Wide-Area 𝐻_∞ Control for Damping Interarea Oscillations with Event-Triggered Scheme

Abstract

An event-triggered scheme is adopted and applied to the design of a centralized wide-area 𝐻_∞ damping controller (WAHDC) of interconnected power systems with external disturbance. Firstly, based on the linearized multimachine system model, the centralized WAHDC with event-triggered scheme design problem is described as time-delay feedback control problem. Then by using Lyapunov functional method and Jensen inequality technique, criteria for stability with an 𝐻_∞ norm bound and for design of the feedback controller are derived. The linear matrix inequality (LMI) is employed to solve the feedback gain. Finally, case studies are carried out based on two-area four-machine power system. Simulations indicate that the proposed WAHDC with event-triggered scheme can damp the interarea oscillation effectively when the external disturbance is bounded and significantly reduce the number of measured states released to WAHDC. Relationships between event trigger parameter 𝜎 and dynamic performance, 𝜎 and network utilization, 𝜎 and time delay, and 𝜎 and disturbance rejection level are investigated and results obtained can be used to choose an appropriate event trigger parameter.

1. Introduction

Interarea oscillations are the manifestation of consequences of small disturbances in weakly interconnected power systems [1]. The phenomenon is not new. However, it has evolved from being a local problem to becoming a global one. As more regional power systems are integrated for higher reliability and economy of scale, low frequency interarea oscillations have become one of the major challenges to the power system operators. This type of oscillations limits the amount of power transfer on the tie lines between the regions containing coherent generator groups [2].

The traditional approach to damping out interarea oscillations is to install power system stabilizers (PSSs) that provide supplementary control action through the generator excitation systems [2]. The input of PSS is often local signal. Such PSS is effective in damping local modes. While for wide-area complex power systems, the effectiveness in damping interarea modes is limited. It has been proved that under certain operating conditions an interarea mode may be controllable from one area and be observable from another [3]. In such cases, local PSSs are not effective for the damping of that mode. Proposed solutions to interarea oscillation damping control range from new PSS design [4], including multiband PSS concept [5] 𝐻_∞-approach-based supervisory-level PSS [6], to control systems that deploy wide-area measurement systems (WAMSs) [7–10]. With the technology of phasor measurement units (PMUs), WAMS allows to synchronized signals measured at remote locations to be available in real time in the control center.

The overall objective of WAMS is to provide dynamic power system measurements. Except for damping control mentioned previously, WAMS can also provide complete monitoring, protection, and control of the power system. Reference [11] presents a sectionalizing method based on WAMS for the build-up strategy in power system restoration. Reference [12] uses globally measured output signals to estimate reduced order state-space models of large power systems with many controllable devices for subspace system identification. Based on the WAMS output of a power system, [13] presents a new methodology to construct power system area load models alongside network reduction. In [14], a fault location method based on principal component analysis is proposed, in which synchronized information data are utilized provided by the WAMS/PMU measurement system. In order to improve the precision, the literature [15] constructs a state estimation objective function combined with the constraints of zero-injection nodes and WAMS.

One can imagine that, with the development of the technology and with a detailed study, WAMS will be used more and more widely in deregulated power systems. It brings a lot of opportunities for improving system stability, but also huge challenges such as the communication pressure to network by remote magnanimity data transmission. However, periodic sample and control strategy will lead to the sending of many redundant signals through network as mentioned in [16–19]. About how to reduce communication requirements on network control systems, other sample and control strategies have appeared in the literature [20–25]. References [26, 27] introduce trigger scheme in excitation control of single machine and state estimation of multiarea power system. As pointed out in [28], although these schemes have different triggering conditions, they are based on a similar concept that the signal will be released to network only “when needed.” Different from those “sampled when needed” strategies, Yue et al. propose a novel event-triggered scheme in [16–19]. It uses a periodic sample scheme, but an evaluation of whether every sample fulfils a certain condition should be taken before the sampled signals are sent to the network. In that case, the number of signals to be transmitted to the controller can be reduced. And in turn, the number of control output calculations can decrease. Therefore the network bandwidth can be used fully and the computation load at the controller can be relieved.

In order to enhance the dynamic performance of power systems and make full use of constrained bandwidth of WAMS at the same time, this paper investigates a wide-area 𝐻_∞ damping controller (WAHDC) design with event-triggered scheme and some related issues about the possible implement method in electrical power engineering. As centralized wide-area damping control provides more efficient solutions due to the availability of large amount of system-wide dynamic data and better observation of interarea modes [2], to improve the damping performance, centralized wide-area full state feedback is introduced based on WAMS. For the feature that machines are distributed widely in multimachine power system, different from [16–19] in which an event generator is located at the sampler side, an event detector sited between PMU and WAHDC at the control center is used to determine whether the newly measured state should be released to WAHDC, but every measured signal by PMU will be sent to the event detector through network. Based on a similar model proposed in [16–19], criteria for the stability with an 𝐻_∞ norm bound and controller design are set and expressed in the form of linear matrix inequality (LMI). For the purpose of dealing with the uncertainty introduced by external disturbance, 𝐻_∞ control method is adopted [29]. Simulation results show the effectiveness of WAHDC with event-triggered scheme and also provide a possible method to choose appropriate event trigger parameter to get trade-off to balance the damping performance, utilization of communication and computation resources, delay margin, and disturbance rejection level.

