**1. Introduction**

Load frequency control (LFC) plays a key role when it comes to measuring the power supply quality of a power system. Ensuring that the frequency is controlled at a fixed value or with small changes in its vicinity is a basic requirement of LFC [1]. In order to maintain the system frequency, some control techniques for power systems have been adopted in LFC such as an adaptive fuzzy logic approach [2], neutral network [3], and robust *H* ∞ control [4]. Di fferent from conventional energy sources, wind energy has the intrinsic intermittence and fluctuation, which will inevitably bring serious influence on the frequency regulation of a power system [5]. Due to the intermittence of wind power, the large-scale wind power grid operation will a ffect the stability and balance of power systems [6]. Recently, the LFC problem with wind power sources has attracted much attention. With more and more wind power integrated in power systems, the LFC issue of power systems has become more di fficult than before. Therefore, designing an advanced LFC strategy for the wind power generations is of significant value to ensure the stable operation of power systems under the stochastic disturbances of wind power and the random load variation. For multi-area power systems in the presence of wind turbines, a LFC design using the model predictive control (MPC) technique is proposed [7]. In [8], a linear active disturbance rejection control method was applied to power systems with high penetration of wind power. Under the condition of wind speed fluctuation, the linear active disturbance rejection technique has a more prominent control e ffect than the traditional control method in the doubly-fed wind turbines, which reduces the adjustment time and overshoot [9]. To solve the nonlinearities in the LFC issue of the interconnected power systems, the hybrid neuro-fuzzy scheme was applied in [10]. A low-frequency damping control strategy of a doubly-fed induction generator based on transient energy function analysis of oscillation was proposed in [11].

With the reform of power marketization, the scale of a power system gradually expands, and LFC needs to carry out wide-area information exchange or experience a large amount of data in non-dedicated communication network, which inevitably brings the problem of time delay [12]. A sliding mode control and a robust predictive control strategy for power systems with time-delay and uncertainties of parameter are presented in [13,14] respectively. The authors of [15,16] studied the impulsive control of a nonlinear dynamic system. Considering the time-varying delays, two impulsive control algorithms were designed for the islanded micro-grids in [17]. The delay correlation stability of a LFC scheme is studied by means of the Lyapunov-theory and linear matrix inequality (LMIs) techniques in [18]. The authors of [19] studied the LFC for power systems with communication delays via an event-triggered control method. In [20], a new criterion for the delay-related stability was proposed when the network multi-area LFC system was subjected to an unknown time variant exogenous load disturbance. Time delay not only reduces the control e ffect of the original LFC, but also makes the controller malfunction, causing the instability of the system and damaging the safe operation of the power grid. Therefore, designing a robust LFC strategy which can perfectly compensate for the influence of time delay becomes an increasingly valuable solution of the wind integrated power system [21].

In this paper, the influence of wind power integration on load frequency of a power system was studied, and the influence of communication delay on the whole system was also considered. An active disturbance rejection control (ADRC) based on equivalent input disturbances (EID) compensation for load frequency control was proposed for a wide integrated power system when communication delays were considered, applied to a two-area power system to dampen its low frequency oscillation. The disturbance information was obtained through the full-order state observer, and the disturbance estimator was designed to compensate for the disturbance. Thus, the disturbance rejection performance of the whole control system was improved.

The remaining sections of this paper are structured as follows: In Section 2, simplified wind turbine models for frequency studies are introduced. In Section 3, ADRC design strategies based on EID are discussed. Some case studies are introduced in Section 4, and the conclusions are drawn in Section 5.

## **2. System Modelling**

Figure 1 shows a two-area interconnected wind integrated power system with two conventional generator units in each area, one aggregated wind turbine model and two controllers based on EID. In the following analysis, the basic parameter description will be listed. The notation Δ indicates the deviation from the normal state.

Currently, the variable speed wind turbine (VSWT) is the most popular type of modern wind turbine. There is a more detailed description of VSWT in [22]. Figure 2 shows a simplified frequency response model of a wind turbine based on doubly-fed induction generator (DFIG).

