Integration of Voltage Dependent Power Injections of Distributed Generators intothe Power Flow by using a Damped Newton Method

Dirk Fetzera,∗, Gustav Lammerta, Stefan Gehlera, Jan Hegemanna, Robert Schmolla, Martin Brauna,b

aDepartment of Energy Management and Power System Operation, University of Kassel, GermanybFraunhofer Institute for Wind Energy and Energy System Technology, Kassel, Germany

Abstract

Voltage dependent active and reactive power injections are used to improve the grid integration of distributed generators,such as photovoltaic systems. In state-of-the-art literature these power injections are considered during power flowcalculations via an external loop, which requires to perform multiple (up to ten or more) power flow calculations in orderto obtain one static operation point. In this paper it is shown how to integrate voltage dependent power injections intothe nonlinear grid equations in order to solve for a static working point by means of performing one Newton power flowcalculation. That makes the external loop unnecessary and save computational time. Furthermore, it is shown for atwo-bus system that the integration of voltage dependent power injections into the nonlinear grid equations can exhibitill-conditioned power mismatch functions. Therefore, a damped Newton method is used in order to avoid numericaloscillations. Simulations of the IEEE 118-bus test system and a real German 234-bus test system show that the dampedmethod converges faster and is computationally more efficient than the external approach currently used in literatureand state-of-the-art simulation tools in power systems. In addition, the implemented power flow algorithm is validatedin the laboratory for a two-bus system and a numerical example of a low-voltage feeder is given.

Keywords: Distributed generation, ill-conditioned system, local voltage control, power flow, damped Newton’s method2017 MSC: 00-01, 99-00

1. Introduction

In recent years, the penetration of medium and lowvoltage grids with inverter coupled distributed generators(DGs) has increased significantly and is expected to keepgrowing [1, 2]. The intermittency and high amount of ac-5tive power infeed of DGs can cause voltage fluctuations[3]. In order to adhere to the respective voltage qual-ity standards like IEC 60038 [4] or EN 50160 [5] DGscan be equipped with voltage dependent active and re-active power injections, also called Volt/VAR control [6],10volt/watt control or local voltage control [7, 8]. The intro-duction of voltage dependent power injections is, amongstothers, motivated by the standard IEEE 1547 [9] in orderto mitigate voltage rises caused by DGs [10, 11, 12, 13].In order to conduct realistic power system studies like grid15planning it is necessary to take voltage dependent powerinjections of DGs into account during the calculation ofstatic operation points, namely during the power flow.

The state-of-the-art method (to take voltage dependentpower injections of DGs into account during the calcula-20tion of a static operation point) consists of performing mul-tiple power flows. This is done by updating the injectedactive and reactive power after each power flow iteratively

∗Corresponding authorEmail address: dirk.fetzer@uni-kassel.de (Dirk Fetzer)

until a convergence is reached [10, 11, 14]. We refer to thissolution method as external algorithm in order to simplify25the further explanations given in this paper.

The goal of this paper is to show how voltage depen-dent power injections of DGs can be integrated into theNewton (or Newton-Raphson) power flow algorithm [15].This is done by incorporating the voltage dependent power30injections into the nonlinear grid equations and using adamped Newton method presented in [16]. We refer tothis solution method as internal algorithm in order to dis-tinguish between the proposed solution method and thestate-of-the art method. The damped (underrelaxed, mod-35ified) method used in this paper is presented in [16] underthe name bisection method. It is based on famous workdone in the 19th and 20th century, e.g., [17, 18, 19]. Arecent overview can be found in [20]. The main contri-bution of this paper is the application of the well-known40damped Netwon method to the problem of incorporatingvoltage dependent power injections of DGs into the powerflow problem. Another contribution of this paper is toshow that voltage dependent power injections can cause ill-conditioned power mismatch functions, which is not very45common for power systems according to [16]. Due to this,the use of the damped Newton method is required in or-der to obtain a converging power flow. Furthermore, theinternal algorithm is compared to the external algorithm.We conduct this comparison to show the benefits of our ap-50

Preprint submitted to International Journal of Electrical Power & Energy Systems January 30, 2018

© 2018 Elsevier Ltd. All rights reserved.

DOI: https://doi.org/10.1016/j.ijepes.2018.01.049

proach compared to the state-of-the-art method. Also, thiscomparison is used to validate the computation results ofthe internal algorithm. Both algorithms are implementedin MATLAB[21], they are stand-alone and can be usedindependently of each other. We use an ill-conditioned55two-bus system to show that the internal algorithm canhave superior convergence properties compared to the ex-ternal algorithm. Furthermore, the internal algorithm isvalidated for the case of a two-bus system with a labo-ratory setup. We apply the proposed internal algorithm60to the IEEE 118-bus test system and a real German 234-bus grid and demonstrate that for these cases, the internalalgorithm is computationally faster than the external al-gorithm.

The paper is structured as follows: We outline the mod-65eling of voltage dependent power injection strategies inSection 2. In Section 3 we describe the state-of-the-art ex-ternal algorithm and describe the internal algorithm. Testcases are presented in Section 4. In Section 5 a numericalexample of a 19 node low-voltage test feeder with five pho-70tovoltaic systems is presented. In Section 6 we summarizethe findings of the paper and highlight the advantages ofthe internal algorithm.

2. Modeling of Voltage Dependent Power Injec-tion75

Power injection methods of DGs can be categorizedinto two groups, as shown in Table 1. Group (a) is charac-terized by a reactive power injection that is either constantor depends on the injected active power. Here, ϕ is theangle of the injected complex power and cos(ϕ) is called80power factor. DGs which employ one of these strategiesare treated as constant power loads (i.e. P,Q constant)during a power flow. Therefore, no further elaboration isneeded for this group from an algorithmic point of view asthese are not voltage dependent. Injection methods from85control group (a) are practically used by transmission anddistribution system operators and are specified in technicalstandards, such as VDE-AR-N 4120 [22].

