded adaptive ETC (UAETC) considering saturation designed based on Corollary 1.

4.1.1 Step power fluctuation scenario

Without losing generality, we assume that the external power disturbance w(t) occurs with a − 0.2 p.u. step change at t = 0 s. The frequency deviations with different control schemes are shown in Fig. 4. The instructions of spinning reserves injected into the MG are shown in Fig. 5, while Fig. 6 illustrates the cumulative triggered times with different ETC schemes. To assess the control performance of the LFC system, the frequency deviation (|Δf|max), peak time (tpeak), settling time (tsettling), final triggered time and the integral absolute error (IAE) of frequency deviation are selected. Table 2 demonstrates the performance indices with different control schemes.

Fig. 4
figure 4

Frequency deviations with different control schemes (step power fluctuation scenario)

Fig. 5
figure 5

Instructions of spinning reserves injected into MG with different control schemes (step power fluctuation scenario)

Fig. 6
figure 6

Cumulative triggered times with different ETC schemes (step power fluctuation scenario)

Table 2 Performance indices of different control schemes (step power fluctuation scenario)

For the same ETC schemes, it can be seen from Figs. 4, 5, 6 and Table 2 that there exist significant differences of the cumulative triggered times and control performance between considering and without considering the spinning reserve constraints. Compared with the case adopting the FTETC scheme considering the spinning reserve constraints, the control center adopting the FTETC scheme without considering the constraints will be triggered an extra four times during the time interval of [0, 60 s]. The corresponding frequency deviation of the FTETC scheme without considering the spinning reserve constraints can be reduced by 2.21% compared with the case considering the constraints. However, opposite conclusions can be obtained for the UAETC schemes considering or without considering the spinning reserve constraints compared with FTETC schemes. The above results show that whether the saturation limitations are taken into consideration has no obvious positive or negative contribution on the cumulative trigger time and control performance when designing the ETC system. This is because the current ETC design methods only focus on the feasibility analysis of ETC parameters while usually ignoring the comprehensive performance improvement of bandwidth occupancy and dynamic control performance.

Moreover, since an optimization algorithm is developed to reconcile the bandwidth occupancy and dynamic control performance, the proposed bounded adaptive ETC scheme has the minimal peak time, settling time and IAE in all the control methods. In addition, according to Fig. 5, the proposed method can approach the active power vacancy faster. Compared with the two periodic triggered control schemes, the frequency deviations with the proposed method have risen by no more than 10.95% while the triggered times are decreased by 95.67%. Compared with the fixed-threshold ETC schemes, the triggered times with the proposed bounded adaptive ETC scheme are reduced by over 13.33% since the threshold value is properly improved after the frequency deviation is effectively damped. In addition, although the triggered times of the proposed method are 30% larger than the unbounded adaptive ETC schemes, the settling times are shortened by 13.68% at least. The reason is that in the unbounded adaptive ETC schemes, the triggered difficulty rises rapidly because of the continuously increased threshold value. In this case, the control center cannot obtain the operating status of the LFC system in time. Therefore, the simulation results show that the proposed method not only saves network resources effectively, but also guarantees a satisfactory recoverability for the frequency deviations.

4.1.2 Random over-saturating power fluctuation scenario

In Sect. 4.1.1, the effectiveness of the proposed method was tested in the case that the external power uncertainties do not exceed the allowable maximum spinning reserve. In this subsection, frequency control performance of each scheme is further simulated and analyzed in a random power fluctuation scenario where the fluctuation amplitude may exceed the spinning reserve constraints. The load demands and active renewable generations are presented in Fig. 7. One can see from Fig. 7 that the changes of the external power fluctuations at t = 20 s and 60 s are about 0.3 p.u. and − 0.65 p.u., respectively, which are larger than the maximum available spinning reserve.

Fig. 7
figure 7

Stochastic power variations

The frequency deviations and spinning reserves injected to the MG with different control schemes are demonstrated in Figs. 8 and 9, respectively. Figure 10 illustrates the triggered times and intervals with different ETC schemes. Variance (\(S_{\Delta f}^2 \)), final triggered times and the IAE of frequency deviations are selected to assess the control performance, as demonstrated in Table 3 for the different control schemes.

Fig. 8
figure 8

Frequency deviations with different control schemes (random power fluctuation scenario)

Fig. 9
figure 9

Instructions of spinning reserves injected into the MG with different control schemes (random power fluctuation scenario)

Fig. 10
figure 10

Triggered intervals with different ETC schemes (random power fluctuation scenario)

Table 3 Performance indices of different control methods (random power fluctuation scenario)

According to Figs. 7 and 8, since the external power uncertainties exceed the maximum spinning reserve, the MG system can only provide the maximum regulation capability for the LFC system during the time intervals as [20 s, 40 s] and [60 s, 80 s]. Correspondingly, the frequency deviations cannot be controlled to zero during the above two intervals. Compared with the periodical triggered control schemes, although the frequency deviation variance with the proposed method rises by 17.7%, the communication triggered times are reduced by 84.4%. In addition, compared with other ETC schemes, since the proposed method takes both control performance and bandwidth utilization into consideration, the frequency deviation variance and the IAE index are reduced by 13.11% and 6.077%, respectively.

