Robust Stabilization of Load Frequency Control System Under Networked Environment

Ashraf Khalil Ji-Hong Wang Omar Mohamed

Robust Stabilization of Load Frequency Control System Under Networked Environment[J]. 国际自动化与计算杂志(英)/International Journal of Automation and Computing, 2017, 14(1): 93-105. doi: 10.1007/s11633-016-1041-z
引用本文: Robust Stabilization of Load Frequency Control System Under Networked Environment
[J]. 国际自动化与计算杂志(英)/International Journal of Automation and Computing, 2017, 14(1): 93-105. doi: 10.1007/s11633-016-1041-z
Ashraf Khalil, Ji-Hong Wang and Omar Mohamed. Robust Stabilization of Load Frequency Control System Under Networked Environment. International Journal of Automation and Computing, vol. 14, no. 1, pp. 93-105, 2017 doi:  10.1007/s11633-016-1041-z
Citation: Ashraf Khalil, Ji-Hong Wang and Omar Mohamed. Robust Stabilization of Load Frequency Control System Under Networked Environment. International Journal of Automation and Computing, vol. 14, no. 1, pp. 93-105, 2017 doi:  10.1007/s11633-016-1041-z

Robust Stabilization of Load Frequency Control System Under Networked Environment

doi: 10.1007/s11633-016-1041-z

Robust Stabilization of Load Frequency Control System Under Networked Environment

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    Author Bio:

    Ji-Hong Wang received the B. Eng. degree from Wuhan University of Technology, China in 1982, the M. Sc. degree from Shandong University of Science and Technology, China in 1985, and the Ph. D. degree from Coventry University, UK in 1995. In 2011 she joined the School of Engineering at the University of Warwick, UK. Her previous post was in the School of Electronic, Electrical and Computer Engineering at the University of Birmingham, where she was professor of control and electrical power and deputy director of the Midlands Energy Graduate School. She has also served as lecturer and senior lecturer at the University of Liverpool from 1998 to 2007. Currently she is professor at the School of Engineering, University of Warwick, UK. She has published over 100 technical papers and gained two best paper awards. Her research has led to several practical innovations. She is a technical editor of the IEEE Transactions on Mechatronics and is associate editor of two other international journals. Her research interests include power system modelling, control and monitoring, including large scale power plant modelling and control, energy efficient actuators and systems, nonlinear system control theory with industrial applications. E-mail:jihong.wang@warwick.ac.uk ORCID iD:0000-0003-1653-6259

    Omar Mohamed received the B. Sc. and M. Sc. degrees in electrical engineering from the University of Benghazi, Libya in 2005 and 2008, respectively, and the Ph. D. degree in electrical engineering from University of Birmingham, UK in 2012. In 2012, he was a lecturer at University of Benghazi, Libya. Currently he is assistant professor at Princess Sumaya University of Technology (PSUT), Electrical Engineering Department, Amman, Jordan. He has published more than 15 refereed journal and conference papers. He is a member of IEEE. His research interests include power system analysis, modelling, identification, planning, and control. E-mail:o.mohamed@psut.edu.jo ORCID iD:0000-0003-0618-2012

    Corresponding author: Ashraf Khalil received the B. Sc. and M. Sc. degrees in electrical engineering from the University of Benghazi, Libya in 2000 and 2004, respectively, and the Ph. D. degree in electrical engineering from University of Birmingham, UK in 2012. In 2012, he was a lecturer at University of Benghazi, Libya, he is assistant professor since 2014. Currently he is the head of the Electrical and Electronic Engineering Department at University of Benghazi, Libya. He published more than 30 refereed journal and conference papers. He participated in curriculum preparation and assisted in establishing and developing a number of universities and high institutes in Libya. In 2012, he participated in preparing and revising curriculum for the Benghazi College of Electrical and Electronics Engineering Technology. In 2014, he assisted in establishing Benghazi University of Technology. He is a member of IEEE. His research interests include networked control systems, renewable energy, smart-grids and the impacts of the time delay on power system stability. E-mail:ashraf.khalil@uob.edu.ly, ORCID iD:0000-0001-5668-9781
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出版历程
  • 收稿日期:  2016-01-15
  • 录用日期:  2016-05-16
  • 网络出版日期:  2016-12-29
  • 刊出日期:  2017-02-01

Robust Stabilization of Load Frequency Control System Under Networked Environment

doi: 10.1007/s11633-016-1041-z
    作者简介:

    Ji-Hong Wang received the B. Eng. degree from Wuhan University of Technology, China in 1982, the M. Sc. degree from Shandong University of Science and Technology, China in 1985, and the Ph. D. degree from Coventry University, UK in 1995. In 2011 she joined the School of Engineering at the University of Warwick, UK. Her previous post was in the School of Electronic, Electrical and Computer Engineering at the University of Birmingham, where she was professor of control and electrical power and deputy director of the Midlands Energy Graduate School. She has also served as lecturer and senior lecturer at the University of Liverpool from 1998 to 2007. Currently she is professor at the School of Engineering, University of Warwick, UK. She has published over 100 technical papers and gained two best paper awards. Her research has led to several practical innovations. She is a technical editor of the IEEE Transactions on Mechatronics and is associate editor of two other international journals. Her research interests include power system modelling, control and monitoring, including large scale power plant modelling and control, energy efficient actuators and systems, nonlinear system control theory with industrial applications. E-mail:jihong.wang@warwick.ac.uk ORCID iD:0000-0003-1653-6259

    Omar Mohamed received the B. Sc. and M. Sc. degrees in electrical engineering from the University of Benghazi, Libya in 2005 and 2008, respectively, and the Ph. D. degree in electrical engineering from University of Birmingham, UK in 2012. In 2012, he was a lecturer at University of Benghazi, Libya. Currently he is assistant professor at Princess Sumaya University of Technology (PSUT), Electrical Engineering Department, Amman, Jordan. He has published more than 15 refereed journal and conference papers. He is a member of IEEE. His research interests include power system analysis, modelling, identification, planning, and control. E-mail:o.mohamed@psut.edu.jo ORCID iD:0000-0003-0618-2012

