Influence of the Grid Impedance on the Operating Range of nParallel Connected Inverters

Jan Reese, Friedrich W. Fuchs

Institute for Power Electronics and Electrical DrivesChristian-Albrechts-University of Kiel

Kaiserstr. 2, 24143 Kiel, GermanyEmail: jre@tf.uni-kiel.de, fwf@tf.uni-kiel.de

Abstract

Parallel operating inverters of distributed energy resources are coupled by the grid impedance.As the grid impedance has an impact on the performance of the inverters, the influence of thiscoupling has to be analyzed. In this paper the effects of the grid impedance on the operatingrange of each inverter is investigated for different grid conditions. The investigation is carriedout analytically based on the model of the equivalent output impedance of a single inverter andis validated by simulation results.

1 Introduction

The constant increasing number of installations of photovoltaic and wind power systems resultsin a grid structure, in which a lot of inverters are connected in parallel to a point of commoncoupling (PCC). This coupling can result in undesired behavior or even instability [1, 2]. Fur-thermore the requirements, set by the grid operators, have to be fulfilled by the connecteddistributed energy systems. For example, distributed energy resources (DER) have to maintainthe quality of the grid voltage by emitting current with a defined maxmim harmonic content.The reduction of the harmonics can be limited either by control or filter design [4]. Since thepassive filter components directs to an increasing volume and losses, usually the filter designis optimized together with the modulation of the output voltage. LCL-filter design is widely usedin industry applications to filter the harmonic content. The drawback of the existing resonancefrequency is well investigated and several damping approaches are published [6, 5]. Further-more DER are requested to be able to operate at power factors between cos(ϕ)=0.95 inductiveand capacitive even at a voltage level of vgrid=1.1 [p.u.] to support the grid voltage at the fun-damental frequency [3]. These requirements will get more comprehensive in future, since theinfluence of DER on the grid voltage grows.

As the density of DERs at the PCC is increasing, the requirements are getting more complex toensure grid stability and a proper grid voltage quality. Thus an accurate model of the couplingand the repercussion gains more importance, taking the operating conditions into account. Ap-proaches for modeling a cluster of inverters have been published in the last years for differentapplications [1, 7]. In this contribution the influence of the grid impedance is investigated withrespect to the operating range of a single inverter. The investigations are based on the modelof the equivalent output impedance.

This contribution is organized as follows: At first the approach of modeling n parallel actinginverters is given. Based on this model the characteristics of the equivalent output impedanceis analyzed for an increasing number of parallel inverters. Secondly the influence of the gridimpedance on the operation range of a single inverter is investigated. Simulation results, given

in the forth section, point out the influence of the number of parallel inverters together with thethe grid impedance on the harmonic content as well as on the operating range. A conclusionsums up the results and the key point of this contribution.

2 Model of n Parallel Inverters

For a single connected voltage source inverter (VSI) the equivalent output impedance can becalculated:

Zeq,single = Zfi +Zfc ·

(Zfg + Zgrid

)Zfc + Zfg + Zgrid

(1)

In eq. (1) Zgrid is the grid impedance and Zfi, Zfg, Zfc are the impedances of the filtercomponents, being inverter side filter inductance, grid side filter inductance and filter capacitor,Fig. 1(a). Since the impedance of the filter capacitor Zfc is very high for the power flow at thefundamental frequency, it can be neglected. The equivalent output impedance is obtained bythe sum of the inductive impedances:

Zeq,single ≈ Zfi + Zfg + Zgrid (2)

In Fig. 1(b) the vector diagram for a single connected inverter is depicted, indicating the requiredoutput voltage of the inverter vinv for two assumed operation points. The shaded area marks the

(a) (b)

Figure 1: (a) Equivalent circuit diagram of a single inverter connected to the grid and (b) vectordiagram of a single connected inverter for two possible operating points: cos(φ)=1(green) and 0<cos(ϕ)<1 inductive (red)

required inverter output voltage, which cannot be provided, since the required output voltageexceeds the limit vinv,max, defined by the DC-link voltage. Hence the operating range is reducedin this case.

The same examination is done for n parallel inverters with the same parameters are connectedto the PCC and acting at the same setpoint, Fig 2(a). By using the superposition principle for nparallel inverters the equivalent ouput impedance Zeq for a single inverter model in this networkcan be modeled. Based on the mentioned assumptions the grid current igrid will be the sum obthe ouput current of each inverter, ifg,j :

igrid =

n∑j=1

ifg,j = n · ifg (3)

(a) (b)

Figure 2: (a) Circuit diagram of parallel inverters connected to the grid and (b) equivalent circuitof a single inverter

Regarding eq. (3) and Fig. 2(a), the voltage drop across Zgrid is scaled by the number ofparallel inverters. Therefore, in case of n parallel inverters the equivalent grid impedance fora single inverter is n times bigger as in the single connection case. Thus, if n inverters areconnected in parallel, the equivalent output impedance for each inverter Zeq is given by:

Zeq = Zfi +Zfc ·

(Zfg + n · Zgrid

)Zfc + Zfg + n · Zgrid

(4)

Using eq. (4) among the above mentioned conditions the impedance characteristic can bestated dependent on Zgrid and the number of parallel inverters n.

