2. Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China
With the rapid development of economy, electrical power demand has become continuously stronger year by year. A vast amount of fossil fuels utilized in power generation results in energy crisis and environmental deterioration around the world [1]. One possible solution is to adopt clean and renewable energy instead of fossil fuel for power generation. Wind power is now the fastest growing energy source around the world because of its zero emission [2]. The percentage of wind generation in power systems increases with years. Wind energy has become one of the central research themes in energy science.
Concerning real applications, a constant supply of electricity in some remote areas cannot be guaranteed by power grids. In these areas, wind energy may be inexhaustible and convenient. Therefore, wind power has been paid more and more attention and some control problems rise up in power systems with wind turbines [3]. An advocacy of wind power is due to its sustainable and renewable status. However, wind power affected by climate changes is intermittent. Its intermittence also has impressive effects on operation and control of renewable power systems.
Consider a power system with wind turbines. The load in the power system is random and the power output of wind power is fluctuating. The poweroutput fluctuation and the load change would pose a reliability supply challenge. The challenge is displayed by power imbalance and frequency deviation in the power system [4]. Consequently, frequency control strategies must be adopted to overcome the challenge.
Load frequency control (LFC) is one of the most profitable auxiliary services to guarantee the stable operation of power systems with the objective of preserving the balance between power generation and power consumption [5]. Recently, the LFC problem of renewable power systems has been paid more and more attention [6], [7]. In order to attack the generation intermittency of renewable power systems, some LFC methods have been investigated, such as fuzzy control [8], [9], predictive control [10], [11], and adaptive control [12], [13].
Invented by Utkin, the sliding mode control (SMC) is recognized as a powerful design tool [14]. On the slidingmode stage, an SMC system is completely insensitive to parametric uncertainties and external disturbances under certain matching conditions, which exhibits better performance than the conventional robust control methods [15]. This property inspires some researches on SMC for the LFC problem [16][22]. However, the SMCbased LFC methods in pervious works [16][22] do not consider the complexity and challenge of renewableenergy sources. The method of terminal sliding mode control (TSMC) [23], [24] guarantee the convergence of the SMC system within finite time. The TSMC method can be considered for the LFC problem of renewable power systems.
Power systems are inherently nonlinear [25]. The two main nonlinear factors are the governor dead band (GDB) and the generation rate constraint (GRC). The existence of GRC has adverse effects on the system robustness, the system performance as well as the system stability [26]. A common technique to deal with the GRC nonlinearity is to design a controller for the linearized nominal model; then the controller is directly imposed on the original nonlinear system. In a sentence, the linearized model is adopted to achieve the control design for the original nonlinear system. Although it is available in most cases, the technique has some potential hazards because there is no theoretical guarantee on the stability of the control system. Concerning the applications of SMC on LFC [16][22], some works [16], [17], [21] only consider linear power systems and other works [18][20], [22] adopt the linearmodelbased design. However, a series of drawbacks may be induced to the linearmodelbased control systems because of lack of theoretical supports.
The methodology of radial basis function neural networks (RBF NNs) has a universal approximating feature [27]. The RBF NNs technology has been widely adopted to solve nonlinearities and uncertainties of complex systems [28]. In [20], RBF NNs are employed to approximate and compensate the GDB nonlinearity of power systems. However, how to conquer the GRC nonlinearity by RBF NNs remains untouched and problematic. To turn the TSMC into practical accounts on the LFC problem of renewable power systems, it is urgent to solve the GRC nonlinearity by the RBF NNs technique.
This paper focuses on the TSMC method for LFC of nonlinear power systems with wind turbines. To deal with the GRC nonlinearity, the linearization method is adopted at first. Then, a terminal sliding mode controller is designed based on the linearized nominal system. The uncertainties of the LFC problem have three components, i.e., the intermittency of wind power, the uncertainties of power systems and the error of linearization. The components mix together and worsen the uncertainties of the LFC problem. To theoretically guarantee the system stability, the slidingmodebased neural networks are designed to suppress the entire uncertainties. Weight update formulas of the neural networks are derived from the Lyapunov direct method. The neuralnetworkbased TSMC scheme is employed to accomplish the LFC problem. To illustrate the feasibility and validity of the presented scheme, some numerical simulations are conducted by a nonlinear interconnected power system with wind turbines.
The remainder of this paper is organized as follows. Section Ⅱ formulates the system configuration. Section Ⅲ presents the TSMC method, the RBF NNs design and the system stability. Simulation results are demonstrated in Section Ⅵ. Finally, conclusions are drawn in Section Ⅴ.
