1 Introduction

Power systems are electrical energy productions and consumption systems composed of power generations, substations, transmissions, and the users. It has an indispensable position in the national economy, defense construction and other fields^[1]. Two basic conditions must be met in the safe operation of the power system: first, all electrical equipment are in the normal state and meet the variety of operating condition; second, system voltage and frequency should be kept within the scope of the provisions, which requires all generators to maintain synchronous operation^[2]. The operation state of power system directly affects the actual safety level of system. Therefore, it is necessary to pay attention to the control problem of safe operation, in order to enhance the reliability of power supply system. The safe control for power system is that all kinds of control measures and methods are taken to make the power system operate in the normal state. In Ref.[3], a large scale power blackout in the domestic and foreign is taken as an example, the security and stability control of power system is proposed based on system response, and the key technology and new research is discussed^[4-8].

After years of research and development, a variety of control methods, such as robust control, adaptive control, fuzzy control, neural network and genetic algorithm have been applied to the power system^[9-11]. In Ref.[12], based on the robust nonlinear control theory, the stability of ship power system containing the propulsion load is analyzed, and the nonlinear control of the ship power system is discussed, in order to enhance the dynamic quality of ship power system. Considered the variable coupling relationship of ship power system, the controller can effectively restrain the influence of the disturbance on the power-angle and the frequency of generator. For the large load of ship electric power system, taking into account the nonlinear impact of the load to the grid some limitations are brought into the design of controller. In Ref.[13], a nonlinear robust adaptive controller is designed to reduce the influence of the interference on the power-angle and the frequency of multi machine power system considered the parameter uncertainty of reactance. In Ref.[14], a robust adaptive excitation controller is designed based on L₂ interference suppression theory of the multi machine power system containing uncertain parameters. In Ref.[15], a kind of intelligent power system stabilizer based on fuzzy control is proposed, it has the advantages that the control can be selected according to the actual operation conditions of power systems, but the fuzzy control rule is needed to be determined by the operation experience, so that in order to carry out fuzzy reasoning and decision, designers need have a certain operation experience of power system. In Ref.[16], a design method based on maximum entropy principle to optimize fuzzy neural network is put forward to realize stable control of power system. However the demand to choose the center parameters of the membership function is high in the scheme, if the center parameters of the membership function are not selected properly, it will affect the learning effect. A fuzzy power system controller adopted genetic algorithm is proposed in Refs.[17-18]. Genetic algorithm is introduced into the design of fuzzy logic controller, which enhances the dynamic performance of system significantly, and it has a good robustness under small disturbance signal. However the overshoot of the system is a little larger, and the transition time is a little longer.

Hamilton system is a kind of conservative nonlinear system derived from classical mechanics, which is one of the important research fields in nonlinear science, and has certain advantages in engineering application. Lyapunov function based on Energy is selected from the Hamilton function of the generalized Hamilton system, so the key problem to solve the nonlinear system with the energy function is that the nonlinear system is translated into a dissipative Hamilton function. The research on generalized Hamilton system in the nonlinear control field of power system is developing rapidly. The Ref.[19] based on the transient energy function of power system, presents a new Lyapunov function. The function is used to stabilize the power system, and the mutual influence among the dynamic processes of generator is considered in the design of the controller, but the external disturbances and load changes have not been discussed. In Ref.[20], it is pointed out that the influence of load nonlinearity must be considered for the control of power system. In Ref.[21], a wide area time-delay damping control of multi machine power system is studied. The stable control of wide area nonlinear power system with transmission signal delay is analyzed by the dissipative Hamiltonian system method. However the adaptive control for power systems with interferences and uncertain parameters is not considered.

In the paper, a Hamilton model with control delay is established for multi machine power system, considered time delay in transmission process. By constructed a Lyapunov-Krasovskii function, an adaptive H_∞ control scheme is designed based on Hamilton function, and the stable control is achieved for nonlinear power system with time-delay, disturbance and uncertain parameters. H_∞ control can restrain the disturbance and the influence of uncertain parameters in the system is overcome by adaptive control. Simulation results show the effective of the control scheme.

2 The Model of Multi-Machine Power System and its Hamilton Model

The power system which is composed of n interconnected generators is considered as follows. Each generator is expressed by a three-order system.