The remainder of the paper is organized as follows. In Section 2, the problem under consideration is formulated after the modeling of multimachine power system. The centralized 𝐻_∞ excitation control performance analysis and synthesis results are studied in Section 3. Case studies are presented in Section 4 using a benchmark two-area four-machine power system controlled by WAHDC with event-triggered scheme. Simulation example is provided to demonstrate the effectiveness of the proposed control scheme. What is more, the relationships between event trigger parameter 𝜎 and damping performance, 𝜎 and delay margin, and 𝜎 and disturbance rejection level are investigated, respectively, in this section. Finally, the conclusions are presented in Section 5.

2. System Modeling and Problem Formulation

2.1. Small Signal Model of Multimachine Power Systems

Generally speaking, low frequency oscillation is a typical problem of small-signal stability. Small-signal (or small-disturbance) stability is the ability of the power system to maintain synchronism under small disturbances. The disturbances are considered to be small and usually will not lead to structure and operating point change of the power systems. For the purpose of analysis, the equations that describe the resulting response of the system may be linearized at certain operating point [30].

Consider a multimachine power system composed of 𝑛 generators. The generator is represented by the third-order model. Assuming the input mechanical torque is constant, that is, neglecting the influence of the governor system, assuming the excitation system is separative stationary controllable silicone excitation system and ignoring the time constants of power unit and excitation regulator, and taking the excitation voltage deviation Δ𝐸_𝑓 as excitation control input, the equation of each generator is expressed as follows [31]. The physical meaning of the corresponding parameters is listed in list of symbols: 𝛿̇_𝑖(𝑡)=𝜔_𝑖(𝑡)−𝜔₀𝜔̇_𝑖(𝑡)=𝜔₀𝑀_𝑖𝑃_𝑚𝑖−𝐷_𝑖𝑀_𝑖𝜔_𝑖(𝑡)−𝜔₀−𝜔₀𝑀_𝑖𝑃_𝑒𝑖𝐸̇^′_𝑞𝑖(𝑡)=−1𝑇^′_𝑑0𝑖𝐸^′_𝑞𝑖(𝑡)+𝐼_𝑑𝑖𝑥_𝑑𝑖−𝑥^′_𝑑𝑖+1𝑇^′_𝑑0𝑖𝐸_𝑓𝑖, (1) where 𝑃_𝑒𝑖=𝐸^′_𝑞𝑖(𝑡)+𝐼_𝑑𝑖𝑥_𝑞𝑖−𝑥^′_𝑑𝑖𝐼_𝑞𝑖,𝐼_𝑑𝑖=^𝑛_𝑗=1𝐸^′_𝑞𝑗(𝑡)𝐺_𝑖𝑗cos𝛿_𝑖𝑗−𝐵_𝑖𝑗sin𝛿_𝑖𝑗,𝐼_𝑞𝑖=^𝑛_𝑗=1𝐸^′_𝑞𝑗(𝑡)𝐺_𝑖𝑗sin𝛿_𝑖𝑗+𝐵_𝑖𝑗cos𝛿_𝑖𝑗. (2)𝑖,𝑗=1,2,…,𝑛.𝛿_𝑖𝑗=𝛿_𝑖−𝛿_𝑗 denotes relative power angle between the 𝑖th and the 𝑗th generators. Other parameters are introduced in list of symbols. Taking 𝑥=Δ𝛿Δ𝜔Δ𝐸^′_𝑞^𝑇 as the state vector, applying Taylor series approximation at the given operating point, and then rearranging appropriately, linearized model of 𝑛-machine power system can be written as [32] 𝑥̇=𝐴₁𝑥+𝐵₁𝑢, (3) where Δ𝛿=Δ𝛿₁Δ𝛿₂⋯Δ𝛿_𝑛^𝑇,Δ𝜔=Δ𝜔₁Δ𝜔₂⋯Δ𝜔_𝑛^𝑇,Δ𝐸^′_𝑞=Δ𝐸^′_𝑞1Δ𝐸^′_𝑞2⋯Δ𝐸^′_𝑞𝑛^𝑇,𝑢=Δ𝐸_𝑓,Δ𝐸_𝑓=Δ𝐸_𝑓1Δ𝐸_𝑓2⋯Δ𝐸_𝑓𝑛^𝑇,𝐴₁=−𝐾0𝐼0₁𝑀−𝐷𝑀−𝐾₂𝑀−1𝑇^′_𝑑0𝐾₃10−𝑇^′_𝑑0𝐾₄,𝐵₁=1000𝑇^′_𝑑0^𝑇, (4)(5)(6) where 𝑀=diag𝑀₁,𝑀₂,…,𝑀_𝑛𝐷=diag𝐷₁,𝐷₂,…,𝐷_𝑛𝑇^′_𝑑0=diag𝑇^′_𝑑01,𝑇^′_𝑑02,…,𝑇^′_𝑑0𝑛𝐾₁=𝜕𝑃_𝑒,𝐾𝜕𝛿₂=𝜕𝑃_𝑒𝜕𝐸^′_𝑞,𝐾₃=𝜕𝐸^′_𝑞+𝑥_𝑞−𝑋^′_𝑑𝐼_𝑑𝜕𝐸^′_𝑞,𝐾₄𝐸=𝜕^′𝑞+𝑥_𝑞−𝑋^′_𝑑𝐼_𝑑.𝜕𝛿 (7) For convenience of excitation controller design, one can choose the 𝑛th machine as the reference one, and the state vector is reselected as 𝑥=𝑥₁𝑥₂⋯𝑥_𝑛^𝑇, where 𝑥_𝑖=Δ𝛿_𝑖𝑛Δ𝜔_𝑖Δ𝐸^′_𝑞𝑖^𝑇, 𝑖=1,2,…,𝑛−1. 𝑥_𝑛=Δ𝜔_𝑛Δ𝐸^′_𝑞𝑛^𝑇. By some mathematical similarity transformation and considering the disturbance entering the excitation winding, one can get the state-space description of 𝑛-machine system as 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝐵₂𝜀(𝑡), (8) where 𝜀(𝑡)=𝜀₁(𝑡)𝜀₂(𝑡)⋯𝜀_𝑛(𝑡)^𝑇 denotes the disturbance signal and 𝜀(𝑡)∈𝐿₂[0,+∞]. 𝐴,𝐵, and 𝐵₂ are constant matrices with appropriate dimensions, and 𝐴,𝐵 can be obtained from (3) by some simple matrix similarity transformation. Note that the order of (8) is one order less than (3); that is, the order of (8) is 3𝑛−1.