Group (b) is characterized by a voltage dependent in-jection of active and/or reactive power. To consider this90group within a power flow calculation, the voltage depen-dency has to be taken into account. The voltage depen-

Table 1: Voltage Dependent Power Injection for Distributed Gener-ators

Group Description Injection Method

(a)

Reactive power injection Qconst.,

is constant or depends cos(ϕ)const.,

on the injected active power. cos(ϕ)(P )

(b)

Active and/or reactive P (V ),

power injection depend Q(V )

on the bus voltage magnitude.

dency can be described with a piecewise function f(V )as shown in Fig. 1. This function represents a unifiedapproach for both voltage dependent active and reactive95power injection. The maximum active and reactive power,depicted as y3, is the maximum possible active and re-active power outputs of the DG in the current operatingstate. The piecewise function f(V ) can be described as

f(V ) =

y1, V ≤ V1y1 + γ1 · (V − V1), V1 < V ≤ V2y2, V2 < V ≤ V3y2 + γ2 · (V − V3), V3 < V ≤ V4y3, V4 < V

(1)

where the gradients are γ1 =y2−y1V2−V1 and γ2 =

y3−y2V4−V3 .100

The method proposed in this paper requires to cal-culate the derivative of (1). The derivative of (1) withrespect to the voltage magnitude is a piecewise constantfunction g(V ) with

g(V ) =∂f

∂V=

0, V < V1

γ1, V1 < V < V2

0, V2 < V < V3

γ2, V3 < V < V4

0, V4 < V.

(2)

The derivative g(V ) is discontinuous and not defined at thepoints V1, V2, V3, V4. To overcome this issue, a smoothingfunction will be introduced in subsection 2.1 .

By choosing appropriate parameters V1, V2, V3, V4, y1,y2, y3 the reactive and active power characteristics can beexpressed by different realizations of f(V ). It is:

P (V ) = fP(V ) (3a)

Q(V ) = fQ(V ). (3b)

V [pu]

f(V )[pu]

1 V3 V4V2V1

y3

y2

y1

Figure 1: Piecewise function f(V ) for P (V ) and Q(V ) voltage de-pendent power injection. This is a generalized depiction of a charac-teristic. Positive and negative slopes are possible.

2

The choice of the parameters is typically done by the105grid operator based on internal grid planning and oper-ation guidelines. These guidelines may vary for differentoperators as well as countries.

2.1. Smoothing function

To avoid the issue of the discontinuity of the derivativeg(V ), the following smoothing function for f(V ) is used

fs(V ) =y1

+γ1k

[ln(1 + ek(V−V1))− ln(1 + ek(V−V2))

]+γ2k

[ln(1 + ek(V−V3))− ln(1 + ek(V−V4))

] (4)where the difference between the smoothing function andthe original function is reduced by increasing values of k.The derivative is

gs(V ) =γ1 [logsig(k(V − V1))− logsig(k(V − V2))]+γ2 [logsig(k(V − V3))− logsig(k(V − V4))]

(5)

with logsig(x) = 11+e−x being the log-sigmoid function.110Fig. 2 shows the original piecewise linear function f

from (1) and its derivative g from (2) together with thesmoothing functions fs from (4) and its derivatives gs from(5) for different values of k. The parameters chosen arey1 = 0.18, y2 = 0, y3 = −0.18, V1 = 0.93, V2 = 0.97, V3 =1151.03, V 4 = 1.07 and thus γ1 = γ2 = −4.5. By analyzingthe curves it can be seen that a choice of k = 600 resultsin a reasonable approximation of (1) as is shown in Fig. 2.

3. Integration into the Power Flow

In Section 3.1 the power flow problem is presented.120More details concerning power flow can be found in vari-

Figure 2: Smoothing function fs and its derivative gs for differentvalues of k compared to the original piecewise linear function f andits derivative g.

ous literature such as [16, 23, 24]. In Section 3.2 the exter-nal algorithm is explained. It is the present state-of-the-artmethod for considering voltage dependent power injectionsin a power flow calculation. Section 3.3 presents the pro-125posed novel internal algorithm. Figures 3 and 4 show thedetails of the implementation of both algorithms.

3.1. Power Flow Formulation

The power flow problem is defined as follows: Assumea power system with n nodes and m DGs with a volt-130age dependent power injection. Without loss of generalityit is assumed that the generators are connected to nodes1, ...,m. It is further assumed, that nodes m+ 1, ..., n− 1are connected to constant power loads and thus are mod-eled as PQ nodes. Node n is chosen as slack node. The135active and reactive power characteristics of the DG con-nected to a node k ∈ {1, ...,m} are described by the piece-wise function Pk = fPk(Vk) and Qk = fQk(Vk), where Vkis the voltage magnitude of node k. A combination of aDG and a constant power load at the same node can be140implemented by adjusting the parameters y1, y2, y3 ap-propriately. If a DG with settings y1, y2, y3 for a P (V )characteristic is present at a bus and a constant power loadof power Pconst is also present at the same bus, then thenew setting for the characteristic is y1new = y1 − Pconst,145y2 new = y2−Pconst and y3 new = y3−Pconst. The parame-ters y1, y2 and y3 denote a generation whereas Pconst is aload, therefore they are subtracted from each other.

For a valid steady-state solution of the system the in-jected power

Sk = Pk + jQk (6)

into each node k ∈ {1, ..., n − 1} has to equal the sum ofthe branch power flows

Sk` = Pk` + jQk` (7)

from node k to all connected nodes `. The power mismatchequation describes the difference between those two quan-tities. It is Sk` = 0 in case that node ` is not connected tonode k. The power mismatch F k at node k is

F k = Sk −n∑`=1` 6=k

Sk`. (8)

As the local voltage characteristic depends on the volt-age magnitude, it is practical to use the polar form. Thevoltage magnitude and angle at node k are denoted withVk and δk. Thus, the state vector is given by

x =(V1, ..., Vn−1, δ1, ..., δn−1

)T(9)

where x ∈ R2n−2. The angle difference between the volt-age at bus k and bus ` is defined as

δk` = δk − δ`. (10)

3

The k`-th element of the bus admittance matrix is definedas

Y k` = Gk` + jBk` (11)

where Gk` is the conductance and Bk` is the susceptanceof the branch connecting nodes k and `. The active andreactive power mismatch at node k is

Re(F k) = Pk(Vk)− Vkn∑`=1` 6=k

V`(Gk` cos δk` +Bk` sin δk`)

(12a)

Im(F k) = Qk(Vk)− Vkn∑`=1` 6=k

V`(Gk` sin δk` −Bk` cos δk`).