    通讯作者: Ashraf Khalil received the B. Sc. and M. Sc. degrees in electrical engineering from the University of Benghazi, Libya in 2000 and 2004, respectively, and the Ph. D. degree in electrical engineering from University of Birmingham, UK in 2012. In 2012, he was a lecturer at University of Benghazi, Libya, he is assistant professor since 2014. Currently he is the head of the Electrical and Electronic Engineering Department at University of Benghazi, Libya. He published more than 30 refereed journal and conference papers. He participated in curriculum preparation and assisted in establishing and developing a number of universities and high institutes in Libya. In 2012, he participated in preparing and revising curriculum for the Benghazi College of Electrical and Electronics Engineering Technology. In 2014, he assisted in establishing Benghazi University of Technology. He is a member of IEEE. His research interests include networked control systems, renewable energy, smart-grids and the impacts of the time delay on power system stability. E-mail:ashraf.khalil@uob.edu.ly, ORCID iD:0000-0001-5668-9781

English Abstract

Robust Stabilization of Load Frequency Control System Under Networked Environment[J]. 国际自动化与计算杂志(英)/International Journal of Automation and Computing, 2017, 14(1): 93-105. doi: 10.1007/s11633-016-1041-z
引用本文: Robust Stabilization of Load Frequency Control System Under Networked Environment
[J]. 国际自动化与计算杂志(英)/International Journal of Automation and Computing, 2017, 14(1): 93-105. doi: 10.1007/s11633-016-1041-z
Ashraf Khalil, Ji-Hong Wang and Omar Mohamed. Robust Stabilization of Load Frequency Control System Under Networked Environment. International Journal of Automation and Computing, vol. 14, no. 1, pp. 93-105, 2017 doi:  10.1007/s11633-016-1041-z
Citation: Ashraf Khalil, Ji-Hong Wang and Omar Mohamed. Robust Stabilization of Load Frequency Control System Under Networked Environment. International Journal of Automation and Computing, vol. 14, no. 1, pp. 93-105, 2017 doi:  10.1007/s11633-016-1041-z
    • The privatization, deregulation and liberalization of the electricity industry coupled with the radical change in communication networks and their low cost are the key driving forces behind adopting new open communication infrastructure in power systems[1]. The load frequency control is one of the classical centralized power system control problems. The main goals of the load frequency control are: 1) to maintain uniform frequency, 2) share the load between the generators, and 3) control the tie-line interchange schedule[2]. Automatic generation control (AGC) and load frequency control (LFC) have been implemented in centralized scheme since the beginning of the interconnected power system[3]. The load frequency is achieved by the AGC where the frequency deviation is used to sense the change in the load demand. In the AGC, a dedicated communication link is used to send the AGC signals. In the case of fault in the dedicated communication link, other communication links are used, usually voice communications through telephone lines[4]. Because of the increased number of ancillary services, the need for a duplex and distributed communication links becomes more pronounced[4].

      The traditional dedicated communication network is unsuitable for the future power system. Future power system needs to be decentralized, integrated, flexible, and open[5]. Communication networks have grown rapidly while power system control centres remain far behind[5]. One of the promising solutions is the migration from traditional supervisory control and data acquisition (SCADA) to transmission control protocol/internet protocol (TCP/IP) and Ethernet. TCP/IP is becoming the de facto world standard for data transmission[1]. The migration from traditional dedicated networks to open and distributed networks such as the Internet has been proposed by several system operators but they are still not adopted because of their non-deterministic characteristics. Open communication networks are reconfigurable and the hardware and the software are well developed. Furthermore they have their simplicity and are adopted worldwide. Although these new technologies are more flexible, reconfigurable, and have high bandwidth, there are some shortcomings. The main issues in the open communication infrastructure are the time delay, data loss and the vulnerability to malicious attack[1]. In this scenario, communication, computing and control will be integrated into future power system. The introduction of the open communication which is a shared network raises concerns about the stability of the LFC system.

      To guarantee that the frequency is within the permissible range the area control error (ACE) and the generator control error (GCE) signals are distributed between the different areas through these shared networks. The ACE and the GCE are used to increase or decrease the generated power. Sending these signals through open communication link will introduce time delay and some of these data will be lost. One of the requirements of the future communication network is to be fault tolerant. The most important part for controller designer is to guarantee the stability of LFC system with the presence of the time delay and the data loss. There are some research works regarding the stability of the LFC system in the presence of the time delay and they are summarized in the following.

