on the centralized nonlinear control of hvdc systems using lyapunov theory

1156 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 28, NO. 2, APRIL 2013

On the Centralized Nonlinear Control ofHVDC Systems Using Lyapunov Theory

Robert Eriksson, Member, IEEE

Abstract—The security region of a power system is an impor-tant and timely issue; different stability criteria may be limiting.Rotor-angle stability can be improved by modulating active powerof installed high-voltage direct current (HVDC) links. This paperproposes a new centralized nonlinear control strategy for coor-dinating several point-to-point and multiterminal HVDC systemsbased on Lyapunov theory. The proposed control Lyapunov func-tion is negative semi-definite along the trajectories and uses theinternal node representation of the system. The proposed controlLyapunov function increases the domain of attraction and, thus,improves the rotor-angle stability. Nonlinear simulations are per-formed on the IEEE 10-machine 39-bus systemwhich shows the ef-fectiveness of the controller. In comparison, simulations using theconventional lead-lag controller are also run.

Index Terms—Control Lyapunov function, coordinated control,energy function, high-voltage direct current (HVDC), multiter-minal dc (MTDC), rotor-angle stability.

I. INTRODUCTION

S TABILITY phenomena such as power oscillations andtransient stability are important in power system stability

analysis. The power oscillation phenomenon consists of syn-chronous generator rotors swinging relative to each other.Under severe disturbances the first swing may be considerablylarge and endanger the stability of the system. If the first swingis handled and reaches transient stability, there might still bea risk of losing synchronism due to low damping torque orvoltage instability.The installed capacity of high-voltage direct-current (HVDC)

links is increasing around the world [1]. Improved technology,such as voltage-source converters (VSCs) and multiterminal dc(MTDC) systems, brings more possibilities than conventionalpoint-to-point HVDC links. The power modulation control ofHVDC links enhances the security [2]–[4]. This, in turn, mayallow more power to be transferred, and has an economic im-pact [3]. When designing the controller for an HVDC link, theaim is to improve the stability; one also has to ensure that it doesnot interact negatively with the system in any situation. Higherdemand comes with having several controllable devices in thesystem since one also has to ensure that negative interactionsbetween the devices do not occur, coordination is therefore nec-essary and highlighted in [5]. Besides one wants to improve the

Manuscript received August 20, 2012; accepted December 15, 2012. Dateof publication February 04, 2013; date of current version March 21, 2013. Thiswork was supported by the Centre of Excellence in Electrical Engineering

at the Royal Institute of Technology. Paper no. TPWRD-00873-2012.The author is with the Department of Electric Power Systems, KTHRoyal In-

stitute of Technology, Stockholm 100-44, Sweden (e-mail: [email protected]).Digital Object Identifier 10.1109/TPWRD.2013.2240021

security region by coordination, utilizing the system in a moreefficient way [3]. In addition, this includes wide-area measure-ment systems (WAMS) put together with different controllabledevices [6], [7].The inherited nature of power systems provides nonlinear dy-

namics. Many well-established analysis and design techniquesexist for linear time-invariant (LTI) systems such as root-locus,Bode plot, Nyquist criterion, state feedback, and pole placement[8]. In general, control systems may not be LTI systems due to,for example, nonlinear dynamics. These linear design and anal-ysis methods cannot necessarily be applied directly without spe-cial consideration.The method of Lyapunov functions plays a central role in

the study of control systems, more specifically nonlinear con-trol systems. Lyapunov introduced a criterion for the stabilityof nonlinear systems, a property where all trajectories of thesystem tend to the origin. This criterion involves the existenceof a certain function, now known as a Lyapunov function. Later,in the classical works of Massera, Barbashin, Krasovskii, andKurzweil, this sufficient condition for stability was also shownto be necessary under various sets of hypotheses. There is nogeneral way of finding a Lyapunov function for nonlinear sys-tems. Faced with specific systems, we have to use experience,intuition, and physical insights, such as system energy, to searchfor an appropriate Lyapunov function [9], [10].The objective of this paper is to develop a new centralized

control Lyapunov function to coordinate point-to-point HVDCsystems or MTDC systems, and aims to improve the rotor-anglestability margin. Much research has been published concerningthe energy function for power systems and single controllablecomponents, and the proposed control Lyapunov function coor-dinates HVDC systems. Related works are presented in the nextsection.