(12b)

With this, the vector of all power mismatches is

F (x) =(Re(F 1), ...,Re(Fn−1), Im(F 1), ..., Im(Fn−1)

)T(13)

where F (x) ∈ R2n−2. The goal of a power flow calcula-tion is to find the bus voltage magnitudes and angles suchthat the system of nonlinear power mismatch equationsbecomes zero. These equations can be written in matrixform as

F (x) = 0. (14)

If the nonlinear equation (14) is well conditioned, it canbe solved by Newton’s method. The system is linearizedat point xν :

0 = F (xν) + Jν∆xν+1 (15)

where the superscript ν denotes the present iteration counterof Newton’s method. Jν ∈ R(n−1)×(n−1) is the Jacobianof F at point xν . The change of the state vector is

∆xν+1 = xν+1 − xν . (16)

After the linear problem (15) is solved, the next approxi-mation point is calculated by

xν+1 = xν + ∆xν+1. (17)

This process is repeated iteratively until the maximumchange of the state vector is below the power flow thresholdεPF:

max|∆xν+1| < εPF. (18)

A value of εPF = 10−5 pu is usually sufficient for power

system applications and also chosen in this paper.150

3.2. External Algorithm

The external algorithm, as used in, e.g., [10, 11, 14],represents the present state-of-the-art to obtain static op-eration points in an electric grid under the presence ofDGs with voltage dependent power injections. To com-155pare the external and internal algorithm, it is beneficial to

look at the external algorithm in some more detail. Themain idea is to sequentially perform a power flow and toupdate the injected active and reactive powers accordingto the voltage characteristics of the DGs until the bus volt-160ages converge. Details of the implementation are shown inFig. 3. The prescript (i) denotes the iteration counter ofthe external loop, which represents the number of powerflows. This avoids confusion with the index ν, which willlater denote the counter of the Newton iteration of the in-165ternal algorithm. For example, (i)Pk is the active powerinjection at bus k during the (i)-th external iteration.

Initially, flat start conditions are chosen, thus the busvoltage magnitudes are set to 1 and the bus voltage anglesare set to 0 (line 3 of Fig. 3). The active and reactive170power injections of all DGs are initially calculated by us-ing the voltage characteristics (3) divided by a filter valueζ (line 4-7). This avoids large overshoots of the voltagesin the first power flow iteration. In practical applications,filter values between 3 and 5 result in a convergence of175the power flow. The higher the filter value the more likelythe external algorithm is to converge. On the other hand,the computational time increases with higher filter val-ues. Thus, the filter value has to be selected manuallyand adjusted if convergence is not reached. Each power180flow is initialized with the previous power flow result (line10). During the power flow the linear problem (15) is setup without considering the voltage dependency of the in-jected powers of the DGs. The DGs are only present atbuses 1, ...,m. The other buses are static. Therefore, an185iteration across all DGs is started (line 11). The bus volt-ages are used to update the injected powers of the DGs(lines 12-13). A filter ζ is applied (lines 14-15). The ex-ternal loop is repeated until the maximum change of thebus voltages and angles is below a threshold εexternal (line19017).

The overall power flow problem is nonlinear and con-

1: procedure External Algorithm2: i = 03: (i)x = (1, ..., 1, 0, ..., 0)T

4: for all k ∈ {1, ...,m} do5: (i)Pk =

fPk((i)Vk)ζ

6: (i)Qk =fQk(

(i)Vk)

ζ

7: end for8: repeat9: i = i+ 1

10: (i−1)x→ Power Flow → (i)x11: for all k ∈ {1, ...,m} do12: (i)Pk = fPk(

(i)Vk)13: (i)Qk = fQk(

(i)Vk)

14: (i)Pk =(i−1)Pk +

(i)Pk−(i−1)Pkζ

15: (i)Qk =(i−1)Qk +

(i)Qk−(i−1)Qkζ

16: end for17: until max|(i)x− (i−1)x| < εexternal18: end procedure

Figure 3: Description of the external algorithm.

4

vergence cannot be guaranteed in all cases. Some powerflow problems without a solution exist. For example this isthe case when the system is in a point of voltage collapse195[25].

To gain more insights into the necessity of the filter it isuseful to assume an example grid with a noticeable amountof DGs with voltage dependent active power injections.Normally, the power flow calculation starts with the full200accessible active power injection of all DGs. That wouldbe the case if lines 4-7 were not included in the algorithm.However, a power flow calculation in which all DGs injecttheir full active power often leads to very high bus voltages.These numerical voltage overshoots can be avoided if the205injected powers of the DGs are not at 100 % during thefirst power flow. This is implemented by line 4-7, where theactive and reactive powers are set to a value which is theinjected active and reactive power, according to the P (V )and Q(V ) characteristic, divided by the filter value. Let210us further assume that the bus voltages computed by thefirst power flow calculation (i = 1) are very low because ofthe presence of the initial filter (line 4-7). In this case, theP (V ) characteristics in line 12 would result in using the100 % injected power in the next power flow calculation215(i = 2). However, using the full active power might leadto very high bus voltages. This will result in a substantialreduction of injected active power in the subsequent powerflow (i = 3), which will then show again high bus voltagesand the cycle will start over again. This is denoted as220numerical oscillation. The oscillation affects the voltages,the currents and the injected powers.

3.3. Internal Algorithm

In this section, it is shown how to include the Q(V ) andP (V ) characteristics directly into the Newton power flowalgorithm. During each Newton iteration ν the injectedpower into bus k ∈ {1, ...,m} is updated according to

P νk = fPk(Vνk ) (19a)

Qνk = fQk(Vνk ). (19b)

As can be seen in (8), the power mismatch consists oftwo voltage dependent terms, namely the injected powersSk and the sum of the branch power flows

∑n`=1, ` 6=k Sk`.

