Issues on Nonlinear Control of Voltage Source FACTS Devices

Ziwen Yao, Senior Member, IEEE, Nicolas Léchevin, Member, IEEE.

Keywords - compensation, reactive power, STATCOM, global linearization, differential geometry, differential algebra, passivity.

Abstract - In this paper, we present three typical nonlinear control strategies for voltage source FACTS devices. The most basic device of this kind is STATCOM (STATatic COMpensators), which is powerful in providing voltage support and improving power system stability. The nonlinear control strategies discussed in the paper include the input-output linearization based on differential geometry and differential algebra as well as passivity approach.

We discuss and compare these control methods from the perspectives of their dynamic performance for local voltage support, impacts on overall power system and implementation.

1. Introduction

OLTAGE source converters are widely used as building blocks of FACTS devices, such as STATCOM, SSSC, UPFC, IPFC, etc. for reactive

power compensation and power flow control because of its obvious advantages over current source converter from the perspectives of economics and performance [1].

Especially, STATCOM (also called Advanced Reactive Power Compensator) is a typical example of this category [2-8], which is increasingly installed in power systems because of its superior characteristics in providing voltage support and improving power system stability.

In order to gain as much as possible the benefits from these devices for local voltage support and stability enhancement of the global systems, various control methods have been studied for these power components [9-17].

Reference [9] proposes a control scheme based on input-output linearization technique to improve transient performance of the compensator. Since the relative degree of the controlled system is one and the system is of third order, the controller based on this technique cannot affect the stability of its internal system. Therefore, the capacitor voltage is not controllable. Even though the internal system of the capacitor is stable, the damping is very weak.

Reference [10] proposed another linearization technique based on the differential algebra theory [18], which obtains

Manuscript received on February 18, 2005. Ziwen Yao is with the British Columbia Transmission Corporation, Suite 1100, Four Bentall Centre, 1055 Dunsmuir Street, Vancouver, BC, V7X 1V5, Canada. Phone: 604-293-5884; Fax: 604-473-2734, e-mail: Michael.Yao@ BCTC.com. N. Léchevin is with Defence Research and Development Canada - Valcartier, 2459 Pie-XI N., Val-Belair, Qc, Canada G3J 1X5.

the direct control on the DC voltage of the capacitor and the output reactive power from the STATCOM.

However, we have to solve a somehow complicated algebraic-differential equation to obtain the control law.

Reference [15] studied the passivity approach for STATCOM, which is aiming at achieving directly the desired invariant passivity property for the subsystem, thus to mitigate the stability of the overall power systems.

The objective of this work is to summarize and compare these three control strategies from the perspectives of their dynamic performance for local voltage support, impacts on overall power system and implementation.

2. Studied System

The studied system (Figure 1) consists of a utility power system and a self commutated inverter that is capable of compensating reactive power.

Self Commutated Inverter

Control System

References of reactive power

and vdc

vs

i

i and vs

VdcC

Utility power system

Figure 1: Functional Diagram of a STATCOM

The inverter essentially comprises six self-commuted thyristors, such as GTOs, IGBTs with anti-parallel diode, and a capacitor C with its intrinsic high resistance. Figure 2 shows the simplified circuit of the system. The control signal is the pulse width modulation (PWM) switching sequence. SPWM [8] is one of the typical PWM schemes in high power applications.

Vdc

VcA

VcB

VcC

1

2

3

4

5

6

N

Powernetwork

Lf

iabc

ic

VsABC

C

Rf

Rc

Figure 2: Proposed power circuit

V

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

WA5.3

0-7803-9354-6/05/$20.00 ©2005 IEEE 1317

The averaged model of the system is presented in (1). It can be seen that this system is nonlinear in the control portion. It is worth pointing out that other severe nonlinearities are not represented in the model, which include the significant time delay due to the low switching frequency in high power applications and harmonics generated by the inverter as well as the constraints on the input signals due to the PWM control schemes. These unmodelled nonlinear characteristics may affect largely the control performance in some circumstances, such as system disturbances.

⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

+

+⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−−

−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

q

d

qd

dcf

dcf

q

d

fdc

q

d

c

f

f

f

f

dc

q

d

iC

iC

vL

vL

v

v

Lvi

i

CR

L

RL

R

vi

i

ρρ

ω

ω

5.15.1

10

01

001001

1

100

0

0

D

D

D

(1)

Rewriting (1) in standard form of nonlinear system used for controller design:

)()()()()()()(

xhywuxgxfx

=++=

•

tttt (2)

and comparing (2) with (1), we obtain the following relationships : f(x) = Ax, [ ]Tqdt ρρ=)(u

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−−

−

=

CR

L

RL

R

c

f

f

f

f

100

0

0

ω

ω

A ⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=q

d

f

f

v

v

L

L

t

00

10

01

)(w

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

=

21

3

3

5.15.1

10

01

)(

xC

xC

xL

xL

f

f

xg

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

dc

q

d

v

i

i

t)(x .

3. Nonlinear Control Strategies

This paper is focused on the following decentralized nonlinear control strategies: 1) input-output linearization based on differential

geometry; 2) linearization based on differential algebra; 3) passivity-aimed control.

3.1 Input-output Linearization Based on Differential Geometry

Using the “Input-Output Linearization Technique” [9], we linearize the system (1) to (3.1)–(3.4).

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=•

•

•

•

),(

),(),(

3

2

1

dq

dqq

dqd

x

x

x

xρηρξρξ

x

xx

(3.1)

where,

f

ddfdqd L

xVxxx

)(),( 3

21ρ

ωσρξ−

+−−= (3.2)

f

qqfdqq L

xVxxx

)(),( 3

12ρ

ωσρξ−

++−= (3.3)

C

xxxx qd

cdq

)(5.1),( 21

3ρρ

σρη+

+−= (3.4)

f

ff L

R=σ ,

CRcc

1=σ

Based on the linearized system (3.1)–(3.4), we apply the linear state feedback control to design the controller [ξd, ξq]T

and obtain the desired damping rate subject to input constraints for the compensator. According to (3.2) and (3.3), the control law [ρd, ρq]T can be easily established as given by (4.1) and (4.2).

])()([121

*11

3dfffdd VLxxLxx

x+−−−= ωσσρ (4.1)

])()([112

*22

3qfffqq VLxxLxx

x++−−= ωσσρ (4.2)

where, σd and σq are specified damping coefficients for the compensator, [x*

1, x*2]T is the desired output of the system.

It can be seen that the third state variable x3 is not directly controllable by ρd and ρq. Therefore, the zero dynamics of the compensator should be examined in order to guarantee the overall stability of the system.

The zero dynamics of a system is defined to be the internal dynamics of the system when the system output is kept constantly at zero by the input. For the system given by (1), the zero dynamics is as follows:

33 xx cσ−=•

where, CRcc /1=σ .Obviously, the zero dynamics of the system is

exponentially stable. However, it is not controllable and its damping is low in real systems given that Rc has a large value.

3.2 Linearization Based on Differential Algebra

Defining a state transformation F(x, w) from x to z, we get an affine system: z = F(x, w) and

21 zz =•

(5.1)

),,,(2 dqddqd vz••

= ρρξ z (5.2)

),,(3 dqdqq vz ρξ z=•

(5.3) where,

z1 = x1 = id,f

d

f

d

Lv

Lx

xxxz +−⋅+⋅−==• ρωσ 3

2112

1318

and z3 = x3 = vdc. Since 0/ ≠xz ∂∂ , the mapping F(x, w)is a diffeomorphism.

The control objective is to achieve a decoupled control that ensures exponential convergence of vdc and id (Figure 2) to reference values with desired damping subject to input constraints. The controller design method is based on the differential algebra theory [18]. From the linearized system (5.1)-(5.3), the linear state feedback control method can be used to design the controller [ξd, ξq]T (see (6.1) and (6.2)) in order to obtain the desired damping rate for the compensator.