Ⅱ. SYSTEM CONFIGURATION A. Component DynamicsThis paper considers the LFC problem of a multiarea interconnected power system. The power system is composed of
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Fig. 1 Dynamic model of the 
According to the LFC objective, not only should the frequency of the control area return to its nominal value, but also the net interchange through the tieline should return to the scheduled values. To achieve the composite goal, a measure, named area control error (ACE), is introduced. In Fig. 1, the measure in the
$ \begin{align} AC{E_i}(t) = \Delta {P_{{\rm tie}, i}}(t) + {B_i}\Delta {f_i}(t) \end{align} $  (1) 
where
$ \begin{align} \Delta {\dot{P}_{{\rm tie}, i}}(t) = 2\pi \left( {\sum^N_{\begin{subarray}{c} j=1, j\ne i\\ \end{subarray}}} {{T_{ij}} }\Delta {f_i}(t)  \Delta {V_i}(t)\right) \end{align} $  (2) 
where
$ \begin{align} \Delta V_i(t) = {\sum^N_{\begin{subarray}{v} j=1, j\ne i\\ \end{subarray}}} {{T_{ij}} }\Delta {f_j}(t). \end{align} $  (3) 
To force the composite measure (1) to zero, the integral of
$ \begin{align} \Delta {E_i}(t) = {K_{Ei}}\int{AC{E_i}(t){d}t} \end{align} $  (4) 
where
Define a vector
$ \begin{align} {\dot{{x}}_i}(t) = {{A}_i}{{x}_i}(t) + {{B}_i}{u_i}(t) + {{F}_i}\Delta {{P}_{di}}(t) \end{align} $  (5) 
where
The doublyfed induction generator (DFIG) system has been proven a proper renewable energy conversion system. A simplified frequency response model of a DFIG based wind turbine unit [10] is illustrated in Fig. 2.
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Fig. 2 Simplified model of DFIG based wind turbine. 
The model of the DFIGbased wind turbine unit can be described by the following equations.
$ {\dot {i}_{qr}}(t) =  \left( {\frac{1}{{{T_1}}}} \right){i_{qr}}(t) + \left( {\frac{{{X_2}}}{{{T_1}}}} \right){V_{qr}}(t) $  (6) 
$ \dot{w}(t) =  \left( {\frac{{{X_3}}}{{2{H_t}}}} \right){i_{qr}}(t) + \left( {\frac{1}{{2{H_t}}}} \right){T_m}(t) $  (7) 
$ {P_e}(t) = w(t){X_3}{i_{qr}}(t) $  (8) 
where
Linearizing the wind turbine model at a certain operating point, we can rewrite (8) as
$ {P_e}(t) = {w_{\rm opt}}{X_3}{i_{qr}}(t) $ 
$ {T_e}(t) = {i_{qs}}(t) =  \frac{{{L_m}}}{{{L_{ss}}}}{i_{qr}}(t) $  (10) 
where
Define
$ \begin{align} {\dot{{x}}_{wi}}(t) = {{A}_{wi}}{{\pmb x}_{wi}}(t) + {{B}_{wi}}{{\pmb u}_{wi}}(t) + {{F}_{wi}}\Delta {{ P}_{{\rm wdi}}}(t). \end{align} $  (11) 
Equation (11) will be employed for the LFC design for wind turbines in the multiarea power system. The details about
Remark 1: Equation (5) presents the mathematical model for the LFC problem of the conventional generating system in control area
From (5) and (11), these system models can be described by a uniform expression. Without loss of generality, the expression has a form of
$ \begin{align} {\dot {{x}}}(t) = { A}{\pmb x}(t) + {B}{\pmb u}(t) + {F}\Delta {P}(t). \end{align} $  (12) 
By considering the parameter uncertainties and the modelling errors, equation (12) can be written as
$ \begin{align} \dot{ {x}}(t) =&\ ({A}' + \Delta {A}){\pmb x}(t) + ({B}' + \Delta {B}){\pmb u}(t) \notag\\ &\, + ({F}' + \Delta {F})\Delta {P}(t) \end{align} $  (13) 
where
It is noted that the above discussions just consider the uncertainties existing in the linear power system. Power systems actually cover the GRC nonlinearity. The existence of GRC has adverse effects on the system stability. Inherently, the GRC nonlinearity acts as a limiter to limit the rate of change in the power generation. Taking the turbine of generating units as an example, the limiter output remains unchanged while reaching its top or bottom. But the turbine output keeps increasing or decreasing at the extreme rate of change. This fact indicates the nonlinear power system becomes linear before the limiter output reaches its limit value. Having been triggered by the two critical points of the limiter, the turbine output is still changing but the change is at its extreme rates determined by the limiter. From Fig. 1, these discussions can be formulated by
$ \begin{align} \Delta {P_{gi}}(t) = \begin{cases} {  \frac{1}{{{T_{gi}}}}\int {\delta {d}t} }, &{\sigma (t) <  \delta } \\ {\frac{1}{{{T_{gi}}}}\int {\sigma (t){d}t} }, &{\sigma (t) \leq \delta } \\ {\frac{1}{{{T_{gi}}}}\int {\delta {d}t} }, &{\sigma (t) > \delta } \end{cases} \end{align} $  (14) 
where
As far as the LFC problem of the nonlinear power system is concerned, the effect of the extreme rates only exists on the outset because
From (13) and (14), the following general model can be derived for the power system.
