Excitation control system for use with synchronous generators

J.Machowski J.W. Bialek S.Robak J.R.Bumby

Indexing terms: Synchronous generators, Excitation control system, Lyapunov S method, Nonlinear system equations

Abstract: An optimal excitation control strategy for a synchronous generator is derived using Lyapunov’s direci: method and the nonlinear system equations. The control strategy requires neither phase compensation nor wash-out circuits characteristic of standard power system stabilisers (PSS). By using a nonlinear system model, the control strategy is optimal over a wide range of rotor angle and swing frequency changes. The excitation control system required to implement the control strategy is hierarchical and has a different structure to the traditional excitation control system with PSS. In the proposed structure a primary controller damps quickly any power swings using the synchronous EMF as the fesdback signal. A secondary controller maintains constant generator voltage by adjusting the reference value of the synchronous EMF fed to the primary controller to that required by the actual operating conditions. The proposed excitation control system has been tested on a single generator- infinite busbar system with the simulation results showing excellent (damping of power swings over a wide range of operating conditions whilst retaining good voli age control.

List of symbols

A bar on top of a symbol denotes a phasor or a com- plex number (e.g. V , 9 Bold face denotes a matrix or a vector (e.g. U) A ‘hat’ on top qf a symbol denotes the final equilib- rium value (e.g. S) E = synchronous internal EMF (i.e. voltage behind

Ed = d-axis component of the synchronous internal the steady-state reactances x d and Xq)

EMF 0 IEE, 1998 IEE Proceedings online no. 19982182 Paper first received 17th Diember 1997 and in revised form 1st June 1998 J. Machowsla and S. Robak are with the Warsaw University of Technol- ogy, Instytut Elektroenergetki, Koszykowa 75, 00-662 Warsaw, Poland J.W. Bialek and J.R. Bumby are With the University of Durham, School of Engineering, Science Site, South Road, Durham DHl 3LE, UK

= q-axis component of the synchronous internal EMF proportional to the field winding self-flux linkages (i.e. propor- tional to the field current itsel0

E4

Ef = excitation voltage Ed, E\

Z, Zd, Zq

K A4 = inertia coefficient pe, PFn = electromagnetic air-gap power and

mechanical power supplied by the prime mover

= open-circuit d- and q-axis transient time constants

= d- and q-axis components of the transient

= armature current and its d- and q-axis

= gain of the controller

internal EMF

components

Tdo, T,,

V = Lyapunov function v, v, = voltage at the generator terminals and

the infinite busbar voltage x d , x d ,

Xq, X q = d- and q-axis synchronous and transient reactance of the generator

xd, x’d, xg, XIq = total d- and q-axis synchronous and tran-

sient reactance between (and including) the generator and the infinite busbar

X,, X,, Xs = reactance of the transformer, transmis- sion line and the system, respectively

U, o s = rotor speed and the synchronous speed Am - o - U$ = rotor speed deviation 6 = rotor angle with respect to thtsynchro-

-

nous reference axis defined by V,

1 Introduction

Synchronous generators are used almost exclusively in power systems as a source of electrical energy. The gen- erator is supplied with real power from a prime mover, usually a turbine, whilst the excitation current is pro- vided by the excitation system shown schematically in Fig. 1. The excitation voltage Er is supplied from the exciter and is controlled by the automatic voltage regu- lator (AVR). Its aim is to keep the terminal voltage V equal to the reference value V r Q ~

Although the AVR is very effective during normal steady-state operation, following a disturbance, the generator is in the transient state and the AVR may

531 IEE Proc.-Gener. Transm. Distfib., Vol. 14S, No. 5, September 1998

have a negative influence on the damping of power swings [l]. As power swings cause the terminal voltage to oscillate, the reaction of the AVR is to force field current changes in the generator which, under certain conditions, may oppose the rotor damping currents induced by the rotor speed deviation Aco. This so-called negative damping may be eliminated by introducing a supplementary control loop, known as the power sys- tem stabiliser (PSS), also shown in Fig. 1. The task of the PSS is to add an additional signal V,,, into the control loop, which compensates for the voltage oscil- lations and provides a damping component that is in phase with Aco.

V Fig. 1 Functional diagram of excitation control

Considerable research effort has been devoted to the problem of PSS design. More detailed information and additional references can be found in [1-31. Generally, the properties of a particular PSS depend on the choice of the input quantities 4, with the most commonly used quantities being: speed deviation Aco, generator real power P, frequency deviation Af, the transient EMF E', and the generator current I. As each of these signals has its advantages and disadvantages, the PSS is often designed to operate on a number (usually two) of these input signals [1, 21. Fig. 2 shows the block diagram of a PSS, which uses real power as the input signal.