2.2. System Description of Wide-Area Damping Control

For the problem of wide-area 𝐻_∞ damping control, one can choose a regulated output vector as 𝑧(𝑡); thus the multimachine power system considered here can be described as 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝐵₂𝜀(𝑡),𝑧(𝑡)=𝐶𝑥(𝑡)+𝐷𝑢(𝑡), (9) where 𝐶 and 𝐷 are weighting matrices and 𝑢(𝑡) is constructed by the linear combination of state signals sampled by remote PMU.

As is known, periodic controlling mechanism may often lead to the sending of many redundant signals through the communication network of WAMS, which will in turn increase the load of network transmission and waste the network bandwidth. Therefore, it is significant to introduce a mechanism to decide which sampled state signals should be taken to calculate the control output. A pretreatment mechanism called event detector (ED) is constructed in the wide-area damping control loop located after the phasor data concentrator (PDC) at the control center. If a signal is sent to the WAHDC, we define that an event happens. The framework of WAMS-based wide-area damping control with event-triggered scheme is shown in Figure 1. Focusing on the closed-loop framework shown in Figure 1, the following functionality is expected.(i)The controlled machines are sampled simultaneously by different PMUs with the sampling period ℎ, which can be obtained by global positioning system (GPS) inducing time function. The time of the 𝑘th sampling is written as 𝑘ℎ,  𝑘=0,1,2,…,∞. For simplifying the analysis, assume all of the measured signals are time stamped and arrive at the event detector at the same time. Meanwhile, we also assume that the delays from the controller to the actuators distributed widely are the same.(ii)If the 𝑘th sampled state signal is taken to calculate the control output, the instant will be marked as 𝑡_𝑚ℎ,𝑚=0,1,2,…,∞. Then the released time at the ED is 𝑡_𝑚ℎ+𝜏_𝑠𝑐(𝑚). The meaning of the notation 𝜏_𝑠𝑐(𝑚) will be depicted later. The ED uses the previously transmitted sampled data 𝑥(𝑡_𝑚ℎ) to decide whether the current sampled data 𝑥((𝑘+𝑗)ℎ)(𝑗∈1,2,…,∞) needs to be released based on the following quadratic condition, that is, the event-triggered release scheme 𝑥𝑘+𝑗ℎ−𝑥𝑡_𝑚ℎ^𝑇Ω𝑥𝑘+𝑗ℎ−𝑥𝑡_𝑚ℎ≤𝜎𝑥^𝑇𝑘+𝑗ℎΩ𝑥𝑘+𝑗ℎ, (10) where Ω is a positive matrix and 𝜎≥0. The current sampled signal 𝑥((𝑘+𝑗)ℎ) will not be released to calculate the control action if condition (10) is satisfied. In other words, only the sampled signals that violate condition (10) will be released by ED to calculate the control output. Assume the signal sampled at the initial time is released, which means 𝑡₀ℎ=0. In particular, when 𝜎=0, all the sampled signals are transmitted to the control center and used to calculate control action. Then the event-triggered scheme reduces to the periodic sampling and controlling scheme.(iii)When the signal has been sampled by PMU, it is forwarded to the control center, introducing a sensor-to-controller delay 𝜏_𝑠𝑐(𝑚). It is noteworthy that 𝜏_𝑠𝑐(𝑚) can be obtained with the time stamp and the synchronization measurement feature of PMU. The controller forwards the actuation signals to the actuators, introducing another communication delay, controller-to-actuator delay 𝜏_𝑐𝑎(𝑚), which includes the time delay introduced by control output computation. The total transmission time delay can be lumped together as 𝜏_𝑚=𝜏_𝑠𝑐(𝑚)+𝜏_𝑐𝑎(𝑚), 𝜏_𝑚∈(0,𝜏], where 𝜏 is a positive real number. Finally, the actuators perform the actuation at the time given by 𝑡_𝑚ℎ+𝜏_𝑚.

