This paper was presented at the 6th Solar Integration Workshop and published in the workshop’s proceedings.

Modelling of Small-Scale Photovoltaic Systems with Active and Reactive Power Control for Dynamic Studies Dirk Fetzer∗ , Gustav Lammert∗ , Kai Fischbach∗ , Manuel Nuhn∗ , Johannes Weide∗ , Dario Lafferte∗ , Tina Paschedag∗ and Martin Braun∗† ∗ University

† Fraunhofer

of Kassel, Germany. Email: [email protected] Institute for Wind Energy and Energy System Technology, Kassel, Germany

Abstract—In this paper an RMS simulation model of a low voltage photovoltaic system for dynamic studies in the range of seconds up to minutes is developed. The model is based on a low voltage photovoltaic system model developed by the Western Electricity Coordinating Council (WECC) Renewable Energy Modeling Task Force. The model is validated with an experimental setup consisting of a photovoltaic emulator, an inverter, a cable and a grid emulator. The laboratory tests show an unexpected dead time of the Q(V ) and P (V ) responses which the authors could not find in published literature. The novelty, compared to state-of-the-art models, is the introduction of a dead time in order to model the measured behavior correctly. The new model is capable of explaining the grid coupled behavior of the investigated photovoltaic system in retrospective. Therefore, this work suggests more laboratory investigations on the dynamic behavior of photovoltaic systems.

The goal of this paper is to develop a model of a PV inverter suited for stability studies in a time range of several seconds up to minutes. The focus of this work lays not only on the modelling of the dynamic Q(V ) behavior, but also on the dynamic P (V ) behavior. The investigation is done by comparing a Matlab/Simulink model against an experimental setup with an off-the-shelf inverter. To show the possibilities of the developed model, a dynamic study of a real German low voltage network with several PV systems is performed. The paper is structured as follows: The modelling is presented in Section II. In Section III the laboratory tests are presented. They contain a steady state and a dynamic investigation. The test case is presented in Section IV and finally a conclusion and an outlook is given in Section V.

Index Terms—Photovoltaic generation, PV, distributed generation, dynamic modelling, simulation, power system stability, renewable energy.

II. PV SYSTEM MODELLING

I. I NTRODUCTION In recent years the complexity of medium and low voltage power grids increased significantly due to a rise of distributed generation such as wind and solar power [1]. PhotoVoltaic (PV) systems have a large share of the total installed capacity. The worldwide installed PV capacity has reached over 180 GW in 2014. Small residential PV systems represent a major portion of the total installed capacity in Europe [2]. For example, in Germany, about 65 % of the installed PV capacity is located in the low voltage grid [3]. The installed PV capacity is expected to increase in the future due to PV cost reduction [2]. An investigation on the improved grid integration of PV systems in Germany can be found in [4]. A techno-economic assessment of Q(V ) and P (V ) control can be found in [5]. An approach for the static simulation of PV systems can be found in [6]. Because of the increasing presence of PV systems in the low voltage level, it becomes important to study the dynamic behaviour of PV systems at this voltage level. Dedicated PV models are required for dynamic studies. Furthermore, aggregated models of large portions of low voltage grids for the use in stability studies of complex grids are required. The dynamic behavior of the voltage dependent reactive power injection of PV inverters according to different Q(V ) characteristics has been thoroughly investigated, e.g., in [7] and [8].

A. Generic PV system model This paper focuses on the dynamic modelling of grid connected small-scale (residential, low voltage level connected) PV systems. Various PV system models can be found in, e.g., [7], [8] and [9]. The simulation model developed in this paper is based on the generic model for distributed and small PV systems provided by the Western Electricity Coordinating Council (WECC) Renewable Energy Modeling Task Force [10]. The advantages of generic models are: They are independent of the manufacturer and the vendor, they are compatible with grid codes, their model structure is open source and the model itself is simulation platform independent [11]. Note that WECC has also developed a model that is suited for large-scale PV systems. This model is more detailed but due to the complex control not of interest in this paper. A DIgSILENT PowerFactory® implementation of a large-scale PV system model can be found in [12]. The model for distributed and small PV systems [10] was specifically developed to represent PV systems which are connected to the distribution grid. Its main features that are of interest in this paper, are: 1) A time constant which specifies the rise time of the the output. This time constant is usually in the range of 20 ms for standard inverters. 2) Furthermore, the model contains a voltage dependent reactive power characteristic for volt/var control. An example of such a Q(V ) characteristic is shown in Fig. 1. Typical parameters for Q(V ) characteristics in

Q(V )[pu]

P (V )[pu]

Q3

P2

P1

Q2

1 V1

V2

1

V3

V4

V5

Fig. 2.

