Unified Modeling of UPFC-SMES in Power System Application

A. Esmaieli

Niroo Research Institute (NRI)

Tehran, Iran

S.A Nabavi Niaki Senior Member, IEEE

University of Mazandaran Babol, Iran

A. Tajfar Niroo Research Institute

(NRI) Tehran, Iran

H. Nademi Damavand Power Plant

Tehran, Iran

Abstract-These instructions give you basic guidelines for

preparing papers for conference proceedings. This paper discusses the modeling issues of incorporation of a Superconducting Magnetic Energy Storage (SMES) coil into UPFC for application in power systems. A UPFC can only absorb/inject reactive power and, consequently, is limited in the degree of freedom and sustained action in which it can help the power grid. The addition of energy storage allows the UPFC to inject and/or absorb active and reactive power simultaneously, providing additional benefits and improvements in the system. Detailed model of UPFC based on switching function and dynamic model of SMES are considered in modeling. Finally a steady-State load flow model of UPFC/SMES is presented.

Index Terms-UPFC, SMES, Modeling, Power Flow Control

I. INTRODUCTION

The UPFC is one of the most potential FACTS devices for power flow control [1]. It can provide independent and simultaneous control of both real and reactive power flow. A UPFC can respond rapidly to transients and assist in maintaining transient and dynamic stability. However it can only absorb/inject reactive power and, consequently, is limited in the degree of freedom and sustained action in which it can help the power grid. The addition of energy storage allows the UPFC to inject and/or absorb active and reactive power simultaneously, providing additional benefits and improvements in the system [2- 4].

In this paper, first the detailed model of UPFC is developed based on the switching function approach. Then this model was extended for UPFC/SMES configuration. The steady-state model is extracted from detailed model based on the average modeling technique. Finally a load flow model for UPFC/SMES is proposed.

II. VOLTGAE SOURCE CONVERTED MODELING The main component of UPFC/SMES configuration is the

voltage source converter (VSC). In this section, the detailed model of a VSC based on the switching function approach is obtained. Figure 1 shows the circuit equivalent model of a VSC. In this figure, if (or if ′ ) is the switching function of switch iS (or iS ′ ) and sr models the switching losses in its ON state. To avoid short circuit, the operation of switches in each converter leg is complementary (i.e. 1a af f ′+ = ). With reference to Fig.1 and applying KVL, the voltage equation of

phase a can be deduced.

a s a MN dc av r i v v f= + + (1)

Also for phase a, the voltage equation can be developed as follows,

aat a L a

div v L R idt

− = + (2)

where L and RL are the leakage inductance and winding resistance of the interface power transformer of VSC. The same equations can be developed for the other phases. In the balance condition, where the sum of phase voltages and

three phases' currents are zero, the voltage MNv is obtained by adding the phase voltage equation of all phases.

( )3dc

MN a b cvv f f f= − + + (3)

LLR

LLR

LLR

aiav

bvcv

atv

btv

ctv

sr sr sr

sr sr sr

aSaf bSbf cScf

aS ′af ′bS ′bf ′

cS ′cf ′

dcC dcv+

−

dci

M

N

Fig. 1, The circuit equivalent model of a voltage source converter

Equation (1) can be rewriting for all phases by substituting

(3) in (1).

( )3dc

i s i dc i a b cvv r i v f f f f= + − + +

(4) where i = a,b,c. Also the general form of (2) can be

obtained by using (4).

( )13

ii it dc i a b c

diL Ri v v f f f fdt

= − + + − + +

(5)

Where s LR r R= + . The current equation can be obtained by applying KCL at

the top node in the circuit of Fig.1.

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dc a a b b c ci i f i f i f= + + (6) Therefore the dc link capacitor equation becomes in the

form of (7).

dca a b b c c

dvc i f i f i fdt

= + + (7)

-1.5

-1

-0.5

0

0.5

1

1.5

0 π 2π

afc onv

Fig.2, Applying average technique to switching function.

Equations (5) and (7) form the basic mathematical model

of voltage source converter. However, because of the switching functions, these equations are in discrete form and are highly nonlinear. To overcome this problem, the average technique is used for switching functions. If SPWM method is used for switching control, the switching frequency is higher than system's fundamental frequency. Then the sinusoidal carrier of the control signal can be assumed constant during one switching period and consider it as a step function [5] (Fig.2).

Envisaging one period of triangle wave and step wave, one can conclude that the switching control pulse is symmetrical with reference to triangle of saw-tooth wave [6].