The structure of this model can be described by the following equations [22]:

$$\mathbf{i}\_{q\mathbf{r}} = -\left(\frac{1}{T\_1}\right)\mathbf{i}\_{q\mathbf{r}} + \left(\frac{\mathbf{X}\_2}{T\_1}\right)V\_{q\mathbf{r}} \tag{1}$$

$$
\dot{w} = -\left(\frac{X\_3}{M\_I}\right)\dot{i}\_{l\bar{l}} + \left(\frac{1}{M\_I}\right)T\_m\tag{2}
$$

$$p\_c = wX\_3i\_{qr} \tag{3}$$

by linearizing, Equation (3) can be rewritten as:

$$p\_{\varepsilon} = w\_{\text{opt}} X\_3 i\_{\text{gr}} \tag{4}$$

$$T\_{\varepsilon} = i\_{\mathbb{q}^{\mathbb{s}}} = -\frac{L\_m}{L\_{\mathbb{s}\mathbb{s}}} i\_{\mathbb{q}^{\mathbb{p}}} \tag{5}$$

where *wopt* is the operating point of the rotational speed, *Te* is the electromagnetic torque, *Tm* is the mechanical power change, ω is the rotational speed, *Pe* is the active power of the wind turbine, *iqr* is the q-axis component of the rotor current, *Vqr* is the q-axis component of the rotor voltage and *Mt* is the equivalent inertia constant of the wind turbine.

The main parameters in Figure 2 can be observed from Table 1.

$$L\_0 = L\_{rr} + \frac{L\_{m}^2}{L\_{ss}}, \quad L\_{ss} = L\_{\star} + L\_{m\star}, \\ L\_{rr} = L\_{rs} + L\_{mr}$$

where *Lm* is the magnetizing inductance, *Rr* and *Rs* are the rotor and stator resistances, respectively.

*Lr* and *Ls* are the rotor and stator leakage inductances, respectively, *Lrr* and *Lss* are the rotor and stator self-inductances, respectively, *ws* is the synchronous speed.

Since each subsystem is connected by power flow through a tie line, a LFC system of each area of the two-area power system should not ignore the control of the interchange power and local frequency with the other control area. Therefore, we take the tie-line power signal into account in the dynamic LFC system model and describes a frequency model for any area *i* of *N* power system control areas with an aggregated generator unit in each area [11].

**Table 1.** Parameters for Figure 1.


**Figure 1.** A two-area wind integrated power system for equivalent input disturbances (EID) based load frequency control (LFC).

**Figure 2.** Simplified frequency responsemodel ofwind turbine base ondoubly-fedinduction generator (DFIG).

The LFC dynamic model of one area of the interconnected power system, shown in Figure 3, can be described as follows:

$$\begin{cases} \Delta \dot{f}\_{l} = -\frac{D\_{l}}{M\_{l}} \Delta f\_{l} + \frac{1}{M\_{l}} \Delta P\_{ml} - \frac{1}{M\_{l}} \Delta P\_{l\acute{e}l,l} + \frac{1}{M\_{l}} \Delta P\_{e} - \frac{1}{M\_{l}} \Delta P\_{\acute{m}l} \\\ \Delta P\_{\acute{m}l} = -\frac{1}{T\_{\acute{e}l}} \Delta P\_{ml} + \frac{1}{T\_{\acute{e}l}} \Delta P\_{\acute{m}l} \\\ \Delta \dot{P}\_{\acute{m}l} = -\frac{1}{K\_{l}T\_{\acute{g}l}} \Delta f\_{l} - \frac{1}{T\_{\acute{g}l}} \Delta P\_{\acute{m}l} - \frac{\Delta E(t-d)}{T\_{\acute{g}l}} + \frac{1}{T\_{\acute{g}l}} \Delta P\_{\acute{m}f} \\\ \Delta \dot{P}\_{l\acute{k}\acute{e},l} = 2\pi \left(\sum\_{j=1, j\neq l}^{N} T\_{lj} \Delta f\_{l} - \sum\_{j=1, j\neq l}^{N} T\_{l\acute{k}} \Delta f\_{l}\right) \\\ \Delta \dot{E}\_{l} = A \Sigma E\_{l} \end{cases} \tag{6}$$

 Additionally, Δ*Ei* is the area control error (*ACE*) integral control.

$$
\Delta E\_l = \int\_0^l ACE(s)ds\tag{7}
$$

For a multi-area LFC system, the *ACE* is essentially composed of a regional frequency deviation and a power deviation of the line, and its calculation formula is as follows:

$$ACE\_l = \Delta P\_{life\,\delta} + \beta\_l \Delta f\_l \tag{8}$$

where β*i* is the frequency deviation coefficient of the control area, Δ*fi* is the frequency deviation of area *i*, Δ*Ptie*,*<sup>i</sup>* is the tie-line power change of area *i*.