$ \begin{align} {\dot {{\pmb x}}}(t) = &\ {A}'{\pmb x}(t) + {B}'{\pmb u}(t) + {F}'\Delta {P}(t) \nonumber\\ & + \Delta {A}{\pmb x}(t) + \Delta{ B}{\pmb u}(t) + \Delta {F}\Delta{ P}(t) + \phi (t) \end{align} $  (15) 
where
From (15), the LFC design of the nonlinear power system with wind turbines can be divided into two parts. One is to design a TSMCbased load frequency controller for the nominal system. The other is to consider how to suppress the system uncertainties.
Ⅲ. CONTROL DESIGN A. Mathematical Descriptions of the System for Terminal Sliding Mode ControlTo aggregate all the uncertain terms in (15),
$ \begin{align} {\pmb d}(t) = {F}'\Delta {P}(t) + \Delta {A}{\pmb x}(t) + \Delta {B}{\pmb u}(t) + \Delta {F}\Delta {P}(t) + \phi (t). \end{align} $  (16) 
Then, the general system model (15) has a form of
$ \begin{align} \dot {{\pmb x}}(t) = {A}'{\pmb x}(t) + {B}'{\pmb u}(t) + {\pmb d}(t). \end{align} $  (17) 
Assumption 1: The uncertain term
According to the principle of matrix controllability decomposition, we adopt the nonsingular transformation of
$ \begin{align} &{T}{A}'{{T}^{  1}} = \left[{\begin{array}{*{5}{c}} {{{A}_{11}}}&{{{A}_{12}}} \\ {{{A}_{21}}}&{{{A}_{22}}} \end{array}} \right], \ \ {T}{B}' = \left[{\begin{array}{*{5}{c}} 0 \\ {{{B}_2}} \end{array}} \right]\notag\\ &{T}{\pmb d}(t) = \left[{\begin{array}{*{20}{c}} 0 \\ {{{B}_2}{H}(t)} \end{array}} \right]. \end{align} $  (18) 
In (18),
According to (18), the following equations (19) can be derived from (17).
$ \begin{align} \begin{cases} {{\dot {{\pmb Z}}}_1}(t) = {{A}_{11}}{{\pmb Z}_1}(t) + {{A}_{12}}{{\pmb Z}_2}(t) \\ {{\dot {{\pmb Z}}}_2} (t) = {{A}_{21}}{{\pmb Z}_1}(t) + {{A}_{22}}{{\pmb Z}_2}(t) + {{B}_2}{\pmb u}(t) + {{B}_2}{H}(t) \end{cases} \end{align} $  (19) 
where
In [24], a kind of sliding surfaces is entitled global fast terminal sliding surface. However, this kind of surfaces in [24] is defined in the scalar form such that it cannot be directly employed for the LFC problem of renewable power systems. To develop a global fast terminal sliding surface for the LFC problem, the general sliding surface vector (20) is extended for multivariable systems [31]. The extended global fast terminal sliding surface vector for the nominal part of (19) is formulated by
$ \begin{align} {\pmb s}(t) = {{C}_1}{{\pmb Z}_1}(t) + {{C}_2}{{ Z}_2}(t) + {{C}_3}{\pmb Z}_1^\frac{q}{p}(t) \end{align} $  (20) 
where
$ {{A}_{11}}  {{A}_{12}}{{C}_2}^{  1}{{C}_1} =  {\rm diag}\{{\alpha _1}, \ \cdots, \ {\alpha _{n  m}}\} $  (21) 
$ {{A}_{12}}{{C}_2}^{  1}{{C}_3} = {\rm diag}\{{\beta _1}, \ \cdots, \ {\beta _{n  m}}\}. $  (22) 
In (21) and (22),
$ {A}_{12}^ + = {A}_{12}^T{({{A}_{12}}{A}_{12}^T)^{  1}}. $  (23) 
From (21) and (22),
$ {{C}_1} = {{C}_2}{A}_{12}^ + \, {\rm diag}\{{\alpha _1}, \ \cdots, \ {\alpha _{n  m}}\} $  (24) 