II. RELATED WORKS

Energy or Lyapunov functions for power systems were de-veloped in the late 1970s by Athay et al. [11], [12]. The useof energy functions or similar to estimate regions of attractionfor stable equilibrium in electric power systems has a long his-tory. The use of energy-based Lyapunov functions for powersystems is well developed in classical literature, such as [13]and the references therein. Energy or energy-like functions areoften used as Lyapunov function candidates. A state-of-the-artpaper by Fouad and Vittal reviews the transient energy method[14].In [15], an automatic voltage regulator (AVR) is included in

the Lyapunov stability of multimachine power systems.

0885-8977/$31.00 © 2013 IEEE

ERIKSSON: ON THE CENTRALIZED NONLINEAR CONTROL OF HVDC SYSTEMS 1157

In [16], Lyapunov control functions are presented for a singleHVDC link to improve transient stability. It is based on thestructure preserving model and uses the frequency deviation ofthe HVDC connecting buses amplified by gain as the input forthe active power modulation. One drawback with this is that thespeed deviation is zero at maximum angle deviation. Also, theHVDC link is modeled by a simple model which neglects thedynamics.In [10] and [13], transient stability analysis is performed via

the energy function method using internal node representationincluding one HVDC link. To find a rigorous energy function,transfer conductances are ignored. The dynamics of the HVDClink consist of one differential equation. Only one HVDC linkis included, and the speed difference of the closest generators isused as the input for the HVDC controller.In [13] and [17]–[20], the internal HVDC links dynamics

are incorporated in the transient energy function. The HVDCdamping controller is included in the transient energy function[17], [20].

III. CONTRIBUTIONS

This paper extends the theory in [13] by developing a newcentralized control Lyapunov function for several point-to-pointHVDC or MTDC systems. This nonlinear centralized controlstrategy coordinates the HVDC systems and improves the rotor-angle stability margin. The proposed control Lyapunov func-tion does not only rely on small changes around an equilibriumpoint, instead it is valid for large disturbances. Remote signalsare used to sense the state in the system and the controller de-rives the active power setpoints change of the HVDC systemsas the control signal. The developed control Lyapunov functionmakes the time-derivative negative along the trajectories to theorigin. The control strategy can be applied to single and coordi-nation of several point-to-point HVDC links or multi-terminalHVDC links in multi-machine systems.To the best knowledge of the author, no publications present

controllers which coordinate several HVDC links based controlLyapunov function method.

IV. REMOTE SIGNALS

Remote signals utilized by the wide-area monitoring and con-trol (WAMC) system applications increase the observability ofrotor-angle stability compared with local measurements [21].The WAMC comes along with how to effectively utilize theWAMS for power state estimation and stability enhancement bypower system stabilizer (PSS), flexible ac transmission system(FACTS) devices and HVDC systems [21]–[24]. The feedbacksignals should have high observability to identify stability is-sues enabling appropriate control actions.In [25], advantages are shown by using remote measure-

ments as feedback signals compared with using local signalsfor stability enhancement. For optimal placement of phasormeasurement units (PMUs) one needs to find the optimallocations by looking at the observability matrix, this matrixcan be constructed in different ways. Observability factorsbased on residues have the drawback of scaling problem when

different measurement signals are involved [26]. Instead ob-servability factors based on geometrical factors are more robustand accurate [27]. The higher observability is observed usingrotor-angle speed of the generators. Furthermore, the control-lability is the measure of the impact seen from the input to theoutput. The controllability is inherited by the location of theinstalled controllable device in the power system. The impactof an input signal to the system’s states is described later on.