$ \left\{ \begin{array}{l} {{\dot \delta }_i} = {\omega _i} - {\omega _0}\\ {{\dot \omega }_i} = \frac{{{\omega _0}}}{{{M_i}}}{P_{mi}} - \frac{{{D_i}}}{{{M_i}}}\left( {{\omega _i} - {\omega _0}} \right) - \\ \;\;\;\;\;\;\;\frac{{{\omega _0}}}{{{M_i}}}\sum\limits_{j = 1}^n {{{E'}_{qi}}{{E'}_{qj}}\left( {{B_{ij}}{S_{{\delta _{ij}}}} + {G_{ij}}{C_{{\delta _{ij}}}}} \right)} \\ {{E'}_{qi}} = - \frac{1}{{{{T'}_{doi}}}}\left[ {{{E'}_{qi}} + \left( {{x_{di}} - {{x'}_{di}}} \right) \cdot } \right.\\ \;\;\;\;\;\;\;\;\left. {\sum\limits_{j = 1}^n {{{E'}_{qi}}\left( {{G_{ij}}{S_{{\delta _{ij}}}} - {B_{ij}}{C_{{\delta _{ij}}}}} \right)} } \right] + \frac{{{u_{fi}}}}{{{{T'}_{doi}}}} \end{array} \right. $

(1)

where subscript i=1, 2, …, n, is corresponding to n-th generator; δ_i is the power-angle of generator in rad; ω_i is the angular velocity in rad/s; ω₀=2πf is the synchronous angular speed; x_di is the d-axis winding self-resistance in per unit; x′_di is the d-axis transient reactance in per unit; T′_doi is the excitation circuit time constant in per unit; E′_qi is the q-axis transient potential in per unit; M_i is the inertia time constant in s; D_i is the damping coefficient in per unit; P_mi is the input mechanical power in per unit; u_fi is the i-th excitation input of generator in per unit; S_δij=sin(δ_i-δ_j); C_δij=cos(δ_i-δ_j).

The Hamilton function of system (1) can be selected as Ref.[2]: The dynamic equation as follows:

$ {{\mathit{\boldsymbol{\dot x}}}_i} = \left[ {{\mathit{\boldsymbol{J}}_i} - {\mathit{\boldsymbol{R}}_i}} \right]{\nabla _{{\mathit{\boldsymbol{x}}_i}}}\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{q}}_i} + {\mathit{\boldsymbol{g}}_i}{\mathit{\boldsymbol{u}}_{fi}} $

(2)

where

$ {\mathit{\boldsymbol{x}}_i} = \left[ {\begin{array}{*{20}{c}} {{{\dot \delta }_i}}\\ {{{\dot \omega }_i}}\\ {{{\dot E'}_{qi}}} \end{array}} \right],{\mathit{\boldsymbol{g}}_i} = \left[ {\begin{array}{*{20}{c}} 0\\ 0\\ {\frac{1}{{{{T'}_{doi}}}}} \end{array}} \right],{\mathit{\boldsymbol{J}}_i} = \left[ {\begin{array}{*{20}{c}} 0&{\frac{{{\omega _0}}}{{{M_i}}}}&0\\ { - \frac{{{\omega _0}}}{{{M_i}}}}&0&0\\ 0&0&0 \end{array}} \right] $

$ {\mathit{\boldsymbol{R}}_i} = \left[ {\begin{array}{*{20}{c}} 0&0&0\\ 0&{\frac{{{D_i}{\omega _0}}}{{M_i^2}}}&0\\ 0&0&{\frac{{{x_{di}} - {{x'}_{di}}}}{{{{T'}_{doi}}}}} \end{array}} \right] $

$ \begin{array}{l} {\mathit{\boldsymbol{q}}_i} = \\ \left[ {\begin{array}{*{20}{c}} 0\\ { - \frac{{2{\omega _0}}}{{{M_i}}}\sum\limits_{1 \le k < i} {{{E'}_{qi}}{{E'}_{qj}}{G_{ik}}{G_{{\delta _{ij}}}}} }\\ { - \frac{{2\left( {{x_{di}} - {{x'}_{di}}} \right)}}{{{{T'}_{doi}}}}\sum\limits_{1 \le k < i} {{{E'}_{qk}}{G_{ik}}{G_{{\delta _{ij}}}}} + \frac{{2\left( {{x_{di}} - {{x'}_{di}}} \right){G_{ii}}}}{{{{T'}_{doi}}}}{\delta _i}{{E'}_{qi}}} \end{array}} \right] \end{array} $