Δ*Vi* is the control area interface:

$$
\Delta V\_l = \sum\_{\substack{l=1,\,l\neq l}}^N T\_{l\bar{l}} \Delta f\_{\bar{l}} \tag{9}
$$

where Δ*f* is the frequency deviation, Δ*Pm* is the generator mechanical power deviation, Δ*Pv* is the turbine value position deviation, Δ*Pd* is the load deviation, *M* and *D* denote inertia moment and damping coefficient of generator, respectively, *Tg*, *Tch* and *R* denote the governor's time constant, turbine's time constant and speed drop, respectively.

Furthermore, the above equations can be combined in the following state-space model:

$$\begin{cases}
\dot{\mathbf{x}}\_i(t) = A\_i \mathbf{x}\_i(t) + A\_{di} \mathbf{x}\_i(t-d) + B\_i u\_i(t) + B\_{wi} w\_i(t) \\
y\_i(t) = \mathbb{C}\_i \mathbf{x}\_i(t)
\end{cases} \tag{10}$$

where

*xi*(*t*) = Δ*fi* Δ*Pmi* Δ*Pvi* Δ*Ei* Δ*Ptie*,*<sup>i</sup>* Δ*iqr* Δ*W T Ai* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ − *Di Mi* 1*Mi* 0 0 −1 *Mi* X3*Wopt Mi* 0 0 −1 *Tchi* 1*Tchi* 00 0 0 − 1 *RiTgi* 0 −1 *Tgi* 00 0 0 β*i* 0 0 01 0 0 2π *N j*=1,*ji Tij* 0 0 00 0 0 0 0 0 00 −1*T*1 0 0 0 0 00 −*X*<sup>3</sup> *Mt* 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, *Adi* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 000 0 000 000 0 000 000 −1 *Tgi* 000 000 0 000 000 0 000 000 0 000 000 0 000 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,*Bi* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0 0 0 0 1*Tgi* 0 0 0 0 0 0 *X*2 *T*1 0 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

$$B\_{\rm{\nu\dot{\imath}}} = \begin{bmatrix} \frac{-1}{M\_{\rm{\dot{\imath}}}} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{M\_{\rm{\dot{\imath}}}}\\ 0 & 0 & 0 & 0 & -2\pi & 0 & 0 \end{bmatrix}^{\rm{T}}$$

$$C\_{\rm{\dot{\imath}}} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}, \newline \boldsymbol{\nu}\_{\dot{\imath}} = \begin{bmatrix} \Delta P\_{\rm{\dot{\imath}}} & \Delta T\_{\rm{\dot{\imath}}} & \Delta V\_{\rm{\dot{\imath}}} \end{bmatrix}^{\rm{T}}$$

**Figure 3.** Dynamic model of one area of the interconnected environment.

#### **3. ADRC Design Based on EID Compensation**

This section is devoted to explaining the proposed EID-based ADRC design for the LFC. The original EID method cannot be used directly for systems with time delays, so we improved the state observer and extended the method to the time-delay system [23]. EID was originally proposed by She [24,25] and further developed by Liu [26–28]. The method is based on the idea that the effect of actual disturbance, *<sup>w</sup>*(*t*), on the output of a plant (Figure 4a) can be replaced by the disturbance, *we*(*t*), on the control input channel (Figure 4b). In Figure 4, *w*(*t*) and *we*(*t*) produced exactly the same outputs. Thus, the disturbance *we*(*t*) is defined as EID.

**Figure 4.** The concept of equivalent input disturbance, (**a**) original plant; (**b**) plant with EID.

Rewriting the plant (Equation (10)) as a plant with EID, we have

$$\begin{cases}
\dot{x}|t = Ax|t + A\_d x|t - d + B[u[t] + w\_c[t]]
\\
y|t = Cx|t
\end{cases} \tag{11}$$

Then, the system (Equation (11)) can be used to design ADRC. We constructed an EID-based closed-loop control system as in Figure 5. The system has five parts: the internal model, the state feedback controller, the disturbance estimator, the modified state observer and the control plant.

#### *3.1. Configuration of the EID-Based Time-Delay System*

In Figure 5, a new EID-based control system structure was established to achieve disturbance suppression of the wind integrated time-delay power system.

The following internal model

$$
\dot{\mathbf{x}}\_R(t) = A\_M \mathbf{x}\_M(t) + B\_M \Big| \Delta f\_{ref}[t] - y[t] \Big| \tag{12}
$$

is still used to ensure accurate tracking of the reference input. When Δ*fre f* is given, *AM* and *BM* can be directly determined.