I VPSS" I

washout phase compensation

Fig.2 Power system stabiliser with realpowev as input signal

To produce the necessary control signal, the PSS usually has a washout block and one or more phase compensation blocks. The washout block is a high-pass filter with a tiine constant high enough to allow signals associated with the speed oscillations to pass through unchanged The phase compensation blocks provide the appropriate lead-lag characteristic to compensate for the phase shift between the exciter output and the generator air-gap torque As the PSS design is usually based on the linearised system model, the phase com- pensation blocks typically provide efficient compensa- tion only for a narrow range of swing frequeiicies Thus the actual settings of the PSS parameters should depend on tlie generator load, impedance of the trans- mission network and the voltage characteristics of the local loads. As these parameters may vary quite consid- erably during system operation, usually a compromise has to be made Moreover, interactions between gener- ators equipped with PSS may cause a particular PSS to damp oscillations in one area of the system but excite oscillations in another area or excite interarea oscilla- tions

Many research centres continue their efforts towards developing improved PSS designs Among the approaches used, it is worth mentioning attempts to use adaptive systems [4] or artificial intelligence tools 15, 61 538

In this paper a novel approach to the design of the control system is proposed, based on Lyapunov's direct method. Traditionally, Lyapunov's direct method has been used in power system analysis to evaluate stability margins and in real-time dynamic security assessment [7, 81. These applications require accurate system mod- elling and careful choice of Lyapunov function to achieve accurate results. Recent work has shown how Lyapunov's direct method can also be used very effec- tively for deriving control strategies for FACTS devices using a nonlinear model of the power system [1, 9, lo]. In such applications Lyapunov's method is used to point the way to the required control situation, and simpler Lyapunov functions can be tolerated than those necessary for assessing stability margins. A simi- lar approach is followed in this paper where Lyapun- ov's direct method is used to derive a control strategy for the excitation control of a synchronous generator.

2 Design method

2. I Design approach using L yapunov's direct method Lyapunov's direct method is concerned with assessing the stability of a dynamic system described by a set of nonlinear equations of the form x = F(x). The point 2 is the equilibrium point if it satisfies the equation F(X) = 0. Lyapunov's direct method is based on finding a suitable scalar function V(x) defined in the state-space of the dynamic system that is positive definite and has a stationary minimum value at the equilibrium point 2 (i.e. for any Ax # 0 it holds that V(2 + Ax) > U ( x ) ) . The point ri is stable if the derivative V = dV/dt is neg- ative semidefinite along the trajectory x(t) of equation x = F(x) (i.e. when 5 0). The point x is asymptoti- cally stable if the derivative i ) is negative definite along the trajectory x(t) (i.e. when i.' < 0).

/ /

Fig. 3 tem

Illustration to direct Lyapunov method for two-dimensional sys-

A negative value of tlie derivative V means that func- tion V(x) decreases with time tending towards its mini- mum value As the minimum value of V(x) is at the equilibrium point, the trajectory ~ ( t ) tends towards the equilibrium point x This is illustrated in Fig 3 where point xo = x(t = 0') denotes a nonzero initial condition lying beyond the equilibrium point ri It is important to note that the higher negative value of V , the faster the trajectory x(t) tends towards the equilibrium point x

IEE Proc -Gener Transm Distrib Vol I45 No 5 Septembei 1558

Consequently, a given control strategy is optimal in the Lyapunov sense if it maximises the negative value of P at each instant of time.

With these observations it is now possible to define a design approach thal comprises three stages: (i) find an appropriate Lyapunov function V(x) for the system that is an explicit function of the control varia- bles, (ii) select a control structure that maximises the nega- tive value of V at all points along the trajectory, (iii) select such loca ily available signals which can be used to execute the chosen control structure. This last point is important as quite often in PSS design a control structure is proposed which relies on measuring the rotor angle 6 in real time. However, measuring the rotor angle with respect to some syn- chronously rotating reference is not easy in a multima- chine system.

I I

I I I

I

2.2 Lyapunov function for generator-infinite busbar system Figs. 4 and 5 show a generator-infinite busbar system and the corresponding phasor diagram. For simplicity, only the series reactance of all the modelled elements is considered. The generator is assumed to have both steady-state and transient saliency (i.e. Xq # Xd and A' * X',) and is represented by the fourth-order modef detailed in the Appendices, Section 8.1 [l]. The trans- former reactance XT, the line reactance X , and the sys- tem equivalent reactance Xs have been added together to form the equivalent transmission reactance X = X , + X , + Xs so that the total d-axis reactances are xd = X + Xd and x2 = X i- A', while the total q-axis reac- tances are xq = X + Xq and x i = X + Yq.