Let u_fi=u_fi1+u_fi2, $ {{u}_{fi2}}=-{{\sigma }_{i}}\frac{\sum\limits_{i=1}^{n}{\nabla _{{{x}_{i}}}^\text{T}\mathit{\boldsymbol{H}}{{\mathit{\boldsymbol{q}}}_{i}}}}{\sum\limits_{i=1}^{n}{\sigma _{i}^{2}}} $ is used to compensate q_i, and it also can be got by Ref.[2].

Eq.(2) is written as follows:

$ {{\mathit{\boldsymbol{\dot x}}}_i} = \left[ {{\mathit{\boldsymbol{J}}_i} - {\mathit{\boldsymbol{R}}_i}} \right]{\nabla _{{\mathit{\boldsymbol{x}}_i}}}\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{g}}_i}{\mathit{\boldsymbol{u}}_{fi1}} $

(3)

Eq.(3) is a kind of dissipative Hamilton form.

3 Adaptive H_∞ Control with Time Delay 3.1 The Establishment of Hamilton Model with Time Delay

Taking 2 machine systems as an example, the Hamilton model with time delay is set up and also can be extended to the multi-machine system. The Hamilton model of 2 machine systems as follows:

$ \left\{ \begin{array}{l} {{\mathit{\boldsymbol{\dot x}}}_1}\left( t \right) = \left[ {{\mathit{\boldsymbol{J}}_1} - {\mathit{\boldsymbol{R}}_1}} \right]{\nabla _{{\mathit{\boldsymbol{x}}_1}}}\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{g}}_1}{\mathit{\boldsymbol{u}}_{f1}}\left( t \right)\\ {{\mathit{\boldsymbol{\dot x}}}_2}\left( t \right) = \left[ {{\mathit{\boldsymbol{J}}_2} - {\mathit{\boldsymbol{R}}_2}} \right]{\nabla _{{\mathit{\boldsymbol{x}}_2}}}\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{g}}_2}{\mathit{\boldsymbol{u}}_{f2}}\left( t \right) \end{array} \right. $

(4)

In the practical system, the feedback control signal of different regions usually has time delay (it is caused by feedback signal transmission, especially for long distant regions), and the feedback control with time delay can be expressed as:

$ \left\{ \begin{array}{l} {\mathit{\boldsymbol{u}}_{f1}}\left( t \right) = {\mathit{\boldsymbol{v}}_1} - {\mathit{\boldsymbol{K}}_{12}}{\nabla _{{\mathit{\boldsymbol{x}}_2}}}{\mathit{\boldsymbol{H}}_{t - \tau }}\\ {\mathit{\boldsymbol{u}}_{f2}}\left( t \right) = {\mathit{\boldsymbol{v}}_2} - {\mathit{\boldsymbol{K}}_{21}}{\nabla _{{\mathit{\boldsymbol{x}}_1}}}{\mathit{\boldsymbol{H}}_{t - \tau }} \end{array} \right. $

(5)

where v₁, v₂ are the controller composed of the local state variables, used to design an adaptive controller. K₁₂, K₂₁ is feedback gain of feedback control signal for different regions. τ is transmission time delay. The time delay of state transfer between different regions is:

$ {\nabla _{{\mathit{\boldsymbol{x}}_i}}}{\mathit{\boldsymbol{H}}_{t - \tau }} = \frac{{\partial H\left[ {{x_i}\left( {t - \tau } \right)} \right]}}{{\partial {x_i}}},i = 1,2 $

Introducing Eq.(5) into Eq.(4):