V. PROBLEM DESCRIPTION USING CONTROLLYAPUNOV THEORY

In control theory, a control Lyapunov function is ageneralization of the notion of Lyapunov function used instability analysis [28]. The system energy is replaced bythis scalar function . The ordinary Lyapunov function isused to test whether a dynamical system is stable(more restrictively, asymptotically stable). That is, whether thesystem starting in a state in some domain will remain in ,or for asymptotic stability will eventually return to . Thecontrol Lyapunov function is used to test whether a system isfeedback stabilizable, that is whether for any state there existsa control such that the nonlinear system

can be brought to the zero state by applying the control.The time derivative of the Lyapunov function is given by

(1)

In a stable power systems, the time derivative of the uncon-trolled system is

(2)

For the controlled system, the time derivative is given by

(3)

To make a control Lyapunov function one needs to find the con-trol vector such that is negative definite (or at least neg-ative semi-definite).

VI. TRANSIENT STABILITY ASSESSMENT

Let be the stable e.p. (equilibrium point) and be theunstable e.p. for is the stability boundaryof the stable e.p. and be the stability manifold of theunstable e.p. Then

(4)

The point where the unstable trajectory crosses the stabilityboundary (i.e., ) is the exit point of this trajectory. Ifthe point is close to then is the true critical energy[14].

assumes its minimum on the stable manifold, implyingthe unstable trajectory passing through the constant energy sur-face

(5)


Fig. 1. Internal control system in the HVDC link.

To find whether a contingency is stable one calculates thevalue of when the disturbance is terminated to thecritical value .The assessment is made by computing the energy margin

given by

(6)

and positive value implies the system is stable [14].Using the system trajectory along the contingency one can

derive the improvement of using the proposed controlLyapunov function. To securely operate the system the systemoperators should keep an adequate margin .

VII. POWER SYSTEM MODELING

This section follows in general the internal node descriptionas in [10]. In [10] the effect of a single HVDC link on eachgenerator is derived, this is extended to include several HVDCsystems.

A. AC Power System Modeling

A power system can be described by a set of differential andalgebraic equations.

(7)

(8)

where contains the state variables and contains the algebraicvariables.Using the internal node representation the power system

modeling can be written as only a set of first-order differentialequations [10]. The dynamics of generator is described by

(9)

(10)

where is the mechanical power, is the moment of in-ertia, is the internal voltage, and and are the rotorangle and speed of the generator , respectively. is thebus admittance matrix, , and is the damping.

is the impact of the HVDC links on generator [10],[29], described in next subsection.

B. HVDC Model

The control method to be developed can be used for both con-ventional-type line commutated converter (LCC) and voltagesource converter (VSC). One may control several point-to-pointHVDC links and MTDC systems.1) LCC HVDC: For the LCC HVDC, the current through the

HVDC link is controllable, and the LCC HVDC link is modeledas follows:

(11)

where

DC current through the HVDC link;

current set-point through the HVDC link;

current change time constant.

For each HVDC link, the setpoint signal is placed in the inputvector which is the actual control signal.Measuring the bus voltage, one can instead control the active

power, similar to VSC. The power through an LCC HVDC linkis controlled by controlling the firing angle . An overview ofthe internal control system is displayed in Fig. 1. In the figure,the control Lyapunov function (CLF) and he power oscillationdamping (POD ) controller also could be installed.2) VSC HVDC: For the VSC, one may control the active and

reactive powers independently of each other. Transient stabilityis mainly affected by active power modulation. Reactive powercontrol focus on locally controlling the connected bus voltage.One may instead model the VSC HVDC link, similar to LCC

HVDC, as follows:

(12)

Tomodel the VSCHVDC, one basically sets the active powersetpoints as the control variable following the same procedureas described in the next section.