G T I I system

FdXd

Fig. 4 sient state

Schematic diagram of generator-infinite busbar system in trun-

Ed I Ed ld Fig. 5 state

Phasor diagram oj' generator-in3nite busbar system in tvunsient

The Lyapunov function used in this paper comprises four terms:

V = v,, + v, + VE:, + VE:, (1)

IEE Proc-Gener. Transm. Distvib., Vol. 145, No. 5, September 1998

where V, is a kinetic energy term, Vp is a potential energy term, and VEk and VEb are two additional terms to account for flux decrement effects on the d and q axes. All four terms are defined in the Appendixes, Sec- tion 8.2, eqns. 28, 29, 30 and 31.

Section 8.2, eqn. 45, shows that the derivative of the Lyapunov function given by eqn. 1 can be expressed as:

( 2 ) where ej = Ef/Tdo and /3, y are the parameters defined in Section 8.1, eqn. 22.

Lyapunov's stability theorem states that the post fault system is stable, provided that the time derivative of the Lyapunov function along the system trajectory is negative semidefinite (i.e. V 5 0). Assuming constant excitation voltage (Ef = constant, i.e. AVR switched off), ef = and the first component of eqn. 2 vanishes giving :

Thus the derivative is negative semidefinite and the system is stable. Whether or not i.' I 0 and the system is stable when the excitation voltage Efis not constant (i.e. when the reaction of the AVR to the disturbance is taken into account) depends on the chosen control strategy. Having obtained a suitable Lyapunov func- tion in eqn. 1, the next stage of the design process is to devise the necessary control strategy that not only ensures that eqn. 2 is negative definite but also maxim- ises the negative value of V at all instants.

3 Control strategy

The control strategy is based on the observation illus- trated in Fig. 3 that the higher the negative value of the derivative P, the faster the system trajectory tends towards the equilibrium point.

3.1 Maximisation of (-I)) by controlling Ef Eqn. 2 shows that the derivative V depends on what is happening in both the direct and quadrature axes. This may be expressed by rearranging eqn. 2 as:

ii = V, + V d

Section 8.2, eqns. 49 and 53, show that: (4)

V d is always negative semidefinite, while Vq may be positive or negative depending on the excitation control strategy chosen. Note that V d does not depend on the excitation control as there is no EMF induced in the d axis due to the rotor excitation current. In other words, V d corresponds to the natural damping produced by currents induced in the rotor 4 axis.

539

Now, let us analyse V,. The first component of eqn. 6 is always negative and, similarly as in eqn. 5, corresponds to the natural damping produced by cur- rents induced in the rotor d axis. The second compo- nent of eqn. 6 is proportional to the change in both the excitation voltage and the synchronous EMF. The con- trol strategy should therefore) aim to make this compo- nent as highly negative as possible at each instant of time during the transient state. This can be achieved by applying the following excitation control strategy:

(E: - Ef) = - K E, - ( where K >> 1 is the controller gain. With this control strategy eqn. 6 becomes:

and the Lyapunov function V decreases at a rate equal to the sum of eqns. 5 and 8, that is:

(9) This derivative is independent of the network reactance X as x, = X, + X and X > = Xd + X so that x, - xb = Xd - Yd. Simlarly, xg - xrq = Xq - X g . This means that the control strategy defined by eqn. 7 assures the same positive damping regardless of the values of the trans- mission network parameters. This is an important fea- ture of the proposed excitation control strategy.

3.2 Structure of the control system The control strategy defined by eqn. 7 can be rewritten as:

E f = Ef + K ( E , - E,) Fig. 6 shows the block diagram of the controller required to execute this control strategy The controller ensures a fast return of the system trajectory to the equilibrium point, which is equivalent to strong damp- ing of the power swngs It is important to note that the controller contains neither a phase compensation block nor a wash-out block typical of traditional PSS systems. This has been achieved by using Eq as the feedback signal rather than other quantities (such as frequency, real power, power angle etc ) used tradition- ally in PSS

I 1

I Eq Fig. 6 Block diagram ofproposed stabilising controlley

It is now necessary to link the stabilising controller, shown in Fig. 6, with the main, steady-state, voltage controller (AVR). Within Gyapunoy’s stability theorem the equilibrium values of Ef and E, correspond to the

540

postfault equilibrium point and, importantly, Eg defines the postfault steady-state synchronous EMF. As this EMF depends on the required terminal voltage, it is natural to assume that the generatof: terminal volt- age is regulated by a proper setting of Eq = Eg ref This can be achieved if the AVR acts with a classical voltage feedback control loop. The resulting excitation control system is shown in Fig. 7. Because Ef is theApostf+t steady-state value of Ef, it is assumed that Ef = Eg = E g ref

I “ref

limiter

I

Fig. 7 Black diagram of proposed excitation controller

In the control system shown in Fig. 7 the output sig- nal is the excitation voltage Ef It is assumed that the exciter contains no significant time lag (as would case for the static exciter). If the exciter has a cant time constant (e.g. it contains cascaded DC gener- ators), an additional control block that coinpensates for this must be connected to the regulator output. This is necessary as the control strategy defined by eqn. 10 requires changes in the excitation voltage Ef (proportional to the changes in the synchronous EMF E,) to be made without any delay.