Active power to voltage characteristic P (V )

Active power control Fig. 6

Q1

Fig. 1.

V [pu]

V6

V [pu]

P

Reactive power to voltage characteristic Q(V )

the high voltage level can be found in dedicated grid codes, e.g., in [13]. For the low voltage level no typical parameters have been suggested in grid codes yet.

V Measurement V’ Fig. 4 Reactive power control Fig. 5

B. Modified generic PV system model The small-scale PV model of WECC lacks some functionalities that are of interest in this paper. Therefore, the following capabilities have been added to the model: 1) A ramp rate limiter for the current output is added. In state-of-the-art inverters these rate limiters can be configured via a web interface of the real inverter. Therefore, it is important to include this feature in the model. 2) An active power to voltage characteristic, also called P (V ) characteristic, is added. In case of a high voltage at the grid coupling point, a P (V ) characteristic allows to reduce the injected active power with regard to the terminal voltage. A typical P (V ) characteristic can be seen in Fig. 2. The reduction in injected active power is realized in the inverter by leaving the Maximum Power Point (MPP). 3) A representation of the voltage measurement is added. The measurement takes place in the time range of up to 5 cycles. The resulting, refined PV model is based on the functionalities of the WECC distributed PV system model. It has to be noted that this model is only suited for small-scale PV systems. During the process of the model development, only the listed functionalities of the WECC model were implemented and then combined with the missing functionalities mentioned above. Fig. 3 shows an overview of the small-scale PV system model. The model was implemented into Matlab/Simulink. It consists of three main blocks: A representation of the voltage measurement and models of the active as well as the reactive power control. These three blocks will be described in detail in the following. For this RMS-model, the active and reactive powers are converted to RMS currents. This is represented by the block on the right in Fig. 3. Since only balanced three-phase systems are considered, this approach is similar to working with DQ0-components.

I=

Fig. 3. grids.

P − jQ I PV V 0∗

Q

Overview of the developed small-scale PV model for low voltage

The block diagram of the measurement can be seen in Fig. 4. It consists of a measuring delay, which is basically a dead time, and a first order time delay. The measuring delay is due to the measurement equipment and signal processing and is approximated with Td,meas = 5 ms. The first order time delay emulates the calculation of the moving RMS value. In recent PV inverters, five grid periods are used for calculation of the RMS value. As a 50 Hz system is considered, a value of Tmeas = 100 ms is chosen in this paper. The block diagram of the reactive power control can be seen in Fig. 5. It consists of a reactive power characteristic, two first order time delays and a rate limiter. The reactive power characteristic Q(V ) is implemented according to Fig. 1 and can be described by a discontinuous function   Q1 , V ≤ V1    Q2 −Q1 Q2 −Q1   V + Q − V , V1 < V < V2 2 2 V2 −V1  V2 −V1 Q(V ) = Q2 , V2 ≤ V ≤ V3   Q3 −Q2 Q3 −Q2  V + Q − V , V3 < V < V4  3 4 V4 −V3 V4 −V3   Q , V4 ≤ V. 3 (1) Measuring delay

V0 e

−Td,meas S

Fig. 4.

RMS calculation

1 1 + Tmeas · s

Block diagram of the measurement.

V

Q(V )

Fig. 5.