ξd(z, d

•ρ , ρdq, vdq) = -a1 z1 - a2 z2 + a1 i0

d (6.1)ξq(z, ρdq, vdq) = -b1 z3 + b1 v0

dc (6.2)Then, the system (5.1)-(5.3) is further simplified to the

linear system (7) if we can find a control law [ρd, ρq]T, which satisfies the differential equation (6.1) with the algebraic equation (6.2).

21 zz =•

0122112

diazazaz +−−=

• (7)

01313

dcvbzbz +−=

•

where i0d and v0

dc are the desired equilibrium points of the reactive current and DC voltage of the capacitor respectively.

Obviously, the original system is decoupled into two subsystems (see (7)). Desired damping for id (z1) and vdc (z3)can be obtained by choosing appropriately the three parameters a1, a2 and b1.

The control law ρdq can be obtained by solving the algebraic-differential equations (6.1) and (6.2).

3.3 Passivity-aimed control

Applications of passivity approach to power system control are drawing more and more attentions of researchers in the areas of power engineering and control [11, 13, 16, 19, 20]. The control method based on passivity consists in establishing a decentralized control strategy such that the controlled device is passive viewed from the power system to which the device is connected. The most important fact is that (i) passive devices always contribute to stability improvement of the overall power systems and that (ii) passivity remains invariant under negative retroaction. It is shown in [11] that a power system represented by means of component-oriented modelling technique is stable if its components are rendered passive or quasi-passive. Passivation of turbo-alternators [19] and of induction motors [20], and passivity analysis [8] of a class of loads (on-load tap changer, large motor, thermostatic heater) are some examples of components that contribute to the (quasi-) passivation and thus stabilization of the power system.

To obtain passivity of STACOM for the pair ( )dqdq vi ~,~

where x~ denote the deviation of x w.r.t. its nominal value, a feedback-plus-feedforward controller is first designed to

achieved efficient set-point regulation of current id and stabilization of vdc. Second, the passivation of the closed-loop system is obtained provided that condition on current trajectory idq is satisfied.

3.3.1 Two Degree of Freedom Controller

We operate a transformation on the inputs, dcdqdq v/µρ = ,

to obtain partial decoupling between dqi and dcv dynamics, which is now expressed as

)()()()()()()(

xhywuxgxfx

=+′+=

•

tttt (8)

where, )(xg′ is equal to g(x) with x3=1. In order to satisfy this transformation, the capacitor voltage has to be non null, which is the case if the initial condition are non null for the voltage and the controller gains presented in the sequel are adequately tuned so that the transient voltage (overshoot) is not too high. Define the following inverse model based control law

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−+=

dt

dLv

odq

fodqdq

odq

iAiµ (9)

where, ⎥⎦

⎤⎢⎣

⎡

−−−

=ff

ff

RLLR

ωω

A ; oqi is the current trajectory to be

determined from the desired trajectories odi , o

dcv and the following constraint derived from (8), considering that set-

point regulation implies that 0/20 =dtdvdc .

0123 2

=− odc

c

odq

Todq v

CRCiµ (10)

Define the feedback loop as ( )o

dqdqfbdqfb iik −−=µ (11)

Using (8), (9), (10) and (11), the closed-loop system obtained with

dqfbodqdq µµµ += is expressed by

( )

⎟⎠⎞

⎜⎝⎛ −+−=

+= −

dqfbTdqdq

Todq

c

dcdc

dqfbfdq

CCRv

dtvd

Ldt

d

ikii

ikAi

µ2322

1(12a)

(12b)

where fbk is a gain matrix defined to arbitrarily locate, in

the left half plane, the poles of ( )fbfL kA +−1 . The voltage tracking error and current tracking error are expressed as

222 odcdcdc vvv −= and o

dqdqdq iii −= .

