Frequency Regulation of Power Systems With a Wind Farm by Sliding-Mode-Based Design

I. Introduction

WITH the increasing depletion of conventional fossil fuels, renewable energy technologies, that provide both sustainability and environmental friendliness, have drawn great attention and have been rapidly developed for several decades. Among them, wind power is one of the most promising technologies. Increasingly more electricity from wind farms has been generated in the last decade, making more of a contribution in the overall power system.

The rising proportion of wind power affects the frequency stability and the operation of power systems to some extent due to some unique features of wind power technology [1]-[3]. Firstly, the randomness and fluctuation of wind energy increases the frequency variations of wind-integrated power systems [4]. Secondly, variable speed wind turbines (VSWTs) have been widely utilized in power systems. VSWTs could reduce the inertia of power systems and result in more severe frequency fluctuations once load variations rise up [5], [6]. Thirdly, the replacement of conventional power generators (e.g., thermal and hydroelectric generators) by wind generators reduces the frequency regulation capability of power systems because of an increase of the regulation constant [3]. However, the VSWTs respond faster in power regulation than conventional power generators [7], [8]. Thus the control methods of VSWTs via inertial emulation control, droop control, deloading control and over-production control have been investigated extensively for the purpose of frequency support in wind-integrated power systems [9]-[11].

Load frequency control (LFC) is an essential auxiliary service to ensure the safety and stability of power systems. For a multi-area interconnected power system [12], the objective of LFC is to balance the load and power generation in all control areas, eliminating the frequency deviations and the tie-line power deviations between different areas [13]. Many methods concerning the LFC problem have been reported such as the integral-based distributed control method [14], the direct-indirect adaptive fuzzy control technique [15], the decentralized resilient H_∞ method [16], and so forth. Recently, the increasing impact of wind power on power systems has drawn great attention. A coordinated approach was designed through the zero dynamics technique for large-scale power systems in [17], which aimed to improve the transient stability of large-scale power systems including wind power systems. The model identification adaptive PID controller in [18] was designed for the LFC of wind-integrated power systems. The cascaded PID controller was designed in [19] for a power system with diverse power sources including wind turbines. However, improving the robustness of the controllers is required due to the uncertainty of wind energy.

Considering wind-integrated power systems, both uncertainties from wind speed and disturbances from load highlight the LFC problem of such power systems substantially. One possible solution to this problem is the control design. The sliding mode control (SMC) methodology is a powerful tool, known for its invariance to system uncertainties and disturbances [20]-[22], which is a potential solution for the LFC problem. Some SMC-based methods have been applied in practice such as the adaptive SMC method [23], the full-order SMC method [24], the nonlinear SMC method [25], and the linear SMC method [26]. At the same time, the impact of renewable energy, especially wind power, has also gotten a lot of attention. In order to regulate the frequency and voltage of an isolated wind-diesel hybrid power system, an adaptive SMC method was developed in [27]. A neural-network-based integral sliding mode controller (NNB I-SMC) was employed in a multi-area interconnected power system integrated with wind farms in [28]. A nonlinear SMC method for the LFC problem of a wind-integrated power system was proposed in [29]. Considering the disturbance induced by integrated renewable energies, a robust H_∞ sliding mode LFC method for multi-area power systems with time delay was investigated in [30].

Compared with other types of LFC methods, the SMC-based LFC methods exhibit stronger robustness. Furthermore, the terminal SMC technology has been utilized for frequency regulation, aiming to guarantee a system with finite-time convergence [2], [31]. Up to now, however, far too little attention has been paid to deal with the singularity problem existing in the LFC based on the terminal SMC technology. In [32], a unified dynamic terminal SMC technology known as the derivative and integral terminal SMC technology was proposed for a class of systems with different relative degrees. Such technology guarantees the system with finite-time convergence and avoids the singularity problem during the control design process. Moreover, it exhibits superior control properties and robustness performance in terms of system uncertainties and external disturbances. Consequently, this study proposes an LFC method for wind-integrated power systems based on the derivative and integral terminal SMC technology.