1 1 + TQ · s

Power electronics

1 1 + TQ,PE · s

Characteristic

Rate limiter

u(s)

V

Q RQ (u(s))

The time constant TQ represents the set-up time of the Q(V ) characteristics. This time is needed for the inverter to settle the output at the dedicated value. It can be configured in the web interface of the inverter. This parameter determines how fast the inverter changes its reactive power injection after a change of the terminal voltage occurred. The first order time delay constant TQ,PE emulates the physical behavior of the power electronics. As stated, e.g., in [14], this time constant can be estimated to be below 20 ms. In this paper a fast measurement equipment was assumed. Therefore, the value TQ,PE = 10 ms was used. The rate limiter RQ (u(s))) limits the positive and negative rate of change of the injected reactive power. Here u(t) is the input to the block. The rate limiter parameters can be adjusted in the inverter’s web interface as well. The block diagram of the active power control can be seen in Fig. 6. Its structure is identically to the block diagram of the reactive power control. The only difference lies in the parameterisation of the P (V ) characteristics and the time constants, which are called TP and TP ,PE . TP represents the set-up time of the injected active power and TP ,PE represents the rise time of the power electronic current output. The rate limiter RP (u(s))) limits the positive and negative rate of change of the injected reactive power. The time constants TP and TQ , that can be manually configured and are typically set in the range of few seconds up to minutes. Therefore, the short time constants of the power electronic (TP, PE and TQ, PE ) can be neglected for studies in the range of several seconds to minutes. However, the detailed eleboration is important to become aware of the different effects that appear in the inverter. III. L ABORATORY TESTS A. System overview The developed PV model was validated by laboratory testing. Fig. 7 shows the general setup. It consists of a programmable AC voltage source, a cable and an inverter connected to two emulators of solar strings. The PV emulators and the programmable AC source are shown in Fig. 8(a) and 8(b). The AC voltage source is capable of performing voltage steps in the half cylce after the command is issued. This can be seen in Fig. 8(c), where a voltage step from V ≈ 247 V to V ≈ 262 V is performed at t = 0 s. The inverter has a nominal power of 5000 VA. It has a convenient configurable web interface through which various parameters can be configured. The cable, that connects the inverter and the grid emulator, has an inductance of L = 2.63 mH and a resistance between R = 700 mΩ and R = 1200 mΩ depending on its temperature. That corresponds to an R/X ration of about 1.15. First of all, a steady state validation of the model is conducted. Thereafter, the dynamic behavior of the Q(V )

Fig. 6. Voltage Source

Rate limiter

u(s)

1 1 + TP,PE · s

1 1 + TP · s

P (V )

Block diagram of the reactive power control.

Power electronics

P T1

P RP (u(s))

Block diagram of the active power control.

1 (Slack)

2 L

I

R

PV plant V slack Fig. 7.

V

Laboratory setup for testing the small-scale PV system model.

(a) Photovoltaic emulator 400

Voltage [V]

V

P T1

200

(b) AC source Fig. 8.

Vsource (t)

0 −200 −400

Active [W] and reactive power [Var]

Characteristic

0

2

4 Time [s]

6

8 ·10−2

(c) AC source performing a voltage step Devices used for the laboratory setup.

2000

0 Q(V ) P (V )

−2000

I 0,95

II

III IV V 1,00

VI 1,05

VII

VIII

1,10

Voltage [V]

Fig. 9. Division of the Q(V ) and P (V ) characteristics into eight sections.

and P (V ) characteristics of the inverter are investigated. Therefore, the response of the inverter to voltage steps at the slack bus is investigated. The voltage steps are chosen such that the operating point of the inverter shifts from one section of the Q(V ) or P (V ) characteristic to another. To refer to the

·10−3

different sections of the characteristics, they are divided into eight segments as seen in Fig. 9. During the tests, different internal time constants of the inverter are varied.