Based on this conclusion and the fact that the switching control pulse duration ( ad ′ ) lasts in one period of switching frequency, the switching function is shown in Fig. 3.

π 2π0

af ′

conv′

t̂riV

4sTont

4sT1t1t

0

( )1 ad π′− ( )1 ad π′+

Fig. 3, One period of switching function of aS ′

With reference to Fig.3, the switching function can be expressed in the form of Fourier series.

( ) ( ) ( )1

1 sin cosna a a s

n

f d n d n tn

π ωπ

∞

=

2′ ′ ′= + −∑ (8)

This equation indicates that the average of the function is equal to ad ′ in the one switching period. For the sake of

fundamental frequency modeling, the higher frequency terms of (8) can be ignored without losing significant accuracy [6].

,a l af d′ ′= (9)

or

,a l af d= (10)

Figure 4 shows the switching function ( af ) and its average model ( ,a lf ) for using in fundamental modeling application.

-2

-1.5

-1

-0.5

0

0.5

1

1.5 con′va,lf af

Fig. 4, One period of switching function of aS ′

The duty function of aS in one switching period can be

obtained with reference to Fig. 3.

1s ona a a

s

T tD d dT− ′= = − = (11)

Also Da can be expressed by:

12 2 4sa

s

t TDT

+= (12)

Using the similar triangles rule in Fig.3, the duty function of aS in one switching period can be expressed by:

0.5ˆ2con

atri

vDV′

= + (13)

It means that the general shape of aD is similar to the step wave conv ′ in Fig.4. Therefore, (9) and (10) yields:

,a l af D= (14)

Equation (14) indicates that function ,a lf is also similar to

conv ′ and conv ′ is similar to conv as well.

By assuming ( )1ˆ coscon conV tω φ= +v , ,a lf can be

expressed in the following form.

( ), 1

ˆ0.5 cos ; ˆ2

a cona l a a

tri

m Vf D t mV

ω φ= + + = (15)

Similar expressions for phases b and c are also deduced. Substituting the switching functions equations in (3) yields:

2dc

MNvv = − (16)

By substituting this result in (1), the phase voltage of VSC at its ac terminal is given by:

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( )1cos2

aa s a dc

mv r i v tω φ= + + (17)

Also, the similar expressions can be deduced for phases b and c. Finally, the dc link capacitor equation is obtained.

( ) ( )( )

1 1

1

[ cos cos 1202

cos 120 ]

dc aa b

c

dv mc i t i tdt

i tω φ ω φ

ω φ= + + + −

+ + + (18)

The fundamental frequency model of VSC is shown in Fig.5 based on (17) and (18).

btv

atv

ctv

dcv+

−dcC

av′+Lai dci

bv′RL

cv′RL

bi

ci

R

Fig. 5, Fundamental frequency model of VSC.

III. UPFC MODELLING The power circuit of a UPFC is composed of an excitation

transformer (in parallel), a boosting transformer (in series), two three-phase GTO based voltage source converter (consist of series and shunt converters) and a dc link capacitor. Figure 6 depicts the schematic diagram of a UPFC. For each VSC of UPFC the same voltage equations as (5) can be deduced. However, the dc link current is a function of both converters' currents.

Etav

EtcvEtbv

dcCdcv+

−

Bav

Btav− +

EavEai Bai

Ei Bi

dci

Fig. 6, Schematic diagram of UPFC.

dc E Bi i i= + (19)

The dc components of each converter current can be expressed as a function of ac currents based on the switching function.

E Ea Ea Eb Eb Ec Eci i f i f i f= + + (20)

B Ba Ba Bb Bb Bc Bci i f i f i f= + + (21)

Therefore the dynamic equations of UPFC can be developed.

( )

( )

1313

EiE E Ei Eti dc Ei Ei Ei Ei

BiB B Bi Bti dc Bi Bi Bi Bi

dcEa Ea Eb Eb Ec Ec Ba Ba Bb Bb Bc Bc

diL R i v v f f f fdt

diL R i v v f f f fdt

dvc i f i f i f i f i f i fdt

= − + + − + + = − + + − + +

= + + + + +

where , ,i a b c= . Finally, applying the same procedure as the VSC

modeling, the fundamental frequency model of UPFC can be developed as shown in Fig. 7.