Additionally, we apply both the thermal generators and wind generators in a wind-integrated power system to provide frequency support. To this end, a wind power system model based on the mechanical dynamics of VSWTs is utilized, which allows the wind power system to flexibly participate in the frequency regulation of the wind-integrated power system.

The remaining part of this paper proceeds as below. In Section II, the models of a thermal power system, a VSWT system and a wind-integrated two-area thermal power system are established and briefly described. In Section III, the controllers are designed for the wind-integrated two-area thermal power system. In Section IV, the control performance of the proposed frequency control method is analyzed by numerical simulations. In Section V, some conclusions are presented eventually.

II. System Configuration

This section first illustrates the system modeling of a single power system and a VSWT system, then, the compositions of a wind-integrated two-area thermal power system.

A Power System

A power system usually consists of multiple control areas interconnected by tie lines. Fig. 1 shows the linearized model of the nth control area of the power system with N areas. It mainly contains a governor, a turbine and the rotating mass and load, which are defined as below [28]:

Figure  1.  Model of the nth control area

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where T_gn, T_tn, and T_pn are the governor time constant, turbine time constant, and electric system time constant, respectively. s is the Laplace variable.

The equations describing the nth area of a multi-area power system are described as follows [29]:

where Δf_n(t), ΔP_mn(t), ΔE_n(t), ΔP_tien(t), and ΔP_vn(t) are the system states representing the deviations of frequency (Hz), generator output (p.u.MW), integral control (p.u.MW), tie-line power flow (p.u.MW), and governor valve position (p.u.MW), respectively. ΔP_dn(t) is the load disturbance (p.u.MW). K_pn, R_n, K_En, and B_n are the power system gain (Hz/p.u.MW), the speed regulation coefficient (Hz/p.u.MW), the integral control gain (Hz/p.u.MW), and the frequency bias factor (Hz/p.u.MW), respectively. T_nk is the interconnection tie-line gain (p.u.MW/rad/s) between the nth and kth area. u is the system input vector. g_n(u) is a function of the system inputs.

B Wind Power System

Wind energy is converted into mechanical torque by the aerodynamic components of the wind power system. Then, the torque is transmitted to the generator by the drive train. The generator converts the torque into electricity and finally outputs it to the power system. Under the ideal condition of neglecting mechanical and electromagnetic losses, the mechanical power of the wind turbine is regarded as the power output to the power system when the wind power system is operating stably. Based on the ideal condition, a simplified VSWT model is applied here as a wind power system. As shown in Fig. 2, it mainly consists of an aerodynamic model and a drive train model. The aerodynamic model is based on the C_p (λ, β) curve of the wind turbine as below:

Figure  2.  Wind power system

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where C_p is the power coefficient; β is the pitch angle; c₀, c₁, c₂, c₃, c₄, c₅, b₀, and b₁ are the constants given in the Appendix; λ is the tip-speed-ratio defined as below:

where ω_t is the rotating speed of the wind turbine rotor (rad/s); R is the rotor radius (m); v_w is the incoming wind speed (m/s). The mechanical power P_m and mechanical torque T_m extracted by the rotor are calculated as follows:

where ρ is the air density (kg/m³). A two-mass system describing the drive train system is as below [29]:

where H_t and D_t are the inertia constant and mechanical damping coefficient of the wind turbine, respectively, while H_g and D_g are those of the wind generator; ω_r is the generator rotating speed; T_tg and K_tg are the speed torque and shaft stiffness of the drive train, respectively; D_tg is the damping coefficient of the flexible coupling; $T _g^{ref}$ is the braking electromotive torque of the generator. The transmission ratio of the gearbox is regarded as a unit.