$ {{C}_3} = {{C}_2}{A}_{12}^ + \, {\rm diag}\{{\beta _1}, \ \cdots, \ {\beta _{n  m}}\}. $  (25) 
Differentiating
$ \begin{align} \dot {{\pmb s}}(t) = &\ {{C}_1}{{\dot{ {\pmb Z}}}_1}(t) + {{C}_2}{{\dot {{\pmb Z}}}_2}(t) + {{C}_3}{G}{{\dot {{\pmb Z}}}_1}(t)\notag \\ = &\ {{C}_1}\left({{A}_{11}}{{\pmb Z}_1}(t) + {{A}_{12}}{{\pmb Z}_2}(t)\right)\notag\\ &\, + {{C}_2}\left({{A}_{21}}{{\pmb Z}_1}(t) + {{A}_{22}}{{\pmb Z}_2}(t) + {{B}_2}{\pmb u}(t) + {{B}_2}{H}(t)\right)\notag\\ &\, + {{C}_3}{G}\left({{A}_{11}}{{\pmb Z}_1}(t) + {{A}_{12}}{{\pmb Z}_2}(t)\right) \end{align} $  (26) 
where
When the system trajectory enters the sliding mode stage and keeps on the sliding surface,
$ \begin{align} {{\pmb u}_{eq}}(t) = &  {({{C}_2}{{B}_2})^{  1}}[({{C}_1}{{A}_{11}} + {{C}_2}{{A}_{21}}){{\pmb Z}_1}(t) + ({{C}_1}{{A}_{12}} \notag\\ &\, + {{C}_2}{{A}_{22}}){{\pmb Z}_2}(t) + {{C}_3}{G}({{A}_{11}}{{\pmb Z}_1}(t) + {{A}_{12}}{{\pmb Z}_2}(t))]. \end{align} $  (27) 
Consider (19) and define the global fast terminal sliding surface (21). Then, the final TSMC law [31] can be formulated by
$ \begin{align} {\pmb u}(t) = \begin{cases} {{{\pmb u}_{eq}}(t)  ({{\bar h}_0} + \eta )\dfrac{{{{({{C}_2}{{B}_2})}^T}{\pmb s} (t)}}{{\{{\pmb s}^T}(t)({{C}_2}{{B}_2})\}}}, &{{\pmb s}(t) \ne {{O}_m}} \\ {{{\pmb u}_{eq}}(t)}, &{{\pmb s}(t) = {{O}_m}} \end{cases} \end{align} $  (28) 
where
Remark 2: Since the TSMC law (28) contains
To fill the gap between the system stability and the boundary value of uncertainties, RBF NNs are employed because such a kind of neural networks owns the ability to approximate complex nonlinear mapping directly from inputoutput data with a simple topological structure. RBF NNs are a kind of threelayer feedforward networks, where the mapping from the input layer to the output layer is inherently nonlinear but the mapping from the hidden layer to the output layer is linear.
Fig. 3 displays the designed RBF NNs. Concerning the LFC problem, each element of the state vector
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Fig. 3 Structure of RBF NNs. 
From Fig. 3, the output of the RBF NNs is determined by
$ \begin{align} {\hat {\bar {h}}}_0\left( {{\pmb x}, ~\pmb \omega } \right) = {\hat {\pmb \omega} ^T}{\pmb h}( {\pmb x}). \end{align} $  (29) 
In (29),
$ \begin{align} {h_\lambda }({\pmb x}) = \exp \left( {  \frac{{\{\pmb x}  {{c}_\lambda }{^2}}}{{2b_\lambda ^2}}} \right) \end{align} $  (30) 
where
Adopting the RBF approximation technology, the control law (28) can be rearranged by
$ \begin{align} {\pmb u}(t) = {\begin{cases} {{{\pmb u}_{eq}}(t)  ({{\hat {\bar {h}}}_0} + \eta )\dfrac{{{{({{C}_2}{{B}_2})}^T}{\pmb s}(t)}} {{\{{\pmb s}^T}(t)({{C}_2}{{B}_2})\}}}, &{{\pmb s}(t) \ne {{O}_m}} \\ {{{\pmb u}_{eq}}(t)}, &{{\pmb s}(t) = {{O}_m}}. \end{cases}} \end{align} $  (31) 