Therefore, the overall Jacobian is a linear combination ofthe Jacobians of these two terms and can be written as

Jν = Jνa + Jνb. (20)

Ja is the Jacobian of the sum of the branch power flowsSk`. Its construction can be found in, e.g., [24]. Jb is theJacobian of the voltage dependent power injection charac-teristics of the injected powers Sk. If, besides DGs withvoltage dependent power injection, no other voltage de-pendent load is present, Jb only exists for the internalalgorithm and is zero in case of the external algorithm. Asthe injected powers are voltage dependent, the Jacobian

Jb may contain nonzero entries. It contains the deriva-tives of the voltage characteristics:

Jνb =

(∂P∂V

∂P∂δ

∂Q∂V

∂Q∂δ

)=

(Jb11 Jb12

Jb21 Jb22

)(21)

where Jb11,Jb12,Jb21,Jb22 ∈ R(n−1)×(n−1). The activeand reactive power vectors are described by

P = (P ν1 , ..., Pνn−1)

T (22a)

Q = (Qν1 , ..., Qνn−1)

T (22b)

where P ,Q ∈ Rn−1. Let gPk(V νk ) and gQk(V νk ) be thederivative of the active and reactive power characteristicsof the DG connected to bus k according to (5), whichis a smoothing function of (2). The injected active andreactive powers do not depend on the bus voltage angle.It is

Jb11 =

gP1(Vν1 )

. . . 0gPm(V

νm)

0 0

(23a)

Jb21 =

gQ1(Vν1 )

. . . 0gQm(V

νm)

0 0

(23b)

Jb12 = Jb22 = 0. (23c)

If only P (V ) (or Q(V )) injections are present, the ma-225trix Jb21 (or Jb11) becomes zero. If both injections areactive, both matrices can have nonzero entries.

According to Section 4.1, the nonlinear voltage char-acteristic can lead to ill-conditioned cases. An explana-tion of ill-conditioned power flow cases is presented in230[16]. For solving those cases robust techniques of the basicNewton’s method have been introduced in literature, e.g.,[26, 27, 28]. These techniques are mainly based on mod-ifying (15). In this paper, we use a modified (damped)Newton method, described as bisection method in [16].235Following the nomenclature of [16],this type of modified(damped) Newton method will be called bisection Newtonmethod in this paper. After the new state vector xν+1

is calculated, it is tested if the power mismatch F (xν+1)decreased. If not, then the change in the state vector ∆xν240is reduced by a factor α. During the first iteration it isα = 1. For an unsolvable case, α will converge towardszero, thus a minimum value of α has to be defined.

The bisection Newton method is similar to the moresophisticated damped Newton-Raphson method presented245

5

in [29] which works with a damping multiplier that corre-sponds to the variable α used in this work. The differ-ence is, that in the damped Newton-Raphson method anoptimum value is found for the damping multiplier by ap-proximating the power mismatch function with a quadratic250function. This method is suitable for obtaining power flowresults in case that the conventional Newton-Raphson isnot converging. This is done by obtaining the minimum ofthe power mismatch. In the modified method used in thispaper, α is simply decreased (divided by 2) during each it-255eration if the power mismatch function does not decrease.

Details of the implementation of the internal algorithmare shown in Fig. 4. The Newton iteration counter ν is ini-tially set to zero (line 2 of Fig. 4) and flat start conditionsare chosen (line 3). At the beginning of each Newton it-260eration the injected powers are updated according to thevoltage characteristics of the DGs (19) (lines 5-8). TheJacobian and the power mismatch function are updated(lines 9-10). In the next step, the modified Newton methodis applied (lines 11-16) and the internal iteration counter265is increased (line 17). This process is repeated until thechange in bus voltages is below the threshold εinternal (line18).

4. Test Cases

To investigate the proposed internal algorithm, a con-270ventional Newton power flow without any voltage depen-dent power injection has been implemented in MATLAB[21]. Subsequently, the internal as well as the external al-gorithm were implemented on top of this basic power flowroutine and they are used for the following test cases. We275used MATPOWER [30] to validate our implementation ofthe conventional Newton power flow. All simulations wereperformed on an Intel i7-3520M 2.90 GHz processor with16 GB RAM and on a Windows 7 operating system. Thevoltage characteristics used in the case studies are based280

1: procedure Internal Algorithm2: ν = 03: xν = (1, ..., 1, 0, ..., 0)T

4: repeat5: for all k ∈ {1, ...m} do6: P νk = fPk(V

νk )

7: Qνk = fQk(Vνk )

8: end for9: Update Jacobian Jν

10: Update F (xν)11: α = 212: repeat13: α = α/214: ∆xν+1 = −α (Jν)−1 F (xν)15: xν+1 = xν + ∆xν+1

16: until max |F (xν+1)| ≤ max |F (xν)| ∨α < 10−217: ν = ν + 118: until max|∆xν+1| < εinternal19: end procedure

Figure 4: Description of the internal algorithm.

on values used by distribution system operators in practi-cal settings.

4.1. Two-Bus System

A two-bus system, as shown in Fig. 5, is used to com-pare the convergence properties of the internal and the285external algorithm. It consists of a DG connected to aninfinite bus via a resistor and an inductor. Furthermore,the simulation results are validated in the laboratory.

4.1.1. Convergence Properties

The parameters of the line and the P (V ) and Q(V )290characteristics are given in Tables 2 and 3. The configu-ration for the characteristics represent realistic values alsoused in distribution grids. The operating point of the DGdepends on the power injection method and can be foundin Table 4. Both internal and external algorithm converge295to the same values, therefore only one power flow resultis shown in Table 4. The voltage at bus 1 is higher thanthe voltage at bus 2 (slack bus) for all injection strategies.This is due to the fact that in all cases active power is fedinto the grid and current is flowing from bus 1 to bus 2.300

The injection strategy ”None” means, that all avail-able active power (in this case 35 kW) is fed into the gridby the DG and that there is no voltage dependent activeor reactive power injection. During the injection strategyQ(V) the DG is feeding reactive power into bus 1 in order305to decrease the voltage. During injection strategy P (V )the DG reduces its active power injection based on thevoltage, again, in order to reduce the voltage at bus 1. In

V slack = 1 ej 0

2

XI

R

V = V ejδ

1

DG

Figure 5: Two-bus system with a distributed generator connected tobus 1.