$ \begin{array}{l} \left[ {\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\dot x}}}_1}\left( t \right)}\\ {{{\mathit{\boldsymbol{\dot x}}}_2}\left( t \right)} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{J}}_1} - {\mathit{\boldsymbol{R}}_1}}&0\\ 0&{{\mathit{\boldsymbol{J}}_2} - {\mathit{\boldsymbol{R}}_2}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\nabla _{{x_1}}}\mathit{\boldsymbol{H}}}\\ {{\nabla _{{x_2}}}\mathit{\boldsymbol{H}}} \end{array}} \right] + \\ \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{g}}_1}}&0\\ 0&{{\mathit{\boldsymbol{g}}_2}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{v}}_1}}\\ {{\mathit{\boldsymbol{v}}_2}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0&{ - {\mathit{\boldsymbol{g}}_1}{\mathit{\boldsymbol{K}}_{12}}}\\ { - {\mathit{\boldsymbol{g}}_2}{\mathit{\boldsymbol{K}}_{21}}}&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\nabla _{{x_1}}}{\mathit{\boldsymbol{H}}_{t - \tau }}}\\ {{\nabla _{{x_2}}}{\mathit{\boldsymbol{H}}_{t - \tau }}} \end{array}} \right] \end{array} $

(6)

Eq.(6) is denoted as follows:

$ \mathit{\boldsymbol{\dot x}} = \left[ {\mathit{\boldsymbol{J}} - \mathit{\boldsymbol{R}}} \right]{\nabla _x}\mathit{\boldsymbol{H}} + \mathit{\boldsymbol{T}}{\nabla _x}{\mathit{\boldsymbol{H}}_{t - \tau }} + \mathit{\boldsymbol{gv}} $

(7)

where

$ \mathit{\boldsymbol{x}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{x}}_1}\left( t \right)}\\ {{\mathit{\boldsymbol{x}}_2}\left( t \right)} \end{array}} \right],\mathit{\boldsymbol{v}} = \left[ {\begin{array}{*{20}{c}} {{v_1}}\\ {{v_2}} \end{array}} \right],{\nabla _x}\mathit{\boldsymbol{H}} = \left[ {\begin{array}{*{20}{c}} {{\nabla _{{x_1}}}\mathit{\boldsymbol{H}}}\\ {{\nabla _{{x_2}}}\mathit{\boldsymbol{H}}} \end{array}} \right] $

$ {\nabla _x}{\mathit{\boldsymbol{H}}_{t - \tau }} = \left[ {\begin{array}{*{20}{c}} {{\nabla _{{x_1}}}{\mathit{\boldsymbol{H}}_{t - \tau }}}\\ {{\nabla _{{x_2}}}{\mathit{\boldsymbol{H}}_{t - \tau }}} \end{array}} \right],\mathit{\boldsymbol{J}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{J}}_1}}&0\\ 0&{{\mathit{\boldsymbol{J}}_2}} \end{array}} \right],\mathit{\boldsymbol{R}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{R}}_1}}&0\\ 0&{{\mathit{\boldsymbol{R}}_2}} \end{array}} \right] $

$ \mathit{\boldsymbol{T}} = \left[ {\begin{array}{*{20}{c}} 0&{ - {\mathit{\boldsymbol{g}}_1}{\mathit{\boldsymbol{K}}_{12}}}\\ { - {\mathit{\boldsymbol{g}}_2}{\mathit{\boldsymbol{K}}_{21}}}&0 \end{array}} \right],\mathit{\boldsymbol{g}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{g}}_1}}&0\\ 0&{{\mathit{\boldsymbol{g}}_2}} \end{array}} \right] $

Multi-machine power system is also obtained as the form of Eq.(7). For convenience hereinafter, the Eq.(7) can be rewritten as follows:

$ \mathit{\boldsymbol{\dot x}} = \left[ {\mathit{\boldsymbol{J}}\left( \mathit{\boldsymbol{x}} \right) - \mathit{\boldsymbol{R}}\left( \mathit{\boldsymbol{x}} \right)} \right]\nabla \mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right) + \mathit{\boldsymbol{T}}\nabla \mathit{\boldsymbol{H}}\left( {{\mathit{\boldsymbol{x}}_\tau }} \right) + \mathit{\boldsymbol{gv}} $

(8)

where x_τ= x(t-τ).

3.2 Adaptive H_∞ Controller Design

Considered external perturbation and parameter perturbation, the adaptive H_∞ control scheme of system (8) is designed. The parameter perturbation of system is small, so that the dissipation of the system is not changed. The Eq.(8) is expressed as a system with unknown constant vector p.

