Networked scheduling for decentralized load frequency control

1.1. Notations

Throughout this paper, $$ \mathbb{R}_n $$ stands for the n-dimensional Euclidean space with vector norm $$ \parallel\cdot\parallel $$. $$ diag\{\cdots\} $$ and $$ col\{\cdots\} $$ denote the block-diagonal matrix and block-column vector, respectively. The superscript $$ T $$ stands for the transpose of a matrix or a vector. $$ I $$ is an identity matrix with an appropriate dimension. Matrix $$ X>0(X\geq0) $$ means that $$ X $$ is a positive definite (positive semi-definite) symmetric matrix. The symbol $$ \ast $$ is the symmetric term in a matrix. $$ w[-h, 0] $$ denotes the Banach type of absolutely continuous functions $$ \phi:[-h, 0]\rightarrow \mathbb{R}_n $$ with $$ \phi^\cdot \in\mathbb{L}_2(-h, 0) $$ with the norm $$ \parallel\phi\parallel_w=\max\limits_{v\in[-h, 0]} \parallel\phi(v)\parallel+[\int_{-h}^0 \parallel\phi^\cdot(v)\parallel^2 ds]^{\frac{1}{2}} $$.

2.1. Multi-area decentralized LFC model

Diagram of the multi-area decentralized LFC under scheduling protocols is shown in Figure 1, where data transmission from sensors to decentralize controllers is scheduled by RR or TOD scheduling protocol.

Figure 1. Multi-area decentralized power systems under scheduling protocols.

The system model is represented as^{[23, 25]}:

where $$ x(t)\in\mathbb{R}_{x} $$ is the state vector, $$ y(t)\in\mathbb{R}_{n_y} $$ is the measurement and $$ \omega(t)\in\mathbb{R}_{\omega} $$ is the disturbance, $$ u(t)\in\mathbb{R}_{u} $$ is the control input vector. The multi-area power systems consist of generators, turbines, and governors, where $$ \Delta P_{vi}, \Delta P_{mi}, \Delta P_{di} $$, and $$ \Delta f_i $$ denote the deviation of valve position, generator mechanical output, load, and the frequency of the $$ ith $$ sub-area, respectively. The area control error $$ ACE_i(s) $$ is

The state and measured output signals are

where $$ x(t)=[x_1^T(t), \cdots, x_n^T(t)]^T $$, $$ \begin{align*} y(t) & = [y_1^T(t), \cdots, y_n^T(t)]^T\in \mathbb{R}_{n_y}, \\ \omega(t) & = [\omega_1^T(t), \cdots, \omega_n^T(t)]^T, \\ A & = [A_{ij}]_{n\times n}, \\ B & = \text{diag}\{B_1, \cdots, B_n\}, \\ C & = [C_1^T, \cdots, C_N^T]^T, \\ F & = \text{diag}\{F_1, \cdots, F_n\}, \\ C_i & = [\underbrace{0, \cdots, 0}_{i-1}, \bar{C}_i, \underbrace{0, \cdots, 0}_{N-i}] \end{align*} $$,

The multi-area power system includes $$ N $$ controlled areas, $$ N $$ decentralized controllers, which are connected via the network. Then, measurement signals are given by $$ y_i(t)=C_i x(t)\in \mathbb{R}_{n_i}, i=1, \cdots, N, \sum\limits_{i=1}^N n_i=n_y $$. Based on the phasor measurement unit, sampling instants of $$ N $$ sensors are synchronized in different areas of power systems. Denote sampling instant by $$ s_q $$, satisfying that

In this paper, we take imperfect network conditions into account, such as data loss and network-induced delay. Denote the sequence after packet loss by $$ \{s_k\}\subseteq\{s_q\} $$; that is, only at sampling instants $$ {s_k} $$, the input of the controller can be updated. For $$ i=1, \cdots, N $$, an uncertain, time-varying delay $$ v_{s_k}^i \in [0, \tau_M^i] $$ is assumed to occur, where $$ \tau_M^i $$ is delay upper bound of the $$ ith $$ channel and $$ \max\limits_{i=1, \cdots, N}\{\tau_M^i\}=\tau_M $$. Buffers are set to store and choose the largest communication delay $$ \tau_{s_k} $$ of power system channels, i.e., $$ \tau_{s_k}=\max\limits_{i=1, \cdots, N}\{v_{s_k}^i\} $$. Then, ZOH updating instant is $$ t_k=s_k+\tau_{s_k} $$. The transmission delay is assumed to be bound, satisfying:

Let $$ \hat{y}(s_k)=[\hat{y}_1^T (s_k) \cdots \hat{y}_N^T (s_k)]^T \in \mathbb{R}_{n_y} $$ denote the output signal transmitted to the scheduler. At instant $$ s_k $$, only one node can be active. Let $$ \iota_k\in \{1, \cdots, N\} $$ be the active output node at $$ s_k $$, which is dependent on the scheduling rule. Then, we have