The feedback signal used in the control strategy giveii by eqn. 10 is Eg, the quadrature component of the synchronous EMF. By definition, E, is propor- tional to the field current is so that field current can be used as a feedback signal instead. However, it can be shown [I] that a disturbance causes also a fundamental frequency component to appear in 9. Thus, to obtain a signal proportional to Eg, it is necessary to filter out the fast 50Hz (or 60Hz) component from the field cur- rent. This can be done by using a low-pass filter (e.g. of the Bessel type).

3.3 Discon tin uo us excitation control The control strategy defined by eqn. 10 has been derived using Lyapunov’s direct method. However, Lyapunov’s stability theory is valid for autonomous system only (i.e. when the system is time-invariant and function F(x) is not explicit with time [7, 81). Conse- quently the system response is due to some nonzero ‘initial condition’ xo # P and no further disturbances are considered. As the system trajectory tends from the initial condition towards the final equilibrium point, the control strategy defined by eqn. 10 maximises the speed with which function V decreases so that the sys- tem returns to the equilibrium point as quickly as pos- sible.

In the case of a power system, the nonzero initial condition may be caused by a short-circuit, tripping of a network element etc. It is important to appreciate that the derived control strategy is optimal after the disturbance has taken place but may not be at all opti- mal during the disturbance itself. Consider, for exam- ple, a short-circuit. A short-circuit causes the generator synchronous EMF to increase suddenly [I], and the controller shown in Fig. 7 will try to reduce the excita-

IEE Proc.-Cener. Tj.ansn?. Distvib., Vol. 145, No. 5, Septenzbeu 1998

tion voltage. Unfortunately, what is really required during this period is an increase in the excitation volt- age [l]. Some traditional PSS systems react in a similar way and, to prevent this, traditional AVR + PSS sys- tems are often equipped with an additional discontinu- ous excitation control circuit [2]. This circuit bypasses the PSS during the ,short-circuit forcing an increase in the excitation voltage up to its ceiling value. A similar solution can be used in the proposed controller. Fig. 7 shows an additional relay SHC, which disconnects the E4 signal during a short-circuit, thereby causing a large positive regulation error AE4 = E4. This error forces the excitation voltage to reach its ceiling value quickly. The relay SHC closes again when the speed deviation Aw reverses its sign (i.e. during the back swing).

4 Hierarchical calntrol

The structure of the proposed control system shown in Fig. 7 is different to the traditional structure of AVR + PSS system shown in Fig. 1. In the traditional solution, the AVR is the main controller, while the PSS is a sup- plementary control loop. The structure of the proposed controller is hierarchical and of the master-slave type. The slave controller (primary level) is the stabiliser based on the derived control strategy. Its task is to damp quickly any power swings, ensuring that the sys- tem reaches the steady-state as quickly as possible. The master controller (secondary level) is the AVR. Its task is to set the reference value Eq ref for the PSS. The refer- ence voltage Vref for the AVR can be set manually or automatically from a tertiary level. The task of this ter- tiary level control is to maintain the optimal system voltage profile as defined by the economic dispatch of reactive power. The overall structure of the proposed hierarchical AVR + PSS system, including the tertiary control level, is shown in Fig. 8.

tertiary level

manual

Hg.8 tem

Proposed excitation controller shown as hierarchical control sys-

This hierarchical excitation control scheme is very similar to the hierarchical structure used in automatic generation control (AGC) [I, 21. In hierarchical AGC the primary controller is the turbine governor. Its task is to control the turbine speed and the mechanical power P, by acting on the control valves. The refer- ence power is set by the secondary controller, whose task is to control frequency and tie-line power flows. The tertiary AGC controller controls the setting of the primary controller to maintain the economic dispatch of generation. Table 1 compares the hierarchical struc- tures of the AGC and the proposed overall generator control system.

Comparison between the generator and the turbine control is quite revealing. In both cases the primary control is concerned with maintaining the stability of the turbine-generator unit, the secondary control is concerned with how the unit impacts the network, while the tertiary control is concerned with the eco- nomic dispatch.

The aim of the primary control to maintain the sta- bility of the generator and the turbine. For the genera- tor, the proposed stabiliser controls Eq to damp quickly power swings which threaten the generator stability. For the turbine, the turbine governor controls the mechanical power P, as a response to the frequency changes, with the droop characteristic of the turbine governor ensuring stable operation [l]. In both cases the controlled variable (E, for the generator and P,,, for the turbine) is controlled indirectly by regulating the control variables, the excitation voltage Ef and the valve position c v, respectively.