Assumption 1: the network voltage dqv is constant. Note

that dqodqdq iii += with o

dqi bounded and piecewise

constant. With Assumption 1, odqµ is constant and system

(12) is time-invariant, therefore, the tracking error system (12) is asymptotically stable [21, p442]. Because of the algebraic structure of the system (8), trajectory o

qi is

1319

determined from odi , o

dcv by substituting (9) in (10) and

solving the differential equation (13) in oqi (set-point

regulation implies 0//2

== dtdvdtdi odc

od ):

c

odco

dfodd

oqq

oqf

oq

f Rv

iRiviviRdt

diL

34222

222

2

−⎟⎠⎞

⎜⎝⎛ −=−+ (13)

Dynamics (13) has two equilibrium points

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−⎟⎠⎞

⎜⎝⎛ −+±=

fc

odco

dfodd

ff

q

f

qoq RR

viRiv

RR

v

R

vi

384

21

22

2

2

2,1 .

For a realizable set point, the discriminant in the expression of o

qi 2,1 must be positive which results in defining a set of

desired operation conditions for odi and o

dcv . Under nominal operating conditions with usual parameter values (see [15]) one of the equilibrium 0

2qi is stable but very low in negative value thus non desirable. The more appropriate (close to zero) equilibrium 0

1qi is unstable. Hence, to impose the

trajectory stirring the system to 01qi (unstable), we avoid to

solve (13) online but we impose oqi 1 as a set point to reach

and then 0/ =dtdi oq in (13).

However, the transient in 2dcv is determined by the

capacitor and resistance value and presents a lack of robustness related to uncertainties in these parameters and a possible low convergence rate. To impose arbitrarily fast dynamics to 2

dcv , the current 0qi is determined in the same

way as previously exposed (cf. (10)) but with an additional term in (14) that consists of a voltage tracking error feedback:

22123

dcoodc

c

odq

Todq vkv

CRC−=−iµ (14)

The closed loop system can now be expressed as:

( )

231 2

2

1

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠

⎞⎜⎝

⎛ +−=

+= −

dqfbTdqdq

Todqdco

dc

dqfbfdq

Cvk

RcCdtvd

Ldt

d

ikii

ikAi

µ

(15a)

(15b)

ok is chosen to impose the convergence rate of 2dcv .

Equation (15) results in the same differential equation as (13) with 2

dco vCk− added to the left member. This

time, odqµ is not constant because of its dependence on 2

dcv .However system (15) is autonomous (time-invariant, non perturbed) considering the variable dependency of

⎟⎠⎞⎜

⎝⎛ 2

dcodq vµ in (15b) and the fact that o

dqi and odcv are constant

on each time interval of the set-point regulation. Also, poles of ( )fbfL kA +−1 can be located arbitrarily in the left half

plane so that dynamics in dqi is rendered dominant, i.e.

2LdqTo

dq ∈iµ . Then the error system (15) is asymptotically

stable [21]. Solution 0qi of (14) and (9) depends

continuously on 2dcv which makes the analysis of 0

qi

trajectories more complex to study. To use a practically

reasonable value of 0qi for the implementation, a steady

state solution is used

⎟⎟⎟

⎠

⎞

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−−⎟

⎠⎞

⎜⎝⎛ −+±

⎜⎜

⎝

⎛= 2

22

2

2

384

21

dcoffc

odco

dfodd

ff

q

f

qoq vk

RC

RRv

iRivRR

v

R

vi (16)

Using (16), equation (15a) becomes

( )⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−+= −

dt

diLdt

d oqdqfbf

dq0

1 ikAi

(17)

where

21

22

2

2

22

6

432

⎟⎟⎟

⎠

⎞

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−−

⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛ −+=

dcoffc

odc

odf

odd

ff

qdco

f

oq

vkRC

RRv

iRivRR

v

dtvd

kRC

dt

di

and dtvd dc /2 is given by (15b). Dynamics in 2dcv and qi are

now coupled. If fbk and ok are such that dynamics in dqi is

much faster than dynamics in 2dcv , it is possible to consider

(17) as a linear time invariant system with forcing term sto

q Cdtdi ≅/ during convergence phase of dqi . Thus

convergence of dqi ensures, afterward, convergence of the

slower 2dcv and hence convergence of o

qi .

