Power System Stabilizer for Comunicationless Parallel Connected Inverters

R. M. Santos Filho

CEFET-MG Belo Horizonte, Brazil

rsantos@deii.cefetmg.br

P. F. Seixas, P. C. Cortizo UFMG

Belo Horizonte, Brazil paulos@cpdee.ufmg.br

Guillaume Gateau LAPLACE

Toulouse, France guillaume.gateau@laplace.univ-tlse.fr

Ernane A. A. Coelho UFU

Uberlândia, Brazil ernane@ufu.br

Abstract— In this paper, small-signal stability issues of low-

voltage parallel-connected voltage source inverters without control interconnections are addressed by means of dynamic analysis and experimentations. The studies performed showed that the parallel system damping is severely reduced when the traditional droop method is employed and the cabling is long. This paper proposes the application of the PSS (Power System Stabilizer) action to the control of parallel inverters in order to increase the stability range concerning the connection impedance composition R/X. It is shown that the PSS action can significantly improve the system damping. Experimental results for two 1kVA inverters in parallel sharing a common load are presented and corroborate the presented ideas.

I. INTRODUCTION Connecting voltage source inverters in parallel is very

important nowadays, as it allows the increase of the total power capacity or the improvement of the system reliability through redundancy. The distributed generation systems and the UPS are two major applications that profit from this technology. In the inverters parallelism, the absence of vital dependence on communication is highly desirable since it strongly improves system reliability. In this sense, the synchronism and the load sharing must be performed based on local data only. Besides, when inverters are connected in parallel, stability problems may arise. Sustained or poorly damped power oscillations may occur in response to disturbances, such as load variations or the connection or disconnection of inverters to the parallel system. The oscillations dynamics depend on the control strategy, line and control parameters, etc.

The existing control strategies for the communicationless parallelism of inverters, commonly called droop methods, rely on the relationships between the inverter output voltage amplitude and frequency and its measured output active and reactive power [1]-[12]. These relationships depend on the composition of the connection impedance Z=R+j⋅X between the inverter output and the common load bus.

However, in contrast to the Electrical Power Systems (EPS) where the lines are mostly inductive, in low voltage installations this particular case may not truly take place due to the cabling impedance composition which adds to the inverter output impedance. The assumption of inductive line may not really take place even if the inverter internal (Thévenin) impedance has been programmed as inductive by control design as proposed in [7],[8]. In fact, low voltage

cabling can be predominantly resistive or not. For instance, the ratio R/X can range from ≈90 for a 1.5mm2 cable down to ≈0.8 for a 300mm2 cable [18]. Therefore, depending on its length, the cabling impedance composition (R/X)c can override the inverter internal impedance composition (R/X)th. This change can increase the unfavorable transmission interaction between the active and reactive power loops [13]-[17]. As a consequence, the system damping may be severely reduced and oblige the entire system to shut down in the worst case scenario. Thus, for analysis and design purposes each inverter can not be seen as comprised of two SISO loops any longer but as a MIMO system instead [3].

Although resulting from more complex mechanisms, the problem of poorly damped power oscillations is well known in the Electrical Power Systems field and has been investigated so far. The PSS (Power System Stabilizer) is a widely employed and very effective countermeasure against undamped power oscillations in the EPS field, and its basic function is “to add damping to the generator rotor oscillations by controlling its excitation using auxiliary stabilizing signals. To provide damping, the stabilizer must produce a component of electrical torque in phase with the rotor speed deviations” [12].

In this paper, we propose the utilization of the PSS action in the context of voltage source inverters parallelism in order to improve the small-signal stability. The PSS action will be adapted to the parallelism of voltage source inverters and the whole system will be modeled in the small-signal sense.

In Section II the parallel inverter system will be modeled and the PSS action will be included in the small-signal model. The cabling effects over system damping with and without the PSS action will be analyzed. In Section III experimental results will be presented and discussed.

II. SYSTEM MODELING AND ANALYSIS The network topology of the parallel inverter system

considered in this paper is depicted in Fig. 1. Taking only one inverter and the load bus leads to the simplified circuit shown in Fig. 2, in which the connection impedance was split in the cabling impedance and in the inverter internal impedance. The inverter output voltage and the load bus voltage have the forms √2⋅E⋅sin(ωi⋅t+θi), and √2⋅V⋅sin(ωv⋅t+θv), respectively. Henceforth the inverter bandwidth will be considered much higher than the overall system bandwidth, and therefore the inverter dynamics will be disregarded.