C Wind-Integrated Two-Area Thermal Power System

The proposed control method is implemented in a wind-integrated two-area thermal power system. Based on the models of the single area power system and the wind power system, the wind-integrated two-area thermal power system is finally established. As shown in Fig. 3, the wind power system is integrated into Area-1. The droop control method and pitch angle control method of VSWTs in [29] are applied to provide frequency support in this study. The pitch angle control system is given as below:

Figure  3.  Wind-integrated two-area thermal power system

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where T_β is a time constant; K_β is a proportionality constant; Δf₁ is the frequency deviation of Area-1; $\omega _r^{opt}$ is the optimized rotating speed of the wind generator (rad/s).

Additionally, u_w and P_g are respectively the control input and power output of the wind power system; $\omega _r^{d}$ and $P _g^{d}$ are the target values of ω_r and P_g, respectively, and the results after taking frequency regulation into account. $P _g^{opt}$ is the optimized wind power output.

The maximum power point tracking (MPPT) method is the most commonly used control method for VSWTs to maximize wind energy capture. According to the MPPT method, $P _g^{opt}$and $\omega _r^{opt}$ are obtained by the following process under different pitch angles:

1) The range of β is set as [0°, 30°]. Taking (9) as the objective function and the range of λ as the constraint, the corresponding maximum power coefficient C_pmax and optimal tip-speed-ratio λ_opt under different values of β are obtained by solving the following performance objective function:

where β₀ is a concrete value between 0° and 30°. The constraint condition about λ is determined by the operation of the wind turbine. By this approach, the performance curves of C_pmax and λ_opt with respect to β are obtained (illustrated in Fig. 4).

Figure  4.  Performance curves for MPPT (a) β-C_pmax curve; (b) β-λ_opt curve.

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2) During the controlling process, the corresponding values of λ_opt and C_pmax are obtained from the performance curves according to the real-time β value. Then ω_r^opt and P_g^opt are calculated by (10) and (11), respectively.

According to (13), the wind power system is represented in the state matrix form as follows:

where u_w = T_g^ref is the control input of the wind power system; A_w$\in{\mathbb{R}}$^3×3, B_w$\in{\mathbb{R}}$^3×1, C_w$\in{\mathbb{R}}$^1×3, D_w$\in{\mathbb{R}}$^3×1 are constants; x_w$\in{\mathbb{R}}$^3×1 is the state vector; y_w = ω_r is the system output. The related parameters and the expressions of A_w, B_w, C_w, D_w, x_w are illustrated in the Appendix.

According to (1)–(8), the two-area thermal power system is also described in the state matrix form as below:

where the output vector of the two-area power system is y = [Δf₁, Δf₂, ΔP_tie1]^T; the control input vector of it is u = [u₁, u₂, u₃]^T; x$\in{\mathbb{R}}$^9×1 is the state vector; d(t)$\in{\mathbb{R}}$^2×1 is the external disturbance vector; A$\in{\mathbb{R}}$^9×9, B$\in{\mathbb{R}}$^9×3, C$\in{\mathbb{R}}$^3×9, D$\in{\mathbb{R}}$^9×2 are the system matrices. The expressions of A, B, C, D, x, d and the relative parameters are also illustrated in the Appendix.

III. Control Design

This section describes the process of designing controllers for the wind-integrated two-area thermal power system based on the derivative and integral terminal SMC technology. The controllers of the two-area thermal power system and the wind power system are designed independently.

According to the definition of the derivative and integral terminal SMC technology, the fractional integral terminal sliding mode control (fractional I-TSMC) method and the approximate derivative-integral terminal sliding mode control (approximate DI-TSMC) method are generated for systems with different relative degrees. The fractional I-TSMC method is designed for the output tracking of relative-degree-one systems. While for high relative degree systems (systems with at least one relative degree greater than 1), the fractional I-TSMC method is extended and simplified to the approximate DI-TSMC method for good convergence and easy implementation [32]. Therefore, the control design in each subsystem should be on the basis of its relative degrees.