Etav

EtcvEtbv

EL ER Eav

Ei BidcCdcv+

−

dciBav

BR BL Bai

Btav− +

Eai

++

Fig. 7, Fundamental frequency model of UPFC

IV. UPFC/SMES MODELING Schematic diagram of UPFC/SMES is shown in Fig. 8.

VSC E− VSC B−

chopper

Et

Bt

dcC

SMES

BiEi

dcv− +esi

dci

Fig. 8, Schematic diagram of UPFC/SMES.

Considering the fact that the SMES current always flows

in one direction and the control of its charge and discharge is applied through its dc voltage, a proper chopper for this application has the same configuration as of Fig.9 [7]. The achievement of this section would be developing a proper model for this chopper to be considered with UPFC equations.

With reference to Fig. 9, the voltage of induction coil can be expressed by:

( )L AN BN a b dcv v v f f v= − = − (22)

Ignoring switches' losses, the instantaneous power balance for the circuit of Fig. 9 is given by:

( )dc es L L es a b Lv i v i i f f i= ⇒ = − (23)

Equation (19) can be revised by considering the chopper current.

dc E B esi i i i= + − (24)

The dynamic of SMES coil is given by: 1L

Les

di vdt L

= (25)

Substituting (22) in (25) results:

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( )a bLdc

es

f fdi vdt L

−= (26)

Therefore the expression for UPFC/SMES can be developed by:

( )

( )

( )( )

1313

EiE E Ei Eti dc Ei Ei Ei Ei

BiB B Bi Bti dc Bi Bi Bi Bi

dcdc Ea Ea Eb Eb Ec Ec

Ba Ba Bb Bb Bc Bc a b L

Les a b dc

diL R i v v f f f fdt

diL R i v v f f f fdtdvC i f i f i fdt

i f i f i f f f idiL f f vdt

= − + + − + + = − + + − + +

= + +

+ + + − −

= −

dcv

+

−

esi

AT BD

ADBT

af

bf ′

Lv+

−

Li

esL

N

A

B

Fig. 9, Proposed chopper for connecting SMES to UPFC.

Using two levels PWM techniques [5] and applying

Fourier method to the above equation, the discontinuity of the equation can be eliminated. Figure 10 shows one period of pulse function created by two levels PWM technique. In this technique, two switches are considered as one pair and have the same switching function. This switching function is obtained by comparison of one triangle wave with a controllable dc voltage.

Considering the similarity between Fig. 10 and Fig. 3, the same equation as (13) and (14) can be developed for Fig. 10.

, 0.5 ; ˆ2a con

a l a atri

m Vf D mV

= = + = (27)

In conjunction with (22) and for the fundamental frequency, we have:

( ) ( ),2 1 2 1L a dc a l dc a dcv f v f v m v= − − = (28)

Substituting (27) in (23) yields:

es a Li m i= (29) The space-state equation of SEMS can be developed by

substituting (28) and (29) in (25). 2

es adc

es

di m vdt L

= (30)

Finally the state-space equation for UPFC/SMES is obtained.

x Ax Bu= + (31) Where

[ ]TEa Eb Ec Ba Bb Bc es dcx i i i i i i i v=

[ ]0 0 TEta Etb Etc Bta Btb Btcu v v v v v v=

The non-zero elements of A and B are given in Appendix. The dynamic, transient and small-signal model of

UPFC/SMES can be developed simply by using (32) and the same procedure as presented in [8].

π 2π0

af

conV

triV

4sTofft 4sT

1t1t

0

Fig. 10, One period of switching function Ta.

V. STEADY-STATE MODEL OF UPFC/SMES To develop the steady-state model of UPFC/SMES, it was

assumed that inductance of SMES's coil is large enough that its current can be considered linear as shown in Fig.11.

0-3

-2

-1

0

1

2

3

( )Li t

0I

LI

Fig.11, Steady-state of SMES's current waveform.

With reference to Fig.11, the SMES current expressed as:

( )( )

m m

0

m m

;

;

dcon

esdcL T a

dc dceson on on

es es

V t t tLVi t i n T mV VL t t t t tL L

≤= + + − − >

(32)

where i0 is the initial value of coil's current

and ( )mm , , mod ,T s T m s

s

t tt n T t n t t TT−= + = = .

At different conditions, the SMES's current is in the one of the following states.

ma = 0 : At this condition, the average current of the coil is constant and the storage system is in the standby state.

ma > 0 : At this condition, the average current of the coil is increasing and the storage system is in the charge state.

ma < 0 : At this condition, the average current of the coil is decreasing and the storage system in the discharge state.