978-1-4244-6391-6/10/$26.00 ©2010 IEEE 1004

A. Open-Loop Model for a Single Inverter The steady-state active and reactive power P and Q in the

circuit shown in Fig. 2 are given by:

( ),sincos1

),,( 222 δδδ XEVREVREXR

VEP +−+

= (1)

( ),sincos1

),,( 222 δδδ REVXEVXEXR

VEQ −−+

= (2)

where δ is the phase-angle between the inverter output voltage and the load bus voltage, given by:

σσωσωθθδ dttt vi

t

vi )()( −=−= ∫0)()()( . (3)

From (1)-(3) it is possible to sketch the open-loop model shown in Fig. 3. From a process control perspective, the manipulated variables are ωi and E, and the process outputs are P and Q. Finally, ωv and V, namely the load bus frequency and voltage are taken as disturbance inputs.

Fig. 1. Network topology of the parallel inverter system.

Fig. 2. Connection of a single inverter to the load bus. The connection impedance is split in inverter Thévenin impedance and in line impedance.

∫dt

Fig. 3. Inverter open-loop model in time domain.

If the connection impedance magnitude |Z|=(R2+X2)1/2 is designed to be small, e.g. |Z| < 5%, as it is the natural practice, then E and δ will be restricted within a small range around the nominal values as the active and reactive powers vary from zero up to the rated values. From (1),(2) it is clear that the connection impedance magnitude acts as the inverse of process gain and hence strongly influences the closed loop stability.

B. The Droop Method One important system characteristic is that it is not self-

regulating, i.e. in open loop it will not reach a steady-state if driven by a constant input (unless ωi≡ωv). The droop method consists in building negative feedback paths so that the loops are closed and the parallel system reaches an equilibrium point. In this case the inverter synchronizes automatically with the load bus (ωi=ωv). As can be seen in Fig. 3, there are two outputs to be controlled by two inputs. The suitable input-output pairing for this plant depends on the connection impedance composition, as discussed in [3],[7]. For a mostly inductive line, the output active power depends mostly on δ (and hence on ωi) and the output reactive power depends mostly on E. In this case the traditional droop control law is to be applied and it is given by

)( omespoi PPk −−= ωω , (4)

)( omesvo QQkEE −−= , (5)

where kp and kv are the droop gains, Pmes and Qmes are the measured inverter output active and reactive power, and Po and Qo are power offsets. Applying these control laws to the open loop model in Fig. 3 results in the system

∫dt

f

f

s ωω+

f

f

s ωω+

Fig. 4. Closed loop model for one inverter controlled by the traditional droop method.

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depicted in Fig. 4. The measurements of the active and reactive power were modelled by first order systems.

C. Multi-Inverter System Model The small-signal model for the multi-inverter system

developed in [1] will be used here in order to allow stability studies and parameters adjustments. Unlike the small-signal model for the parallelism of one inverter with the stiff bus, in this model the load bus voltage is allowed to react from changes in any inverter output. In this model the dynamics of each inverter is described by

⎥⎦

⎤⎢⎣

⎡ΔΔ

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΔΔΔ

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΔΔΔ

i

i

iq

id

i

iq

id

i

Q

P

e

e

e

edtd

iiCM

ωω, (6)

where Δωi is the frequency deviation of the inverter output voltage; Δedi and Δeqi are the small-signal deviations of the components of the inverter output voltage phasor E=edi +j⋅eqi; and finally the state matrix Mi and the gain matrix Ci are comprised of constant coefficients that result from the linearization around the operating point p0=(Ed0, Eq0), filters cutoff frequencies and droop gains. The small-signal deviations ΔSi=[ΔPi ΔQi]T of the inverter output active and reactive power are taken as the inputs of (4). This way, all inverters can be coupled through the system network admittance matrix Ys, as illustrated in Fig. 5. The system order will be 3⋅N, where N is the number of inverters in the system. After concatenating and reorganizing the matrixes in Fig. 5, one finds the dynamic small-signal model in the form

AXX =& . (7)

There will always exist one null eigenvalue in this model, what prevents the determination of the absolute stability. However, it can be ignored since it is an artifact that occurs because none inverter voltage angle was chosen as the reference for the computation of the other inverters phase-angles. The existence of the null eigenvalue due to this reason is well known in the EPS field [12]. For more details on this

small-signal model see [1].