Remark 3: The term
Assumption 2: There exists an optimal weight vector
$ \begin{align} {\pmb \omega ^ * }^T{\pmb h}( {\pmb x})  {\bar {h}_0} = \varepsilon ({x}) < {\varepsilon _1}. \end{align} $  (32) 
Assumption 3: The boundary value
$ \begin{align} {\bar h_0}  \{H}(t)\ > {\varepsilon _0} > {\varepsilon _1}. \end{align} $  (33) 
Theorem 1: Take Assumptions 13 into account, consider the system model (17), define the global fast terminal sliding surface (20), and adopt the control law (31). Then, the TSMCbased LFC system is of asymptotic stability if and only if the update law of the network weight vector has a form of
$ \begin{align} \dot {\hat {\omega}} = \xi \{{\pmb s}^T}({{C}_2}{{ B}_2})\{\pmb h}( {\pmb x}) \end{align} $  (34) 
where
$ \begin{align} \xi = {\varepsilon _0}  {\varepsilon _1} > 0. \end{align} $  (35) 
Proof: Consider the candidate Lyapunov function
$ \begin{align} V = \frac{1}{2}{{\pmb s}^T}{\pmb s} + \frac{1}{2}{\xi ^{  1}}{\pmb{\tilde{\omega}}}^T{\pmb{\tilde{\omega}}} \end{align} $  (36) 
where
$ \begin{align} \pmb{\tilde {\omega}} = {\pmb{\omega ^ * }}  \pmb{\hat{\omega}}. \end{align} $  (37) 
Differentiate
$ \begin{align} \dot {V} = {{\pmb s}^T}\dot {{\pmb s}}  {\xi ^{  1}}{\pmb{\tilde {\omega}}}^T{\dot{\pmb{ \hat {\omega}}}}. \end{align} $  (38) 
When
$ \begin{align} \dot {V} = &\ {{\pmb s}^T}\dot{ {\pmb s}}  {\xi ^{  1}}{{\tilde {\pmb \omega} }^T}\dot{ \hat{{\pmb \omega}}} \notag \\ = &\ {{\pmb s}^T}[{{C}_1}({{A}_{11}}{{\pmb Z}_1} + {{A}_{12}}{{\pmb Z}_2}) + {{C}_2}({{A}_{21}}{{\pmb Z}_1} + {{A}_{22}}{{\pmb Z}_2} \notag \\ & + {{B}_2}{u} + {{B}_2}{H}(t)) + {{C}_3}{G}({{A}_{11}}{{\pmb Z}_1} + {{A}_{12}}{{\pmb Z}_2})] \notag \\ & {\xi ^{  1}}{{\tilde {\pmb \omega} }^T}\dot {\hat {\pmb \omega}} \notag \\ = &\ {{\pmb s}^T}({{C}_2}{{B}_2})[({{\hat {\bar {h}}}_0} + \eta )\frac{{{{({{C}_2}{{B}_2})}^T}{\pmb s}}}{{\{{\pmb s}^T}({{C}_2}{{B}_2})\}} + {H}(t)]\notag \\ &  {\xi ^{  1}}{{\tilde {\pmb \omega} }^T}\dot {\hat {\pmb \omega}} \notag \\ \leq &  \{{\pmb s}^T}({{C}_2}{{B}_2})\({{\hat {\bar {h}}}_0} + \eta ) + \{{\pmb s}^T}({{C}_2}{{B}_2})\ \cdot \{H}(t)\\notag \\ &  {\xi ^{  1}}{{\tilde {\pmb \omega} }^T}\dot {\hat {\pmb \omega}} \notag \\ = &  \{{\pmb s}^T}({{C}_2}{{B}_2})\[({{\hat {\bar {h}}}_0} + \eta ) + {{\bar {h}}_0}{{\bar {h}}_0}]\notag \\ & + \{{\pmb s}^T}({{C}_2}{{B}_2})\ \cdot \{H}(t)\  {\xi ^{  1}}{{\tilde{\pmb \omega} }^T}\dot {\hat {\pmb \omega}} \notag \\ = &  \{{\pmb s}^T}({{C}_2}{{B}_2})\({{\hat {\bar {h}}}_0} + \eta  {{\bar {h}}_0}) \notag \\ &  \{{\pmb s}^T}({{C}_2}{{B}_2})\({{\bar {h}}_0}  \{H}(t)\)  {\xi ^{  1}}{{\tilde{\pmb \omega }}^T}\dot {\hat {\pmb \omega}} \notag \\ = &  \{{\pmb s}^T}({{C}_2}{{B}_2})\[{{\hat {\pmb \omega} }^T}{\pmb h}({\pmb x}) + \eta ({\omega ^ * }^T{\pmb h}({\pmb x})\varepsilon ({\pmb x}))]\notag \\ &  \{{\pmb s}^T}({{C}_2}{{B}_2})\({{\bar {h}}_0}  \{H}(t)\) \notag \\ &  \{{\pmb s}^T}({C_2}{B_2})\({\pmb \omega ^ * }^T  {{\hat{\pmb \omega}^T}}){\pmb h}({\pmb x})\notag \\ = &  \eta \{{\pmb s}^T}({{C}_2}{{B}_2})\  \{{\pmb s}^T}({{C}_2}{{B}_2})\\varepsilon ({\pmb x})\notag \\ &  \{{\pmb s}^T}({{C}_2}{{B}_2})\({{\bar {h}}_0}  \{H}(t)\) \notag \\ \leq &  \eta \{{\pmb s}^T}({{C}_2}{{B}_2})\ + \{{\pmb s}^T}({{C}_2}{{B}_2})\[\varepsilon ({\pmb x})\notag \\ &({{\bar {h}}_0}\{H}(t)\)]. \end{align} $  (39) 
Taking Assumptions 2 and 3 into account, we can conclude that the inequality
$ \begin{align} \dot V < &  \eta \{{\pmb s}^T}({{C}_2}{{B}_2})\  ({\varepsilon _0}  {\varepsilon _1})\{{\pmb s}^T}({{C}_2}{{B}_2})\ \notag \\ = &  \eta \{{\pmb s}^T}({{C}_2}{{B}_2})\  \xi \{{\pmb s}^T}({{C}_2}{{B}_2})\\notag \\ = &\ (  \eta  \xi )\{{\pmb s}^T}({{C}_2}{{B}_2})\ \notag \\ \leq &\ (  \eta  \xi )\{{C}_2}{{B}_2}\ \cdot \{\pmb s}\ \leq 0. \end{align} $  (40) 
In (40), it is obvious that
From (36) and (40), we have
$ \{\pmb s}\ = \sqrt 2 {V^{\frac{1}{2}}}. $  (41) 
Define
$ \dot V <  \rho {V^{\frac{1}{2}}}. $  (42) 
Integrating both sides in (42) yields
$ \int_{{V_0}}^0 {{V^{  \frac{1}{2}}}dV} <  \rho \int_0^\tau {dt}. $  (43) 
The time
$ \tau < \frac{2}{\rho }{V_0}^{\frac{1}{2}}. $  (44) 
In the sense of Lyapunov,
On the sliding surface, there is
$ {\pmb s} = {{C}_1}{{\pmb Z}_1} + {{C}_2}{{\pmb Z}_2} + {{C}_3}{\pmb Z}_1^\frac{q}{p} = {{O}_m}. $  (45) 
Since
$ {{\pmb Z}_2} =  {{C}_2}^{  1}{{C}_1}{{\pmb Z}_1}  {{C}_2}^{  1}{{C}_3}{\pmb Z}_1^\frac{q}{p}. $  (46) 
Substituting (46) into (19) yields
$ {\dot{{\pmb Z}}_1} = {{A}_{11}}{{\pmb Z}_1}  {{A}_{12}}{{C}_2}^{  1}{{C}_1}{{\pmb Z}_1}  {{ A}_{12}}{{C}_2}^{  1}{{C}_3}{\pmb Z}_1^\frac{q}{p}. $  (47) 
By selecting
$ \begin{align} {{\dot {{\pmb Z}}}_1} =& \left[{\begin{array}{*{20}{c}} {{{\dot z}_1}} \\ {{{\dot z}_2}} \\ \vdots \\ {{{\dot z}_{nm}}} \end{array}} \right]\notag\\ = &  {\rm diag}\{{\alpha _1}~ \cdots~{\alpha _{n  m}}\}{{\pmb Z}_1}  {\rm diag}\{{\beta _1}~\cdots~{\beta _{n  m}}\}{{\pmb Z}_1}^\frac{q}{p}\notag \\ = & \left[{\begin{array}{*{20}{c}} {{\alpha _1}{z_1}{\beta _1}{z_1}^\frac{q}{p}} \\ {{\alpha _2}{z_2}  {\beta _2}{z_2}^\frac{q}{p}} \\ \vdots \\ {  {\alpha _{n  m}}{z_{n  m}}  {\beta _{n  m}}{z_{n  m}}^\frac{q}{p}} \end{array}} \right]. \end{align} $  (48) 
From (48), each element in both
Consequently, the closedloop LFC control system is asymptotically stable in the sense of Lyapunov. In this regard, not only can the designed update law (34) guarantee the convergence of the RBF NNs, but also the whole LFC system can earn the asymptotic stability by such a control scheme.