A full-order time-delay observer is used to estimate the EID and reconstruct the state of the controlled object, we write the state-space representation of the observer as

$$\begin{cases} \dot{\overline{x}}|t = \Phi \overline{x}|t + A\_d \overline{x}|t - d + \Psi[y[t] - C\overline{x}[t]] + \Gamma u\_f|t \\ \overline{y}|t = T^{-1}\overline{x}|t \end{cases} \tag{13}$$

where %*x*(*t*) is the reconstruction state of *<sup>x</sup>*(*t*). The gain of this full-order time-delay observer is *L*.

Then, we design the state-feedback controller as

$$
\Lambda u\_f(t) = \mathbb{K}\_{\mathsf{M}} \mathsf{x}\_{\mathsf{M}}(t) + \mathsf{K}\_{\mathsf{N}} \overline{\mathfrak{x}}(t) \tag{14}
$$

From the above equation, we yield the disturbance estimation of the EID *w*<sup>ˆ</sup>(*t*) in Figure 5 as

$$\hat{w}(t) = B^{+}T^{-1}\Psi C[\mathbf{x}[t] - \overline{\mathbf{x}}[t]] + \boldsymbol{u}\_f(t) - \boldsymbol{u}(t) \tag{15}$$

where *B*<sup>+</sup> = *BTB* <sup>−</sup>1*B<sup>T</sup>*

Since the output contains measurement noises, we used a low-pass filter to select angular frequency bandwidth for disturbance estimation. The state-space equation of the filter is described as

$$\begin{cases}
\dot{x}\_N[t = A\_N x\_N] t + B\_N \dot{w}[t] \\
\overleftarrow{w}[t = C\_N x\_N] t
\end{cases} \tag{16}$$

where *xN*(*t*) is the state of filter, *w*%(*t*) is the filter disturbance estimation.

Thus, the new control law of the closed-loop control system is

$$
\mu(t) = \mu\_f(t) - \overline{w}(t) \tag{17}
$$

**Figure 5.** Configuration of the EID-based closed-loop control system.

#### *3.2. Optimal Design of Controller Parameters*

Time delay will influence the stability of the system. On account of inherent time delay, the characteristic equation of the system becomes infinitely dimensional. Thus, we propose the parameter of a controller design based on the LMI in this section.

Let Δ*fre f*(*t*) = 0, *w*(*t*) = 0. Then, the time-delay model (Equation (10)) is

$$\begin{cases}
\dot{x}(t) = Ax(t) + A\_d x(t-d) + Bu(t) \\
y(t) = Cx(t)
\end{cases} \tag{18}$$

As shown in Figure 4, there are four states, %*<sup>x</sup>*(*t*), <sup>Δ</sup>*x*(*t*), *xN*(*t*) and *xM*(*t*). Define

$$
\Delta \mathbf{x} = \mathbf{x}(t) - \overline{\mathbf{x}}(t) \tag{19}
$$

and describe the closed-loop system as

$$\boldsymbol{\varphi}(t) = \begin{bmatrix} \ \overleftarrow{\mathbf{x}}^T(t) & \Delta \mathbf{x}^T(t) & \mathbf{x}\_N^T(t) & \mathbf{x}\_M^T(t) \end{bmatrix}^T \tag{20}$$

Substituting Equation (19) into (13) yields

$$
\widetilde{\mathbf{x}}(t) = A\widetilde{\mathbf{x}}(t) + LC\Delta\mathbf{x}(t) + Bu\_f(t) + A\_d\widetilde{\mathbf{x}}(t-d) \tag{21}
$$

Combining Equations (13), (17)–(19), yields

.

$$
\Delta \dot{\mathbf{x}}(t) = [A - LC] \Delta \mathbf{x}(t) - B \mathbf{C}\_N \mathbf{x}\_N(t) + A\_d \Delta \mathbf{x}(t - d) \tag{22}
$$

Combining Equations (15) and (17), the filter is described as

$$\dot{\mathbf{x}}\_N(t) = B\_N B^+ L C \Delta \mathbf{x}(t) + (A\_N + B\_N \mathbf{C}\_N) \mathbf{x}\_N(t) \tag{23}$$

Substituting Equation (19) into (12), the internal model is obtained as:

$$\dot{\mathbf{x}}\mathbf{x}(t) = B\_N \mathbf{B}^+ L \mathbf{C} \Delta \mathbf{x}(t) + (A\mathbf{v} + B\mathbf{v}\mathbf{C}\mathbf{v})\mathbf{x}(t) \tag{24}$$