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{\dot x}} = \left[ {\mathit{\boldsymbol{J}}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{p}}} \right) - \mathit{\boldsymbol{R}}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{p}}} \right)} \right]\nabla \mathit{\boldsymbol{H}}\left( {{\mathit{\boldsymbol{x}}_\tau }} \right) + \\ \;\;\;\;\;\mathit{\boldsymbol{T}}\nabla \mathit{\boldsymbol{H}}\left( {{\mathit{\boldsymbol{x}}_\tau }} \right) + \mathit{\boldsymbol{gv}} + \mathit{\boldsymbol{dw}}\\ \mathit{\boldsymbol{y}} = {\mathit{\boldsymbol{d}}^{\rm{T}}}\nabla \mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right)\\ \mathit{\boldsymbol{z}} = \mathit{\boldsymbol{r}}{\mathit{\boldsymbol{g}}^{\rm{T}}}\nabla \mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right) \end{array} \right. $

(9)

where y is the output; w is the disturbance input; z is the penalty signal; d is disturbance input matrix; r is the weighted matrix which the column is full rank.

The adaptive H_∞ control problems of system (9) are described as following: an interference attenuation level is given γ > 0, to find a state feedback controller:

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{v}} = \alpha \left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{\hat \theta }}} \right)\\ \mathit{\boldsymbol{\dot {\hat \theta} }} = \mathit{\boldsymbol{\beta }}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{\hat \theta }}} \right) \end{array} \right. $

(10)

It will made the L₂ gain (from w to z) of the closed-loop system less than or equal to γ as follows.

$ \int_0^T {{{\left\| z \right\|}^2}{\rm{d}}t} \le {\gamma ^2}\int_0^T {{{\left\| w \right\|}^2}{\rm{d}}t} $

(11)

In Eq.(10) θ is an unknown parameter vector, $ \mathit{\boldsymbol{\hat \theta}} $ is an estimate of θ, ∀w∈L₂[0, T].

The closed loop system under the control scheme (10) is asymptotically stable when w=0.

Various items of uncertain parameters in the system (9) can be further written as:

$ \begin{array}{l} \mathit{\boldsymbol{J}}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{p}}} \right) = {\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}_J}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{p}}} \right) + \mathit{\boldsymbol{J}}\left( \mathit{\boldsymbol{x}} \right)\\ \mathit{\boldsymbol{R}}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{p}}} \right) = {\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}_R}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{p}}} \right) + \mathit{\boldsymbol{R}}\left( \mathit{\boldsymbol{x}} \right) \end{array} $

where Δ_J(x, p), and Δ_R(x, p) are uncertainties.

To facilitate the design of adaptive H_∞ controller, some assumptions are given:

Assumption 1  There exists a suitable dimension matrix Ψ(x) so that

$ \left[ {\mathit{\boldsymbol{J}}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{p}}} \right) - \mathit{\boldsymbol{R}}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{p}}} \right)} \right]{\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}_H}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{p}}} \right) = \mathit{\boldsymbol{g}}{\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}\left( \mathit{\boldsymbol{x}} \right)\mathit{\boldsymbol{\theta }} $

(12)

where, θ is a constant vector about p. Assumption 1 is called the matching condition and it is also a general assumption in the Hamilton system control. In most cases, there are such Ψ(x) and θ, making the Eq.(12) be true.

Assumption 2  R(x, p)≥ A, A ≥0 is a constant matrix.

Considering system (9), and supposing assumption 1 and 2 satisfied, if there are matrices P = P^T > 0, Q = Q^T > 0, Γ = Γ^T > 0 such that

$ \mathit{\boldsymbol{ \boldsymbol{\varLambda} }} = \left[ {\begin{array}{*{20}{c}} {{{\bar \Lambda }_{1,1}}}&{{{ \mathit{\bar\Lambda} }_{1,2}}}&{\frac{1}{2}d}\\ * &{{{\mathit{\bar \Lambda }}_{2,2}}}&0\\ *&* &{ - \frac{1}{2}{\gamma ^2}\mathit{\boldsymbol{I}}} \end{array}} \right] < 0 $