Figure 1 

General structure of wide-area damping control system with event-triggered scheme.

Based on the previous analysis, considering the effect of the transmission delay, the system model with event-triggered scheme can be described as 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢𝑡_𝑚ℎ+𝐵₂𝜀(𝑡),𝑧(𝑡)=𝐶𝑥(𝑡)+𝐷𝑢𝑡_𝑚ℎ,𝑡∈𝑡_𝑚ℎ+𝜏_𝑚,𝑡_𝑚+1ℎ+𝜏_𝑚+1. (11) For the ease of analysis, for 𝑡∈[𝑡_𝑚ℎ+𝜏_𝑚,𝑡_𝑚+1ℎ+𝜏_𝑚+1), the system (11) can be converted to a time-delay system using similar methods in [17, 19] 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝐾𝑥(𝑡−𝜏(𝑡))+𝐵𝐾𝑒_𝑚(𝑡)+𝐵₂𝜀(𝑡)𝑧(𝑡)=𝐶𝑥(𝑡)+𝐷𝐾𝑥(𝑡−𝜏(𝑡))+𝐷𝐾𝑒_𝑚(𝑡)𝑥(𝑡)=𝜙(𝑡),𝑡∈−𝜏_𝑀,0, (12)(13)(14) where 𝜙(𝑡) is the initial time of 𝑥(𝑡).  𝜏_𝑀=ℎ+𝜏, and the definition of 𝑒_𝑚(𝑡) is the same as 𝑒_𝑘(𝑡) in [19]. Correspondingly, the event-triggered condition is converted as 𝑒^𝑇_𝑚(𝑡)Ω𝑒_𝑚(𝑡)≤𝜎𝑥^𝑇(𝑡−𝜏(𝑡))Ω𝑥(𝑡−𝜏(𝑡)). (15)

2.3. Problem Formulation

The purpose of the paper is to develop techniques to deal with 𝐻_∞ event-triggered control problem for 𝑛-machine power system (12)–(14) based on WAMS. For a given scalar 𝛾>0, the performance of system (12)–(14) is defined to be 𝐽(𝜀)=^∞₀𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝜀^𝑇(𝑡)𝜀(𝑡)𝑑𝑡. (16) Then the 𝐻_∞ event-triggered control problem addressed in this paper can be stated so as to design a state feedback controller such that system (12)–(14) under the event detector with event triggering condition (10) satisfies the following two requirements.(1)The closed-loop system (12)–(14) should be asymptotically stable under the condition 𝜀(𝑡)=0.(2)𝐽(𝜀)<0 for all nonzero 𝜀(𝑡) under the zero initial condition and a given 𝛾>0.

3. 𝐻_∞ Performance Analysis and Controller Design

3.1. Some Lemmas

Before giving the main theorems, we need the following two lemmas.

Lemma 1 (see [33]). For any constant matrix 𝑄∈ℝ^𝑛×𝑛, 𝑄>0, scalars 𝜏₁≤𝜏(𝑡)≤𝜏₃, and vector function 𝑥̇∶[−𝜏₃,−𝜏₁]→ℝⁿ such that the following integration is well defined, then it holds that −𝜏₃−𝜏₁^𝑡−𝜏1_𝑡−𝜏3𝑥̇^𝑇(𝑠)𝑄𝑥̇(𝑠)𝑑𝑠≤𝜁^𝑇₁(𝑡)𝜁−𝑄∗∗𝑄−2𝑄∗0𝑄−𝑄¹(𝑡), (17) where 𝜁^𝑇₁(𝑡)=𝑥^𝑇(𝑡−𝜏₁)𝑥^𝑇(𝑡−𝜏(𝑡))𝑥^𝑇(𝑡−𝜏₃).

Lemma 2 (see [34]). For matrices 𝑅>0 and 𝑋^𝑇=𝑋, one has −𝑋𝑅⁻¹𝑋≤𝜌²𝑅−2𝜌𝑋, (18) where 𝜌 is any chosen constant.

3.2. 𝐻_∞ Performance Analysis and Controller Design

Based on the Lyapunov functional method, we conclude the following results. Different from [17–19] in which free weighing matrix method is used, we use Jensen inequality approach to 𝐻_∞ performance analysis and 𝐻_∞ controller design. Reference [35] has theoretically proved that both of the methods are equivalent. However, using Jensen inequality can decrease decision variables and is easy for computation of LMI.