      The methods reported in the literature focus on estimating the maximum time delay margin. The maximum time delay margin is defined as the time delay that LFC system can withstand before it becomes unstable. In the published research works the methods for linear time delay are applied to estimate the maximum time delay margin for LFC systems. There are mainly two types of methods that deal with linear time delay systems; the indirect methods based on Lyapunov stability theory and the direct methods which are based on tracking the eigenvalues of the characteristics equation. In [4] the authors used the simulation to study the impact of the time delay on the load frequency control system stability and discuss the communication requirement for LFC system. They considered both constant and random time delays. Although they investigated the impacts of the packets loss on the stability of LFC system, they did not propose any method to estimate the control system requirements in terms of the maximum time delay margin. Their simulations show that the increased number of packet drop-outs can result in system instability. Yu and Tomsovic[6] used a simple stability criterion but rather conservative that was reported in the 1980s and introduced in [7, 8]. In their work the problem of the load frequency control is formulated as a general time delay problem which is then solved in linear matrix inequality (LMI) form, but it is only applicable to constant time delays. In [9] the time delay margin for LFC system is calculated using stability criteria for linear time delay systems with variable time delay for one-area and multi-area LFC system. The effects of the proportion integration (PI) controller, $K_{P}$ and $K_{I}$ , on the time delay margin are investigated for one-area and multi-area LFC system. In [10], the genetic algorithms are used to tune the controllers and the authors considered generation rate constraint (GRC), dead band, and time delay. The sliding mode control is used in [11] to design robust LFC system against uncertainty and time delay where the $H_{\infty}$ optimal control is used to derive the sliding surface parameters. The general predictive control is investigated in [12] to compensate the effects of the bounded random time delay. The time delay is treated as uncertainty in [13]. A decentralized control algorithm is proposed and $H_{2}/H_{\infty}$ control synthesis is used to derive the PI controller gains for three-area LFC system. In [14] the direct method for estimating the maximum time delay margin is presented. Rekasius substitution is used to eliminate the transcendency in the characteristic equation and to convert it to polynomial. The imaginary roots for positive delays are tracked and then Routh-Hurwitz criterion is used to determine the time delay margin. However the results of the maximum time delay margin reported in [14] are less conservative than the results of the method reported in [9], the drawback of the method in [14] is its limitation to constant time delays and the complexity increases in the case of multi-area LFC system.

      The two area load frequency control with time delay and data loss has been modelled as Markovian jump linear system in [15, 16]. The particle swarm optimization is used to derive the PI controller gains. The Markov chain is used to estimate the data drop-outs and a lower and upper bound for the time delay are determined. In [17] the problem of stabilizing LFC system using $H_{\infty}$ controller is investigated. Lyapunov-Krasovskii functional approach is used to design two-term $H_{\infty}$ controller for two area LFC system. The $H_{\infty}$ controller design problem is then formulated as a set of LMIs. The conventional LFC has been applied to Microgrids and reported in a number of publications[18, 19]. The model predictive control and smith predictor are used in [18] to compensate the communication network effects on the frequency in the Microgrid. MPC is compared with the conventional droop control method and its found that the performance with MPC is superior to the droop control even with the presence of the communication delay. In [19] the stability analysis of LFC system in Microgrid is investigated. The Microgrid is formed by distributed diesel generator and photovoltaic generator. The Rekasius substitution along with the sum of squares decomposition are used to find the time-delay stability margin. The effect of the stochastic nature of the photovoltaic generator on the LFC system is not considered. $H_{\infty}$ control can be used to further design a robust LFC system that handles the uncertainties in the LFC system model. Ahmadi and Aldeen[20] report the design of robust $H_{\infty}$ control to handle the uncertainties in the LFC system model. The communication delay is assumed to be bounded and parametric uncertainties are norm bounded. The design problem is formulated into LMIs framework. An improved delay-dependent stability of LFC system is presented in [21]. The unknown exogenous load disturbance has been considered where $H_{\infty}$ criterion is added as a concentrate to the delay-dependent stability problem. It is reported that the number of decision variables in [21] are reduced compared to the number of decision variables in [9], and this will lead to less conservative results for the maximum allowable delay bound. The delay-distribution-dependent LFC system stability analysis with probabilistic time delay is reported in [22]. The TCP/IP-based communication delay is assumed to have non-uniform distribution characteristics. The time delay is modelled as stochastic process and the LFC system is transformed into stochastic time-delay system. $H_{\infty}$ control design is used to derive the PI controller gain with a given probabilistic delay. The Lyapunov-Krasovskii functional along with convex combination technique are used to derive criteria for delay-distribution-dependent stability analysis. Then the control synthesis is formulated in terms of linear matrix inequalities.

      Most of the published research works in the literature focus on constant and varying delays. The random delay has not been considered in many papers and some of them treat LFC system under open communication as normal time delay system where the sampling rate, the time delay and the data drop-outs are not considered. Additionally the methods assume a pre-designed LFC system then the maximum time delay margin is calculated. Furthermore, most of the constant time delay criteria cannot be used to determine the degree of stability. The random time delay and the data loss should be considered during LFC system design stage. The method presented in this paper is based on designing the controller while taking the stochastic nature of the network into consideration[23]. Under the dedicated conventional closed communication links the assumptions of constant or varying time delay is justified, however, the time delay and data loss in open communication network are random and in many cases can be modelled using Markov chains. The paper deals with the stochastic stability of the LFC system when the time delays and packets loss are governed by Markov chains.

      This paper focuses on the stochastic stabilization of LFC system under networked environment. The paper starts with the modelling of LFC system with random time delay and packet loss. LFC system with the random time delay and packet loss is modelled as standard Markovian discrete-time jump linear system. The stochastic stability of this type of systems is briefly discussed. Then the stability criterion is formulated as bilinear matrix inequalities. The bilinear matrix inequalities are solved using V-K iteration method. In the V-K iteration method, the problem is divided into three linear matrix inequalities which are solved using Matlab. With the V-K iteration method the range of the PI controller that satisfies the stochastic stability of the LFC system is determined. Finally, the range of the PI controllers that achieve the robust stochastic stability is determined using the decay rate as a measure of the system robustness. Simulation results show the merit of the proposed method.