(13)

where * represents symmetric term, and

$ {{\mathit{\boldsymbol{ \boldsymbol{\bar \varLambda} }}}_{1,1}} - \mathit{\boldsymbol{A}} - \mathit{\boldsymbol{g}}{\mathit{\boldsymbol{g}}^{\rm{T}}}\left( {\mathit{\boldsymbol{P}} + \mathit{\boldsymbol{Q}}} \right) + \mathit{\boldsymbol{P}} + \frac{1}{2}\mathit{\boldsymbol{g}}{\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{r}}{\mathit{\boldsymbol{g}}^{\rm{T}}} $

$ {{\mathit{\boldsymbol{ \boldsymbol{\bar \varLambda} }}}_{1,2}} = \frac{1}{2}\mathit{\boldsymbol{T}}\left( \mathit{\boldsymbol{x}} \right) $

Λ_{2, 2}=-P, the adaptive control law of the Eq.(14) can be used to make the system (9) asymptotically stable.

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{v}} = - {\mathit{\boldsymbol{g}}^{\rm{T}}}\left( {\mathit{\boldsymbol{P}} + \mathit{\boldsymbol{Q}}} \right)\nabla \mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right) - {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}\left( \mathit{\boldsymbol{x}} \right)\mathit{\boldsymbol{\hat \theta }}\\ \mathit{\boldsymbol{\dot {\hat \theta} }} = \mathit{\boldsymbol{ \boldsymbol{\varGamma} \boldsymbol{\varPsi} }}\left( \mathit{\boldsymbol{x}} \right){\mathit{\boldsymbol{g}}^{\rm{T}}}\nabla \mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right) \end{array} \right. $

(14)

Proof  Introducing Eq.(12) and Eq.(14) into Eq.(9)

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{\dot x}} = \left[ {\mathit{\boldsymbol{J}}\left( \mathit{\boldsymbol{x}} \right) - \mathit{\boldsymbol{R}}\left( \mathit{\boldsymbol{x}} \right)} \right]\nabla \mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right) + \mathit{\boldsymbol{T}}\left( \mathit{\boldsymbol{x}} \right)\nabla \mathit{\boldsymbol{H}}\left( {{\mathit{\boldsymbol{x}}_\tau }} \right) - \\ \;\;\;\;\;\;\mathit{\boldsymbol{g}}{\mathit{\boldsymbol{g}}^{\rm{T}}}\left( {\mathit{\boldsymbol{P}} + \mathit{\boldsymbol{Q}}} \right)\nabla \mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right) + \mathit{\boldsymbol{dw}} + \mathit{\boldsymbol{g}}{\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}\left( \mathit{\boldsymbol{x}} \right)\left( {\mathit{\boldsymbol{\theta }} - \mathit{\boldsymbol{\hat \theta }}} \right)\\ \mathit{\boldsymbol{\dot {\hat \theta} }} = \mathit{\boldsymbol{ \boldsymbol{\varGamma} \boldsymbol{\varPsi} }}\left( \mathit{\boldsymbol{x}} \right){\mathit{\boldsymbol{g}}^{\rm{T}}}\nabla \mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right)\\ \mathit{\boldsymbol{y}} = {\mathit{\boldsymbol{d}}^{\rm{T}}}\nabla \mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right)\\ \mathit{\boldsymbol{z}} = \mathit{\boldsymbol{r}}{\mathit{\boldsymbol{g}}^{\rm{T}}}\nabla \mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right) \end{array} \right. $

(15)

The Lyapunov-Krasovskii function is chosen as follows:

$ \begin{array}{l} \mathit{\boldsymbol{V}}\left( {\mathit{\boldsymbol{x}},{\mathit{\boldsymbol{x}}_\tau },\mathit{\boldsymbol{\theta }}} \right) = \mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right) + \int_{t - \tau }^t {{\nabla ^{\rm{T}}}\mathit{\boldsymbol{H}}\left( {\mathit{\boldsymbol{x}}\left( s \right)\mathit{\boldsymbol{P}}\nabla \mathit{\boldsymbol{H}}\left( {\mathit{\boldsymbol{x}}\left( s \right)} \right){\rm{d}}s + } \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{2}{{\mathit{\boldsymbol{\tilde \theta }}}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}^{ - 1}}\mathit{\boldsymbol{\tilde \theta }} \end{array} $

(16)

where $ \mathit{\boldsymbol{\tilde \theta}}=\mathit{\boldsymbol{ \theta}}-\mathit{\boldsymbol{\hat \theta}} $.