C. Voltage dependent reactive power injection Q(V) The P (V ) and Q(V ) characteristics according to Table I are configured in the inverter. Furthermore, a set-up time for each characteristic is configured. For this experiment, set-up times of 5 s and 20 s are chosen. For validating the dynamic inverter behavior of the Q(V ) characteristic, a step of the slack bus voltage is applied. Its magnitude is chosen such that the terminal voltage of the inverter jumps from section IV to section V as described in Fig. 9. The relation between slack bus voltage V slack and inverter terminal voltage V can be derived from Fig. 7. It is V = V slack + (R + jX) · I

(2)

with R and X representing the line parameters and I is the terminal output current of the inverter. The measurement results can be seen in Fig. 12 and 13. Both figures are divided into two parts. The upper part shows the slack bus voltage V slack as well as the measured and simulated terminal voltages V meas and V sim of the inverter. The lower part shows the measured and simulated injected active and reactive power at the the inverter terminal, namely Pmeas , Psim , Qmeas and Qsim . In both cases, a voltage step at t = 0 s is performed such that the operating point moves from section IV to section V as described in Fig. 9. In the measurement results in Fig. 12, it can be seen that the injection of reactive power starts with a short time delay

Qconfig (V )

Reactive power [Var]

1

0 1,01

1,02

1,03

1,04

1,05

Voltage magnitude [pu]

(a) Comparison of the Q(V ) characteristic configured in the inverter and the measured characteristic in the laboratory.

·10−3 Qmeas (V ) Qadj (V )

2 Reactive power [Var]

The P (V ) and Q(V ) characteristics are configured into the inverter. The configured setpoints according to Fig. 1 and Fig. 2 are shown in Table I. In order to validate the steady state behavior of the inverter, the P (V ) and Q(V ) characteristics have been measured and compared to the configured characteristic. Hence, the inverter’s injected active and reactive powers have been measured for various terminal voltages. Fig. 10(a) shows the measured and configured reactive power injection of the inverter against the terminal voltage magnitude. It can be seen that there is a small offset between both curves. As the simulation model needs to emulate the correct steady state behavior the settings for the Q(V ) characteristic were adjusted in the simulation model to match the measured values. Fig. 10(b) shows the adjusted characteristic together with the measured characteristic. It can be seen that they coincide. The same procedure is done for the P (V ) characteristic. Table I shows the adjusted parameters used in the simulation model for both, Q(V ) and P (V ) characteristic. In the next step, the steady state operation points of the inverter are compared with the simulation model after the P (V ) and Q(V ) characteristics were adjusted according to Table I. Fig. 11 shows a quasi-static time variation of the slack bus voltage and the measured and simulated terminal voltages of the inverter. It can be seen that both, the Q(V ) and P (V ) characteristics are active. Furthermore, the measured and simulated values coincide. The steady state behavior was successfully validated at voltages above 1 pu.

1

0 1,01

1,02

1,03

1,04

1,05

Voltage magnitude [pu]

(b) Comparison of the Q(V ) adjusted characteristic used in the simulation and the measured characteristic in the laboratory. Fig. 10.

Steady state investigation of the inverter’s Q(V ) characteristic.

1,10 Voltage magnitude [pu]

B. Steady state validation

Qmeas (V )

2

P (V )

1,05 Q(V ) Vslack Vmeas Vsim

1,00

10

20

30

40

50

Time [s]

Fig. 11. Bus voltage at the inverter terminals after a quasi static variation of the slack bus voltage. The adjusted Q(V ) and P (V ) characteristics are used.

of about 0.4 s after the voltage step occurs. Furthermore, it can be seen that the set-up time, which the inverter needs to impose the final reactive power value, is much less than the

TABLE I O RIGINAL AND ADJUSTED SETPOINTS OF THE ACTIVE AND REACTIVE POWER CHARACTERISTICS Parameters Configured in inverter Adjusted in simulation

V1 [pu] 0.960 0.964

V2 [pu] 0.980 0.985

V3 [pu] 1.020 1.024

V4 [pu] 1.040 1.045

Q1 [kVar] −2.500 −2.337

Vslack Vmeas Vsim

1,06

V5 [pu] 1.080 1.085

V6 [pu] 1.100 1.103

P1 [kW] 2.800 2.800

P2 [kW] 0.800 0.800

Vslack Vmeas Vsim

1,04 Voltage [pu]

Voltage [pu]