Figure 3: Characteristic of the equivalent output impedance of a single inverter operating inparallel for a variation of Zgrid;|Zgrid| ∈ 0, 09...0.45Ω; (a) magnitude of Zeq and (b)phase of Zeq against number of parallel inverters and phase angle of Zgrid

Fig. 3 shows the magnitude and phase characteristics of the equivalent output impedanceat the fundamental frequency for a given filter setup (table 1). As depicted in Fig. 3(a) themagnitude of Zeq is increasing with the number of parallel inverters, in which the increasingcharacteristic strongly depends on the phase and amplitude of Zgrid. Depending on the phaseof Zgrid the increasing characteristic is changing from linear to exponential. This is because ofthe existing resonance frequency fres of Zeq, which is moving towards lower frequency for anincreasing number of inverters:

fres =1

2π·

√1

Cf·(

1

Lfi+

1

Lfg + n · Lgrid

)(5)

For low voltage systems the ratio Xgrid/Rgrid < 1, and therefore the impedance angle is be-tween 0 < ∠(Zgrid) < π/4. Especially in the case of resistive grid impedance, the equivalentoutput impedance tends to be capacitive for parallel inverters. This affects the control of dis-tributed energy resources and has to be considered for the control design of grid connectedinverters.

3 Influence of the Grid Impedance on the Operating Range

The operating range is defined by the setpoints cos(φ) of an inverter, which can be achieved bythe modulation of the PWM. In this contribution the PWM it is limited to the linear modulation.The modulation index M , given by eq. (6), is defined for the fundamental voltage 1vinv,ref andis limited to a theoretical value Mmax = 2√

3[8].

M =1vref,phase

vDC2

=1Zeq ·1 ii +1 vgrid

vDC2

≤ 2√3

(6)

In the following the operating range of a single inverter operating in parallel is analyzed. Asmentioned before, the inverters connected to the grid are supposed to be able to deliver ca-pacitive reactive power (cos(φ)<0) aswell as inductive reactive power (cos(φ)>0). Thus, in thiscontribution the constraints of the requested operation range are defined to cos(φ)=0.8 induc-tive and capacitive. For analysis, the filter parameters are set to those given in Table 1.

Table 1: parameters for analysis and simulation

Quantity Parameter Valuegrid rated voltage Vrated 400 V

(setup 1) resistance Rgrid 300 mΩinductance Lgrid 60µ H

grid short-circuit power SSC 532 kVA(setup 2) resistance Rgrid 0.03 mΩ

inductance Lgrid 950µ Hgrid short-circuit power SSC 532 kVA

inverter rated power Srated 6.25 kVArelative power SSC/Srated 85.12

filter converter side inductance Lfi 4 mH (0.04 [p.u.])line side inductance Lfg 4 mH (0.04 [p.u.])

filter capacitor Cf 4µ F (31 [p.u.])

Fig. 4 shows the required modulation indices for these operating conditions. As depicted inFig. 4(a), the more resistive the grid impedance Zgrid the higher the modulation index M for alloperating points. Furthermore, with an increasing number of parallel inverters the maximum ofthe requested modulation index is shifting towards minor angles φ, since Zeq tends to be moreresistive and the voltage drop across Zgrid is in phase with the active current component. Byincreasing the numbers of parallel inverters even more, the characteristic of Zeq will turn to becapacitive as shown in fig. 3(b).

−0.8−0.9

10.9

0.8

2

4

6

8

10

1

1.05

1.1

1.15

1.2

1.25

# Inverter

Cos( φ )

M

(a)

−0.8−0.9

10.9

0.8

2

4

6

8

10

0.95

1

1.05

1.1

1.15

1.2

1.25

# Inverter

Cos( φ )M

(b)

Figure 4: Required modulation index M against the number of connected inverters and cos(φ);(a) Lgrid = 60µH and a variation of Rgrid = 20..300mΩ (b) Rgrid = 20mΩ and avariation of Lgrid = 60..600µH

Fig. 4(b) shows the required modulation degree for an increasing inductive component of thegrid impedance. The supply of inductive reactive power to the grid results to a voltage dropacross Zgrid in phase with the active current component. On the other hand, the supply ofcapacitive power is eased due to the inductive consumption of Zeq which is increasing with thenumber of inverters as well. In both cases the operating range of a single inverter is reduceddue to parallel acting inverters.

4 Simulation Results

To validate the effects of parallel inverters the simulation results in Fig. 5 and Fig. 6 showthe control behavior of a single inverter in time domain under different conditions. Besides thevoltages at the PCC, vPCC , and the injected currents iinv of an inverter the required modulationindex is depicted to show the influence of the grid impedance and of the operating point of theinverter. Furthermore the number of parallel acting inverters is given as well as the injectedpower of each inverter.

Each figure shows three different operating points: At first a single inverter is injecting nominalactive power to the grid. At t = 0.15 s additional 20 inverters are switched on and acting at thesame setpoint as the first inverter. Finally, at t = 0.2 s the setpoint of all parallel inverters ischanging.