Remark 4: From (19),
$ \begin{align} {{\pmb u}_{eq}}(t) = &  {({{C}_2}{{B}_2})^{  1}}[({{C}_1}{{A}_{11}} + {{C}_2}{{A}_{21}}){{\pmb Z}_1}(t) \nonumber\\ & + ({{C}_1}{{A}_{12}} + {{C}_2}{{A}_{22}}){{\pmb Z}_2}(t) + {{C}_3}{G}{{\dot {{\pmb Z}}}_1}]. \end{align} $  (49) 
In (49), the
Remark 5: An inherent drawback of the sliding mode control technique is chattering [14]. To reduce the chattering, the smooth processing should be adopted for the relay characteristics in (31) [32]. The amendment has the form of
$ \begin{align} {\pmb u}(t) = {\begin{cases} {{{\pmb u}_{eq}}(t)  ({{\hat {\bar {h}}}_0} + \eta ) \dfrac{{{{({{C}_2}{{B}_2})}^T}{\pmb s}(t)}}{{\{{\pmb s}^T}(t)({{C}_2}{{B}_2})\ + \delta }}}, &\!{{\pmb s}(t) \ne {{O}_m}} \\ {{{\pmb u}_{eq}}(t)}, &\!{{\pmb s}(t) = {{O}_m}} \end{cases}} \end{align} $  (50) 
where
Remark 6: According to the designed control scheme, the GRC nonlinearity is transformed into a part of system uncertainties. The RBF NNs can be treated as a compensator to approximate and compensate the entire system uncertainties. The compensator and the controller cooperate to overcome the LFC problem for the nonlinear power system with wind turbines.
Remark 7: The generating unit with GRC in Fig. 1 and the wind turbine in Fig. 2 simultaneously exist in a control area. Both of them can affect the frequency of the considered control area. Hence, two load frequency controllers and their compensators in one control area should be designed to achieve the LFC task according to (5) and (11), respectively.
Ⅳ. SIMULATION RESULTSConsider the LFC problem of renewable power systems. An interconnected power system with wind turbines is employed to illustrate the effectiveness and feasibility of the proposed TSMC scheme. The power system is composed of two control areas. Each control area has an aggregated generating unit with GRC in Fig. 1 and an aggregated wind turbine unit in Fig. 2. As mentioned above, the generating unit and the wind turbine mean all generator units and all wind turbines in the control area are aggregated together. The proposed control scheme will be carried out by the LFC solution of such a nonlinear interconnected power system, where the system schematic is illustrated in Fig. 4.
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Fig. 4 Block diagram of the considered twoarea power system. 
Some parameters and data from the interconnected power system [10] are listed in Table Ⅰ. The power system consists of two
As mentioned above, some controller parameters are predefined. Concerning the TSMCbased controllers in Areas 1 and 2, the transformation matrices
$ \begin{align*} & {{T}_1} = {{T}_3} = \left[{\begin{array}{ccccc} 0&0&1&0&0 \\ 0&1&0&0&0 \\ 0&0&0&1&0 \\ 0&0&0&0&1 \\ 1&0&0&0&0 \end{array}} \right] \\ & {{T}_2} = {{T}_4} = \left[{\begin{array}{ccccccc} 0&0&0&0&0&1&0 \\ 0&0&0&0&0&0&1 \\ 0&0&1&0&0&0&0 \\ 0&1&0&0&0&0&0 \\ 0&0&0&1&0&0&0 \\ 0&0&0&0&1&0&0 \\ 1&0&0&0&0&0&0 \end{array}} \right]. \end{align*} $ 
For Controllers 1 and 3,
$ \begin{align*} &{{ C}_1} = {{C}_3} = \left[{\begin{array}{ccccc} {1.1}&{1.2}&{1.3}&{1.4}&{1.5} \\ {1.1}&{1.2}&{1.3}&{1.4}&{1.5} \end{array}} \right]\\ &{{C}_2}=\left[{\begin{array}{cc} 1&0 \\ 0&1 \end{array}} \right].\end{align*} $ 
Other controller parameters are determined by
Concerning the designed RBF NNs, some network parameters should also be set up. For RBF NNs
To show the performance of the presented method, two step load disturbances
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Fig. 5 Simulation results of the interconnected power system with and without RBF NNs. (a) Frequency deviation 
From Fig. 5,
Fig. 6 illustrates the control inputs in the two control areas, where the red solid lines indicates the LFC system without RBF NNs and the blue solid lines indicates the LFC system with RBF NNs. To guarantee the system stability, the LFC system without RBF NNs needs to predefine a large
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Fig. 6 Simulation results of control inputs. (a) Control inputs in Area 1. (b) Control inputs in Area 2. 
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Fig. 7 Simulation results of RBF NN outputs. (a) RBF NN outputs in Area 1. (b) RBF NN outputs in Area 2. 