A Control Design For the Two-Area Thermal Power System

The two-area thermal power system of (16) is changed into the following form for convenience:

where f(x(t), d(t)) = Ax(t)+Dd(t) is a dynamic function of the disturbances and states; the output vector is h(x(t)) = C×x(t).

The relative degree vector of the two-area thermal power system is r = [r₁, r₂, r₃] = [3, 3, 4]. Thus the approximate DI-TSMC method is applied to the control design of the two-area thermal power system. r_i (i = 1, 2, 3) is the relative degree of the ith system output y_i and is defined as the smallest number such that $\sum\nolimits_{j = 1}^3 {{L_{bj}}} L_f^{ri - 1}{h_i}\left( {x,d} \right){u_j} \ne 0$ is satisfied in (18).

where y_i^(ri) is the r_ith derivative of y_i. h_i is the ith row of h(x(t)); L_f, L_bj are Lie derivatives; j = 1, 2, 3.

The target values of Δf₁, Δf₂, and ΔP_tie1 are as follows:

Meanwhile, ΔE₁^d and ΔE₂^d are both 0. Combining (4)–(8) and (19), the target values of the other states can be obtained.

According to (18), the input-output dynamics of the two-area thermal power system are expressed as

where G$\in{\mathbb{R}}$^3×3 consists of ${L_{bj}}L_f^{{r_i} - 1}{h_i}(x,d)$ as defined in (20), and is independent of the system states and disturbances; Z(x, d)$\in{\mathbb{R}}$^3×1 is also directly defined in (20).

Assumption 1: Z(x, d) is composed of the nominal part $\hat Z(x)$ and the uncertain part ΔZ(d)

where $\hat Z{,_{}}\;\Delta Z \in{{\mathbb{R}}^{3 \times 1}}.$ ΔZ(d) is generated by system disturbances. There exists a known positive function θ_i(d) (i = 1, 2, 3) as the upper bound of ΔZ_i(d)

where ΔZ_i(d) is the ith component of ΔZ(d). ||·||_∞ is the infinity norm.

Assumption 2: $\hat Z\left( x \right)$ is locally Lipschitz for a proper discussion control region $x \in \Omega$. The following inequality is held [32]:

where s = [s₁, s₂, s₃]^T is the terminal sliding function vector of the two-area thermal power system; ς and ρ_g are constants satisfying ς > 0 and ρ_g > 0; κ = 0, $1,\ldots,$ ς; e = x−x^d is the tracking error vector of the system states, e$\in{\mathbb{R}}$^9×1; x^d$\in{\mathbb{R}}$^9×1 is the target value of the state vector x. ||·|| in this paper represents any norm of the same type. Without loss of generality, the 2-norm is applied in this study for the simulation analysis.

According to the approximate DI-TSMC method, the recursive structure of s_i is as below:

with ${e_{Ili}}\left( 0 \right) = - \left( {{{\dot e}_{D(l - 1)i}}\left( 0 \right) + {\beta _{li}}{e_{D(l - 1)i}}\left( 0 \right)} \right)/{\alpha _{li}},$ where s_i (i = 1, 2, 3) is the ith term of s; l = 1, …, r_i −1; α_li, μ_li and β_li are parameters respectively satisfying α_li > 0, μ_li > 0 and β_li > 0; y_i^d is the ith component of the objective output vector y ^d, which takes values according to (19), y ^d$\in{\mathbb{R}}$^3×1; e_i is the tracking error of y_i. tanh(·) is a hyperbolic tangent function that is continuous and smooth.

The definition of the initial value e_Ili(0) ensures that the sliding function s_i starts from the plane of s_i(0) = 0, which eliminates the reaching time of the sliding mode. e_i converges to 0 when the system keeps operating on the plane of s_i(t) = 0. As mentioned above, the approximate DI-TSMC method achieves a near-finite-time convergence. Some researchers have studied the estimation of the convergence time [21], [22], which is out of the scope of this study.