From Equations (21)–(24), the state-space representation of the control-loop system reconstructed according to EID in Figure 4, is as follows:

$$
\dot{\varphi}(t) = \overline{A}\varphi(t) + \overline{B}u\_f(t) + \overline{A}\_d \varphi(t-d) \tag{25}
$$

where

$$
\overline{A} = \begin{bmatrix}
 A & LC & 0 & 0 \\
 0 & A - LC & -BC\_N & 0 \\
 0 & B\_N B^+LC & A\_N + B\_N C\_N & 0 \\
 -B\_M C & -B\_M C & 0 & A\_M
\end{bmatrix},
$$

$$
\overline{A}\_d = \begin{bmatrix}
 A\_d & 0 & 0 & 0 \\
 0 & A\_d & 0 & 0 \\
 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0
\end{bmatrix}, \overline{B} = \begin{bmatrix}
 B^T & 0 & 0 & 0
\end{bmatrix}^T
$$

The state-feedback controller law is:

$$
\mu\_f = \overline{\mathbb{K}} \wp(t) \tag{26}
$$

where

> *K* = [ *KN* 0 0 *KM* ]

Substituting Equation (26) into (25) yields the general form of time-delay system:

$$
\dot{\varphi}(t) = \dot{A}\varphi(t) + \overline{A}\_d \varphi(t-d) \tag{27}
$$

where

$$A = \begin{bmatrix} A + BK\_N & LC & 0 & BK\_M \\ 0 & A - LC & -BC\_N & 0 \\ 0 & B\_N B^+ LC & A\_N + B\_N \mathbb{C}\_N & 0 \\ -B\_M \mathbb{C} & -B\_M \mathbb{C} & 0 & A\_M \end{bmatrix}$$

## *3.3. Stability Analysis*

Using the following Lemma 1, the time-delay power system stability analysis was carried out.

**Lemma 1.** *If there is a positive definite matrix Q and P, which makes the following LMI feasible, then the time-delay system (27) is asymptotically stable* [29,30].

$$\begin{bmatrix} P\hat{A} + \hat{A}^T P + Q & P\overline{A}\_d \\ \overline{A}\_d \,^T P & -Q \end{bmatrix} < 0 \tag{28}$$

Based on Lemma 1, the controller gain and a sufficient condition for power system stability are obtained as follows.

**Theorem 1.** *If there is a positive definite matrix X1, X11, X22, X3, X4, Y1, Y2, Y3 and Y4, and suitable dimension matrices Z1, Z2, and Z3, the following LMI is feasible. For a given positive parameter* α *and* γ*, the time-delay system (25) is asymptotically stable under the control law (26).*

$$
\begin{bmatrix}
\Phi & \Psi & X \\
& \Psi^T & -Y & 0 \\
X^T & 0 & -Y
\end{bmatrix} < 0\tag{29}
$$

where

$$\Phi = \begin{bmatrix} \Phi\_{11} & Z\_2 \mathbb{C} & 0 & \Phi\_{14} \\ \mathbb{C}^T Z\_2^{\top} & \Phi\_{22} & \Phi\_{23} & -X\_2 \mathbb{C}^T B\_{M^\top} \\ 0 & \Phi\_{23} \mathbb{T} & \Phi\_{33} & 0 \\ \Phi\_{14} \mathbb{T} & -B\_{\text{M}} \mathbb{C} X\_2^\top & 0 & \Phi\_{44} \end{bmatrix}.$$

$$\mathbb{T} = \begin{bmatrix} aA\_d Y\_1 & 0 & 0 & 0 \\ 0 & A\_d Y\_2 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}.$$

$$X = \begin{bmatrix} aX\_1 & 0 & 0 & 0 \\ 0 & X\_2 & 0 & 0 \\ 0 & 0 & X\_3 & 0 \\ 0 & 0 & 0 & \gamma X\_4 \end{bmatrix}, Y = \begin{bmatrix} Y\_1 & 0 & 0 & 0 \\ 0 & Y\_2 & 0 & 0 \\ 0 & 0 & Y\_3 & 0 \\ 0 & 0 & 0 & Y\_4 \end{bmatrix}$$

$$\Phi\_{11} = aAX\_1 + aX\_1A^\top + aBZ\_1 + aZ\_1^\top B^\top,$$

$$\Phi\_{12} = \gamma BZ\_3 - aX\_1\mathbb{C}^T B\_{M^\top}$$

$$\Phi\_{22} = AX\_2 + X\_2A^\top - Z\_2\mathbb{C} - \mathbb{C}^T Z\_2^\top,$$

$$\Phi\_{23} = -B\mathbb{C}X\_3 + \mathbb{C}^T Z\_2^\top B^\top + \mathbb{C}^T B\_{M^\top}$$