C. Internal Node Description With HVDC

The effect of the HVDC link given as canbe expressed as [10] and [29]

(13)


(14)

(15)

where is element in B, derived for all HVDC links andgenerators [10], [29]. Including the HVDC system dynamics,the following is achieved:

(16)

(17)

(18)

where (18) is a vector containing the first-order differentialequations of the HVDC links. and contain the actualand setpoints values, respectively, and is the vector oftime constants.The system is now described by first-order differential

equations.

VIII. ENERGY FUNCTION

The system energy is treated mathematically by thescalar function . To formulate the energy function oneneeds to use the approach of center of inertia. It is given as

The energy function for the power system without HVDC isgiven by [10]

(19)

(20)

Including the dynamics of the HVDC links, the proposed en-ergy function is given by

(21)

where is the number of point-to-point HVDC links or MTDCsystem. An MTDC system is said to have controllable links if

it has converters, one of the converters acts as the slack.The time derivative of the energy function along the trajectoriesof system (21) is given by

(22)

The following equation is achieved:

(23)To make (23) a Lyapunov candidate, it needs to satisfy theasymptotic stability condition. For this function, the timederivative needs to be negative along the trajectories. If theHVDC links are not controlled, (23) is identically zero. Formu-lating a control Lyapunov function that implies a negative partof the first term in (23) is a control Lyapunov function. Thisfunction is proposed in next section.

IX. CONTROL STRATEGY

To make (23) negative definite, the new control Lyapunovfunction can now be formulated as

...

... ...

(24)

This implies the time derivative of the energy function to benegative semi-definite along the trajectories. It is not negativedefinite because for , where and arepositive matrices.The time derivative is given by

(25)

Naturally, to decrease the second term in (25), needs tobe different from zero.To improve transient stability it is reasonable to make the

input set-points dependent on angle deviation from steady-state.Using only speed deviation would have less impact. In a stablefirst swing at the largest angle separation, the speed deviationis zero, using only speed deviation would then not apply any


Fig. 2. Overview of the CLF controller.

power change of the HVDC links. By including the angle devi-ation power change is applied along the first swing until it goesback. Once the first swing is over one may only use speed devia-tion as input to dampen the remaining oscillations. An overviewof the Lyapunov control is depicted in Fig. 2 where the sign of

decides by the sign of .

X. NONLINEAR SIMULATIONS

Simulations are performed on the IEEE 10-machine 39-bussystem, data for this system are given in [10], [11]. The 39-bussystem is well-known and represents a greatly reduced modelof the power system in New England. It has been used by nu-merous researchers to study both static and dynamic problemsin power systems. The system has 10 generators, 19 loads, 36transmission lines, and 12 transformers. Generator 1 representsthe aggregation of a large number of generators. In addition,three HVDC systems are installed, and the system is shown inFig. 3. The power change during the contingencies of the HVDClinks is limited to 200 MW.The new control Lyapunov function is compared to no control

and lead-lag control using local inputs, frequency deviation ofthe HVDC connecting buses.

A. Contingency 1

A solid three-phase-to-ground fault occurs at the middle ofthe line between Buses 16 and 17. The fault is cleared by dis-connection of the faulted line after 150 ms. The result is shownin Fig. 4 for Generator 5, which is representative for the system’sbehavior. The result clearly shows the effectiveness of the de-veloped controller.Especially, large reduction of the transient angle separation is

seen and one can conclude that the transient stability is highlyimproved. Increased damping can also be seen which is also animportant aspect. Nevertheless, the damping is also improvedusing lead-lag power modulation control. The main differenceis the reduction of the transient which hardly is affected bylead-lag control. Fig. 5 plots the normalized energy functioncomparing different controls. The effectiveness can clearly beseen of the proposed controller since it is in the lower peak valueand it decays much faster.

Fig. 3. IEEE 10-machine 39-bus system including three HVDC systems.

Fig. 4. Contingency 1—rotor angle Generator 5.