The secondary level is concerned with how the tur- bine-generator unit impacts the network. The aim of control is to keep constant the network quantities (i.e. the generator terminal voltage V and the frequency J). These network quantities are controlled indirectly by setting the reference values for the primary controlled quantities: E, ref for the generator and P, ref for the tur- bine. The tertiary level is concerned with the economic dispatch. The economic dispatch of reactive power is achieved by setting the reference value at the secondary level (i.e. the voltage at the generator busbars Vyef). The economic dispatch of real power is achieved by setting the reference value at the primary level P,

5 Simulation results

Simulation results have been obtained for a generator- infinite busbar system using the data detailed in Appendix 10. When deriving the control strategy given

Tablo 1: Comparison of hierarchical control of generator and turbine

Generator Turbine Control level

Goal of control

Primary maintain stability of the unit synchronous EMF E, mechanical power P, by controlling:

by regulating excitation voltage €f valve position cv

the network quantity:

by regulating the reference

Secondary maintain reference value of terminal voltage V frequency f

the primary level E4 ref the primary level P, ref value of:

Tertiary enforce economic dispatch o f reactive power real power

by regulating the reference value of:

the secondary level Vref the primary level P,,, ref

- IEE Proc.-Gener. Trans,. Dislrib., Vol. 14S, No. 5, September 1998 54 1

by eqn. 10, the fourth-order generator model was used with all the system resistances neglected. For the simu- lation, the sixth-order generator model has been used, thereby including subtransient effects [ 11. Generator and network resistances have also been included. Also included in the generator niodel is a thyristor-control- led static exciter, having both a buck and boost facility with a ceiling voltage of 28pu. As the time constant of the firing circuits is very short, this exciter is simply modelled by a constant gain and its excitation limits. The exciter is also assumed to be supplied from a com- pensated supply so that its excitation function is not impaired during the fault period. The block diagram of the proposed controller is shown in Fig. 7 with the gain of the primary level stabilising controller set at K = 15.

The secondary level AVR controller is responsible for setting the reference value Eq and in the simula- tions a PID-based secondary regulator has been used. The parameters of this controller have been selected to ensure an almost aperiodic settling of the generator voltage to a new steady-state value. As the design prin- ciples involved in selecting the parameters for this new control structure are very different to those used in tra- ditional AVR + PSS systems, the principles of optimis- ing the parameters of this secondary regulator will be the subject of a future paper.

I 0 2 4 6

t, s

c .

"'y "

0 2 4 6 t, s

Fig.9 _ _ _ _ constant Er

~ pi-oposed controller

Response of suniple system to nonzero initial rotor angle deviation ........ slandard PSS

To study the performance of the proposed controller, simulations have been performed that cover a wide range of possible generator loading conditions. Com- parisons have also been made with a conventional exciter equipped with the standard PSS shown in Fig. 2. The values of the PSS parameters used are given in Appendix 10. This PSS is tuned to give 'good' performance over a wide range of operating conditions,

842

0 8 t

0.6 I I I I 0 2 4 6

t, $4

l o t

-10 0 2 4 6

t, s

Fig. 10 tion

Response of sample system to nonzero iniiial rotor angle devia-

constant E

~ proposed colitloller

_ _ _ _ ........ standard h S

I5O t I A

"

I I I

0 2 4 6 t, s

1.5 r

I Y , I I

0 2 4 6 t, s

Fig. 11 constaiit E

~ proposed controller

Response of sample system to short-circuit in network ~~-~ ........ standard $SS

IEE Puoc.-Genev. Tuunsm. Distuib., Vol. 145, No. 5, September 1998

rather than being tightly tuned to one specific operat- ing condition. In all the tests performed, the proposed excitation control system produced fast damping of power swings and small oscillations in the terminal voltage following tlhe disturbance. For illustration, Figs. 9 and 10 show the effect of a nonzero initial rotor angle deviation, whilst Figs. 11 and 12 show the controller performance following a short-circuit in the

1.2 1

1 .o =! Q

5 > h

0.8

0.6 1 I

0 2 4 6 t, s

I

2 4 6 -10 I 1

t, s Fig. 12 _ _ _ _ constant E - _ _ _ ___. standard $sS ~ proposed controller

Response of sample system to short-circuit in network

network. The dashed lines correspond to the case when the excitation voltage was held constant (AVR switched off), the solid lines to the proposed controller and the dotted lines to the standard PSS.

For the initial nonzero rotor angle deviation, Figs. 9 and 10, the proposed controller brings the system into the steady-state after only one swing with almost no backswing. For the short-circuit in the network, Figs. 10 and 11, the situation is similar. During the short-cir- cuit, the discontinuous control (switch SHC in Fig. 6) forces the excitation voltage up to its ceiling. After fault clearing, the proposed controller takes over bring- ing the system very quickly to the steady-state, again in an almost aperiodic way. The backswing of power, angle and voltage is almost negligible. In both cases the postfault performance of the proposed controller is substantially better than the standard PSS. The worst performance aspect of the proposed controller is the initial recovery of the terminal voltage following the fault, Figs. 10 and 12, and, in this respect, the perform- ance of the conventional AVR with standard PSS is slightly better. However, this voltage recovery depends primarily on the control action taken during the fault itself. As explained in Section 3.3, the proposed con- troller is optimal only after the fault has been removed. What action the controller should take during the fault period is a different question, and the introduction of the relay SHC in Fig. 7 was a first step in improving this part of the control action.