Table 2: Parameters of the simulated two-bus system

Description Symbol Value Unit

Base apparent power Sbase 100 [kVA]

Base voltage Vbase 230 [V]

Line resistance R 0.383 [pu]

Line reactance X 0.190 [pu]

Table 3: Voltage characteristic of the simulated two-bus system

Injection [V1, V2, V3V4] [y1, y2, y3]

method [pu] [pu]

P (V ) [1.08, 1.08, 1.08, 1.09] [0.35, 0.35, 0.20]

Q(V ) [0.93, 0.97, 1.03, 1.07] [0.18, 0.0,−0.18]

6

Table 4: Power flow results of the simulated two-bus system

Injection Voltage Voltage Active Reactive

strategy magnitude angle power power

[pu] [degree] [pu] [pu]

None 1.118 3.392 0.35 0.00

Q(V ) 1.085 7.161 0.35 -0.18

P (V ) 1.087 2.477 0.25 0.00

P (V ) & Q(V ) 1.081 7.062 0.34 -0.18

the strategy P (V ) & Q(V ) the DG feeds in reactive powerand reduces its active power in order to reduce the voltage310at bus 1. Active and reactive power injection takes placedepending on the voltage at bus 1 according to the voltagecharacteristic given in Table 3.

A more detailed investigation of the convergence be-havior of the external and internal algorithm is presented315for the case of P (V ) injection. It is assumed that no reac-tive power is injected. Fig. 6 shows the maximum powermismatch and the voltage magnitude after each iterationof the external algorithm for three different filter values.For a filter value of ζ = 2 the voltage magnitude shows320an oscillatory behavior and does not converge. The reasonis that the injected active power keeps on jumping fromits highest (y1) to its lowest value (y3). A more detailedexplanation and a plot of the injected power can be foundin [14]. For ζ = 3 and ζ = 4 the voltage magnitude con-325verge towards the same final value and thus both powermismatches converge towards zero. For ζ = 3 the overall

2 4 6 8 10 12 141

1.05

1.1

Iteration i of external algorithm

Volt

ageV

[pu

]

2 4 6 8 10 12 14

10−5

10−3

10−1

Iteration i of external algorithm

Pow

erm

ism

atc

hF

[pu

]

ζ = 2

ζ = 3

ζ = 4

Figure 6: Active power mismatch (top) and bus voltage magnitude(bottom) against the power flow iteration i of the external algorithmwith P (V ) injection for different filter values ζ for the simulatedtwo-bus system.

convergence speed is lower and the numerical oscillationsare higher than for ζ = 4.

The convergence behavior of the internal algorithm is330shown in Fig. 7 for both Newton’s method and bisectionNewton’s method. The maximum power mismatch (top)and the voltage magnitude (bottom) are shown for eachNewton iteration ν. It can be observed that the bisectionmethod converges and the normal Newton method without335bisection algorithm does not converge. It has to be pointedout that the power mismatch is in the order of 10−6 aftereight Newton iterations. For reaching the same level ofpower mismatch the external algorithm requires about 14external iterations which corresponds to 30 Newton iter-340ations. Thus, the internal algorithm required much lessNewton iterations than the external algorithm (8 insteadof 30) in this case. Furthermore, the filter value needs tobe set in the external algorithm, which is a crucial dis-advantage that our proposed internal algorithm does not345suffer from. The speedup in terms of actual computationaltime is not very significant for the two-bus system. Theexternal algorithm needs about 13 ms and the internal al-gorithm about 5 ms of computational time. The reason is,that for the two-bus system, the Newton step is actually350less time consuming than the other parts of the algorithmlike, for example, the initialization. As the matrices in-volved are only of size [2 × 2] for a two-bus system, thebenefit in computational time becomes more evident incase of larger grids. This is shown in Section 4.2.355

To illustrate the ill-conditioned power mismatch func-tion, Fig. 8 (top) shows the P (V ) characteristic and theinverse nose curve. The latter is derived in the Appendix

1 2 3 4 5 6 7 81

1.05

1.1

Newton iteration ν of internal algorithm

Volt

ageV

[pu

]

1 2 3 4 5 6 7 8

10−5

10−3

10−1

Newton iteration ν of internal algorithm

Pow

erm

ism

atc

hF

[pu

]

Newton method

Bisection Newton method

Figure 7: Active power mismatch (top) and bus voltage magnitude(bottom) for the internal algorithm of the Newton method and thebisection Newton method for the simulated two-bus system.

7

for zero injected reactive power. The nose curve describesthe active power injection depending on the voltage mag-360nitude V . The operating point is the intersection betweenthe two curves. The intersection coincides with the valuein Table 4 for the injection strategy P (V ). The powermismatch function is the difference between the inversenose curve and the P (V ) characteristic and is shown in365Fig. 8 (bottom). It can be seen that the power mismatchfunction is ill-conditioned. The reason is the piecewiselinear definition of the P (V ) characteristic and its steepdecrease relative to the inverse nose curve in the area of1.08 < V < 1.09. Such a system can only be solved by the370conventional Newton method in case the initial value forthe voltage V is located in the area of 1.08 < V < 1.09.A more detailed explanation on ill-conditioned power flowcases can be found in, e.g., [16].

To guarantee convergence of the internal algorithm in375those ill-conditioned cases the bisection method as ex-plained in Section 3.3 is used.

The iterative process of the internal and external al-gorithm is now addressed in detail. Table 5 shows theinterim results for the external algorithm for a filter value380of ζ = 4. Note that (i)P1 is the injected power of theDG into bus 1. During initialization (i = 0) the injectedpower of the DG is set to (0)P1 = 0.35/4 = 0.087 pu. Thefirst power flow calculation (i = 1) results in a voltage of1.0324∠0.918◦. With this voltage the new injected power385is fP1(1.0324) = 0.35. The application of the filter resultsin (1)P1 = 0.087 + (0.35 − 0.087)/4 = 0.153 pu. This in-jected power leads to a voltage of 1.0553∠1.572◦ in thesecond iteration (i = 2). The process is repeated until asteady state is reached in the 12th external iteration.390

1.07 1.08 1.09 1.1−1

0

1

2·10−2

Voltage V [pu]

Pow

erm

ism

atc

hF

[pu

]

1.07 1.08 1.09 1.1

2

3

4·10−2

Voltage V [pu]

Act

ive

pow

erP

[pu

]

P (V ) characteristic

Nose curve

Figure 8: Inverse nose curve and P (V ) characteristic (top) and re-sulting active power mismatch function (bottom) for the simulatedtwo-bus system.