The derivative of V (x, x_τ, θ) to time is calculated as follows:

$ \begin{array}{l} \mathit{\boldsymbol{\dot V}}\left( {\mathit{\boldsymbol{x}},{\mathit{\boldsymbol{x}}_\tau },\mathit{\boldsymbol{\tilde \theta }}} \right) - \frac{1}{2}\left( {{\gamma ^2}{{\left\| \mathit{\boldsymbol{w}} \right\|}^2} - {{\left\| \mathit{\boldsymbol{z}} \right\|}^2}} \right) \le - {\nabla ^{\rm{T}}}\mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right)\left[ {\mathit{\boldsymbol{A}} + } \right.\\ \;\;\;\;\;\;\;\left. {\mathit{\boldsymbol{g}}{\mathit{\boldsymbol{g}}^{\rm{T}}}\left( {\mathit{\boldsymbol{P}} + \mathit{\boldsymbol{Q}}} \right) - \mathit{\boldsymbol{P}}} \right]\nabla \mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right) + {\nabla ^{\rm{T}}}\mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right)\mathit{\boldsymbol{T}}\nabla \mathit{\boldsymbol{H}}\left( {{\mathit{\boldsymbol{x}}_\tau }} \right) - \\ \;\;\;\;\;\;\;{\nabla ^{\rm{T}}}\mathit{\boldsymbol{H}}\left( {{\mathit{\boldsymbol{x}}_\tau }} \right)\mathit{\boldsymbol{P}}\nabla \mathit{\boldsymbol{H}}\left( {{\mathit{\boldsymbol{x}}_\tau }} \right) + {\nabla ^{\rm{T}}}\mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right){\rm{d}}\mathit{\boldsymbol{w}} - \frac{1}{2}{\gamma ^2}{\mathit{\boldsymbol{w}}^{\rm{T}}}\mathit{\boldsymbol{w + }}\\ \;\;\;\;\;\;\;\frac{1}{2}{\nabla ^{\rm{T}}}\mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right)\mathit{\boldsymbol{g}}{\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{r}}{\mathit{\boldsymbol{g}}^{\rm{T}}}\nabla \mathit{\boldsymbol{H}}\left( \mathit{\boldsymbol{x}} \right) = {\mathit{\boldsymbol{\xi }}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{ \boldsymbol{\varLambda} \xi }}\left( t \right) \end{array} $

(17)

where ξ(t)=[∇^TH (x), ∇^T H (x_τ), ω^T]^T, from inequality(13) Λ < 0, it can be seen that:

$ \mathit{\boldsymbol{\dot V}}\left( {\mathit{\boldsymbol{x}},{\mathit{\boldsymbol{x}}_\tau },\mathit{\boldsymbol{\tilde \theta }}} \right) - \frac{1}{2}\left( {{\gamma ^2}{{\left\| \mathit{\boldsymbol{w}} \right\|}^2} - {{\left\| \mathit{\boldsymbol{z}} \right\|}^2}} \right) < 0 $

(18)

The Eq.(9) indicates that the asymptotically stability is achieved.

4 Simulation Study

The simulation research for double machine power system is done. Select generator parameters:

M₁=23.64 s, M₂=6.4 s, x_d1=0.146 p.u., x_d2=0.895 8 p.u., x′_d1=0.060 8 p.u., x′_d2=0.119 p.u., D₁=0.31 p.u., D₂=0.535 p.u., P_m1=0.7157 p.u., P_m2=1.629 5 p.u., T′_d01=8.96 s, T′_d02=6 s, $ \mathit{\boldsymbol{B}}=[{{\mathit{\boldsymbol{B}}}_{ij}}]=\left[ \begin{matrix} -2.988\ 2&1.513\ 0 \\ 1.513\ 0&-2.723\ 8 \\ \end{matrix} \right] $, $ \mathit{\boldsymbol{G}}=[{{\mathit{\boldsymbol{G}}}_{ij}}]=\left[ \begin{matrix} 0.854\ 3&0.287\ 0 \\ 0.287\ 0&0.419\ 9 \\ \end{matrix} \right] $, τ=1.2 s.