If the storage system is in the standby state and considering (33) and the fact that the coil is fully charged, the standby current can be expressed in (34).

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212es esW L I= (33)

2 nomsn

es

WIL

= (34)

where nomW is the rated power of SMES. Assuming that i0 is the standby current of SMES, the

average current of SMES is given by:

2a T

L L sn dces

m n TI i I VL

= +

(35)

The average input current to SMES is equal to:

es a LI m I= (36)

And the dc current at the steady-state is equal to:

dc E B esI I I I= + − (37)

At the steady-state, it can be assumed that the voltage and current of SMES are equal to their average value.

L L a dcv V m V= = (38)

To find SMES’s current at steady-state, it is assumed that at steady-state its average power is constant.

ESS dc esP V I cte= = (39)

Since dcV is constant, therefore the average currant ( esI ) should be constant as well. From (35) and (37) we have

( )2

a Tes a L a sn dc

es

m n TI m I m I V cteL

= = + =

Therefore the following constraint condition should be held at the steady-state condition.

2 02

dc ESSTa a sn

es dc

V Pn Tm m IL V

+ − =

(40)

To satisfy the above constraint, the following condition must be maintained.

( )2 2 0ESSsn T

es

PI n TL

+ ≥ (41)

The above condition could be rewriting in the following form when the storage system is in the charging state ( 0ESSP > ).

( ) 2

2es nom

T snESS ESS

L Wn T IP P

≥ − = − (42)

And in the discharging state, (42) could be rewriting in the following form.

( ) nomT

ESS

Wn TP

≤ − (43)

Equation (44) indicates that during the discharge period, the operating limit reaches at the point where all the system’s energy is used. Then there is no solution for am in (40) beyond this point. Equation (40) also indicates that the

coefficient am not only controls the speed of system’s charging and discharging time but also is able to control between the charge state and discharge state.

Based on the above argument, at the steady-state condition and as long as there is a solution for (41), it can be assume that:

ESSes es

dc

Pi IV

= = (44)

Finally the steady-state model of UPFC/SMES can be shown in Fig. 12.

0-3

-2

-1

0

1

2

3

( )Li t

0I

LI

Fig. 12 Steady-state model of UPFC/SMES

At the steady-state, (20) and (21) can be rewritten in the

following form. 3 cos( )43 cos( )4

E E Ea E E

B B Ba B B

I m I

I m I

θ φ

θ φ

= −

= − (45)

To have constant dcv , it is necessary that the dc current ( dcI ) be zero. Therefore the values of , , ,a E B Em mφ φ and

Bm should be selected such a way that satisfy (37). Among these five parameters, four parameters , ,E B Emφ φ and Bm could be considered as independent parameters for the sake of controlling system’s parameters and keeping am for satisfying (37). In other words, am can be adjusted to balance the real power exchanged between two voltage source converters of UPFC.

Multiplying dcV in (37) yields:

3 cos( )4

3 cos( )4

dc dc E Ea dc E E

B Ba dc B B a L dc

I V m I V

m I V m I V

θ φ

θ φ

= −

+ − − (46)

The power exchanged at the shunt component is given by: 33Re{ } cos( )4E Ea Ea E dc Ea E EP V I m V I θ φ∗= = − (47)

Also the power exchanged at the series component is given by:

33Re{ } cos( )4B Ba Ba B dc Ba B BP V I m V I θ φ∗= = − (48)

Therefore the average power of storage system is equal to:

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ESS dc es a L dcP V I m I V= = (49)

Substituting (48), (49) and (50) in (47) yields: 0dc dc E B ESSI V P P P= + − = (50)

From (51) one can conclude that the capacitor power is equal to the sum of series and parallel ac sources’ power, and also at the steady-state condition, the net power exchanged between converters and the power system is provided by storage system. In other words, the average power exchanged between dc capacitor and the power system becomes zero and as a result its dc voltage remains constant.

Based on the above discussion, the single line equivalent diagram of UPFC/SMES at the steady-state is shown in Fig.13.

EV

EjXEI

ER

EtVBV

BRBjXBI

B E ESSP P P− =

+

+

Fig. 13, Single line equivalent diagram of UPFC/SMES at the steady-state.