Remarks: Transient of 2dcv imposes additional constraints on

2odcv in (16) to assure a realizable set point. The control

design implies asymptotic stability of 2dcv . The capacitor

voltage that is initially positive will tend to the desired

voltage2o

dcv if it remains positive, i.e. if gains are tuned such that the overshoot is not too high. As already mentioned, the control problem is solved as a set-point regulation problem because of instability of the dynamical equation used to define the full trajectory for the desirable current o

qi .However, it is possible to interpolate between each constant reference with a second order trajectory to obtain a little improvement (faster response) of the system state transient.

3.3.2 Passivity-aimed strategy

General dynamical system expressed as

1320

),()(),,()( uxhyuxfx ==•

tt (18)

where, nRXx ⊂∈ , pRYy ⊂∈ and mRUu ⊂∈ , is said to be dissipative with supply rate s if there exists a non-negative function V(x): RX → such that the inequality:

∫≤−t

tdyustxVtxV

00 ))(),(())(())(( τττ (19)

holds for all admissible u, all initial values Xx ∈0 and all 0tt ≥ . Furthermore, the system (18) with m = p is said to be (i) passive if it is dissipative with supply rate uTy, (ii) strictly input (resp. output) passive if it is dissipative with

supply rate 2uyuT α− (resp. 2yyuT β− ) for a suitable α, β > 0. A lossless passive system is also called conservative.

Different input-output pair selections may result in different control schemes. In the case of STATCOM, reference [11] chooses the bus voltage in dq frame (vdq) as the input u of the subsystem and the current injected into the power network in dq frame (idq) as the output y of the subsystem. Especially, when we are only interested in bringing the system back to an equilibrium point after system disturbances the error model is used for controller design.

Hence, the passivity-aimed control for STATCOM consists in establishing a controller of ρ

dq such that:

2~~~)~,~( dqdqT

dqdqdq vivivs β−= (20)

with error signals *~dqdqdq vvv −= and *~

dqdqdq iii −= and

where *dqv and *

dqi stand for the voltage and current at equilibrium or nominal operating condition, respectively. Another variant of this category of control method is Passivity-Based Control (PBC) [16], which involves the techniques of energy function reshaping and damping injection. This control strategy provides an elegant solution to improve the dynamical performance and robustness to parameter uncertainties of STATCOMs.

Passivity inequality (19) with (20) is obtained by deriving appropriate conditions on the trajectory o

dqi such that

( ) 2* ~~dq

Tt

t dqodq

Tdq vdtiiv

per

per

β≥−∫+

, 0>dcv (21)

stand for all [ )perperper ttT δ+∈ , where tper and δper

denote the occurrence time of a perturbation and its duration, respectively. Consider the scalar matrix

⎥⎦

⎤⎢⎣

⎡=

43

21

aa

aaaa and determined the current trajectory to track

as *dq

odq ii aa= . Thus, conditions (21) are satisfied if

( )( )

( )22

*2

*1

~~

~~~

~~~

yxRyvxv

ivvyv

ivvxv

fqd

qqqq

dddd

+≥+

+≥

+≥

α

α

(22)

where *2

*1 qd iaiax += , *

4*

3 qd iaiay += and ),min( 21 ααβ ≤ .

Conditions on x and y thus on odqi are devised by

considering sign of dv~ and qv~ . See [13] for complete

characterization of passivating current trajectory odqi , which

has to fulfill specific sector conditions; that is, sufficient conditions concerning o

dqi are drawn so that the STATCOM

remains passive for the pair ( )( )*,~dq

odqdq iiv − . If the

feedback-plus-feedforward controller is tuned sufficiently fast, the pair of interest, )~,~( dqdq iv , is passive [13]. In other words, after or during the occurrence of exogenous disturbances the STATCOM dissipates the amount of energy caused by deviations in

dqv~ and dqi~ and thus contributes to the stability of the power system provided (quasi-) passivity framework is used for the controller design of each component or aggregate of components of the network [11].