Table 5: Interim results for each external iteration i of the externalalgorithm for P (V ) characteristic of the simulated 2-bus system

i V i1 [pu](i)P1 [pu]

0 1∠0◦ 0.0871 1.0324∠0.918◦ 0.1532 1.0553∠1.572◦ 0.2023 1.0718∠2.045◦ 0.2394 1.0838∠2.392◦ 0.2535 1.0881∠2.516◦ 0.2476 1.0862∠2.4609◦ 0.2497 1.0870∠2.485◦ 0.2488 1.0866∠2.474◦ 0.2499 1.0868∠2.479◦ 0.24810 1.0867∠2.477◦ 0.24911 1.0868∠2.478◦ 0.24812 1.0868∠2.477◦ 0.249

Table 6 shows the iteration results of the internal al-gorithm. P ν1 is the injected power of the DG into bus1 and gP1(V

ν1 ) is the slope of the P (V ) characteristic at

V ν1 . During initialization (ν = 0) the injected power isset to P 01 = 0.350 which is the total power accessible. In395the first Newton iteration (ν = 1), the resulting voltageis 1.1342∠3.791◦ which is on the far right of the P (V )characteristic and thus the injected power is P 11 = 0.200.During the next Newton iteration (ν = 2) the voltagedrops again to 1.0746∠2.108◦ which is left of the point400V3 of the P (V ) characteristic according to Table 3. Thusthe injected power is increased to its maximum value ofP 21 = 0.35. During the third iteration (ν = 3) the voltageis 1.0974∠2.774◦ and thus the injected power is set to itsminimum value of P ν3 = hlm0.2. During these first 3 iter-405ations, the Jacobian Jb is always zero because the voltagewas not on the slope of the P (V ) characteristic. In thefourth iteration (ν = 4) the voltage is 1.0845∠2.407◦ andfor the first time, the voltage is such that the operatingpoint is on the slope of the P (V ) characteristic. Therefore410the injected power is set to P 41 = 0.282. The Jacobian isthus nonzero. It is

Table 6: Interim results for each internal iteration ν of the internalalgorithm for P (V ) characteristic of the simulated 2-bus system

ν V ν1 Pν1 gP1(V

ν1 )

[pu] [pu] [pu]

0 1∠0◦ 0.0350 0.0

1 1.1342∠3.791◦ 0.0200 0.0

2 1.0746∠2.108◦ 0.0350 0.0

3 1.0974∠2.774◦ 0.0200 0.0

4 1.0845∠2.407◦ 0.0282 -1.50

5 1.0869∠2.482◦ 0.0246 -1.50

6 1.0868∠2.477◦ 0.0249 -1.50

8

Jb11 =

(−1.5 0

0 0

)(24)

Jb21 =

(0 00 0

). (25)

Finally, the Jacobian due to the DG is

Jνb =

−1.5 0 0 0

0 0 0 00 0 0 00 0 0 0

. (26)The Jacobian Ja of the system is computed as usually

during a power flow algorithm. The process is repeateduntil sufficient accuracy is reached.415

4.1.2. Experimental Validation

The algorithm was verified with an experimental setupof the two-bus system as shown in Fig. 9. A DC sourcewhich mimics the properties of a string of solar panelswas connected to a 4.6 kW SMA Sunny Boy inverter. The420inverter was connected to an AMETEK grid emulator viaa cable. The resistance and inductance of the cable wasmeasured as R = 0.41 Ω and L = 672µH.

The voltage V 1 was varied. Due to this variation, theQ(V ) and P (V ) injection of the inverter was activated.425The measured Q(V ) and P (V ) characteristics of the in-verter are shown in Fig. 10. Fig. 11 shows the measure-ment values and the simulated values of V1 and V2. It canbe observed that the measured and the simulated valuescoincide. It is crucial to investigate new inverter capa-430bilities like voltage dependent active and reactive powerinjection also in a practical lab setup.

4.2. Performance Analysis

A performance analysis of the internal and externalalgorithm was done with two test systems.435

The first test system is the IEEE 118-bus system whichrepresents a portion of the American Electric Power Sys-tem. The data of this test system is given in [31]. In thissystem, 54 synchronous generators with power and volt-age control are present. As the goal of the investigation440is to test voltage dependent power injections with P (V )and Q(V ) characteristics, all synchronous generators wereremoved and replaced with constant power injections. In

Grid emulator CableInverter

DC Source

2

V 1

1

V 2

I

Figure 9: Experimental setup of the two-bus system.

1 1.02 1.04 1.06 1.08 1.1

1

2

3

Voltage V1 [pu]

Act

ive

[kW

]an

dre

act

ive

[kV

Ar]

pow

er

P (V )

Q(V )

Figure 10: Measured Q(V ) and P (V ) characteristics of the inverter.

0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12

1

1.05

1.1

Voltage V2 [pu]

Volt

ageV1

[pu

]

Measurement

Simulation

Figure 11: Voltage at the point of common coupling at bus 1 againstgrid voltage at bus 2 for P (V ) and Q(V ) injection strategies. Thebase voltage is 230V .

the next step, 50 DGs with P (V ) and Q(V ) characteris-tics were placed randomly in the grid. The total nomi-445nal power of the DGs was chosen to be 10% of the totalpower of the original generators. This value was chosenin order to obtain significantly high voltages in the rangeof 1.03 to 1.1 pu, where the voltage characteristics of theDGs have an effect. The description of the voltage charac-450teristics is given in Table 7. The table contains the P (V )and Q(V ) characteristics for all generators where Pn is thenominal power of the DG and d as well as e are parametersthat are specified for each case. By varying these param-eters, different characteristics are obtained. The simula-455tions were performed in two setups S1 and S2. In setupS1, all DGs were given the same P (V ) and Q(V ) charac-teristics. Specifically we chose Pn = 0.0875 pu, d = 0 ande = 0. In setup S2 all characteristics are chosen differentlyin order to investigate the influence of different character-460istics on the runtime. We have set d = 0.02/50 · (i−1) ande = 0.01/50 · (25.5 − i) with i = 1, ..., 50 being the indexof the DG.

