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

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 proposedWAHDCwith eventtriggered 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.


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]  ∞ -approach-based supervisory-level PSS [6], to control systems that deploy wide-area measurement systems (WAMSs) [7][8][9][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 Mathematical Problems in Engineering 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][17][18][19].About how to reduce communication requirements on network control systems, other sample and control strategies have appeared in the literature [20][21][22][23][24][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][17][18][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 widearea  ∞ damping controller (WAHDC) design with eventtriggered 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 systemwide dynamic data and better observation of interarea modes [2], to improve the damping performance, centralized widearea full state feedback is introduced based on WAMS.For the feature that machines are distributed widely in multimachine power system, different from [16][17][18][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][17][18][19], criteria for the stability with an  ∞ 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,  ∞ 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  ∞ 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  and damping performance,  and delay margin, and  and disturbance rejection level are investigated, respectively, in this section.Finally, the conclusions are presented in Section 5.

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  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 Δ  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: where ,  = 1, 2, . . ., .   =   −   denotes relative power angle between the th and the th generators.Other parameters are introduced in list of symbols.Taking  = [Δ Δ Δ   ]  as the state vector, applying Taylor series approximation at the given operating point, and then rearranging appropriately, linearized model of -machine power system can be written as [32] where where For convenience of excitation controller design, one can choose the th machine as the reference one, and the state vector is reselected as

𝑇
. By some mathematical similarity transformation and considering the disturbance entering the excitation winding, one can get the state-space description of -machine system as  where denotes the disturbance signal and () ∈  2 [0, +∞]., , and  2 are constant matrices with appropriate dimensions, and ,  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 3 − 1.

System Description of Wide-Area Damping Control.
For the problem of wide-area  ∞ damping control, one can choose a regulated output vector as (); thus the multimachine power system considered here can be described as where  and  are weighting matrices and () 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 th sampling is written as ℎ,  = 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 th sampled state signal is taken to calculate the control output, the instant will be marked as   ℎ,  = 0, 1, 2, . . ., ∞.Then the released time at the ED is   ℎ +   ().The meaning of the notation   () will be depicted later.The ED uses the previously transmitted sampled data (  ℎ) to decide whether the current sampled data ((+)ℎ) ( ∈ 1, 2, . . ., ∞) needs to be released based on the following quadratic condition, that is, the event-triggered release scheme where Ω is a positive matrix and  ≥ 0. The current sampled signal (( + )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  0 ℎ = 0.In particular, when  = 0, all the sampled signals are transmitted to the control center and used to calculate control action.Then the eventtriggered 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 sensorto-controller delay   ().It is noteworthy that   () 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-toactuator delay   (), which includes the time delay introduced by control output computation.The total transmission time delay can be lumped together as where  is a positive real number.Finally, the actuators perform the actuation at the time given by   ℎ +   .
Based on the previous analysis, considering the effect of the transmission delay, the system model with eventtriggered scheme can be described as For the ease of analysis, for  ∈ [  ℎ +   ,  +1 ℎ +  +1 ), the system (11) can be converted to a time-delay system using similar methods in [17,19] ẋ () =  () +  ( −  ()) +   () +  2  () (12) where () is the initial time of ().  = ℎ + , and the definition of   () is the same as   () in [19].Correspondingly, the event-triggered condition is converted as 2.3.Problem Formulation.The purpose of the paper is to develop techniques to deal with  ∞ event-triggered control problem for -machine power system ( 12)-( 14) based on WAMS.For a given scalar  > 0, the performance of system ( 12)-( 14) is defined to be Then the  ∞ 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.

𝐻 ∞ Performance Analysis and Controller Design
3.1.Some Lemmas.Before giving the main theorems, we need the following two lemmas.

𝐻 ∞ Performance Analysis and Controller Design.
Based on the Lyapunov functional method, we conclude the following results.Different from [17][18][19] in which free weighing matrix method is used, we use Jensen inequality approach to  ∞ performance analysis and  ∞ 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.
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  and , the system (12)-( 14) under the event-triggered scheme (15) with Ω =  −1 Ω −1 can keep on the internal stability and also the external disturbance rejection index , if there exist positive definite matrices , , , Ω, and  of appropriate dimensions such that the following matrix inequality is feasible.Moreover, the gain matrix of such  ∞ damping controller can be obtained as where Proof.Defining  =  −1 , pre-and postmultiplying (19) with diag{, , , , ,  −1 , }, making the following changes in the variables:  = ,  = , Ω = Ω,  = , and using Lemma 2 and Schur complement, ( 31) is derived from (19).