Assuming that the uncertain parameters as follows:

$ {{T'}_{d0i}} \to {{T'}_{d0i}} + {p_{i1}},{B_{ii}} \to {B_{ii}} + {p_{i2}},i = 1,2 $

Let p_i=[p_i1  p_i2], p =(p₁^T  p₂^T)^T, The uncertain dissipation Hamilton system can be achieved as Eq.(9).

Let

$ \mathit{\boldsymbol{\theta }} = {\left( {\begin{array}{*{20}{c}} {\frac{{0.0852{p_{12}} + {p_{11}}}}{{8.96 + {p_{11}}}}}&{\frac{{0.7768{p_{22}} + 2{p_{21}}}}{{6 + {p_{21}}}}} \end{array}} \right)^{\rm{T}}} $

$ \mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( x \right) = \left( {\begin{array}{*{20}{c}} {{{E'}_{q1}}}&0\\ 0&{{{E'}_{q2}}} \end{array}} \right) $

It is easy to test that Assumption 1 is satisfied.

After many tests, let

$ {K_{12}} = 25,{K_{21}} = 1,\mathit{\Gamma } = 1 $

$ \mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} {0.5}&0\\ 0&1 \end{array}} \right],\mathit{\boldsymbol{P = }}\left[ {\begin{array}{*{20}{c}} {22.0331}&0\\ 0&{2.0331} \end{array}} \right] $

$ \mathit{\boldsymbol{Q}} = \left[ {\begin{array}{*{20}{c}} {2.1021}&{0.3214}\\ {0.3214}&{2.0331} \end{array}} \right] $

satisfying the inequality (13), and set w=0.25 sin t

During the steady operation of system, and assuming that one sudden three phase short circuit fault takes place at 4 s, x′_d2=0, and is resected after 1.2 s, the simulation results are shown in Figs. 1-5.

The goal of power system transient stability control is to make each generator tend to the rated synchronous speed^[22]. As can be seen from the simulation results shown in Figs. 1-5, after a period of adjustment, the system can reach a stable operating point again. The difference of angular speed and synchronous speed of generator converges to zero, and the rotor running angle of each generator converges to a fixed angle, and the stability is good. The system has a strong robustness to external disturbance, and the ideal control effect is obtained.

In order to test the control effect, the controller designed in this paper is compared with the controller based on DIEB (Damping Injection Energy Balancing) in Ref.[19].

Comparing Fig. 1 and Fig. 6, it can be seen that the turbulence amplitude in the paper is less than that in Ref.[19], and the response time is also less than that in Ref.[19]. The controller designed in the paper is better than that in Ref.[19].

To verify the adaptive ability of power system, the parameter variations of power system are considered. Let

$ {{T'}_{d01}} = 8.96\;{\rm{p}}.{\rm{u}}.\;,{{T'}_{d02}} = 6.32\;{\rm{p}}.{\rm{u}}. $

$ \mathit{\boldsymbol{B}} = \left[ {{\mathit{\boldsymbol{B}}_{ij}}} \right] = \left[ {\begin{array}{*{20}{c}} { - 2.8522}&{1.5130}\\ {1.5130}&{ - 2.9138} \end{array}} \right] $

Assuming that the sudden three phase short circuit fault takes place at 4 s again, x′_d2=0, and is resected after 1.2 s, the simulations are shown in Figs. 7-11.

As can be seen from Figs. 6-10, the controller has a strong adaptive ability to the parameter perturbation of system. When the system parameters have changed, the system tends to be stable, and the good control effect is obtained.

5 Conclusions

In this paper, the adaptive H_∞ control scheme of multi-machine power system with control time delay and parameter uncertainty is studied based on Hamiltonian function. Considered the time delay effect in the transmission process of control, a Hamilton model with control delay is established. Selected the appropriate Lyapunov-Krasovskii function, an adaptive robust H_∞ state feedback control law is derived which can guarantee the asymptotic stability of the system. H_∞ controller can realize the disturbance suppression, and adaptive control can improve the robustness of system to uncertain parameters. The simulation results show that the designed controller makes the power system to be stable, and the effect of the proposed controller is tested.