The derivative of s_i with respect to time is as follows:

Substituting the derivative of e_Dli into (25), another form of it is obtained as below:

with, ${\gamma _i}\left( {e_{D(l - 1)i}^{({r_i} - l)},e_{Ili}^{({r_i} - l)}} \right) = \sum\nolimits_{l = 1}^{{r_i} - 1} {\left( {{\beta _{ki}}e_{D(l - 1)i}^{({r_i} - l)} + {\alpha _{li}}e_{Ili}^{({r_i} - l)}} \right)} ,$ where Z_i and G_i are the ith terms of Z and G, respectively. To drive the vector s to zero, the control law of the two-area thermal power system is designed as

where y_d^(r) = [y₁^d(r1), y₂^d(r2), y₃^d(r3)]^T, γ = [γ₁, γ₂, γ₃]^T; $K \in{{\mathbb{R}}^{3 \times 3}}$ is a positive-defined diagonal matrix; ρ_g is a positive constant; $\bar G$$\in{\mathbb{R}}$^3×3 is the pseudo-inverse matrix of G.

Theorem 1: The tracking error vector e(t) = [e₁, e₂, e₃]^T is guaranteed with asymptotical stability and near-finite-time convergence when K satisfies

where θ₁, θ₂, θ₃ are the upper bounds of ΔZ₁, ΔZ₂, ΔZ₃, respectively.

Proof: The Lyapunov function is selected as $V = {s^T}G\bar Gs/2$. $G\bar G$ is a 3×3 positive definite symmetric matrix. According to the property of the pseudo-inverse matrix, G and $\bar G$ satisfy $\bar G = \bar GG\bar G$. Therefore, the following formula is obtained by taking the derivative of V and then substituting (26)–(28) into it:

It is easy to present the following conclusions from Assumption 1 and Assumption 2:

Thus, $\dot V \leq 0$ is held true, and the equation holds only when s = 0. This illustrates that the approximate DI-TSMC method with the control law in (27) ensures the two-area thermal power system with asymptotic stability. Besides, the tracking error e(t) converges to zero in an approximate super twisting way [32].

B Control Design for the Wind Power System

Similar to (17), the wind power system of (15) is also changed into the following form:

where the disturbance term is d_w = T_m; The relative degree of y_w is r_w = 1. Thus, the fractional I-TSMC method is applied to the wind power system. The input-output dynamics of the wind power system is also expressed in the following form similar to (18):

where Z_w and G_w are constants directly defined in (30). Z_w does not contain the disturbance term.

Assumption 3: Z_w(x_w) is locally Lipschitz for a proper discussion control region ${x_w} \in {\Omega _w}$. The following inequalities can be held [32]:

where s_w is the sliding function of the wind power system; ρ_w is a constant satisfying ρ_w>0; $\underline {e_w}$ = x_w−x_w^d, x_w^d is the target value of x_w. $\underline {e_w}$$\in{\mathbb{R}}$^3×1, x_w^d$\in{\mathbb{R}}$^3×1.

The control objective of the system of (29) is to force ω_r to track its target value ω_r^d. From Fig. 3, ω_r^d and the tracking error e_w are calculated as below:

According to the fractional I-TSMC method, the sliding function of the wind power system is designed as follows:

where α is a positive parameter; p and q are positive odd numbers satisfying p > q, which avoids the singularity problem; the nonlinear term e_w^q/p = sign(e_w)|e_w|^q/p, |·| is the absolute value function. e_wI(0) is defined to eliminate the reaching time of the sliding mode. From (34), the following equation is obtained when on the surface of s_w(t) = 0:

From (35) and (36), the following equation is derived:

where T_w is the convergence time of e_wI(t). Moreover, the time for e_w(t) to converge to 0 is also T_w, according to (34).