Consider the scheduling error between output $$ y_i(s_k) $$ and the last available measurement $$ \hat{y}_i(s_{k-1}) $$:

where $$ \hat{y}_i(s_{-1})\triangleq 0, \; i=1, \cdots, N $$. In this paper, controllers and actuators in the $$ ith $$ area are event-driven. Let $$ L_i>0, i=1, \cdots, N $$ be node weighting matrices. Under TOD scheduling,

Under RR scheduling protocol, the active node $$ \iota_k $$ is selected periodically:

In the following, we will design the decentralized LFC law under the communication network. Similar to^[23], a PI controller is used in this paper:

Therefore, dynamic model of the scheduled power systems under the decentralized LFC Equation (8) and imperfect network environments can be formalized as

where $$ u(t)=\sum\limits_{i=1}^N K_i y_i(t)=Ky(t), \; K=[K_1, \cdots, K_N] $$, $$ K_i=[K_{Pi}, K_{Ii}] $$.

2.2. Impulsive model and study objective

From Equation (5), one can obtain that

Define an artificial delay $$ \tau(t)=t-s_k $$, from which one arrives at

From Equation (5) and Equation (9), the impulsive power system model can be

For system Equation (12), the initial condition of $$ x(t) $$ on $$ [-\tau_M, 0] $$ is supplemented as $$ x(t)=\phi(t), t\in[-\tau_M, 0] $$, with $$ \phi(0)=x_0 $$, where $$ \phi(t) $$ is a continuous function on $$ [-\tau_M, 0] $$.

Using scheduling scheme Equation (6) and Equation (7), this paper is to design the decentralized controller Equation (8) such that system Equation (12) is exponentially stable with a prescribed $$ H_\infty $$ performance $$ \gamma $$.

3.1. Stability analysis under the TOD scheduling scheme Equation (6)

Construct Lyapunov-Krasovskii functional candidate:

where

Remark 1$$ V_H $$is introduced to cope with reset conditions, which is continuous on $$ [t_k, t_{k+1}) $$, and does not grow in the jumps when $$ t=t_{k+1} $$ since

Theorem 1Under TOD scheduling scheme Equation (6), for given scalars $$ \tau_M, \alpha>0, \gamma>0 $$, impulsive system Equation (12) is exponentially stable with a prescribed $$ H_\infty $$ performance $$ \gamma $$, if there exist appropriate dimensions $$ Y $$ and real matrices $$ O>0, S>0, T>0, H_i>0, J_i>0, L_i>0, i=1, \cdots, N $$ such that

are feasible for $$ i, j=1, \cdots, N $$, where

Proof: Differentiating $$ V(t) $$ along Equation (12) and applying Wirtinger-based integral inequality^[26], we can obtain

where $$ \xi_i(t)=col\{x(t), x(t-\tau(t)), x(t-\tau_M), \bar{\xi}_i(t), \omega(t)\} $$.

We use the reciprocally convex approach^[27] to deal with the cross item in Equation (18). By using Schur's complement, one can get

In the following, we will prove that

(ⅰ) with $$ \omega(t)=0 $$, the impulsive system Equation (12) is exponentially stable.

Since $$ \frac{d}{dt}e^{2\alpha t}V(t)=e^{2\alpha t}(\dot{V}(t)+2\alpha V(t)) $$ and $$ V(t) $$ is continuous in $$ t\in[t_k, t_{k+1}) $$, the integration of Equation (19) yields

It follows from $$ \int_{t_k}^t e^{2\alpha(s-t)}ds\leq \tau_M $$ that

where

Then taking Equation (17) and Equation (22) into account, denoting $$ \kappa_i=[e_i(t_k), C_i[-x(s_{k+1})+x(s_k)]]^T $$, we have

Hence, one can conclude that

From Equation (24), we have

which implies that

It follows from $$ \Omega_i<0 $$ that for $$ t\geq t_0 $$

For $$ t<t_0 $$, the system takes the form $$ \dot{x}=Ax(t)+F\omega(t) $$, then its corresponding solution is given by

Clearly, there exists the maximum of $$ |x(t)| $$ for $$ t\in[t_0-\tau_M, t_0] $$. Thus, $$ V(t_0)\leq \mu \|x_{t_0}\|_W^2+\sum\limits_{i=1}^N e_i^T(t_0)L_ie_i(t_0) $$ for some $$ \mu>0 $$.

Therefore, the exponential stability of the system Equation (12) with $$ \omega(t)=0 $$ is guaranteed.

Next, we will show that

(ⅱ) under the zero initial condition, the inequality $$ \|y\|_{L_2}\leq\gamma\|\omega\|_{L_2} $$ holds for any nonzero $$ \omega\in L_2[0, +\infty) $$.