    • The one-area LFC system is shown in Fig. 1. The main assumption is that all the generators are equipped with non-reheat turbines. The state-space linear model of one-area LFC system is expressed as[9]

      $ \dot{x}_{c}(t) =A_{c}x_{c}(t)+B_{c}u_{c}(t)+F_{c}\Delta P_{d} $

      (1)

      $ y_{c}(t)=C_{c}x_{c}(t) $

      (2)

      Figure 1.  Dynamic model of one-area LFC scheme

      where

      $ A_{c}=\begin{pmatrix} \dfrac{-D}{M}&\dfrac{1}{M}&0\\[2mm] 0&\dfrac{-1}{T_{ch}}&\dfrac{1}{T_{ch}}\\[2mm] \dfrac{-1}{RT_{g}}&0&\dfrac{-1}{T_{g}} \end{pmatrix} B_{c}=\begin{pmatrix} 0\\ 0\\ \dfrac{1}{T_{g}} \end{pmatrix} F_{c}=\begin{pmatrix} \dfrac{-1}{M}\\ 0\\ 0 \end{pmatrix} \nonumber $

      $ C_{c}=\begin{pmatrix} \beta&0&0 \end{pmatrix} x_{c}(t)=\begin{pmatrix} \Delta f&\Delta P_{m}&\Delta P_{v} \end{pmatrix}^{\rm T} \nonumber $

      $ y_{c}(t)=ACE.\nonumber $

      The parameters are given as: $\Delta P_{d}$ is the load deviation, $ \Delta P_{m}$ is the generator mechanical output deviation, $\Delta P_{v}$ is the valve position deviation, $\Delta f$ is the frequency deviation. M is the moment of inertia, D is the generator damping coefficient, $T_{g}$ is the time constant of the governor, $T_{ch}$ is the time constant of the turbine, R is the speed drop, and $\beta$ is the frequency bias factor. For one-area LFC system, the area control error ACE is given as

      $ ACE=\beta \Delta f $

      (3)

      where AGC has two components; the first is updated every 5 minutes for economical dispatch and the second is updated in the order of 1-5 s[6]. The later signal delay is the one considered in the paper. The conventional PI controller for stabilizing LFC system is given as

      $ u_{c}(t)=-K_{P}ACE-K_{I} \int{ACE} $

      (4)

      where $K_{P}$ is the proportional gain, $K_{I}$ is the integral gain and $\int ACE$ is the integration of the area control error. With PI controller, the closed-loop system is expressed as follows:

      $ \dot{x}(t) =Ax(t)+Bu(t-\tau (t))+F\Delta P_{d} $

      (5)

      $ y(t)=Cx(t) $

      (6)

      where

      $ A=\begin{pmatrix} \dfrac{-D}{M}&\dfrac{1}{M}&0&0\\[2mm] 0&\dfrac{-1}{T_{ch}}&\dfrac{1}{T_{ch}}&0\\[2mm] \dfrac{-1}{RT_{g}}&0&\dfrac{-1}{T_{g}}&0\\[2mm] \beta&0&0&0 \end{pmatrix}, B=\begin{pmatrix} 0\\ 0\\ \dfrac{1}{T_{g}}\\ 0 \end{pmatrix}, F=\begin{pmatrix} \dfrac{-1}{M}\\ 0\\ 0\\ 0 \end{pmatrix}, \nonumber $

      $ C=\begin{pmatrix} \beta&0&0&0\\ 0&0&0&1 \end{pmatrix}, x(t)= %\begin{pmatrix} (\Delta f~\Delta P_{m}~\Delta P_{v}\small{\int{ACE}}) %\end{pmatrix} ^{\rm T}. \nonumber $

      Discretizing system (5) and the controller (4) with sampling time, $T_{s}$ , (5) and (4) become

      $ x(k+1)=A_{d}x(k)+B_{d}u(k)+F_{d}\Delta P_{d}(k) $

      (7)

      $ u(k)=K(r_{s}(k))x(k-r_{s}(k)) $

      (8)

      where $\tau(k)=r_{s}(k) \times h$ , h is the sampling period and $r_{s}(k)$ is a bounded random integer sequence governed by Markov chain with $0 \leq r_{s}(k)\leq d_{s} < \infty$ , and $d_{s}$ is the finite delay bound. $A_{d}$ , $B_{d}$ and $F_{d}$ are matrices with appropriate size and depend on the sampling rate. Before we proceed the following assumptions are made: 1) The random time delay in the network is bounded. 2) The number of data drop-outs is finite. 3) All the data are sent as single packet. 4) Time stamping is used where the old data are discarded. Introducing the augmenting state variable given as

      $ \overline{x}(k)=\begin{pmatrix} x(k)^{\rm T}&x(k-1)^{\rm T}&\cdots&x(k-d_{s})^{\rm T} \end{pmatrix}^{\rm T} \nonumber $

      where $\overline{x}(k)\in {\bf R}^{(d_{s}+1)n}$ , applying the controller (8) into (7) the closed-loop system becomes

      $ \overline{x}(k+1)=(\overline{A}+\overline{B}K(r_{s}(k))\overline{C} (r_{s}(k)))\overline{x}(k) $

      (9)

      where

      $ \overline{A}=\begin{pmatrix} A_{d}&0&\cdots&0&0\\ I&0&\cdots&0&0\\ 0&I&\cdots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&I&0 \end{pmatrix} \overline{B}=\begin{pmatrix} B_{d}\\ 0\\ 0\\ \vdots\\ 0 \end{pmatrix} \nonumber $

      $ \overline{C}(r_{s}(k))=\begin{pmatrix} 0&\cdots&0&C_{d}&0&\cdots&0 \end{pmatrix} \nonumber $

      where $\overline{C}(r_{s}k)$ has all elements being zero except for the $r_{s}(k)$ -th block which equals $C_{d}$ . Notice that the time delay and the data loss are incorporated into $\overline{C}(r_{s}(k))$ . The closed-loop system (9) can be rewritten as

      $ \overline{x}(k+1)=A_{cl}(r_{s}(k))\overline{x}(k). $

      (10)

      The system represented by (10) is the standard discrete-time Markovian jump linear system (DTMJLS). The Markovian jump system is mostly used to study the stability and stabilization of system with abrupt changes due to the variations in the system structure or partly system failure[23]. The open communication network is modelled as a finite state Markov process with the following properties:

      $ P[r_{s}(k+1)=j|r_{s}(k)=i]=p_{ij} \nonumber $

      $ 0\leq i,j\leq d_{s},\,0\leq p_{ij}\leq 1,\,\sum_{j=0}^d p_{ij}=1 $

      (11)

      where $d_{s}$ is the finite delay and represents the number of modes. It should be noted that (11) incorporates the packets drop-outs. $p_{ij}$ is the transition probability from mode i to mode j. The general transition probability matrix is given by

      $ P=\begin{pmatrix} p_{00}&p_{01}&0&0&\cdots&0\\ p_{10}&p_{11}&p_{12}&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&0\\ p_{d0}&p_{d1}&p_{d2}&p_{d3}&\cdots&p_{d_{s}d_{s}} \end{pmatrix}. $

      (12)

      The constraint (11) means that the summation of the probabilities in every row is one. The assumption made is that the old data are discarded. Suppose that at instant k we received $x(k)$ , at $k+1$ if there is no new data then the old data will be used by the controller, but if we receive $x(k-1)$ at $k+1$ then it will be older than $x(k)$ and hence $x(k-1)$ must be discarded, this can be interpreted as

      $ p[r_{s}(k+1)>r_{s}(k)+1]=0. $

      (13)

      From (13), the time delay can increase only at one step but it can decrease at many steps as can be seen from (12). The diagonal elements in (12) represent the probability of successive equal time delays or in other words the probability that the network remains in the same state. The upper diagonal elements represent the possibility of receiving longer delays or increasing the network load. The zero elements represent the discard of the old data.

    • LFC system (10) is a stochastic hybrid system. The stochastic stability, mean square stability and the exponential mean square stability are all equivalent and every condition implies the almost sure (asymptotic) stability[24].

      Definition 1. [25] The system (10) is mean square stable if for every initial condition state, $(\overline{x}_{0}, r_{0}$ )

      $ \lim_{k\longrightarrow\infty}{\rm E}(\Vert\overline{x}(k)\Vert^{2})=0. $

      (14)

      Definition 2. [25] The system (10) is mean square stable with decay rate $\beta$ [26] if for every initial condition state, $(\overline{x}_{0}, r_{0})$

      $ \lim_{k\longrightarrow\infty}\beta^{k} {\rm E}(\Vert\overline{x}(k)\Vert^{2})=0. $

      (15)

      The necessary and sufficient conditions for mean square stability for jump system are given in the following theorem.

      Theorem 1. [25] The mean square stability of (10) is equivalent to the existence of symmetric positive definite matrices $Q_{0}, \cdots, Q_{d}$ satisfying any one of the following 4 conditions:

      $ A_{j}^{\rm T}(\sum_{i=0}^{d}p_{ji}Q_{i})A_{j}<Q_{j} \nonumber $

      $ A_{i}(\sum_{j=0}^{d}p_{ji}Q_{j})A_{i}^{\rm T}<Q_{i} \nonumber $

      $ \sum_{j=0}^{d}p_{ji}A_{j}Q_{j}A_{i}^{\rm T}<Q_{i} \nonumber $

      $ \sum_{j=0}^{d}p_{ji}A_{i}^{\rm T}Q_{j}A_{i}<Q_{j} \nonumber $

      where $i=0, \cdots, d$ represents the number of modes. The conditions 1-4 are equivalent for studying the stability of the DTMJLS but for the controller design each condition will lead to a different optimum controller.

      Choosing condition (4) in Theorem 1 and replacing $Q_{i}$ by $\alpha Q_{i}$ (where the decay rate or Lyapunov exponent, $\beta =\frac {1}{\alpha}$ and $\lim_{k\rightarrow\infty}\beta^{k} M(k)=0$ ) on the right hand side, the closed-loop system becomes

      $ \sum_{j=0}^{d}p_{ji}(A_{i}+B_{i}K_{i}C_{i})^{\rm T}Q_{j}(A_{i}+B_{i}K_{i}C_{i})<\alpha Q_{j} $

      (16)

      where $i=0, \cdots, d$ . The coupled (16) are bilinear matrix inequalities (BMIs) which are non-convex and finding a global optimal solution is very difficult. However many control problems are formulated as BMIs, whereas there are a few methods for solving the BMIs. For example the path-following linearisation method reported in [27] can be used where each matrix is perturbed and the higher order terms are neglected. The most widely used techniques for the solution is by iteration methods such as D-K, G-K and V-K iteration algorithms[28]. If we fix $K_{i}$ ( $i=0, \cdots, d$ ) then we have a generalized eigenvalue problem (GEVP) and if we fix $Q_{i}$ ( $i=0, \cdots, d$ ) then we have eigenvalue problem (EVP)[25]. Both of these problems can be solved very efficiently using Matlab LMI toolbox. Equation (16) can be written as

      $ \alpha Q_{j}-\\ \begin{pmatrix} \widehat{A}_{0}^{\rm T}&\widehat{A}_{1}^{\rm T}&\cdots&\widehat{A}_{d}^{\rm T} \end{pmatrix} \begin{pmatrix} p_{j0}Q_{0}&\cdots&0\\ \vdots&\ddots&\vdots\\ \vdots&\cdots&p_{jd}Q_{d} \end{pmatrix} \begin{pmatrix} \widehat{A}_{0}\\ \widehat{A}_{1}\\ \vdots\\ \widehat{A}_{d} \end{pmatrix}>0 $

      (17)

      where $i=0, \cdots, d$ , $\widehat{A}_{i}=A_{i}+B_{i}K_{i}C_{i}$ . Using Schur complement to (17) then we have

      $ \begin{pmatrix} \alpha Q_{j}&\widehat{A}_{0}^{\rm T}&\cdots&\widehat{A}_{d}^{\rm T}\\ \widehat{A}_{0}&p_{j0}^{-1}Q_{0}^{-1}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ \widehat{A}_{d}&0&\cdots&p_{jd}^{-1}Q_{d}^{-1} \end{pmatrix} > 0. $