Theorem 3. For given parameters 𝜎,  𝛾 and matrix 𝐾, the system (12)–(14) under the event-triggered scheme (15) can keep on the internal stability and also the external disturbance rejection index 𝛾, if there exist positive definite matrices 𝑃,  𝑄,  𝑅, and Ω, such that the following matrix inequality holds: Φ₁₁Φ∗∗∗₂₁−𝛾²Φ𝐼∗∗₃₁𝜏_𝑀𝑅𝐵₂Φ−𝑅∗₄₁00−𝐼<0, (19) where Φ₁₁=𝑃𝐴+𝐴^𝑇𝐾𝑃+𝑄−𝑅∗∗∗_𝑇𝐵^𝑇𝐾𝑃+𝑅𝜎Ω−2𝑅∗∗0𝑅−𝑅−𝑄∗_𝑇𝐵^𝑇Φ𝑃00−Ω₂₁=𝐵^𝑇₂Φ𝑃000₃₁=𝜏_𝑀𝑅𝐴𝜏_𝑀𝑅𝐵𝐾0𝜏_𝑀Φ𝑅𝐵𝐾₄₁=𝐶𝐷𝐾0𝐷𝐾. (20)

Proof. Construct a Lyapunov functional as 𝑉(𝑡)=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+^𝑡_{𝑡−𝜏𝑀}𝑥^𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠+𝜏_𝑀^𝑡_{𝑡−𝜏𝑀}^𝑡_𝑠𝑥̇^𝑇(𝑣)𝑅𝑥̇(𝑣)𝑑𝑣𝑑𝑠, (21) where 𝑃>0,  𝑄>0, and 𝑅>0. For 𝑡∈[𝑡_𝑚+𝜏_𝑚,𝑡_𝑚+1+𝜏_𝑚+1), taking the time derivation on (21), 𝑉̇(𝑡)=2𝑥^𝑇(𝑡)𝑃[𝐴𝑥(𝑡)+𝐵𝐾𝑥(𝑡−𝜏(𝑡))+𝐵𝐾𝑒_𝑚(𝑡)+𝐵₂𝜀(𝑡)+𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑥^𝑇𝑡−𝜏_𝑀𝑄𝑥𝑡−𝜏_𝑀+𝜏²_𝑀𝑥̇^𝑇(𝑡)𝑅𝑥̇(𝑡)−𝜏_𝑀^𝑡_{𝑡−𝜏𝑀}𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠. (22) Applying Lemma 1, we have −𝜏_𝑀^𝑡_{𝑡−𝜏𝑀}𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠≤𝜁^𝑇(𝑡)𝜁(𝑡),−𝑅∗∗𝑅−2𝑅∗0𝑅−𝑅 (23) where 𝜁^𝑇(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−𝜏(𝑡))𝑥^𝑇(𝑡−𝜏_𝑀). Considering (22) and (15), we can obtain that 𝑉̇(𝑡)≤𝜉^𝑇(𝑡)Π₁𝜉(𝑡)+𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝜀^𝑇(𝑡)𝜀(𝑡)−𝑧^𝑇(𝑡)𝑧(𝑡)+𝛾²𝜀^𝑇(𝑡)𝜀(𝑡), (24) where 𝜉^𝑇(𝑡)=𝜁^𝑇(𝑡)𝑒^𝑇_𝑚(𝑡)𝜀^𝑇(𝑡)Π₁=Φ₁₁+Φ^𝑇₃₁𝑅⁻¹Φ₃₁∗Φ₂₁+𝜏_𝑀𝐵^𝑇₂𝑅Φ₃₁𝜏²_𝑀𝐵^𝑇₂𝑅𝐵₂. (25) Combining (13) and (24), one can conclude that 𝑉̇(𝑡)≤𝜉^𝑇(𝑡)Π₂𝜉(𝑡)−𝑧^𝑇(𝑡)𝑧(𝑡)+𝛾²𝜀^𝑇(𝑡)𝜀(𝑡), (26) where Π₂=Π₁+Φ^𝑇₄₁Φ₄₁∗0−𝛾²𝐼. (27) From (19) and by Schur complement, one can conclude from (26) that, for 𝑡∈[𝑡_𝑚+𝜏_𝑚,𝑡_𝑚+1+𝜏_𝑚+1), 𝑉̇(𝑡)≤−𝑧^𝑇(𝑡)𝑧(𝑡)+𝛾²𝜀^𝑇(𝑡)𝜀(𝑡). (28) Since ∪^∞_𝑚=0[𝑡_𝑚+𝜏_𝑚,𝑡_𝑚+1+𝜏_𝑚+1)=[0,∞), 𝑉(𝑡) is continuous in 𝑡 since 𝑥(𝑡) is continuous in 𝑡. Therefore, on both sides of (26), the integral from 0 to ∞ with respect to 𝑡 yields 𝑉(∞)(0)≤−𝑉_∞0−𝑧^𝑇(𝑠)𝑧(𝑠)+𝛾²𝜀^𝑇(𝑠)𝜀(𝑠)𝑑𝑠. (29) Under zero initial conditions, the following can be obtained: ^∞₀𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝜀^𝑇(𝑠)𝜀(𝑠)𝑑𝑠≤0. (30) That is 𝐽(𝜀)<0. This completes the proof.