To demonstrate the control performance of the presented control scheme, the comparisons between the presented scheme and the SMC method with RBF NNs are shown in Fig. 8. In Fig. 8, the RBF NNs parameters in the two LFC control systems make no difference. The control performance is just decided by the two control methods.
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Fig. 8 Comparisons between TSMC with RBF NNs and SMC with RBF NNs. (a) Frequency deviation 
Compared with the results of the SMC method with RBF NNs, the performance of the presented control scheme is better without doubt. Especially, the overshoot of the presented method in Fig. 8 is zero which shows that the presented control scheme is more robust against load disturbances in power systems with wind turbines.
Ⅴ. CONCLUSIONSThis article has addressed the LFC problem for renewable power systems in the presence of GRC. The control scheme is designed by means of TSMC. To suppress the uncertainties of the LFC problem, RBF NNs are adopted. The theoretical analysis in the sense of Lyapunov proves that the TSMCbased LFC system is asymptotically stable. The presented control scheme has solved the LFC problem of an interconnected renewable power system composed of two control areas. Some numerical simulation results have demonstrated the performance of the presented method against uncertainties of nonlinear power systems with renewable sources.
APPENDIX PARAMETER MATRICES$ \begin{align*} &{{A}_i} = \left[{\begin{array}{*{20}{c}} {\frac{1}{{{T_{gi}}}}}&0&{\frac{1}{{{R_i} \cdot {T_{gi}}}}}&0&0 \\ {\frac{1}{{{T_{ti}}}}}&{\frac{1}{{{T_{ti}}}}}&0&0&0 \\ 0&{\frac{{{K_{pi}}}}{{{T_{pi}}}}}&{  \frac{1}{{{T_{pi}}}}}&{  \frac{{{K_{pi}}}}{{{T_{pi}}}}}&0 \\ 0&0&{2\pi \sum\limits^N_{\begin{subarray}{l} j = 1, j \ne i \end{subarray}} {{T_{ij}}} }&0&0 \\ 0&0&{{K_{Ei}}{B_i}}&{{K_{Ei}}}&0 \end{array}} \right]\\ &{{B}_i} = \left[{\begin{array}{*{20}{c}} {\frac{1}{{{T_{gi}}}}} \\ 0 \\ 0 \\ 0 \\ 0 \end{array}} \right], \ \ {{F}_i} = \left[{\begin{array}{*{20}{c}} 0&0 \\ 0&0 \\ {\frac{{{K_{pi}}}}{{{T_{pi}}}}}&0 \\ 0&{2\pi } \\ 0&0 \end{array}} \right]\\ &\Xi={\frac{{{K_{pi}}{X_3}\Delta {w_{\rm opt}}}}{{{T_{pi}}}}}\\ &{{A}_{wi}} = \left[{\begin{array}{ccccccc} { \frac{1}{{{T_{gi}}}}}&0&{ \frac{1}{{{R_i}{T_{gi}}}}}&0&0&0&0 \\ {\frac{1}{{{T_{ti}}}}}&{ \frac{1}{{{T_{ti}}}}}&0&0&0&0&0 \\ 0&{\frac{{{K_{pi}}}}{{{T_{pi}}}}}&{ \frac{1}{{{T_{pi}}}}}&{ \frac{{{K_{pi}}}}{{{T_{pi}}}}}&\Xi&0&0 \\ 0&0&{2\pi \sum\limits_{\begin{subarray}{l} j = 1, j \ne i \end{subarray}} ^N {{T_{ij}}} }&0&0&0&0 \\ 0&0&0&0&{ \frac{1}{{{T_1}}}}&0&0 \\ 0&0&0&0&{ \frac{{{X_3}}}{{2{H_{ti}}}}}&0&0 \\ 0&0&{{K_{Ei}}{B_i}}&{{K_{Ei}}}&0&0&0 \end{array}} \right]\\ &{{B}_{wi}} = \left[{\begin{array}{*{20}{c}} {\frac{1}{{{T_{gi}}}}}&0 \\ 0&0 \\ 0&0 \\ 0&0 \\ 0&{\frac{{{X_2}}}{{{T_1}}}} \\ 0&0 \\ 0&0 \end{array}} \right]\\ &{{F}_{wi}} = \left[{\begin{array}{*{20}{c}} 0&0&0 \\ 0&0&0 \\ {\frac{{{T_{pi}}}}{{{K_{pi}}}}}&0&0 \\ 0&{2\pi }&0 \\ 0&0&0 \\ 0&0&{\frac{1}{{2{H_{ti}}}}} \\ 0&0&0 \end{array}} \right].\end{align*} $ 
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