The second test system is a real German 234-bus grid.In this system, 108 DGs with P (V ) and Q(V ) character-465istics are included. Similar to the test with the IEEE 118-bus system, two setups S1 and S2 are investigated. Forboth setups we have set Pn = 0.0153 pu. For setup S1,all generators were given the same characteristics, thusd = 0 and e = 0. For setup S2, different characteristics470were selected for all DGs, specifically d = 0.02/108 · (i−1)and e = 0.01/108 · (54.5 − i) with i = 1, ..., 108 being theindex of the DG. The values chosen are similar to practi-

9

Table 7: Voltage characteristics for the performance test of the IEEE 118-bus test system and the real German 234-bus grid.

Injection V1 V2 V3 V4 y1 y2 y3

method [pu] [pu] [pu] [pu] [pu] [pu] [pu]

P (V ) 1.08 − d 1.08 − d 1.08 − d 1.09 − d Pn + e Pn + e (Pn + e) · 0.2Q(V ) 0.92 − d 0.97 − d 1.03 − d 1.08 − d (Pn + e) · 0.5 0.000 −(Pn + e) · 0.5

cal settings that are used by German distribution systemoperators.475

Simulations of each setup where performed for 40 dif-ferent random locations of DGs. For each setting, tensimulation runs were performed. The average values andvariances of all simulation run times were computed. Theresults for the complex bus voltages of the internal and480external algorithm were compared to each other after eachsingle power flow calculation. For each bus voltage, theabsolute value of the complex voltage difference betweeninternal and external algorithm was smaller than 10−4 pu.Thus, both algorithms converged to the same values. To485obtain this, a filter value of ζ = 5 was chosen for the exter-nal algorithm. The filter value was chosen such that it wasthe minimal value for which all simulations still converged.Lower filter values resulted in a significant amount of sim-ulations that did not converge. Higher filter values only490increased the computational time. Therefore, the chosenfilter value was the best possible choice concerning com-putational speed and convergence. The results for the runtime are displayed in Table 8. It shows the average com-putation time, the total number of Newton iterations and495the number of power flows for each algorithm. For ex-ample in case of the IEEE 118-bus system the externalalgorithm requires an average number of 44 power flows toconverge. This results in an average total number of 102Newton iterations. This means that on average, a power500flow calculation needed 102/44 = 2.3 iterations to con-verge. The internal algorithm needed only one power flowiteration which took an average of 6 Newton iterations.The advantage of the internal algorithm can be found in

Table 8: Comparison of external and internal algorithm (averagevalues and variances.

System Algo- Time Sum of Number

rithm [ms] Newton of Power

iterations flows

118 Bus Ext. 132 ± 72 102 ± 11 44 ± 3(S1) Int. 9 ± 1 6 ± 1 1 ± 0

118 Bus Ext. 136 ± 7 106 ± 58 45 ± 16(S2) Int. 10 ± 1 7 ± 2 1 ± 0

234 Bus Ext. 169 ± 43 93 ± 1 40 ± 1(S1) Int. 19 ± 2 8 ± 1 1 ± 0

234 Bus Ext. 165 ± 8 92 ± 1 41 ± 1(S2) Int. 18 ± 1 8 ± 1 1 ± 0

both the run time and the average total number of New-505ton iterations. The internal algorithm is between 8 and14 times faster than the external algorithm and the exter-nal algorithm needs between 11 to 17 times more Newtoniterations.

5. Numerical Example510

In this section a numerical example of a low-voltagefeeder with photovoltaic systems is presented. The feedertopology can be seen in Fig. 12. It consists of 19 nodesand 5 PV systems. The branch data is given in Table 9 inper unit. The base power is 1 MV A and the base voltage515is 230 V. The slack bus is located at bus 1 and has a fixedvoltage of 1.04∠0◦ pu. A total of seven loads, each havinga power of 3 kW are connected to the feeder. This is alsolisted in Table 10. The PV systems are located at buses 9,11, 13, 15, 17, having an available active power infeed be-520tween 30 and 38 kW. Their voltage dependent active andreactive power characteristics are depicted in Table 14. Apower flow is conducted for three cases: In the first case,all photovoltaic systems are disconnected from the feeder.This cases serves as a reference scenario in order to verify525the bus voltages for the case in which only the loads areconnected. In the second case the photovoltaic systemsfeed in their full active power without voltage dependentpower injection. In the third case all photovoltaic systemsare connected and the voltage dependent active and reac-530tive power injection is active.

The results for the bus voltages for each case can beseen in Table 11, the injected active powers are depictedin Table 12 and the injected reactive power are depictedin Table 13. The example is designed such that voltages535

123

456

78

910

1112

1314

1516

171819

Figure 12: Topology of the example Grid. Photovoltaic systems arehighlighted in red.

10

Table 9: Branch data of the example feeder in per unit notation.The base power is 1 MVA and the base voltage is 230 V.

From To r x

1 2 0.017 0.094

2 3 0.432 0.067

18 16 0.018 0.007

16 14 0.018 0.007

14 12 0.018 0.007

12 10 0.018 0.007

10 8 0.018 0.007

8 6 0.018 0.007

6 5 0.018 0.007

18 19 0.060 0.008

6 7 0.060 0.008

16 17 0.060 0.008

14 15 0.060 0.008

12 13 0.060 0.008

10 11 0.060 0.008

8 9 0.060 0.008

4 3 0.060 0.008

5 3 0.224 0.035

Table 10: Load data of the test feeder.

Bus Load [kW]

5 3

7 3

8 3

11 3

12 3

15 3

18 3

above 1.1 pu occur for the first case. Therefore, the volt-age dependent active and reactive power injection, whichis activated in the third case, results in an infeed of reac-tive power and a reduction in active power. As expected,the voltage dependent active and reactive power injection,540activated in the third case, result in a mitigation of thevoltage rise caused by the photovoltaic systems. There-fore the bus voltages for the third case are between thebus voltages of the first and second case.

6. Conclusion and Outlook545

In this paper, we presented a method to integrate volt-age dependent power injections of DGs into the powerflow. The method was named internal algorithm in order

Table 11: Bus voltages in per unit for the three cases of the example.In the first case all photovoltaic systems are disconnected from thesystem. In the second case, the photovoltaic systems feed in theirmaximum active power and in the third case voltage dependent activeand reactive power injection is activated.