Q3 [kVar] 2.500 2.210

1,06

1,04 1,02 1

1,02 1 0,98

0,98 2

4 Time [s]

6

8

3000 Pmeas Psim Qmeas Qsim

2500 2000 1500 1000 500 0 0

2

4 Time [s]

6

8

0

10

10

Fig. 12. Step response of the small-scale PV model for Q(V ) control after a step of the slack bus voltage. Top: Slack bus voltage V slack , measured and simulated voltages V meas and V sim at the inverter terminals. Bottom: Measured and simulated injected active powers Pmeas and Psim as well as measured and simulated injected reactive powers Qmeas and Qsim of the inverter. The configured set-up time for the Q(V ) characteristic is configured in the inverter’s web interface to 5 s. The adjusted time constants in the simulation are TP = 1.6 s and TD,P = 0.4 s.

set-up time of 5 s configured in the inverter’s web interface. In the measurement results in Fig. 13 it can be seen that, similar to Fig. 12, the injection of reactive power starts with a time delay after the voltage step occurs. The delay is approximitely 1.4 s. Furthermore, the set-up time which the inverter needs to impose the final reactive power value is much less than the set-up time of 20 s configured in the web interface of the inverter. To model the delay that occurs, after the voltage step took place, a dead time TD,Q is included after the Q(V ) characteristic in the reactive power control section of the PV model. This can be seen in Fig. 14. To make the simulation results match the measurement results, the dead time TD,Q and the time constant TQ are adjusted manually until sufficient overlapping is achieved. D. Voltage dependent active power injection P(V) The investigation of the dynamic P (V ) behavior is done similar to the investigation of the Q(V ) behavior presented in Sec. III-C. The P (V ) and Q(V ) characteristics were configured according to Table I and the set-up times were configured to 5 s and 20 s in the inverter’s web interface.

Active [W] and reactive power [Var]

0

Active [W] and reactive power [Var]

Q2 [kVar] 0.000 −0.050

2

4 Time [s]

6

8

10

3000 Pmeas Psim Qmeas Qsim

2500 2000 1500 1000 500 0 0

2

4 Time [s]

6

8

10

Fig. 13. Step response of the small-scale PV model for Q(V ) control after a step of the slack bus voltage. Top: Slack bus voltage V slack , measured and simulated voltages V meas and V sim at the inverter terminals. Bottom: Measured and simulated injected active powers Pmeas and Psim as well as measured and simulated injected reactive powers Qmeas and Qsim of the inverter. The configured set-up time for the Q(V ) characteristic is configured in the inverter’s web interface to 20 s. The adjusted time constants in the simulation are TP = 5.8 s and TD,P = 1.4 s. Characteristic Dead time

V Q(V )

Fig. 14.

e

−TD,Q s

P T1

1 1 + TQ s

Power electronics

1 1 + TQ,PE s

Rate limiter

Q RQ (u(s))

Block diagram of the reactive power control with dead time.

The magnitude of the voltage step is chosen such that the terminal voltage of the inverter jumps from section VI to section VII as described in Fig. 9. The measurement results can be seen in Fig. 15 and 16. The graphs are structured identically to Fig. 12 and 13. For a detailed description of the structure, see Sec. III-C. In the measurement results in Fig. 15 it can be seen that the injection of active power starts with a time delay of about 0.5 s after the voltage step occurs. Furthermore, the time that is needed for the reduction of active power injection is less than the setup time of 5 s configured in the inverter’s web interface. In Fig. 16 it can be seen that the injection of active power starts with a time delay of about 2.2 s after the voltage step occurs. Furthermore, the time that

Vslack Vmeas Vsim

Active [W] and reactive power [Var]

Voltage [pu]

1,09

1,08

1,07

0

2

4 Time [s]

6

8

Pmeas Psim Qmeas Qsim

2600 2400 2200 2000

0

2

4 Time [s]

6

8

10

Fig. 15. Step response of the small-scale PV model for P (V ) control after a step of the slack bus voltage. Top: Slack bus voltage V slack , measured and simulated voltages V meas and V sim at the inverter terminals. Bottom: Measured and simulated injected active powers Pmeas and Psim as well as measured and simulated injected reactive powers Qmeas and Qsim of the inverter. The configured set-up time for the P (V ) characteristic is configured in the inverter’s web interface to 5 s. The adjusted time constants in the simulation are TP = 2.5 s and TD,P = 0.5 s.