B. Contingency 2

In this contingency, a solid three-phase-to-ground fault oc-curs at the middle of the line between Buses 2 and 25, and thefault is clear after 130 ms by disconnecting the faulted line.Fig. 6 shows the angle versus time for Generator 5. The rotor-angle stability is significantly improved for this contingency too.

C. Contingency 3

A similar fault occurs as in Contingency 2; however, the faultis cleared after 130 ms without disconnecting the line. In Fig. 7,the phase plane for Generators 9 and 7 is plotted. Clearly, thetransient stability is improved and the damping is increased.


Fig. 5. Normalized energy function.

Fig. 6. Contingency 2—rotor-angle Generator 5.

The figure shows the phase portrait which gives an overviewof speed deviation versus angle.

D. Contingency 4

For this contingency, a solid three-phase-to-ground fault oc-curs at the middle of the line between Buses 16 and 21; further-more, the fault is clear after 200 ms. In Fig. 8, the result applyingthis fault is shown, comparing Lyapunov control, lead-lag con-trol, and no power modulation. The coordinating controller alsoimproves the stability margin for this contingency.

E. Critical Clearing Time

The stability margin indicates the margin to theminimum boundary of instability. To visualize the transientstability improvement of the proposed controller the criticalclearing time is found through simulations for the appliedcontingencies. The critical clearing times for each contingency,obtained from nonlinear simulations, are shown in Table I. Itclearly shows improved results and improved transient stability.

F. Loadability Surface

Instead of looking at the stability improvement, loadabilitylimits are studied. We look at how much we can increase thepower transfers but keep the rotor-angle stability margin as in

Fig. 7. Phase plane for Contingency 4.

Fig. 8. Fault III—rotor angle Generator 5.

lead-lag control. The loadability surface is made up ofall loadability limit points in parameter space. We let

(26)

where is the load at the present operating point. We increasein one direction in load space until we reach , which


TABLE ICRITICAL CLEARING TIME

TABLE IIINCREASED POWER TRANSFER

TABLE IIICRITICAL CLEARING TIME

is the margin in the uncontrolled case. Then, we apply the de-veloped controller in nonlinear simulations, increasing the loaduntil reaching the new . Table II shows the amount ofpower we may increase in direction . This, inturn, means that we can transfer more power between differentregions in the system. By keeping a certain stability margin(e.g., a certain value of ), we may rate the economicimpact of the proposed control Lyapunov function. The direc-tion in load space has an impact of .

G. Signal Latency

Signal processing and remote measurements contribute tosignal latency which should be kept to a minimum. Largesignal latency has an impact on the control action duringtransients. In power oscillation damping, one may apply properphase compensation to compensate for reasonably small-signallatency. Signal latency has a minor impact on power oscillationdamping if it is compensated correctly [30].In this case, we apply signal latency of 50ms to see the impact

on the transient behavior; an overview of the result is shownin Table III. It can be seen that the controller performance isslightly reduced compared to no signal latency; however, thecontroller performance is promising.

XI. CONCLUSION

This paper derives a new control Lyapunov function to coor-dinate the power modulation of MTDC or several point-to-pointHVDC systems in order to increase the domain of attraction.The proposed centralized controller uses wide-area measure-ments to derive the coordinated action of the HVDC systems.The proposed controller decreases the postdisturbance energywhich, in turn, improves the rotor-angle stability margin.Nonlinear simulations are run in the IEEE 10-machine 39-bus

system; the centralized control Lyapunov function improves the

rotor-angle stability for different contingencies and loadings.The influence of signal latency is also demonstrated and is han-dled well by the controller.

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Robert Eriksson received the M.Sc. and Ph.D. de-grees in electrical engineering from the KTH RoyalInstitute of Technology, Stockholm, Sweden, in 2005and 2011, respectively.Currently, he is a Postdoctoral Researcher in the

Division of Electric Power Systems, KTH RoyalInstitute of Technology. His research interests in-clude power system dynamics and stability, HVDCsystems, dc grids, and automatic control.

on the centralized nonlinear control of hvdc systems using lyapunov theory

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