Taking the derivative of the sliding function (34) with respect to time, and then substituting (33) and (35) in it, the following equation is obtained:

The control law of the wind power system is set as

where K_w and δ_w are constants.

Theorem 2: If the parameters K_w and δ_w in the control law satisfy the following conditions:

then the control errors e_wI(t) and e_w(t) are guaranteed with finite-time convergence stability.

Proof: The Lyapunov function is selected as ${V_w} = s_w^T{s_w}/2$. Taking the derivative of V_w and then substituting (38) and (39) in it:

According to Assumption 3 and Theorem 2, the following inequalities are held true:

Thus, ${\dot V_w} \leq 0$ is held true, and the equation holds only when s_w = 0. This illustrates that the fractional I-TSMC method with the control law in (39) ensures the wind power system with asymptotic stability. In addition, as s_w(0) = 0 is achieved by (35), the tracking errors e_wI(t) and e_w(t) are ensured to converge to 0 within the time of T_w.

IV. Numerical Simulation

To evaluate the effectiveness of the proposed control method, corresponding numerical simulations are carried out on the wind-integrated two-area thermal power system. The parameters of the wind power system controller are set as q = 13, p = 15, α = 2, δ_w = 2, K_w = 0.15, ρ_w = 300, L_w = 10, ς = 1. The parameters of the two-area thermal power system controller are set as β₁₁ = β₁₂ = β₁₃ = β₂₁ = β₂₂ = β₂₃ = β₃₃ = 15, α₁₁ = α₁₂ = α₁₃ = α₂₁ = α₂₂ = α₂₃ = α₃₃ = 8, μ₁₁ = μ₁₂ = μ₁₃ = μ₂₁= μ₂₂ = μ₂₃ = μ₃₃ = 0.1, K = diag(50, 50,300), ρ_g = 0.2, L = 10.

A Comparisons

For the first simulation experiment, comparisons among the LFC method proposed in this study and the LFC methods based on the nonlinear SMC method [29] and the NNB I-SMC method [28] are carried out.

For comparison, the wind speed and load disturbances are set to be the same constants under different LFC methods. The wind speed of the wind power system is set to v_w = 10 m/s, and the load disturbances of the two-area thermal power system are set to ΔP_d1 = 0.01 and ΔP_d2 = 0.01, respectively.

The simulation results of the three methods are displayed in Fig. 5. Fig. 5(a) shows the responses of the power output P_g of the wind power system under the three methods. Though P_g converges stably under different methods, it converges significantly faster by applying the proposed method. Figs. 5(b), 5(c), and 5(d) exhibit the responses of Δf₁, Δf₂, and ΔP_tie1 of the two-area thermal power system, respectively. The nadir of the dot-dash line in Fig. 5(b) is −1.84×10⁻³ Hz at 0.38 s and that of the dot-dash line in Fig. 5(c) is −1.91×10⁻³ Hz at 0.49 s. To compare the responses of Δf₁ and Δf₂ more clearly, Figs. 5(b) and 5(c) only show the signals in the ordinate range from −2×10⁻⁴ Hz to 2×10⁻⁴ Hz. Δf₁, Δf₂, and ΔP_tie1 converge to 0 in the shortest time under the LFC method proposed in this study. Moreover, the oscillations under the proposed method are significantly smaller than those under the other two methods. Thus, the LFC method proposed in this study have better effectiveness in solving the LFC problem of the wind-integrated power system.

Figure  5.  Response comparison between the proposed LFC method, the nonlinear SMC-based method and the NNB I-SMC-based method (a) Power output of the wind power system P_g; (b) Frequency deviation Δf_1; (c) Frequency deviation Δf₂; (d) Deviation of tie-line power flow ΔP_tie1.

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B Dynamic Response Analysis

For the second simulation experiment, the dynamic robustness of the wind-integrated power system when applying the proposed method is validated under variable external disturbances. Fig. 6 shows the signal settings of the external disturbances, including the wind speed fluctuations and load disturbances.