From Equation (19), we obtain that for $$ t\in[t_k, t_{k+1}) $$,

Integrating Equation (27) on $$ t $$ from $$ t_k $$ to $$ t_{k+1}^- $$ yields

Then, by summing Equation (28) on $$ k $$ from $$ 0 $$ to $$ \rho $$, where $$ \rho\rightarrow +\infty $$, we have

Under the zero initial condition, $$ \int_0^{+\infty} y^T(s)y(s)ds\leq \int_0^{+\infty}\gamma^2\omega^T(s)\omega(s)ds $$. Therefore, one can derive that under the zero initial condition, $$ \|y\|_{L_2}\leq\gamma\|\omega\|_{L_2} $$ for any nonzero $$ \omega\in L_2[0, +\infty) $$. This completes the proof.

Remark 2The feasibility of the linear martix inequalities has been sufficiently explained in^{[23, 28]}. Due to page limitations, this part is omitted here.

3.2. Stability analysis under the RR scheduling scheme Equation (7)

Construct the following Lyapunov-Krasovskii functional candidate:

where

Similar to Theorem 5.2 in^[29], we establish the following result.

Theorem 2Under RR scheduling scheme Equation (7), given $$ \tau_M>0 $$ and $$ \alpha>0 $$, assume that there exist matrices $$ O>0, S>0, T>0, L_i>0, i=1, \cdots, N $$ and $$ Y $$ with appropriate dimensions such that Equation (15) and Equation (16) are feasible with $$ J_i=\frac{L_i}{N-1} $$, where $$ G_i $$ is given by Equation (30). Then, system Equation (12) is exponentially stable with a prescribed $$ H_\infty $$ performance $$ \gamma $$.

Proof: The detailed derivation process can refer to^[29] and the proof of Theorem 1, which is omitted here due to the limited pages.

3.3. Controller Design under the TOD scheduling scheme Equation (6)

Theorem 3.3.Under TOD scheduling scheme Equation (6), for given matrices $$ A, B, C, F $$, and scalars $$ \tau_M, \alpha>0, \gamma>0, \zeta>0 $$, system Equation (12) is exponentially stable with a prescribed $$ H_\infty $$ performance $$ \gamma $$, if there exist real matrices

is solvable, and the controller gain is given by $$ K=\digamma\beth^{-1} $$, where $$ Equation (32)^* $$ is equivalent to Equation (32) by replacing $$ -H_i^{-1} -Z^T\tilde{T}_r^{-1} Z $$ with $$ -h_i, \rho^2 \tilde{T}_r-2\rho Z $$, and

Proof: Let $$ Z=O^{-1}, \tilde{J}_i=ZJ_iZ, \tilde{L_i}=ZL_iZ, i=1, \cdots, N $$. Pre- and post-multiplying Equation (15) and Equation (16) with

and their transposes, respectively. For the nonlinear terms $$ -Z\tilde{T}_r^{-1}Z, r=1, 2 $$ and $$ -H_i^{-1} $$, using inequalities $$ -Z\tilde{T}_r^{-1}Z\leq\rho^2 \tilde{T}_r-2\rho Z $$ and the cone complementary linearization algorithm in^[27], one can obtain $$ Equation (32)^* $$. By solving the minimization problem Equation (31), system Equation (12) is exponentially stable with a prescribed $$ H_\infty $$ performance $$ \gamma $$ and $$ K=\digamma\beth^{-1} $$.

3.4. Controller design under the RR scheduling scheme Equation (7)

Similar to Theorem 5.2 in^[29], we establish Theorem 4.

Theorem 4Under the RR scheduling scheme Equation (7), for given matrices $$ A, B, C, F $$, and scalars $$ \tau_M, \alpha>0, \gamma>0, \zeta>0 $$, system Equation (12) is exponentially stable with a prescribed $$ H_\infty $$ performance $$ \gamma $$, if there exist real matrices $$ Z>0, \tilde{S}>0, \tilde{T}>0, \tilde{L}_i>0, i=1, \cdots, N $$ and $$ \tilde{Y}, \beth, \digamma $$ with appropriate dimensions such that Equation (31) is solvable with $$ J_i=\frac{L_i}{N-1} $$, where $$ H_i $$ is given by Equation (30). The controller gain is given by $$ K=\digamma\beth^{-1} $$.

Proof: The detailed derivation process can refer to^[29] and the proof of Theorem 3, which is omitted here.

Authors' contributions

Made substantial contributions to the research, idea generation, algorithm design, performed critical review, commentary and revision, as well as provided administrative, technical, and material support: Peng C

Conceived and designed the experiments, analyzed simulation results, wrote and edited the original draft: Yang H

Financial support and sponsorship

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

Availability of data and materials

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Conflicts of interest

All authors declared that there are no conflicts of interest.

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Copyright

© The Author(s) 2022.