is needed for the reduction of active power injection is less then the setup time of 5 s configured in the inverter’s web interface. As done in Sec. III-C, the model of the active power control is modified in order to simulate the physical behavior correctly. Therefore a dead-time TD,P was included after the P (V ) characteristics. The modified model of the active power control can be seen in Fig. 17. In order to make the simulation results match the measurement results, the dead time TD,P and the time constant TP are adjusted until sufficient overlapping is attained. E. Discussion of measurement results The three main observations from the measurements are: First, the P (V ) and Q(V ) characteristics configured in the inverter have a slight offset, compared to the measurements. The offset is smaller than 0.005 pu. Second, the time delay, configured in the inverter, did not correspond to the time constant of the PT1 block, used in the simulation model. Third, the measurements showed a dead time that was not expected. This dead time is not known before the measurement is done and it varies with the set-up time that is configured in the inverter’s web interface. To match the simulation and the measurement results, the characteristics, the time constant and the dead time of the simulation model had to be adapted manually. After adaptation of these parameters, a good match between measurements and simulations was obtained. P (V ) and Q(V ) controls are not yet part of low voltage grid codes. Therefore the behavior of the inverter in the investigated situations is not standardized yet. Also the internal control loops of the inverter are not known in detail. Therefore, the

1,09

1,08

1,07

10

3000 2800

Vslack Vmeas Vsim

1,1

Active [W] and reactive power [Var]

Voltage [pu]

1,1

0

2

4 Time [s]

6

8

10

3000 Pmeas Psim Qmeas Qsim

2800 2600 2400 2200 2000

0

2

4 Time [s]

6

8

10

Fig. 16. Step response of the small-scale PV model for P (V ) control after a step of the slack bus voltage. Top: Slack bus voltage, measured voltage at the inverter terminals. Bottom: Measured and simulated active and reactive power injection of the inverter. Configured inverter hardware setting for the P (V ) set-up time is 20 s. Adjusted parameters in the simulation are TP = 10.6 s and TD,P = 2.2 s. Characteristic Dead time

V P (V )

Fig. 17.

e

−TD,P S

P T1

1 1 + TP s

Power electronics

1 1 + TP,PE s

Rate limiter

P RP (u(s))

Block diagram of the active power control with dead time.

presented model can only be an approximation of the real inverter behavior. IV. S IMULATION TEST CASE To show a possible application of the investigated PV model, a larger test case was composed in Matlab/Simulink. It consists of a real German 0.4 kV low voltage grid that is connected via a transformer to a 20 kV medium voltage grid. The general setup can be seen in Fig. 18. The LV grid consists of 234 nodes. Altogether 20 PV systems with a total installed apparent power of 250 kVA were included in the system. The P (V ) and Q(V ) characteristics were implemented for all PV systems. For the investigation, a low load setup was selected, where the total injected PV power is much higher than the total load consumption. The dynamic behavior of the low voltage grid was investigated for two configurations, as described in Table II. In the first configuration, the PV systems were placed as far away from the slack bus as possible, namely at the end of the feeders. In the second configuration, the PV systems were placed near the slack bus, namely at the beginning of the feeders. Fig. 19 shows the reverse power flow from the low voltage grid into the medium voltage grid after a voltage step at

TABLE II D ESCRIPTION OF THE TWO SIMULATED CONFIGURATIONS Configuration 1 2

V. C ONCLUSION

Description PV systems are far from slack node PV systems are close to slack node

Slack bus

I LV grid

V slack

Voltage V [pu]

Fig. 18.

Active power [pu]

Vslack

1,1 1,0 0,9

0

5

10 Time [s]

15

20

Pcase1 Pcase2

2,0 1,8 1,6 0

Reactive power [pu]

Setup of the low voltage grid test case.