$$\Phi\_{33} = (A\_N + B\_N \mathbb{C})X\_3 + X\_3(A\_N + B\_N \mathbb{C})^\top,$$

$$\Phi\_{44} = \gamma A\_N X\_4 + \gamma X\_4 A\_M^\top.$$

and the singular-value decomposition of *X*2 is

$$X\_2 = V d\|q\_{\overline{\chi}}(X\_{11}, X\_{22} \| V^T)$$

The gain of the state feedback controller and the observer is

$$K\_{\rm N} = Z\_1 X\_1^{-1}, \\ K\_{\rm M} = Z\_3 X\_4^{-1}, \\ L = Z\_2 \, \text{LIS} X\_{\rm II}^{-1} \text{S}^{-1} \text{d}I^T \tag{30}$$

Additionally, *U* and *V* can be obtained from

$$\mathbf{C} = \mathbb{U}[\mathbf{S}, \mathbf{0}]V^T \tag{31}$$

where Equation (31) is a singular value decomposition expression of matrix *C*.

A detailed proof of Theorem 1 is given in [23], so we omitted the process of proof in this paper.

## **4. Case Studies**

In this section, the proposed ADRC design based on EID compensation is evaluated using MATLAB/SIMULINK software. The basic parameter description of the model is listed in Appendix A [31]. Furthermore, the different profiles of load variation are considered to test the performance of ADRC allocated to each area.

Firstly, the effectiveness of the proposed ADRC method is verified. We study the case that both areas are without ADRC to witness the impact of time delays on two areas of power system and the other case that both areas are equipped with ADRC to confirm the effectiveness of this method. Two-area time-delay power systems with wind farm are considered. It is worth noting that the communication delay is set to be 0.2 s [32,33]. When the hysteresis power system is subjected to random load disturbance as shown in Figure 6a, from Figure 6b we can see that the frequency deviation is large and oscillating, the frequency fluctuation is beyond [−1 1] Hz after 2.2 s since the fundamental frequency of a power network is 50 Hz, and even diverges. However, the frequency fluctuations can be damped in the range of [−0.5 0.5] × 10−<sup>3</sup> pu as shown in Figure 6c by ADRC based on EID compensation.

Next, we verify the superiority of the proposed method. The dynamic responses of wind farm time-delay power system equipped with PID controller and the proposed EID-based ADRC are shown in Figures 7 and 8. Figure 7 shows the dynamic response of frequency in Area 1 and Figure 8 shows the dynamic response of frequency in Area 2. By PID controller, the frequency is varied between [−1 1] × 10−<sup>3</sup> pu. However, compared with PID controller, the ADRC method based on EID compensation proposed in this paper has higher stability and faster speed. As shown in Figure 7c, Δ*f* is controlled within [−2 2.5] × 10−<sup>6</sup> pu. Therefore, EID-based ADRC method has better robustness than the PID method.

**Figure 6.** *Cont.*

**Figure 6.** Dynamic response of two areas, (**a**) random load variation; (**b**) frequency response of two areas without ADRC; (**c**) frequency response of two areas with ADRC.

**Figure 7.** *Cont.*

**Figure 7.** Dynamic response of Area 1, (**a**) load variation; (**b**) frequency deviations of Area 1 with different controller; (**c**) enlarged view under EID.

**Figure 8.** Dynamic response of Area 2, (**a**) load variation of Area 2; (**b**) frequency deviations of Area 2 with different controller; (**c**) enlarged view under EID.

When the system is disturbed by different random load as shown in Figure 9a, the dynamic performances are shown in Figure 9b–d. It can be found that the proposed strategy based on EID can enhance the frequency stability of each area and the control performance is better the PID control strategy.

**Figure 9.** Tie-line power response of a time-delay power system with wind farm under random load, (**a**) load variation; (**b**) tie-line power deviation; (**c**) enlarged view under EID; (**d**) system control signals of EID-LFC under random load disturbance.

Figure 10 shows the simulation results with wind farm participation or without wind farm participation. It has been shown that the control system with the wind farm participation is more stable compared to the system without wind farm participation under the proposed control method.

**Figure 10.** Dynamic response of EID with wind farm participation and EID without wind farm participation (**a**) response of Area 1; (**b**) response of Area 2.