      (18)

      In the V-K algorithm the BMI is divided into two LMIs and by solving these two LMIs a local optimal solution can be found. The problem solution process is divided into three basic problems which are: feasibility problem (FP), eigenvalue problem (EVP), and generalized eigenvalue problem (GEVP). These problems can be solved using the Matlab LMI toolbox. In the V-K algorithm, the problem is iterated between the EVP and the GEVP. The proof of the algorithm convergence is given in [28]. The detailed algorithm is shown in the flowchart in Fig. 2. The algorithm starts with the initialization, and then if the solution is feasible EVP and GEVP are iterated until the desired transition matrix is reached. In this improved algorithm the decay rate is maximized in both EVP and GEVP iterations. A brief on formulating the three problems and solving them using Matlab LMI toolbox is given below:

      Figure 2.  V-K iteration algorithm

      1) Feasibility problem (FP)

      The first step is to find an initial feasible solution to (16). The initial solution is to set: $\alpha =1$ , $K_{0}=K_{1}, \cdots, K_{d}=K$ and the initial transition probability, $P=P_{0}$ , then solve for $Q_{0}, \cdots, Q_{d}$ . The Matlab feasp function in Matlab LMI Toolbox is used to solve this problem[29].

      2) Eigenvalue problem (EVP)

      Fix $Q_{i}'s$ , $i=0, \cdots, d$ , then (18) becomes LMIs in $\alpha$ and $K_{i}'s$ , $i=0, \cdots, d$ , but (18) is not in the solvable form and by changing the variables it can be solved efficiently using Matlab LMI Toolbox[29], introducing the new variables; $Y_{i}=K_{i}C_{i}$ , then (18) becomes

      $ \begin{pmatrix} \alpha Q_{j}&A_{0}^{\rm T}+Y_{0}^{\rm T}B_{0}^{\rm T}&\cdots&A_{d}^{\rm T}+Y_{d}^{\rm T}B_{d}^{\rm T}\\ A_{0}+B_{0}Y_{0}&p_{j0}^{-1}Q_{0}^{-1}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ A_{d}+B_{d}Y_{d}&0&\cdots&p_{jd}^{-1}Q_{d}^{-1} \end{pmatrix}>0 $

      (19)

      where $j=0, \cdots, d$ . This problem can be solved using mincx Matlab function in LMI Toolbox.

      3) Generalized eigenvalue problem (GEVP)

      If we fix $K_{i}'s$ , $i=0, \cdots, d$ , then we have a set of BMIs in $Q_{i}'s$ , $i=0, \cdots, d$ , and $\alpha$ . In [25], this problem is solved by setting $\alpha\ = 1$ and solving the GEVP for $Q_{i}'s$ , $i=0, \cdots, d$ . The problem can be solved using gevp Matlab function in LMI Toolbox[29]. Another easier solution is to use the iteration to decrease $\alpha$ in every iteration and solve the feasibility problem.

      The initial transition probability matrix is chosen to be

      $ P_{0}=\begin{pmatrix} 1&0&\cdots&0\\ 1&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 1&0&\cdots&0\\ \end{pmatrix}\approx \begin{pmatrix} 1-n \times \varepsilon&\varepsilon_{1}&\cdots&\varepsilon_{n}\\ 1-n \times \varepsilon&\varepsilon_{1}&\cdots&\varepsilon_{n}\\ \vdots&\vdots&\ddots&\vdots\\ 1-n \times \varepsilon&\varepsilon_{1}&\cdots&\varepsilon_{n}\\ \end{pmatrix}. \nonumber $

      It should be noted that the initial controller is designed for the delay free system and hence the initial solution is feasible. To get an initial feasible solution we have to start from small time delays and perturb the transition probability matrix toward longer time delays. The perturbation $\varepsilon$ should be very small positive number in the order of 0.005. An example of the perturbation matrix is given as

      $ \Delta P_{0}=\begin{pmatrix} -s&s&\cdots&0\\ 0&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&0\\ \end{pmatrix} \Delta P_{1}=\begin{pmatrix} 0&0&\cdots&0\\ -s&-s&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&0\\ \end{pmatrix} \nonumber $

      As can be seen the sum of the perturbation through any row is zero. More aggressive initial transition probability matrix can be used. In [25, 30], the perturbation is around 0.01 but even with this small perturbation sometimes the problem is divergent and we need to use further smaller perturbation, for example around 0.005. Also for the two modes, the two probability matrices are perturbed at the same time while in our algorithm they are perturbed separately.

    • The parameters of the LFC system shown in Fig. 1 are given as: $T_{ch}=0.3$ , $T_{g}=0.1$ , $R=0.05$ , $D=1.0$ , $\beta =21.0$ and $M=10$ . Under open communication network the remote terminal unit (RTU) sends the signals to the central controller through the shared network, and then the controller sends the commands back. The two delays are defined as feed forward and feedback delays. In most of the studies these two delays are aggregated into a single delay and this assumption is made in this paper[4-6]. In power systems the data collection is in the order of 1-5 s[1] and in the US the ACE signal is transmitted every 4 s[4]. The time delay is in the range of 80-200 milliseconds if the telemetered signals are transmitted via dedicated channel in synchronous digital hierarchy (SDH) or IP switch mode[12]. The control signal is sent to the power unit every 1 s[11]. In the light of these facts two different scenarios are investigated with different sampling rates. The sampling time is chosen to be 0.3 s and 1 s. The shared network is chosen to have three different states; namely: Low, Medium and High as shown in Fig. 3. The network is represented as Markov process in Fig. 4. These three states represent the network at low, medium and high load. At higher network load the time delay is high. The network state jumps between these three states in stochastic manner with the following transition probabilities: $P_{_{HL}}$ , $P_{_{HM}}$ , $P_{_{MH}}$ , $P_{_{ML}}$ , $P_{_{LM}}$ and $P_{_{LH}}$ .