Figure  6.  Wind speed fluctuations and load disturbances (a) Wind speed v_w; (b) Load disturbance ΔP_d1; (c) Load disturbance ΔP_d2.

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The responses of the wind power system are shown in Fig. 7. With changes in the external disturbances, u_w, P_g, and ω_r respond quickly (see Figs. 7(a)–7(c)). Both P_g and ω_r steadily track their corresponding target values. The responses of u_w, P_g, and ω_r are consistent with the variation of external disturbances. β remains almost at 0 after the initial over-shoot (see Fig. 7(d)).

Figure  7.  Responses of the wind power system (a) Control input u_w; (b) Power output of the wind power system P_g; (c) Generator rotating speed ω_r; (d) Pitch angle β.

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Meanwhile, Fig. 8 shows the responses of the two-area thermal power system. The control inputs u₁, u₂, and u₃, respond rapidly to the changes in external disturbances (see Figs. 8(a), 8(c), and 8(e)). In this process, Δf₁, Δf₂, and ΔP_tie1 almost remain at their respective target values (see Figs. 8(b), 8(d), and 8(f)). Moreover, every time the wind speed and load disturbances change, system fluctuations are created. Those fluctuations can be observed by the response signals in Fig. 8. There are enlarged views in Fig. 8 showing the system fluctuation at the time around 45 s. From the enlarged views, it can be seen that fluctuation is produced and smoothed down in a short time. Indeed, the system converges to a new stable state after the fluctuation. Under the stable state, the frequency and tie-line power flow deviations are almost eliminated.

Figure  8.  Responses of the tow-area thermal power system (a) Control input u₁; (b) Frequency deviation Δf₁; (c) Control input u₂; (d) Frequency deviation Δf₂; (e) Control input u₃; (f) Deviation of tie-line power flow ΔP_tie1.

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According to the responses in Figs. 7 and 8, the whole system responds well to the changes both in the load disturbances and wind speed, where the wind speed fluctuations have the most apparent impact on the system. The proposed LFC method enables the wind-integrated two-area thermal power system with stability and robustness to external disturbances.

V. Conclusions

In this study, the LFC problem of a two-area thermal power system integrated with a wind power system is solved by applying derivative and integral terminal SMC technology. According to the relative degrees of different subsystems, the fractional I-TSMC method and the approximate DI-TSMC method are applied to the wind power system and the two-area thermal power system, respectively. In this way, the suitable sliding mode controllers with corresponding sliding surface functions are designed for the two-area thermal power system and the wind power system to stabilize the whole system. System stability is proven in the sense of Lyapunov. With the help of droop control and pitch angle control, the wind power system also participates in frequency regulation. Numerical simulations are carried out to test both the effectiveness and the dynamic performances of the proposed control method. The simulation results indicate that the presented control method is feasible.

Appendix

1) System matrices and parameters of the wind power system

where K_β = 1, T_β = 0.5, H_t = 2.49 s, H_g = 0.9 s, D_tg = 1.5, D_t = D_g = 0, K_tg = 296.7 p.u.

Parameters of the wind power system: rotor radius, 50 m; rated capacity, 3.5 MW; rated generator revolution speed, 2 rad/s.

Parameters of the C_p (λ, β) curve: c₀ = 0.5176, c₁ = 116, c₂ = −0.4, c₃ = −5, c₄ = −21, c₅ = 0.0068, b₀ = 0.08, b₁ = −0.035.

2) System matrices and corresponding parameters of the two-area thermal power system

where T_ch1 = 0.3, T_ch2 = 0.17, K_p1 = 1, K_p2 = 0.67, T_p1 = 10, T_p2 = 8, T_g1 = 0.1, T_g2 = 0.4, R₁ = R₂ = 0.05, B₁ = 41, B₂ = 81.5, K_E1 = 0.5, K_E2 = 0.5, T₁₂ = 3.77 (taken from [29]).