With reference to the model of Fig.13, the same decupled

load flow model of UPFC [8] can be extended for UPFC/SMES. Generally based on the selected control strategy, the real power at the UPFC/SMES terminal and the power of the storage system can be set (i.e. as known parameters). Therefore, the load flow model of UPFC/SMES can be expressed in one of the following forms as shown in Fig.14.

Et

Et

PV

−

Et ESS

Bt

P PQ

− −−

B Bus−E Bus−

Et

Et

PV

−

Et ESS

Bt

P PV

+

B Bus−E Bus−

Et

Et

PQ

Et ESS

Bt

P PQ

−−

B Bus−E Bus−

Et

Et

PQ

Et ESS

Bt

P PV

+

B Bus−E Bus−

Fig. 14, Different presentations of UPFC/SMES for load flow studies.

After performing load flow studies, the unknown

parameters of the buses are obtained. Then the internal unknown parameters of UPFC/SMES are calculated based on model of Fig.14 [2].

VI. CONCLUSION The range of operation of a UPFC could be dramatically

improved if it be equipped with a Superconducting Magnetic Energy Storage (SMES). An adequate chopper for connecting SMES to UPFC at its dc terminal is proposed. To evaluate the advantage of this configuration, a proper model based on the switching function for UPFC is developed and then is extended to UPFC/SMES. The average technique is used to extract the fundamental frequency model of UPFC/SMES.

The small-signal and load flow model can directly be developed from this model. Finally, different presentations of UPFC/SMES for load flow studies are proposed.

With reference to the model of Fig.13, the same decupled load flow model of UPFC [8] can be extended for UPFC/SMES. Generally based on the selected control strategy, the real power at the UPFC/SMES terminal and the power of the storage system can be set (i.e. as known parameters). Therefore, the load flow model of UPFC/SMEC can be expressed in one of the following forms as shown in Fig.14.

VII. APPENDIX Non-Zero elements of matrixes A and B in (31).

11 22 33 44 55 66

18 81 1

28 82 1

38 83 1

18 81 1

28 82 1

38 83

R R;

cos( )2

cos( 120)2

cos( 120)2

cos( )2

cos( 120)2

E B

E B

EE dc E

EE dc E

EE dc E

BB dc B

BB dc B

B dc

a a a a a aL Lma L a C tma L a C tma L a C tma L a C tma L a C t

a L a C

ω φ

ω φ

ω φ

ω φ

ω φ

= = = − = = = −

= = − +

= = − + −

= = − + +

= = − +

= = − + −

= = 1

2

78

cos( 120)2

BB

a

es

m t

maL

ω φ− + +

=

11 22 33 44 55 66

77 88

1 1;

1 1;E B

es dc

b b b b b bL L

b bL C

= = = = = =

= =

VIII. REFERENCES [1] L. Gyugyi, “A Unified Power Flow Control concept for Flexible AC Transmission Systems”, IEE Proceedings-C, Vol. 139, No 4, July 1992. [2] A. Esmaeili, "Improving UPFC Performance incorporation of Energy Storage System", M.Sc. dissertation, Dept. Electrical Eng. Univ. Mazandaran, 2004 [3] Paulo F. Ribeiro, Brian K. Johnson, Mariesa L. Crow, Aysen Arsoy, Yilu Liu, "Energy Storage Systems for Advanced Power Applications", proceedings of the IEEE, vol. 89, no. 12, 2001. [4] A. Esmaeili, S.A. Nabavi Niaki, "The Impact of Energy Storage System (ESS) on UPFC Operation at Steady-State Condition", in Proc. 2004 Power System Conference, Tehran, Iran, 98-F-PQA-425 [5] N. Mohan, T. M. Undeland, W. P. Rpobbins, “Power electronics; Converters, Applications and Design”, John Wiley & Sons, 1995. [6] R. Wu, S. B. Dewan, G. R. Slemon, “Analysis of an ac-to-dc Voltage Source Converter Using PWM with Phase and Amplitude control”, IEEE Trans. On Industry Applications, Vol. 27, No. 2, 1991 [7] R.L. Kustom, J. Skiles, J. Wang, K. Klontz, T. Ise, K. Ko, F. Vang, "Research on Power Conditioning Systems for SMES", IEEE trans. On Magnetic, Vol. 27, No. 2, 1991 [8] S.A. Nabavi-Niaki, M.R. Iravani, “Steady State and Dynamic Models of Unified Power Flow Controller (UPFC) for Power System Studies”, IEEE trans. on power systems, Vol. 11, No. 4, November 1996.

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