      Figure 3.  Time delay distribution in shared network with three states

      Figure 4.  Network with three states represented as finite state Markov chain

      Firstly, the stochastic stability range of the PI controller gains as function of sampling rate is investigated in the paper and is shown in Fig. 5. For positive values of the PI controller gains the stochastic stability region of the LFC system is semicircle. Fig. 5 is obtained by changing the values of the PI control gains ( $K_{P}$ and $K_{I}$ ) and solving the feasibility problem. The transition probability matrix for the network is given by

      $ P=\begin{pmatrix} 0.5&0.5&0.0\\ 0.1&0.65&0.25\\ 0.1&0.65&0.25 \end{pmatrix}. \nonumber $

      Figure 5.  Stability region with different sampling rates

      Increasing the sampling rate extends the stability region of the LFC system. This means with high sampling rates we can select $K_{P}$ and $K_{I}$ to be large which improves the LFC system performance. On the other hand increasing the sampling rate increases the computation and the load on the network which increases the time delay and the data loss. With low sampling rates the stability region becomes small. Lower sampling rates degrade the performance of the system and a compromise between the performance of the LFC system and the sampling rate should be made. It should be noted that the size of the stability region has strong dependence on $K_{I}$ .

      Scenario 1. The sampling rate is chosen to be 0.3 s[12]. Our goal is to design stabilizing controller for LFC system under the network represented by P. The initial PI controller is chosen to have $K_{P}=1.0$ and $K_{I}=1.0$ , solving the feasibility problem shows that the initial controller with the given transition probability matrix fails to stabilize LFC system. The initial transition probability matrix with high probability of short delays or packet drop-outs given is as

      $ P_{0}=\begin{pmatrix} 0.5&0.5&0.0\\ 0.4&0.5&0.1\\ 0.4&0.5&0.1 \end{pmatrix}. \nonumber $

      The initial controller stabilizes the system with this initial transition probability matrix. After 60 iterations the new values of the PI controller gains are: $K_{P}=-0.034 2$ and $K_{I}=0.479 1$ . The Matlab/Simulink implementation of the LFC system with random time delay is shown in Fig. 6. In the simulation a change of 0.1 pu in the load demand occurs at T=10 s. The frequency deviation of LFC system with and without random time delay compensation controller is shown in Fig. 7. The area control error with and without random time delay compensation controller is shown in Fig. 8. The random time delay is shown in Fig. 9. It should be noted that the average time delay in the network is 0.314 1 s. This can be explained with the aid of the histogram of the random delays as shown in Fig. 10 where 0.3 s delay (or loss of one packet) has the highest probability. Under constant time delay with $K_{P}=1.0$ and $K_{I}=1.0$ , using Theorem 1 in [31] the maximum time delay margin is 0.255 9 s. With the final controller ( $K_{P}=-0.034 2$ and $K_{I}=0.479 1$ ) the maximum time delay margin is 1.660 9 s. Clearly the final controller improves the delay margin.

      Figure 6.  Simulink implementation of LFC system with random time delay

      Figure 7.  Frequency deviation, Δf

      Figure 8.  Area control error, ACE

      Figure 9.  Random time delay

      Figure 10.  Histogram of the random delay in Fig. 9

      It should be noted that the final controller compensates the random delay completely as clarified in Figs. 7 and 8. For comparison the response of LFC system with the initial and the final controllers is shown in Fig. 11 for the delay free LFC system. From Fig. 11 the performance of the initial controller ( $K_{P}=1.0$ and $K_{I}=1.0$ ) is better than the performance of the final controller ( $K_{P}=-0.034 2$ and $K_{I}=0.479 1$ ) without time delay, however under random time delay or data loss the initial controller fails to stabilize LFC system. To explain this, further analysis of the system eigenvalues is carried out. Fig. 12 shows the eigenvalues of the LFC system with the initial and the final controller. We noticed the small change in the real roots and a considerable change in the imaginary roots which shows a reduction in the oscillation mode of the system. With the final controller we noticed; higher overshoot, less oscillation and no change in the settling time.

      Figure 11.  Frequency deviation, Δf

      Figure 12.  Roots of the LFC system

      Scenario 2.

      Sampling with 1 s[11] and choosing the initial PI controller as: $K_{P}=0.8$ and $K_{I}=0.8$ . Using the same initial transition probability matrix in Scenario I with the following perturbation matrix:

      $ \Delta P=\begin{pmatrix} 0.0&0.0&0.0\\ -0.005&0.002\,5&0.002\,5\\ -0.005&0.002\,5&0.002\,5\\ \end{pmatrix}. \nonumber $

      After 60 iterations the final controller gains are $K_{P}=0.005 7$ and $K_{I}=0.178$ . through using Theorem 1 in [31] the time delay margin for the constant time delay is increased from 0.351 9 s to 5.232 9 s. The average time delay in the network is 1.16 s. The frequency deviation with and without random time delay compensation controller is shown in Fig. 13. Although the system without random time delay compensation is stable with this random time delay, it becomes unstable with different random seed. It should be noted that this observation has been reported in [4] through simulation studies where the LFC system becomes unstable with random packet drop-outs in certain situations. The random time delay increases the oscillation in the frequency deviation and in the accumulative area control error as can be seen in Fig. 14.