In order to ensure the damping control performance under the circumstance of time delay of wide-area state feedback with event-triggered scheme and external disturbance, the main result is summarized in the following theorem, which can be derived based on Theorem 3.

Theorem 4. For given parameters 𝜎 and 𝛾, the system (12)–(14) under the event-triggered scheme (15) with Ω=𝑋⁻¹Ω𝑋⁻¹ can keep on the internal stability and also the external disturbance rejection index 𝛾, if there exist positive definite matrices 𝑋, 𝑄, 𝑅, Ω, and 𝑌 of appropriate dimensions such that the following matrix inequality is feasible. Moreover, the gain matrix of such 𝐻_∞ damping controller can be obtained as 𝐾=𝑌𝑋⁻¹: Φ₁₁∗∗∗Φ₂₁−𝛾²𝐼∗∗Φ₃₁𝜏_𝑀𝐵₂𝑅−2𝑋∗Φ₄₁00−𝐼<0, (31) where Φ₁₁=𝐴𝑋+𝑋𝐴^𝑇+𝑄−𝑌𝑅∗∗∗_𝑇𝐵^𝑇+𝑅𝜎Ω−20𝑅∗∗𝑅−𝑅−𝑌𝑄∗_𝑇𝐵^𝑇00−ΩΦ₂₁=𝐵^𝑇₂000Φ₃₁=𝜏_𝑀𝐴𝑋𝜏_𝑀𝐵𝑌0𝜏_𝑀𝐵𝑌Φ₄₁=𝐶𝑋𝐷𝑌0𝐷𝑌. (32)(33)(34)(35)

Proof. . Defining 𝑋=𝑃⁻¹, pre- and postmultiplying (19) with diag{𝑋,𝑋,𝑋,𝑋,𝐼,𝑅⁻¹,𝐼}, making the following changes in the variables: 𝑅=𝑋𝑅𝑋,  𝑄=𝑋𝑄𝑋,  Ω=𝑋Ω𝑋,  𝑌=𝐾𝑋, and using Lemma 2 and Schur complement, (31) is derived from (19).

4. Results of Case Studies

This section aims to validate the proposed wide-area 𝐻_∞ damping controller design method with event-triggered scheme. There are different performance indices to evaluate the wide-area 𝐻_∞ damping controller, and there are two parameters, that is, event-triggering parameter 𝜎 and disturbance rejection index 𝛾 that should be chosen appropriately. In this section, we focus on the time response of the power system with small signal, release times of sampled state signals during a certain simulation time, maximum allowable delay time (delay margin), and disturbance rejection level. To find the appropriate 𝜎 and 𝛾, a series of numeric experiments is carried out through the application of WAHDC to the four-machine two-area test system, which is also the benchmark system for the oscillation damping study [30]. Figure 2 shows the four-machine two-area system. The detailed parameters can be found in [30].

Figure 2 

Two-area four-machine system.

In the following simulations, the state-space description of the power system is based on model (8), that is, the relative power angle synchronized reference frame model. The matrices 𝐴 and 𝐵 can be established by the modeling method mentioned previously at the given operating point. For regulated output, we set the weighting matrices 𝐶=diag{10,1.0,10,1,10,1,10,0.5,0.5,0.5} and 𝐷 as 𝐷=.00000000000000000000000000001000010000100001 (36) And also we choose 𝐵₂=𝐷. For the convenience of analysis, the state variables are reordered as𝑥=Δ𝜔₁Δ𝛿₁₄Δ𝜔₂Δ𝛿₂₄Δ𝜔₃Δ𝛿₃₄Δ𝜔₄Δ𝐸^′_𝑞1Δ𝐸^′_𝑞2Δ𝐸^′_𝑞3Δ𝐸^′_𝑞4^𝑇. (37)The duration of all the following simulations is always 30s.

4.1. Case Study about the Influence of 𝜎 on System Time Response and Network Utilization

A time domain analysis method of the closed-loop system is carried out to reveal the performance of the proposed controller in terms of improving the damping of interarea oscillation. The exogenous disturbance is assumed to be 𝜀₁(𝑡)=0.1sign(sin𝑡),if𝑡∈[0,5]0,otherwise𝜀_𝑖(𝑡)=0,𝑖=2,3,4. (38) The damping action of the 𝐻_∞ controller is examined under different event-triggering parameter 𝜎. According to the time-delay analysis of WAMS in [36, 37], we set 𝜏_𝑀=0.08s, ℎ=0.02s, and then the transmission delay 𝜏 is 0.06s correspondingly. For 𝛾=20, the dynamic response with different 𝜎 chosen from the feasible range, which is shown in Figure 7 and will be described in detail later, is displayed in Figures 3 and 4, and the corresponding network utilization is reported in Table 1. The data transmission rate 𝛼=𝑁/𝑁₀×100% is also listed in Table 1, where 𝑁₀ denotes the number of total measured data release times with 𝜎=0, and 𝑁 indicates the number of release times with 𝜎≠0.