3.3.2 Simulation results

In this section, we validate the control design applied to the averaged model (2) and verify its robustness with respect to resistance uncertainties. We focus on the simulation of the system with the proposed passivity-aimed controllers.

The system is simulated (fig. 4) with detailed commutation behavior of the inverter modeled by the switching function approach with a frequency of 4kHz. Due to the commutations, a small steady state error has been observed in dcv . Term 2

dco vk is replaced by

∫+t

dcidco dtvkvk0

22 with 2=ik . Simulations of a 1kVA

STATCOM are performed with the following parameters: Rf=1 Ω, Lf=40 mH, C=132 µF, Rc=10 kΩ, f=60 Hz, and voltage magnitude of 120 V. The controller gains are ok =

70 and fbk =⎥⎦

⎤⎢⎣

⎡

−−−

35753773778975 . The sequence for the current

reference in di is -8, -4, 3, 0 A. System simulations are started with 0i =dq A and dcv =250 V. The following sequence of parameter uncertainties is considered in the simulations. Note that with Lf = 1.3 pu, a large line impedance (0.4 pu) is taken into account with the next variations of Lf : (i) Condition Co : nominal case; (ii) Condition C1: nominal case with the following uncertainty sequence: o

ff LL 3.1= ( 1.00 <≤ t ); off RR 3.1= ;

( 2.01.0 <≤ t ); oCC 3.1= ( 3.02.0 <≤ t ); occ RR 3.1=

( 4.03.0 <≤ t ); (iii) Condition C2: nominal case with the following uncertainty sequence : o

ff LL 7.0= ( 1.00 <≤ t );off RR 7.0= ( 2.01.0 <≤ t ); oCC 7.0= ( 3.02.0 <≤ t );

occ RR 7.0= ( 4.03.0 <≤ t ). The response in di (fig. 4(a))

1321

remains quite unchanged relatively to parameter uncertainties because of the use of sufficiently high gains and small steady state error is obtained. However, the transient in qi (fig. 4(b)) is more important, and a slight

overshoot is observed for the response in di (fig. 4(a)) when off LL 3.1= is considered. A null steady state error for the

dc voltage is maintained (fig. 4(c)) due to the added integrator. The error in dcv causes the variations of o

qi and

then qi . The perturbations in dcv depends on dqi transient and can be decreased by increasing the capacitance C. Gain

ok is tuned to obtain a desired convergence rate for 2dcv response. In steady state, the modulating signal 1ρ of

phase A (fig. 4(d)) used in the PWM is kept inferior or close to one, which facilitates the controller implementation. Responses in case of a 4% amplitude variation on phase Ashow small oscillations for qi and dcv though di remains quite unchanged due to the sufficient high gain controller that tends to decrease the effect of the voltage unbalance (fig. 5).

Time (s)

(A)di

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-10

-8

-6

-4

-2

0

2

4

Co(-);C1(..);C2(- -)

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-10

-8

-6

-4

-2

0

2

Iq (A)

Time (s)

(A)qi 4

Co(-);C1(..);C2(- -)

(b)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4242

244

246

248

250

252

Time (s)

(Volt)dcv254

Co(-);C1(..);C2(- -)

(c)

Time (s)

aρ

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-1.5

-1

-0.5

0

0.5

1

1.5

(d)Figure 4: Switching function model responses: (a) current

di (Co, C1, C2); (b) current qi (Co, C1, C2); (c) voltage dcv

(Co, C1, C2); (d) Modulating signal aρ

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-12

-10

-8

-6

-4

-2

0

2

4

6

Times (s)

idq (A)

id

iq

(a)

1322

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4242

244

246

248

250

252

254vdc (Volt)

Time (s) (b)Figure 5: Switching function model with 4% voltage abcv

unbalance: (a) current dqi ; (b) voltage dcv

4. Discussions

From the summary above, it can be seen that the input-output linearization technique based on differential geometry has a simple controller structure, thus easy to design and implement. It is also robust to parameter uncertainty since the controller’s parameters are not sensitive to the device parameters. Moreover, this controller has another robustness to the switching frequency of the voltage source inverter. However the voltage of the capacitor that provides the voltage source is not controllable even though the zero dynamics of the subsystem is stable. This uncontrollable DC voltage may drift to an uncertain level. A potential solution to this drawback is to add an outer PI loop acting on the iqcomponent to control the DC voltage within an acceptable range.