Note that, although the standard PSS brings the real power to its steady-steady value relatively quickly, the regulation of the rotor angle and the voltage takes longer and there is still a small error after 6 seconds. This is due to using real power as the only input signal for the PSS. By comparison, the proposed controller brings all three signals quickly to their steady-state val- ues.

To further quantify the benefits of the proposed con- troller, simulations have been performed over a range of inductive and capacitive loading conditions. Table 2 shows the results of integrating the absolute value of error in power angle, real power and terminal voltage

Table 2: Simulation results for a short-circuit in the network -

Integral of deviation of: Prefault Type of loading of the generator

Transmission line A"' + PSS rotor angle real power voltage

system 6 P V

P = P, 50 km conventional Q = 0, proposed

100 km conventional

proposed

P = 0.7 P, 50 km conventional

Q = 0.70, proposed

IOOkm conventional

proposed

P = 0.7 P, 50 km conventional

0 = 0.45Q, proposed

100km conventional

proposed

P = O.6Pn 50 km conventional Q=-O.lQ, proposed

100km conventional

orooosed

0.434

0.240

0.538

0.349

0.310

0.150

0.293

0.184

0.344

0.157

0.330

0.183

0.605

0.206

0.444

0.205

0.573

0.315

0.564

0.222

0.328

0.21 1

0.308

0.193

0.336

0.193

0.3 19

0.173

0.306

0.149

0.295

0.136

0.122

0.108

0.126

0.162

0.113

0.079

0.095

0.082

0.114

0.083

0.102

0.089

0.150

0.095

0.125

0.101

IEE Proc.-Gener. Transm. Distiib., Vol. 145, No. 5, September 1998 543

over a 5 second period for two transmission line lengths of 50km and lOOkm (the base case). In all cases the lower value of integrated error indicates the better performance. In this Table the loading condition P = P,, Q = Q,, and 100 km long transmission line cor- respond to Figs. 11 and 12. In all cases, the proposed controller outperforms the conventional AVR + PSS except for the voltage error corresponding to Figs. 10 and 12. As explained above, it is expected that adjust- ments to the control during the fault period will improve this situation.

6 Conclusions

In this paper an optimal excitation control strategy has been derived Tor the nonlinear generator-infinite busbar system. By using the Lyapunov’s direct method in con- junction with an energy-type Lyapunov function, the optimal control strategy has been derived, which max- imises the speed with which Lyapunov function decreases, thus maximising energy dissipation in the system. The resulting proportional controller dispenses with the need for lead-lag correcting circuits or wash- out circuits characteristic of the standard PSS system. By using the nonlinear system model, the control strat- egy is optimal over a wide range of rotor angle and swing frequency changes. It also achieves the same pos- itive damping, regardless of the values of the transmis- sion network parameters. The resulting overall structure of the excitation control system is completely different to that of the standard AVR + PSS system. In the standard systems the main voltage controller is the AVR and the damping of power swings is achieved by a supplementary control loop (PSS). In this paper a master-slave structure for the excitation control system is proposed, where the damping of power swings is achieved by the fast slave controller controlling the generator synchronous EMF. The master voltage con- troller is much slower and its task IS to set the reference value of the synchronous EMF.

Simulation results on a single-machme-infinite-bus- bar system have shown that the proposed excitation control is very effective in damping power swings caused by a variety of disturbances in the generator- infinite busbar system.

7

1

2

3

4

5

6

7

8

9

References

MACHOWSKI, J., BIALEK, J W , and BUMBY, J R ‘Power system dynamics and stability’ (John Wiley, Chichester, 1997) KUNDUR, P ‘Power system stability and control’ (McGraw Hill, 1994) Task Force 07 of Advisory Group 01 of Study Committee 38 Analysis and control of power system oscillations’ CIGRE 1996 CHEN, G P , MALIK, O P , HOPE, G S , QIN, Y H , and XU, G Y ‘An adaptive power system stabiliser based on the self- optimising pole shifting control strategy’, IEEE Trans , 1993, EC- 8, (4), pp 639-645 EL-METWALLY, K A ~ and LIK, 0 P ‘Fuzzy logic power system stabiliser’, IEE Proc 7 Transm Dwtrib ~ 1995, 142, j3), pp 277-281 ABDEL-MAGID, Y L , BETTAYEB, M , and DAWOUD, M M ‘Simultaneous stabilisation of power systems using genetic algorithms’, IEE Proc Gener Tvansrn Dzstrib , 1997, 144, (l), pp 39-44 PAI, M A ‘Power system stability, Analysis by the direct method of Lyapunov’ (North Holland Publishing Company, Amsterdam, 1981) PAVELLA, M, and MURTHY, P G ‘Transient stability of power systems, theory and practice’ (John Wiley, 1994) MACHOWSKI, J , and NELLES, D ‘Power system transient stability enhancement by optimal control o f static VAR compen- sators’, Int J Elect Powev Energy Syst , 1992, 14, (5) , pp 411- 42 1