Table 1 

Network utilization for different 𝜎.

Figure 3 

Dynamic response of relative angle between 𝐺1 and 𝐺4 for different 𝜎.

Figure 4 

Dynamic response of angular velocity deviation of 𝐺3 for different 𝜎.

To illustrate the quality of the systems response, we introduce two indices, that is, the integrated absolute error (IAE) between the system response of periodic release (or time-based) strategy and event-triggered scheme [38] and the time response of the error (TRE) between the time-based strategy and the event-triggered scheme, where IAE=^𝑡₀𝑥_time-based(𝑡)−𝑥_{event-triggered}(𝑡)𝑑𝑡,TRE=𝑥_time-based(𝑡)−𝑥_{event-triggered}(𝑡). (39)(40) The TREs and IAEs for different 𝜎 are illustrated in Figures 5 and 6 and Table 2, respectively. In Figure 5, Δ𝛿₁₄₀ denotes the relative angle between 𝐺₁ and 𝐺₄ with time-based strategy, and Δ𝛿_14𝑖 represents the relative angle between 𝐺₁ and 𝐺₄ with different 𝜎_𝑖, 𝑖=1,2,3, where 𝜎₁=0.1, 𝜎₂=0.2, and 𝜎₃=0.3. In Figure 6, Δ𝜔_3𝑖, 𝑖=0,1,2,3 has similar meaning.

Table 2 

IAEs for different 𝜎.

Figure 5 

Time response of the error between time-based strategy and event-triggered scheme (Δ𝛿14).

Figure 6 

Dynamic response of the error between time-based strategy and event-triggered scheme (Δ𝜔3).

Figure 7 

Upper bound of 𝜎 for different 𝜏𝑀.

The curves depicted in Figures 3 and 4 show that the oscillation between Area 1 and Area 2 is damped quickly for different event-triggering parameter 𝜎. Even when 𝜎=0.3, the oscillation is damped well. It shows that the WAHDC with event-triggered scheme can provide effective damping of interarea oscillations when the power system is subjected to external interference and communication time delay exists in the damping control loop. Moreover, from Table 1, it is found that the times when the sampled signals are released to be used to calculate the control output at the control center decrease with the increase of 𝜎. It can be seen from the last row of Table 1 that, for 𝜎=0.1,0.2,0.3, with event-triggered scheme, only 21.45%, 16.19%, and 12.59% of measured states need to be released to the WAHDC, respectively. In other words, the network utilization by the event-triggered scheme can be obtained by 78.55%, 83.81%, and 87.41% improvement, respectively. Thus the burden of the network communication between control center and the actuators widely distributed is reduced and the communication bandwidth is saved. That validates the effectiveness of the design approach.

However, according to the definition of IAE we can see a lower value of IAE meaning that the control strategy provides a better damping performance. The data in Table 2 show that the dynamic response is degraded to a certain degree with increase of 𝜎; that is, it is expected that larger 𝜎 will yield larger IAE with less network utilization. From the point of view of WAHDC design, the relationship between event trigger parameter 𝜎 and network utilization can provide an effective guide for the design of WAHDC, and the event-triggering parameter can be chosen to achieve trade-off between the network utilization and damping performance for certain delay margin and perturbation attenuation index.

As stated previously, for certain delay margin and perturbation attenuation index, event trigger parameter 𝜎 can be chosen in a feasible range. The following studies are aimed at finding the feasible range.

4.2. Case Study about the Relationship between 𝜎 and Time Delay

In this section, simulation studies are carried out to find a guiding method to choose an appropriate 𝜎 for certain 𝛾. For given time delay and 𝛾, a feasible range of 𝜎 can be found by manually increasing the value of 𝜎 and checking the feasibility of (31). The upper bound of 𝜎 is calculated by selecting different sets of 𝜏_𝑀 and 𝛾. In other words, the delay margin that resulted from different 𝜎 and 𝛾 can also be obtained in similar way. Results of the upper bound of 𝜎 with respect to different 𝜏_𝑀 and the delay margin for different 𝜎 are shown in Figure 7 and Table 3 separately.

Table 3 

Delay margin for different 𝜎.

Results in Figure 7 show that, for a certain 𝛾, the curve declines slowly at first and after a turning point it declines sharply with the increase of 𝜏_𝑀. The trend of the relationship between 𝜏_𝑀 and the upper bound of 𝜎 for different 𝛾 is similar except that with the increase of 𝛾, the value of the upper bound of 𝜎 decreases for the same 𝜏_𝑀. Namely, with the increase of 𝜏_𝑀, one can choose a 𝜎 in an increasing feasible range. From another perspective, from Table 3 we can also conclude that for given 𝛾, increasing the value of 𝜎, delay margin of WAHDC decreases. Thus for the full use of network bandwidth one can choose a larger 𝜎. For example, for 𝛾=30, one can choose 𝜎 from 0 to about 0.6 when the maximum delay in damping control loop is 0.08s. However, as addressed previously, considering the damping performance and delay margin of the controller, the value of 𝜎 cannot be too large.