D. PSS Action for Parallel Connected Inverters As stated before, due to the cabling impedance, the

assumption of inductive line may not really take place even if the inverter internal impedance has been programmed as inductive by control design. In this case, as will be shown in the next section, the system stability can be seriously compromised. To cope with this, we propose the application of the PSS action in conjunction to the droop method to control each inverter.

Translated from the EPS field to the present matter, the PSS action is the utilization of the inverter frequency deviations to generate voltage deviations, as shown in Fig. 6. The high-pass filter with cutoff frequency ωh aims to avoid steady-state voltage deviation due to loading. The new inverter output voltage will be

PSSi

drpio EEEE ++= , (8)

where

ωω

Δ+

= sh

PSSi k

ssE , (9)

where ks is the PSS gain. As the high-pass filter is first order, the order of the

inverter model will be increased by one and will now be fourth order. Hence the system order will now be 4⋅N. Substituting (8) and (9) in (6) it is possible to obtain the small-signal model that describes the behavior of a single inverter with PSS action:

⎥⎦

⎤⎢⎣

⎡ΔΔ

+

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

ΔΔΔΔ

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

ΔΔΔΔ

i

i

xPSSi

iq

id

xPSSi

iq

id

i

Q

P

E

e

e

E

e

e

dtd

i

)24()44(

PSSi

PSSi

CM

ωω

, (10)

where PSSiM and PSS

iC are given by (11) and (12), respectively.

NNNNNuCxMx +=&

NNxIy =

11111 uCxMx +=&

11xIy =

11xIy =

2222uCxMx

2+=&

Fig. 5. Modeling developed in [1] for a multi-inverter parallel system.

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∫dt

hssω+

p

p

s ωω+

q

q

s ωω+

Fig. 6. Droop control for a single inverter together with the PSS action.

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−

−

−−−

+

−

−

−−−

+

−

=

hfs

qddq

fhd

qddq

fqd

qddq

fdd

qddq

fsdd

dqqd

fhq

dqqd

fqq

dqqd

fdq

dqqd

fsqq

f

k

nmnm

m

nmnm

nm

nmnm

nm

nmnm

kmn

nmnm

m

nmnm

nm

nmnm

nm

nmnm

kmn

ωω

ωωωωω

ωωωωω

ω

00

)(

)(

000

PSSi

M (11),

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−−

−−

−

=

0

0

fps

qddq

dfv

qddq

dfps

dqqd

qfv

dqqd

qfps

fp

kk

nmnm

mk

nmnm

mkk

nmnm

mk

nmnm

mkk

k

ω

ωω

ωωω

PSSiC (12)

Following the scheme shown in Fig. 5, it is possible to obtain the small-signal model for the whole parallel inverter system with PSS action:

PSSPSSPSS XAX =& . (13)

E. Dynamic Analysis The parameters given in Table I were used in the dynamic

analysis of the parallel system comprised of two inverters, in two situations: without PSS and with the PSS action. In both cases the line composition R/X was swept in the range (10-

2,102) so that the behavior of the system regarding the line composition could be evaluated. The models presented above were used to compute the eigenvalues.

Fig. 7 shows the results obtained when the PSS action was turned off and on. As can be seen, the eigenvalues run to the right half plan as the connection impedance composition R/X increases. This fact was investigated in [3]. However, the eigenvalues loci for the system with PSS are different, as they tend to be more damped and to run to the right half plan slowly as R/X increases. This theoretical result suggests that the PSS can improve the system damping and can avoid instability.

In a real system, the increase of the connection impedance composition R/X with regard to the design value is due to the cabling. Thus, the damping of the dominant eigenvalues as a function of cabling length |Zc| and composition (R/X)c was

Table I System Parameters

Parameter Symbol Value Load-bus nominal voltage Vnom 127 VRMS Load-bus nominal frequency fnom 50 Hz Inverter nominal apparent power Snom 1.0 kVA Connection impedance Z 0.01 pu Active power loop gain kp 1.57⋅10-3 rad/s/W (0.005pu) Reactive power loop gain kv 6.35⋅10-3 VRMS/VAr (0.05pu) Voltage offset Eo 130.175 1.025 pu Frequency offset ωo 314.9447 rad/s (1.0025 pu) Power LPF cutoff frequency ωf 2.0 Hz PSS HPF cutoff frequency ωh 0.5 Hz calculated for the system parameters shown in Table I and |Zth|=1%, (R/X)th=0.1, ks=10V/rad/s. Fig. 8 shows the curves obtained. As can be seen, theory predicts that the system damping is greatly improved by the PSS action.