Although Table 3 shows that the trigger times with unbounded adaptive ETC schemes are reduced by over 49% compared with the proposed method, the frequency deviations during [40 s, 60 s] are not controlled effectively with the unbounded adaptive ETC scheme and exceed the allowable range (0.2 Hz) during [80 s, 100 s]. This is because the triggered values in the unbounded adaptive ETC methods increase rapidly when the frequency deviations start to be damped. In this case, the triggered difficulty will be improved which causes the monitoring sensitivity of MG control center for the frequency deviations to be degenerated. Therefore, the proposed bounded adaptive ETC scheme has a better recovery ability for the frequency deviation than existing ETC methods.

4.2 Influence of ESSs and delays on control performance

In this subsection, the influence of ESSs and delays on control performance are further discussed from physical and cyber perspectives.

4.2.1 Influence of ESSs on frequency stability

To verify the contributions of VSG-controlled ESSs to the frequency stability of the islanded MG system, the following three scenarios are simulated and discussed.

Scenario 1 Only the diesel generators are used as the frequency regulation device.

Scenario 2 The diesel generators and ESSs with constant-power control scheme [16] are used as the frequency regulation devices.

Scenario 3 The diesel generators and VSG-controlled ESSs are used as the frequency regulation devices (the framework adopted in this paper).

Assuming that the LFC systems in the above three scenarios adopt the proposed bounded adaptive ETC scheme, all the above three scenarios are tested under the random external power fluctuations as shown in Fig. 6. Figure 11 shows the frequency responses in different scenarios. The triggered intervals are presented in Fig. 12, and Table 4 demonstrates the corresponding control performance indices.

Fig. 11
figure 11

Frequency deviations in different scenarios

Fig. 12
figure 12

Triggered intervals in different scenarios

Table 4 Performance indices in different scenarios

According to Fig. 11 and Table 4, there exists no significant difference of triggered times in the three scenarios. Compared with Scenario 1, the frequency deviation variance and IAE index with auxiliary VSG-controlled ESSs participated in LFC are reduced by 27.40% and 15.87%, respectively. Compared with Scenario 2, the above two performance indices with the proposed LFC framework are reduced by 13.11% and 6.67%, respectively. In addition, from Fig. 10, it can be seen that the frequency deviations in Scenario 1 exceed the allowable range during [20 s, 40 s] and [80 s, 100 s]. Similarly, in Scenario 2, the frequency over-limitation occurs during [80 s, 100 s]. For the proposed LFC framework, the frequency deviations are always within the permitted range (i.e., [− 0.2 Hz, 0.2 Hz]). This is because the VSG-controlled ESSs provide not only adjustable regulation capability but also inertia support for the LFC system. Therefore, the simulation results show that the proposed method has a stronger frequency stability in the presence of external power uncertainties.

4.2.2 Influence of delays on frequency stability

The robustness of the proposed method with different transmission delays is simulated and discussed. Assuming that the external power disturbance w(t) has a − 0.2 p.u. step change a t = 0 s, Figs. 13 and 14 illustrate the frequency deviations and triggered intervals with different bounded transmission delays, respectively.

Fig. 13
figure 13

Frequency deviations with different transmission delays

Fig. 14
figure 14

Triggered intervals with different transmission delays

It can be seen that with the increase of transmission delays, the frequency deviation restoration time is gradually prolonged. In addition, the IAE of frequency deviations rises by 22.65% when the maximum delay increases from 0 to 100 ms. This means that the transmission delays can lead to decreased frequency stability. However, since the optimization algorithm proposed in this paper takes minimizing the frequency deviation amplitude as one of the optimization objectives, the maximum frequency deviation under each delay scenario does not exceed the allowable range. The simulation results show that the proposed bounded adaptive event-triggered LFC scheme and the corresponding optimization algorithm can guarantee satisfactory control performance in heavy network load scenarios.

5 Conclusions

For the LFC issue in an islanded MG, a bounded adaptive event-triggered control scheme considering the spinning reserve constraints has been proposed in this paper. By designing a bounded adjustment mechanism for the triggered threshold, sensitive monitoring ability for the LFC systems and low dependence on network resources can be guaranteed simultaneously. The analytical relations among spinning reserve constraints, uncertain communication delays and ETC parameters have been deduced via constructing the Lyapunov function. Furthermore, an ETC parameter optimization algorithm has been developed to provide optimal balance and compromise between network bandwidth utilization and control performance. Simulation results demonstrate that frequency stability using the proposed method can be enhanced. In the future, the adaptive event-triggered LFC strategy considering spinning reserves, cyber-attacks and more control performance constraints will be further investigated.

Availability of data and materials

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Abbreviations

MG:

Microgrid

ETC:

Event-triggered control

LFC:

Load frequency control

VSG:

Virtual synchronous generator

ESS:

Energy storage system

BLMI:

Bilinear matrix inequality

PTC:

Periodic triggered control

FTETC:

Fixed-threshold ETC

UAETC:

Unbounded adaptive ETC

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Acknowledgements

Not applicable.

Funding

This work was supported in part by the National Natural Science Foundation of China under Grants 62173218, 61833011, 62103254, and the International Corporation Project of Shanghai Science and Technology Commission under Grant 21190780300.

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YZ and CP contributed to the conception and design of the study. All authors listed have made a substantial, direct and intellectual contribution to this paper for publication.

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Correspondence to Chen Peng.

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Zhang, Y., Peng, C. Adaptive H event-triggered load frequency control in islanded microgirds with limited spinning reserve constraints. Prot Control Mod Power Syst 8, 30 (2023). https://doi.org/10.1186/s41601-023-00303-z

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