4.3. Case Study about the Relationship between 𝜎 and the Disturbance Rejection Level 𝛾

Different selection of disturbance rejection level 𝛾 also influences the feasible range of 𝜎. The relationship between 𝜎 and the disturbance rejection level 𝛾 is investigated in this section. By applying Theorem 4, we can get the upper bound of 𝜎 for different set of 𝛾 and time delays and also obtain the smallest 𝛾 for certain combination of 𝜎 and time delay. Results of upper bound of 𝜎 with respect to different 𝛾 and the smallest 𝛾 for different 𝜎 and time delay are shown in Figure 8 and Table 4 separately.

Table 4 

Smallest 𝛾 for different 𝜎.

Figure 8 

Upper bound of 𝜎 for different 𝛾.

As is well known, for 𝐻_∞ control problem, smaller 𝛾 means better disturbance rejection level. However, from Figure 8 and Table 4 we can know that, when 𝛾 is set to less than 40, the feasible range of 𝜎 decreases rapidly for both 0.06s and 0.08s time delay. Since larger 𝜎 means less network overheads generally, for reduction of network overheads, the value of 𝛾 cannot be set too small. Moreover, when 𝜎=0 we can know the smallest value of 𝛾 for given time delay. Therefore, the findings can provide an effective guide for the WAHDC design for certain disturbance rejection performance and delay margin to achieve full use of network resource.

5. Conclusion and Future Work

A centralized 𝐻_∞ wide-area damping controller for multi-machine power system is proposed in this paper. Besides using the wide-area signals from remote generators to improve the damping of the interarea oscillation in the power system, to reduce the amount of data transfer, an event-triggered scheme is introduced. Moreover, an 𝐻_∞ control criterion is obtained for the controller design based on the Jensen inequality and LMI approach. By solving a set of LMIs, Theorem 4 provides a useful way of design for feedback gain with given trigger parameter 𝜎. Then to validate the presented WAHDC with event-triggered scheme and to find the method to choose an appropriate event trigger parameter 𝜎, case studies are carried out based on the two-area four-machine power system. From the simulation results, the following conclusion can be drawn.(i)The presented WAHDC with event-triggered scheme can damp the interarea oscillation effectively when the disturbance is energy bounded. Increasing the value of 𝜎 in the feasible range can deduce the traffic volume between WAHDC and machines. But the dynamic response of power system under WAHDC is degenerated to some extent.(ii)The introduction of event-triggered scheme has some influence on system dynamic performance, network utilization, delay margin, and disturbance rejection level. In view of WAHDC design, increasing the value of 𝜎 can decrease the delay margin and disturbance rejection performance. Findings from previous simulations give a guiding method to select appropriate 𝜎 to get compromise between damping performance, network utilization, delay margin, and disturbance rejection level.

The previous researches are promising, but also have some limits. The real power system is nonlinear and of high dimensionality, and it is neither practical nor necessary to design the full state feedback controller based on the linearized model. Further research is still needed to combine the model reduction theory and the event-triggered scheme presented in this paper. Furthermore, the proposed WAHDC with event-triggered scheme only reduces the data transfer from control center to distributed machines. Another direction of our future research is to construct distributed event detectors at the PMUs side to reduce the whole network overheads. In addition, the algorithm and method proposed in [39, 40] may be used to find the optimal value of 𝜎.

Symbols

𝛿𝑖(𝑡):	Power angle of the ith generator,  in deg
𝜔𝑖(𝑡):	The angular velocity of the ith generator,  in p.u.
𝜔0:	Synchronous machine speed,  in p.u.
𝐸′𝑞𝑖(𝑡):	q-axis transient potential,  in p.u.
𝐸𝑓(𝑡):	Control input,  excitation voltage,  in p.u.
𝑥𝑑𝑖:	d-axis synchronous reactance,  in p.u.
𝑥′𝑑𝑖:	d-axis transient reactance,  in p.u.
𝑌𝑖𝑗=𝐺𝑖𝑗+𝑗𝐵𝑖𝑗:	Modulus of the transfer admittance between the ith and jth generator,  in p.u.
𝑃𝑚:	Mechanical input power,  in  p.u.,  which is a constant
𝑃𝑒:	Electrical power,  in p.u.
𝐷𝑖:	Damping coefficient of ith generator
𝑀i:	Stored energy at rated speed,  inertia constant,  in seconds
𝑇′𝑑0:	d-axis open circuit transient timeconstant,  in seconds
𝐼𝑑𝑖:	Direct axis current,  in p.u.
𝐼𝑞𝑖:	Quadrature axis current,  in p.u.

Acknowledgments

The authors would like to thank the anonymous reviewers for providing their valuable comments and constructive suggestions that have improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China (nos. 60834002 and 61074025) and the Ph. D. Programs Foundation of Ministry of Education of China (no. 20110142110036).