      Figure 13.  The frequency deviation, Δf

      Figure 14.  Area control error, ACE

      With the final controller the accumulative ACE is reduced from $37.98\times10^{-4}$ pu to $4.18\times10^{-4}$ pu where the accumulative area control error is given by

      $ J=\int_{0}^{\rm T}(ACE_{i})^2 {\rm d}t. $

      (20)

      The random time delay and its histogram are shown in Figs. 15 and 16, respectively. The frequency deviation with the initial and the final controller without time delay is shown in Fig. 17. As can be seen the performance of the initial controller without time delay is better than the performance of the system with the final controller. For the system with the random time delay the final controller compensates the effects of the random time delay without degrading the system performance. This can be examined by looking at Fig. 18. We noticed that with the initial and the final controller there are slight differences between the real roots while the change in the imaginary roots is small.

      Figure 15.  Random time delay

      Figure 16.  Histogram of the random delay in Fig. 15

      Figure 17.  Frequency deviation, Δf

      Figure 18.  Roots of the LFC system

      Most of the studies focus on finding the delay margin, in practice it is important to measure the degree of the system stability and ascertain the stability of LFC system with the presence of the imperfections caused by the communication network. In this work the Lyapunov exponent (decay rate) is used as a measure of the robustness of the LFC system against the random time delay. The inverse Lyapunov exponent is plotted as function of the PI controller gains, ( $K_{P}$ and $K_{I}$ ), and is shown in Fig. 19.

      Figure 19.  Inverse decay rate (Lyapunov exponent)

      Practically the PI gains are tuned online by experience based trial-and-error methods. In the case of open communication network Fig. 19 can be used as a guide for choosing the appropriate values of $K_{P}$ and $K_{I}$ . As can be seen from Fig. 19 the Lyapunov exponent varies with both $K_{P}$ and $K_{I}$ . For small values of $K_{P}$ the variation of the Lyapunov exponent with $K_{I}$ is very large. For large values of $K_{P}$ and $K_{I}$ the inverse Lyapunov exponent is large and as $K_{P}$ and $K_{I}$ increase the LFC system becomes stochastically unstable. There is small region where the inverse Lyapunov exponent is minimum as clarified in Fig. 20.

      Figure 20.  2-dimensional representation of the Lyapunov exponent

      The inverse Lyapunov exponent as function of the integral gain for different values of $K_{P}$ is shown in Fig. 21. For example if we select $K_{P}=0$ as it is usually the case in practice the optimum value of $K_{I}$ that achieves the largest Lyapunov exponent is 0.2 as shown in Fig. 22. It should be noted that the PI controller derived through the V-K iteration lies in the optimum region, $K_{P}=0.005 7$ and $K_{I}=0.178$ , this fact proves that the V-K iteration achieves the optimum controller. In this paper we consider only non-switching controller where the controller gains are not changing with the time delay. In the studies reported for LFC system with constant and varying time delay the results show that the maximum time delay decreases with increasing the integral gain while our results for the random time delay show that there is an optimum value for $K_{I}$ . The frequency deviation of the LFC system with different values of $K_{P}$ and $K_{I}$ are shown in Fig. 23. The response shows that the optimum values of $K_{P}$ and $K_{I}$ are achieved. Increasing the integral gain beyond 0.2 increases the oscillations because of the random time delay. Decreasing the integral gain below 0.2 makes the response of LFC system slower and hence increases the frequency deviation and the ACE.

      Figure 21.  Inverse Lyapunov exponent

      Figure 22.  Inverse Lyapunov exponent, Kp=0

      Figure 23.  Frequency deviation, Δf

      From Fig. 22 for $K_{P}=0$ and under random time delay $K_{I}$ should be less than 0.74. For the delay free LFC system with $K_{P}=0$ , $K_{I}$ should be less than 2.192 for the system to be stable and this clearly shows large reduction in the margin for selecting $K_{I}$ with the presence of the random time delay. The controller derived through the V-K iteration decreases the oscillation resulting from the random time delay dramatically. These large oscillations may damage the equipment, additionally they degrade the LFC system performance, cause overload on transmission lines and cause interference to the protection system[32]. They can even lead to power system collapse. Practically the PI controller gains are chosen to be large in order to achieve faster time response, however under random time delay these values must kept low. The work in this paper shows the feasibility of achieving the stability of LFC system with random time delay which mimics the control of LFC system under networked environment. The challenge for open communication networks is to guarantee the reliability, availability and immunity against malicious attack. The conventional SCADA availability is more than 99.995[1]. Enhancing the availability, reliability and security of the shared networks will make the migration from conventional dedicated communication links to open communication network possible. In this paper the conventional PI controller is used and in future studies different controllers can be used. It should be noted that the quantization error, the uncertainty and the nonlinearity in the discrete-time LFC system are not considered in this paper, however the decay rate is used to compensate for both the random time delay and the uncertainty in the model. For example very low value of the decay rate, $\beta$ could allow more robustness against the time delay and the uncertainties. Recently some research work on stabilizing networked control systems with these imperfections has been reported see e.g., [33-36] and the references therein.

    • In this paper the stochastic stabilization of one-area LFC system is investigated. Under open communication network the time delay and data loss are usually random and in many cases can be modelled using Markov chains. The conventional LFC system with Markovian random time delay is modelled as standard Markovian linear jump system. The stochastic stability of the LFC system is formulated as bilinear matrix inequalities which are non-convex. The BMIs are divided into three LMIs and the V-K iteration algorithm is used to solve the three LMIs and derive the stabilizing controller. With the derived stability criteria the range of the PI controller gains that achieve the stochastic stability is determined. It is found that the stochastic stability region is semicircle and depends on the sampling rate. The relationship between the robustness in terms of the Lyapunov exponent and the PI controller gains is investigated. We noticed a small part in the stochastic stability region that achieved the optimum performance with the presence of the time delay. The stabilizing controller shows good performance despite the existence of the random time delay. Furthermore the derived controller achieves the optimum performance as indicated in the stability region and proved through simulations. The controller design method is to be extended to multi-area LFC system. Also $H_{\infty}$ performance could be used in the V-K iteration algorithm to account for the uncertainties in the LFC system model.

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