In Fig. 5 the control bahavior of an inverter is given, connected to the grid through a resistivegrid impedance with the parameters of setup 1, table 1. Due to the increasing number of par-allel inverters, the amplitude of vPCC is increasing. As the injected power of a single inverter

is constant, the amplitude of the injected current is dropping. While supply capacitive reactivepower to the grid the modulation index is dropping, as Zeq is inductive. On the other side, therequired modulation index for a single inverter is increasing for an injection of inductive reactivepower.

0.05 0.1 0.15 0.2 0.25

−1

0

1

v PC

C/ v

PC

C,N

0.05 0.1 0.15 0.2 0.25

−1

0

1

i inv/ i

inv,

N

0.05 0.1 0.15 0.2 0.250.4

0.6

0.8

1

1.2

M =

2*v

inv,

req/U

DC

0.05 0.1 0.15 0.2 0.250

10

20

# in

vert

er

0.05 0.1 0.15 0.2 0.25−5

0

5

P,Q

[kV

A]

t [s]

(a)

0.05 0.1 0.15 0.2 0.25

−1

0

1

v PC

C/ v

PC

C,N

0.05 0.1 0.15 0.2 0.25

−1

0

1

i inv/ i

inv,

N

0.05 0.1 0.15 0.2 0.250.4

0.6

0.8

1

1.2

M =

2*v

inv,

req/U

DC

0.05 0.1 0.15 0.2 0.250

10

20#

inve

rter

0.05 0.1 0.15 0.2 0.25−5

0

5

P,Q

[kV

A]

t [s]

(b)

Figure 5: Simulation results: required modulation index of a single inverter for a resistive gridimpedance (setup 1); supply (a) capacitive reactive power (b) inductive reactive power

Fig. 6 shows the control behavior of an inverter in case of an inductive grid impedance withthe parameters given in table 1 - setup 2. For this grid impedance the resistive component ofZeq is negligible even for a higher number of parallel acting inverters. This is shown in fig. 3.Thus the required modulation index is not affected during the injection of nominal active power.Nevertheless, due to the voltage drop across Zgrid, which is not in phase with the voltage vPCC

is distorted.

For an injection of capacitive reactive power, this effect gains more importance. In contrast,while injecting inductive reactive power vPCC returns to a better sinusoidal shape, Fig. 6(b).Additionally the resonance of the LCL-filter is damped worse, because of the lower resistivecomponent of Zeq. Finally, comparing the settling time for the reactive power injection in fig.5 and 6, the dynamics for an resistive grid impedance increase with the number of paralleledinverters, because of the changing phase characteric of Zeq for parallel inverters.

Summing up the simulation results, the control of a single inverter in terms of the operatingrange can be described with the equivalent output impedance for parallel application as well.

0.05 0.1 0.15 0.2 0.25

−1

0

1

v PC

C/ v

PC

C,N

0.05 0.1 0.15 0.2 0.25

−1

0

1

i inv/ i

inv,

N

0.05 0.1 0.15 0.2 0.250.4

0.6

0.8

1

1.2

M =

2*v

inv,

req/U

DC

0.05 0.1 0.15 0.2 0.250

10

20

# in

vert

er

0.05 0.1 0.15 0.2 0.25−5

0

5

P,Q

[kV

A]

t [s]

(a)

0.05 0.1 0.15 0.2 0.25

−1

0

1

v PC

C/ v

PC

C,N

0.05 0.1 0.15 0.2 0.25

−1

0

1

i inv/ i

inv,

N

0.05 0.1 0.15 0.2 0.250.4

0.6

0.8

1

1.2

M =

2*v

inv,

req/U

DC

0.05 0.1 0.15 0.2 0.250

10

20

# in

vert

er

0.05 0.1 0.15 0.2 0.25−5

0

5 P

,Q [k

VA

]

t [s]

(b)

Figure 6: Simulation results: required modulation index of a single inverter for an inductive gridimpedance (setup 2); supply (a) capacitive reactive power (b) inductive reactive power

The influence of the grid impedance on the operating range gains more importance if the num-ber of connected inverters increases. As the phase characteristic of the equivalent outputimpedance is changing with an increasing number of inverters, this has to be considered incontrol.

5 Conclusion

In this contribution the influence of the grid impedance on the operating range of parallel con-nected inverters is analyzed. Based on a model for the equivalent output impedance of aninverter the influence of parallel connected inverters is investigated for different grid conditions.For a resistive grid impedance the characteristic of the output impedance is changing for theconnection of parallel inverters. Thus, for a resistive grid impedance the amount of connectedinverters affects the operating range as well as the dynamics of the control. For an inductivegrid impedances, the number of parallel connected inverters has an impact on the resonancefrequency and therefore on the stability. The influence on the operating range is less, sincethe operating range is limited to cos(φ)=0.8. The analytical investigations are validated by sim-ulation results. Nevertheless, the assumptions made in this contributions may differ from realconditions, as the setpoint for each inverter may be different. The operating range of a singleinverter can be described with the equivalent output impedance for parallel application and fordifferent grid conditions.

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