III. EXPERIMENTAL RESULTS Experimental results were carried out on a system

comprised by two equal single-phase half-bridge inverters with LC output filtering (L=500μH, C=40μF). Each inverter was controlled by a multi-purpose platform based on the DSP TMS320C6713 running at 225MHz. The PWM frequency was set to 15750Hz. The whole control algorithm took only 22% (13.8μs) of the switching period. The system parameters are the same as those shown in Table I.

1007

-100 -80 -60 -40 -20 0 20-40

-30

-20

-10

0

10

20

30

40im

ag

real

ZL=2%

-200 -150 -100 -50 0 25-50

-40

-30-20

-100

10

2030

4050

imag

real

ZL=1%

- w/o PSS - with PSSXX

Fig. 7. Root loci for the system comprised of two inverters in parallel and R/X sweeping the range (10-2 102). On the left the connection impedance magnitude ZL=2%. On the right ZL=1%. In each graph two situations are considered: without PSS and with PSS (ks=10V/rad/s). The symbol ‘’ indicates the end of the range of R/X.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

ζ

| Zc | (%)

(R/X)c = 0

(R/X)c = 1

(R/X)c = 2

(R/X)c = 5

(R/X)c = 20

Fig. 8. Damping of the dominant eigenvalues as a function of the cabling length. Two situations are considered: without PSS (continuous lines) and with PSS (ks=10V/rad/s) (dotted lines). (R/X)th=0.1, Zth=1%.

The inverter internal impedance at line frequency was Zth≈(0.0012+j⋅0.01)pu, (R/X)th≈0.12. The load is resistive with RL=15.9Ω. The tests were performed with one inverter supplying a full resistive load while the other inverter was running synchronized to the load bus before the parallel connection. One oscilloscope for each inverter was used to register the variables that were output from the DSP through a D/A converter. Later the results were superimposed using the scope software Wavestar.

Fig. 9 shows the transient response for the system controlled by the traditional droop method only (PSS turned off). As can be seen, damping is very poor.

If the line impedance is changed from resistive to inductive, but maintaining its magnitude, damping is improved as expected. Fig. 10 shows this result.

Fig. 11 shows the transient response for the same system but now with the PSS turned on with ks=10V/rad/s. Line is resistive [R=0.18Ω (1.1%)] for each inverter. Damping is clearly improved as predicted by the dynamic analysis.

P1/P2: 0.3kW/div – Q1/Q2: 0.15kVAr/div – 250ms/div

Fig. 9. Experimental transient response of the system without PSS and resistive line [R=0.18Ω (1.1%)]. Top traces: active power. Bottom traces: reactive power.

P1/P2: 0,3kW/div Q1/Q2: 0,15kVAr/div 100ms/div

Fig. 10. Experimental transient response of the system without PSS and inductive line [L=500μH (1.0%)]. Top traces: active power. Bottom traces: reactive power.

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P1/P2: 0,3kW/div Q1/Q2: 0,3kVAr/div 100ms/div

Fig. 11. Experimental transient response of the system with PSS and resistive line [R=0.18Ω (1.1%)]; ks=10V/rad/s.Top traces: active power. Bottom traces: reactive power.

IV. CONCLUDING REMARKS In this paper, the PSS action, which is widely applied in the

Electrical Power Systems, was adapted to the parallel connection of voltage source inverters without control interconnections. The PSS action was modeled and included in the small-signal model for the whole parallel system, which allowed stability studies.

It was shown that the damping of a parallel inverter system can be significantly improved by applying the PSS action to the control strategy of each inverter. The sensitivity of system damping to cabling impedance composition and length was greatly reduced.

Care must be taken on the adjustment of the PSS gain ks because large output voltage deviations may result during transient conditions. The saturation of the inverter output voltage reference or the PSS output must be considered. Moreover, as the PSS modifies the inverter voltage reference, distortion in the output voltage may result due to the residual ripple present in the active power filter or during transient conditions when using high PSS gains.

Experiments were carried out on a DSP platform controlling two power inverters. The practical results achieved validated the proposed ideas.

V. ACKNOWLEDGEMENTS The authors are grateful to CAPES, CNPq and CEFET-

MG for supporting this work.

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