Results of Case Studies
This section aims to validate the proposed wide-area  ∞ damping controller design method with event-triggered scheme.There are different performance indices to evaluate the wide-area  ∞ damping controller, and there are two parameters, that is, event-triggering parameter  and disturbance rejection index  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  and , a series of numeric experiments is carried out through the application of WAHDC to the fourmachine 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].
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  and  can be established by the modeling method mentioned previously at the given operating point.For regulated output, we set the weighting matrices  = diag{10, 1.0, 10, 1, 10, 1, 10, 0.5, 0.5, 0.5} and  as And also we choose  2 = .For the convenience of analysis, the state variables are reordered as The duration of all the following simulations is always 30 s.

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 The damping action of the  ∞ controller is examined under different event-triggering parameter .According to the time-delay analysis of WAMS in [36,37], we set   = 0.08 s, ℎ = 0.02 s, and then the transmission delay  is 0.06 s correspondingly.For  = 20, the dynamic response with different  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  = / 0 × 100% is also listed in Table 1, where  0 denotes the number of total measured data release times with  = 0, and  indicates the number of release times with  ̸ = 0. 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 The TREs and IAEs for different  are illustrated in Figures 5  and 6 and Table 2, respectively.In Figure 5, Δ 14 0 denotes the relative angle between  1 and  4 with time-based strategy, and Δ 14  represents the relative angle between  1 and  4 with different   ,  = 1, 2, 3, where  1 = 0.1,  2 = 0.2, and  3 = 0.3.In Figure 6, Δ 3  ,  = 0, 1, 2, 3 has similar meaning.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 .Even when  = 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 .It can be seen from the last row of Table 1 that, for  = 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 ; that is, it is expected that larger  will yield larger IAE with less network utilization.From the point of view of WAHDC design, the relationship between event  trigger parameter  and network utilization can provide an effective guide for the design of WAHDC, and the eventtriggering 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  can be chosen in a feasible range.The following studies are aimed at finding the feasible range.

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  for certain .For given time delay and , a feasible range of  can be found by manually increasing the value of  and checking the feasibility of (31).The upper bound of  is calculated by selecting different sets of   and .In other words, the delay margin that resulted from different  and  can also be obtained in similar way.Results of the upper bound of  with respect to different   and the delay margin for different  are shown in Figure 7 and Table 3 separately.
Results in Figure 7 show that, for a certain , the curve declines slowly at first and after a turning point it declines sharply with the increase of   .The trend of the relationship between   and the upper bound of  for different  is similar except that with the increase of , the value of the upper    overheads generally, for reduction of network overheads, the value of  cannot be set too small.Moreover, when  = 0 we can know the smallest value of  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.

Conclusion and Future Work
A centralized  ∞ wide-area damping controller for multimachine 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 eventtriggered scheme is introduced.Moreover, an  ∞ 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 .Then to validate the presented WAHDC with event-triggered scheme and to find the method to choose an appropriate event trigger parameter , case studies are carried out based on the two-area fourmachine 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  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  can decrease the delay margin and disturbance rejection performance.Findings from previous simulations give a guiding method to select appropriate  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 .

Figure 1 :
Figure1: General structure of wide-area damping control system with event-triggered scheme.

Figure 5 :
Figure 5: Time response of the error between time-based strategy and event-triggered scheme (Δ 14 ).

Figure 6 :
Figure 6: Dynamic response of the error between time-based strategy and event-triggered scheme (Δ 3 ).

Table 1 :
Network utilization for different .

Table 3 :
Delay margin for different .

Table 4 :
Smallest  for different .decreasesfor the same   .Namely, with the increase of   , one can choose a  in an increasing feasible range.From another perspective, from Table3we can also conclude that for given , increasing the value of , delay margin of WAHDC decreases.Thus for the full use of network bandwidth one can choose a larger .For example, for  = 30, one can choose  from 0 to about 0.6 when the maximum delay in damping control loop is 0.08 s.However, as addressed previously, considering the damping performance and delay margin of the controller, the value of  cannot be too large.4.3.Case Study about the Relationship between  and the Dis-turbance Rejection Level .Different selection of disturbance rejection level  also influences the feasible range of .The relationship between  and the disturbance rejection level  is investigated in this section.By applying Theorem 4, we can get the upper bound of  for different set of  and time delays and also obtain the smallest  for certain combination of  and time delay.Results of upper bound of  with respect to different  and the smallest  for different  and time delay are shown in Figure8and Table4separately.As is well known, for  ∞ control problem, smaller  means better disturbance rejection level.However, from Figure8and Table4we can know that, when  is set to less than 40, the feasible range of  decreases rapidly for both 0.06 s and 0.08 s time delay.Since larger  means less network (): Power angle of the ith generator, in deg   (): The angular velocity of the ith generator, in p.u.  0 : Synchronous machine speed, in p.u.    (): q-axis transient potential, in p.u.   (): Control input, excitation voltage, in p.u.   : d-axis synchronous reactance, in p.u.     =   +   : Modulus of the transfer admittance between the ith and jth generator, in p.u.   : M e c h a n i c a li n p u tp o w e r ,i np .u.,wh i c h is a constant   :E l e c t r i c a l p o w e r ,i n p .u .