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11 AYLETT, P D ‘The energy integral function of transient stabil- ity limits of power system’, Pvoc IEE, 1958, 105, (2)

12 GLESS, G E ‘Direct method of Liapunov applied to transient power system stability’, IEEE Tvans , 1966, PAS-85, (2)

13 KAKIMOTO, N , OHSAWA, Y , and HAYASHI, M ‘Tran- sient stability analysis of multimachine power systems with field flux decays via Lyapunov’s direct method’, IEEE Tvans , 1980, PAS-99, (5), pp 1819-1827

8 Appendices

8. I System model The system model is that of the single-generator-infi- nite busbar shown in Fig. 4, with a fourth-order gener- ator model being used to formulate the Lyapunov function Full mathematical details of this model can be found in [l] and only those equations used in the evaluation of the Lyapunov function are quoted here. The model takes Into account transient effects in both d and q axes with the differential equations determining the generator behaviour being.

where

- = AW dd dt

= P, - P, d a w M- d t

and

P, = PE‘(6)

The co-ordinates of the equilibrium point are denoted by using a ‘hat’ on top of a variable so that:

Ai; = 0; Pe(8) = P,; Eq = E f ; E d = 0 (18) Dividing both sides of eqns. 13 and 14 by the corre- sponding time constants Tb, and TIqs and substituting Ed and Eq given by eqns. 15 and 16 yields:

where

544 IEE Proc-Gener. Transm. Distrib., Vol. 145, No. 5, September 1998

Eqns. 19 and 20 together with eqns. 11 and 12 form the nonlinear system state model where the state variables are (Aw, 6, Eq, Eh).

At the equilibrium point, both sides of eqns. 19 and 20 are equal to zero:

when adding eqns. 19 and 23, and eqns. 20 and 24, gives:

dE’ Q A= [ ‘ f - ; f ] - - d t x:,

These equations are equivalent to eqns. 19 and 20 but additional values corresponding to the equilibrium point have been introduced. This formulation will be useful when applying, Lyapunov’s stability theory.

8.2 Developmerrt of the Lyapunov function If flux decrement cffects are neglected, the system model is simplified to the classical second-order model in which the generator is modelled by the swing eqns. 11 and 12 and a constant transient EMF E’ behind the transient reactance X d . An energy-based Lyapunov function for such a 2,ystem was derived in [l 1, 121 and corresponds to the total system energy made up from the sum of the system kinetic and potential energy with respect to the equilibrium point. For the third-order model (i.e. when eqn. 14 is neglected), a Lyapunov function was derived in [13], where the additional com- ponent ad!ed to the total system energy is proportional to (E: - E‘ )2. In this paper the fourth-order model is used and t i e Lyapiinov function requires a further component proportional to (Eh - l?’d)2 giving the pro- posed Lyapunov function as:

V = ’Llk + Vp + VEh + VEL (27) where

t Ald v k = / M ( g ) d t = / Md(Aw) = - M ( ~ w ) ~ 1

2

6

respect to the equilibrium point ( t = 03, 8, Ah = 0, Pq, Ed). The first two components, Vk and V,, correspond to the kinetic and potential energy of the system at any point (Am, 6) caiculated with respect to the equilibrium point (Ah = 0, 6) with their sum being the well-known Lyapunov function for the classical system model [ l l , 121. The two additional components given by eqns. 30 and 31 take into account the changes in the tran- sient EMF in both the d and q axes.

To use Lyapunov’s stability theorem it is necessary to: (i) prove that function V defined by eqn. 27 is positive definite in the vicinity of the equilibrium point; (ii) derive a control strategy which makes the time derivative V negative semidefinite (V s 0) along the sys- tem trajectory. At the equilibrium point, the right-hand sides of eqns. 28, 29, 30 and 31 are all equal to zero. The components Vk, VEq and VEk are all positive definite as they are pro- portional to the state variables squared. To prove that the remaining component, V, givep by eqn. 29, has a minimum at the equilibrium p_oint 6 note that as dV,ld$ = -(Pm - 9.) and P, = Pe(6), dv,/d6 = 0 at 6 = 6. Whether this is a maximum or minimum depends on the sign of the second derivative:

which is equal to the generator synchronising power coefficient. The synchronising power coefficient must be positive below the maximum, pull-out, value of the power-angle characteristic, otherwise the steady-state stability conditions are not met. This prpves that V,. has a minimum value equal to zero at 6 = 6 and is positive definite around it. Consequently, the function defined by eqn. 27 is positive definite around the equilibrium point, thereby satisfying the first condition of the Lya- punov stability theorem.