The control strategy based on differential algebra is advantageous over the previous one in DC voltage control since it can achieve a decoupled control for vdc and id(corresponding to reactive power injection) that ensures exponential convergence of all of the three variables with desired damping subject to input constraints. However, the controller structure is much more complex, thus difficult to design and implement such controllers in real systems. Furthermore, another major drawback of the method is the lack of robustness to system parameter uncertainties, switching frequency and feedback noise of harmonics generated by the inverter. Especially, its sensitivity to feedback noise makes it almost unfeasible in practice given the harmonics produced by the converters exists constantly. An extra observer must be added to filter out the noise in the feedback signals, which considerably complicates the controller structure. Although this method is theoretically elegant and attractive, it is almost unfeasible in practice.

The focus of passivity approach is to improve power system stability by achieving the invariant passivity property

for the subsystem viewed from the power system to which the device is connected.

The passivity theory provides a solid framework for decentralized controller design for power system devices, which has the following interesting features: 1) the decentralized controller can be designed using

only the localized model and local feedback signals; 2) a passive device always enhances the stability of the

overall power system; 3) an “over” passive device can compensate for the lack

of passivity of the others in its neighbouring area.

This approach seems pretty promising in establishing a unified way of decentralized controller design for devices of power systems. However, due attention must be paid to a reasonable compromise between the invariant passive property for power systems and required dynamical performance for the individual device.

References

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[10] Z. Yao, V. Donescu, P. Kesimpar, N. Léchevin, V. Rajagopalan, “Nonlinear Control for STATCOM Based on Differential Algebra”,

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IEEE Power Electronics Specialists Conference-PESC’98, 17-22 May 1998, Fukuoka, Japan.

[11] N. Léchevin, P. Sicard and Z. Yao: “Stability Analysis of Radial Power Systems: A Passivity Approach”, IEEProceedings on Control Theory Applications, Vol. 151, No. 3, May 2004, pp. 264-270.

[12] K. V. Patil, J. Senthil, J. Jiang and R. M. Mathur: “Application of STATCOM for Damping Torsional Oscillations in Series Compensated AC Systems”, IEEE Transactions on Energy Conversion Vol. 13, No. 5, September 1998, pp. 237-243.

[13] N. Léchevin, “Approche de Passivité pour la Stabilisation des Réseaux Électriques”, PhD. Thesis, University of Quebec at Trois-Rivières, 2000.

[14] Jain, A.K. Behal, A. Zhang, X.T. Dawson, D.M. Mohan, N.: “Nonlinear controllers for fast Voltage regulation using STATCOMs”, IEEE Control Systems Technology, Nov. 2004, pp. 827- 842, Vol. 12, No. 6.

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[17] B. Lu: “Nonlinear control of center-node UPFC and VSC-based FACTS controllers”, Ph.D. Thesis, McGill University 2004.

[18] M. Fliess, “Generalized Controller Canonical Forms for Linear and Nonlinear Dynamics”, IEEETransactions on Automatic Control, Vol. 35, No. 9, pp. 994-1001, Sept. 1990.

[19] N. Léchevin , P. Sicard and Z. Yao , “PSS design for passivity-aimed control of turboalternators: an LMI approach”, Proceedings of the American control conference, June 2000, Chicago, USA.

[20] N. Léchevin, Z. Yao Z. and P. Sicard, “Passivity property of induction motors from the power supply: towards a new framework of stability analysis and control for power distribution systems,” Proceedings of the IEEE Conference on decision and control, December 1998, Tampa, USA.

[21] H.K. Khalil, Nonlinear Systems, Prentice Hall, 2nd

Edition, 1996.

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