5

1,0

10 Time [s]

15

20

ACKNOWLEDGMENT Qcase1 Qcase2

0,8

0,6

0

5

10 Time [s]

In this paper a small-scale RMS model for low voltage PV systems is developed, based on a distributed PV system model proposed by WECC. The main focus of the work is to emulate the dynamic behavior of the voltage dependent active and reactive power injection via P (V ) and Q(V ) characteristics. The developed simulation model is validated against an experimental setup with an off-the-shelf inverter in the power range of 5000 VA. The new contribution of this paper is to show that there are dead times in the measurement data that change significantly with the set-up time configured in the inverter. However, no connection between these two variables can be found. As far as the authors know, measurements of such dead times have not been reported in literature yet. By including these dead times into the PV model, the simulation and the measurement data match. However, to achieve that, the parameters of the model have to be tuned manually after the measurement was performed. Thus, the behavior of the inverter can not be determined by simulation prior to conducting the experiment. The main outcome of this work is, that the generic models, used as a basis for this work did not give adequate results for the specific inverter used. As the cause of the dead time is not known by the authors, it will be necessary to conduct future work on validating PV system models. Also it will be beneficial to test more scenarios in the laboratory.

15

20

Fig. 19. Step response of the slack bus active and reactive power after a voltage step at the slack bus at t = 0 s.

the medium voltage side. The voltage step at the slack bus occurs at t = 0 s. The voltage rise at the slack bus results in voltage rises at each bus in the system. The voltage rise at the terminals of the PV systems is such that their operating points shift into sections where the active power injection is reduced and where more reactive power is injected. Thus, the operating point of some inverters shift into the sections VII and VIII (see Fig. 9). Due to this, the injected active power is reduced. In the investigated grid, the voltage at each feeder is highest at its end points (for configurations of high PV power and low load). Therefore, the active power injection in configuration 1 is generally lower than the one in configuration 2. This can also be seen in Fig. 19. For the same reason, the injected reactive power is higher than in configuration 2. The high peak in active and reactive power at t = 0 s is only due to the simulation method and can therefore be neglected. The dynamic behavior of the slack bus active and reactive powers is stable in both configurations.

We thank EnergieNetz Mitte, Kassel, for providing the grid data for the simulation test case. The authors thank Bernd Gruß from University of Kassel for his dedicated support during the lab experiments. Also, the authors thank Florian Sch¨afer from University of Kassel for the fruitful discussions. This work was supported by the German Federal Ministry for Economic Affairs and Energy and the Projekttr¨ager Julich GmbH (PTJ) within the framework of the project OpSim (FKZ: 0325593B). R EFERENCES [1] A. Monti and F. Ponci, “Power grids of the future: Why smart means complex,” in Complexity in Engineering, 2010. COMPENG ’10., Feb 2010, pp. 7–11. [2] F. T. Manoel Rekinger, “Global Market Outlook for Solar Power / 2015 - 2019,” SolarPower Europe, Technical Report, 2014. [3] J. von Appen, M. Braun, T. Stetz, K. Diwold, and D. Geibel, “Time in the sun: The challenge of high pv penetration in the german electric grid,” IEEE Power and Energy Magazine, vol. 11, no. 2, pp. 55–64, March 2013. [4] T. Stetz, F. Marten, and M. Braun, “Improved low voltage gridintegration of photovoltaic systems in germany,” IEEE Transactions on Sustainable Energy, vol. 4, no. 2, pp. 534–542, April 2013. [5] T. Stetz, K. Diwold, M. Kraiczy, D. Geibel, S. Schmidt, and M. Braun, “Techno-economic assessment of voltage control strategies in low voltage grids,” IEEE Transactions on Smart Grid, vol. 5, no. 4, pp. 2125–2132, July 2014. [6] D. Fetzer, G. Lammert, S. Gehler, J. Hegemann, and M. Braun, “A bisection newton power flow algorithm for local voltage control strategies,” Power Systems, IEEE Transactions on, submitted for publication. [7] R. W. Marco Lindner, “A dynamic rms-model of the local voltage control system q(v) applied in photovoltaic inverters,” 23rd International Conference on Electricity Distribution, Lyon, 15-18 June 2015.

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