The time derivative $’ is equal to the sum of the time derivatives of the components given by eqns. 28, 29, 30 and 31, each of which can be calculated using:

. dV aV dxl dV dx aV dxn dt 3x1 d t 8x2 d t ax, d t v = - = - -+- 2+ . . . +--

= [gradVIT x = [gradVIT F(x) (33)

The derivative of eqn. 28 is: * avk d a w d a w V k = --=Maw-

daw dt d t

= [ M T ] a w = + [P, - P,] aU (34)

where eqn. 12 has been used in the last transformation, For eqn. 29 the use of eqn. 33 yields:

eqn. 17 is used to obtain the partial derivatives:

(36) av, - = - [P, - P,] as

In these equations the values taken at any point along the trajectory ( t , 6, Am, Etq, Eh) are calculated with

IEE Proc-Gener. Transm. Dis!rib., Val 145, No. 5, September 1998 545

6

~ av?J = + 1 z d 6 = [sin 6 - sin 81 (38) 3E:, n

which substituted into eqn. 35 gives:

V p = - [P, -P ,]Aw-- cos6-cos6 -1 - d 2 x'd vs [

(39) Following a similar procedure for the functions given by eqns. 30 and 31 gives:

To further transform the components of eqn. 40, an auxiliary equation is derived by multiplying both sides of eqn. 25 by 1//3 dE',/dt to give:

As the second component in this equation is identical to the right-hand side of eqn. 40:

A similar procedure is used to transform eqn. 41 but now the auxiliary equation is obtained by multiplying both sides of eqn. 26 by l l y dEhldt to finally give:

Importantly, the derivative Vp expressed by eqn. 39 contains identical components as the derivatives expressed by eqns. 34, 43 and 44. This is due to there being an exchange of energy during transient state between the potential energy Vp, the kinetic energy V,, and the remaining components V,,, VEd. Summing all four components gives:

. . v = vi, + ii, + lj,; + ii,,

(45) and the system is stable provided that in eqn. 45 V I 0. This depends on the sign of the first term in eqn. 45.

Eqn. 45 can be further transformed by analysing sep-

9 = ii, + V d (46) where

2 1 dEh V d = -7

Substituting y from eqn. 22 and the derivative dE',Idt from eqn. 14 into eqn. 48 gives:

This term corresponds entirely to the natural damping produced by electromagnetic effects in the rotor quad- rature axis.

In contrast V,, eqn. 47, depends explicitly on the excitation control via the term [es - e;rl and implicitly via the derivative dE,ldt. Substituting values of ef and j3 from eqns. 21 and 22 into eqn. 47 gives:

At the equilibrium point dE',Idt = 0 and eqn. 13 can be rewritten as:

0 = Ef - E, (51) which, when subtracted from eqn. 13 gives:

dE' dt TAo> = ( E f - kf) - (Eq - f i q ) (52)

Substituting this equation for dE',ldt into eqn. 50 finally gives:

(53)

8.3 Simulation data All the reactances and resistances are in p.u. and the time constants in seconds. For per unit calculations S,,,,, = 426MVA was assumed System: Short-circuit MVA equal to 15000MVA at 220 kV giving the equivalent system impedance 2, = 0 0024 + J O 024 Transmission line: length 100km, zL = 0.0384 + ~0.285 Step-up transformer: S, = 426MVA, V, = 22/250kV, ZT = 0.0006 + j0.12 Generatov: S, = 426MVA, V, = 22kV, cos@, = 0.85, T, = 6.45 s, M = TmIcos = 0.0205pu R = 0.0016, Xd = 2.6, X ' d = 0.33, X)h = 0.235, T'dO = 9.2, T",O = 0.042 X , = 2.48, X', 0.53, ;rb = 0.29, T,o = 1.095, Tqo = 0.065 PSS: Tp = 0 .03~ , T,. = 5 ~ , TI = 0.05~, T2 = 7 ~ , Kpss = 120, Vp,ySmln = -1.75, VpSSmax = 1.75 Predisturbance generator loading was assumed to be: S = 426MVA, V = 1.05, cos@ = 0.85, all measured at the upper-voltage terminal of the transformer.

Short-circuit clearing time was assumed to be 150ms. The discontinuous relay SHC was assumed to open at the instant of fault and close after 180ms.

546 IEE Proc.-Gene,.. Transm. Distrib., Vol. 145, No. 5, September 1998