Bus Case 1 Case 2 Case 3(no PV) (Full) (P(V) and Q(V))

1 1.040∠0◦ 1.040∠0◦ 1.040∠0◦

2 1.040∠− 0.1◦ 1.042∠0.7◦ 1.038∠0.4◦3 1.031∠− 0.2◦ 1.098∠1.2◦ 1.066∠1.6◦4 1.031∠− 0.2◦ 1.098∠1.2◦ 1.066∠1.6◦5 1.027∠− 0.2◦ 1.128∠1.4◦ 1.081∠2.2◦6 1.027∠− 0.2◦ 1.131∠1.5◦ 1.082∠2.2◦7 1.026∠− 0.2◦ 1.131∠1.5◦ 1.082∠2.2◦8 1.026∠− 0.2◦ 1.133∠1.5◦ 1.084∠2.3◦9 1.026∠− 0.2◦ 1.135∠1.5◦ 1.085∠2.3◦10 1.026∠− 0.2◦ 1.135∠1.6◦ 1.084∠2.3◦11 1.026∠− 0.2◦ 1.137∠1.6◦ 1.085∠2.4◦12 1.026∠− 0.2◦ 1.137∠1.6◦ 1.085∠2.4◦13 1.026∠− 0.2◦ 1.138∠1.6◦ 1.086∠2.4◦14 1.026∠− 0.2◦ 1.138∠1.6◦ 1.086∠2.4◦15 1.026∠− 0.2◦ 1.140∠1.6◦ 1.086∠2.4◦16 1.026∠− 0.2◦ 1.138∠1.6◦ 1.086∠2.4◦17 1.026∠− 0.2◦ 1.140∠1.6◦ 1.087∠2.4◦18 1.026∠− 0.2◦ 1.138∠1.6◦ 1.086∠2.4◦19 1.026∠− 0.2◦ 1.138∠1.6◦ 1.086∠2.4◦

Table 12: Injected active power of the photovoltaic systems for thethree cases of numerical example.

Bus Case 1 Case 2 Case 3

(no PV) (Full) (P(V) and Q(V))

9 0 30 21.0

11 0 32 20.2

13 0 34 19.7

15 0 36 19.4

17 0 38 19.4

Table 13: Injected reactive power of the photovoltaic systems for thethree cases of numerical example.

Bus Case 1 Case 2 Case 3

(no PV) (Full) (P(V) and Q(V))

9 0 0 8.0

11 0 0 8.0

13 0 0 8.0

15 0 0 8.0

17 0 0 8.0

to distinguish it from the state-of-the-art method, whichis called external algorithm in this work. We have com-550pared the internal algorithm with the commonly used ex-ternal algorithm. The comparison was done with respectto convergence properties for an ill-conditioned two-bussystem and with respect to computational complexity forthe IEEE 118-bus test system and for a real German 234-555

11

Table 14: Voltage dependent active and reactive power characteristics for the photovoltaic systems connected to the example low voltagefeeder.

Bus V1, V2, V3 V4 y1, y2 y3 V1 V2 V3 V4 y1 y2 y3

[pu] [pu] [kW] [kW] [pu] [pu] [pu] [pu] [kVA] [kVA] [kVA]

9 1.08 1.09 30 10 0.92 0.97 1.03 1.08 -8 0 8

11 1.08 1.09 32 10 0.92 0.97 1.03 1.08 -8 0 8

13 1.08 1.09 34 10 0.92 0.97 1.03 1.08 -8 0 8

15 1.08 1.09 36 10 0.92 0.97 1.03 1.08 -8 0 8

17 1.08 1.09 38 10 0.92 0.97 1.03 1.08 -8 0 8

bus network. Furthermore, we performed a laboratory ex-periment to verify the internal algorithm and provided anumerical example.

The main contribution of this paper is the integrationof voltage dependent power characteristics into the power560flow. From a mathematical point, this approach is obvious.However, to the knowledge of the authors, this has notbeen reported in literature before. Furthermore, this workshows that the integration of voltage dependent power in-jections into the power flow can lead to ill-conditioned565power mismatch functions. Therefore, a damped Netwonmethod is used in order to solve the power flow problem.

In the paper it is demonstrated that the incorporationof voltage dependent power injections into the power flowhas major advantages over the external algorithm. First570of all, it allows integration of voltage dependent powerinjections with minimal effort in analyses where the Ja-cobion matrix is required, e.g. voltage stability problems.A further advantage is the improved robustness comparedto the external algorithm. The filter ζ of the external al-575gorithm needs to be tuned and it is a trade-off betweenspeed and robustness. In case of the internal algorithm,ill-conditioned cases can be treated with robust power flowmethods and no manual tuning is required. For this, thebisection Newton method was used. Another advantage is580the improved computational efficiency. This allows signif-icant speed-ups in cases where thousands of power flowshave to be solved, e.g., in power system planning and op-eration.

Appendix585

Derivation of inverse nose curve: The single phase com-plex power consumption at bus 1 in Fig. 5 is

S = P + jQ = V I∗. (27)

The current can be calculated as I = (V − V slack)Y =(V − 1)Y with Y = (R + jX)−1 = G + jB being theline admittance. The capacitance of the line is neglected.Taking real and imaginary part of (27) results in activeand reactive powers. After reordering we obtain:

P −GV 2 = −GV cos(δ)−BV sin(δ) (28a)Q+BV 2 = BV cos(δ)−GV sin(δ). (28b)

Taking the square of (28a) and (28b) and summing themup removes the angle dependency. Solving it for the activepower finally leads to the inverse nose curve

P (V ) = GV 2 ±√

(G2 +B2 − 2BQ)V 2 −B2V 4 −Q2,(29)

which is approximately linear in the range where it is plot-ted in Fig.8.

Acknowledgement

This work was supported by the German Federal Min-istry for Economic Affairs and Energy and the Projek-590tträger Julich GmbH (PTJ) within the framework of theproject Smart Grid Models (FKZ: 0325616). The authorsare solely responsible for the content of this publication.The authors thank Bernd Gruß, University of Kassel, forthe support during the laboratory experiments. Further-595more, the authors thank Wolfram Heckmann, FraunhoferInstitute for Wind Energy and Energy System Technology,and EnergieNetz Mitte GmbH for providing the grid data.The authors gratefully acknowledge the suggestions of Dr.Federico Milano, University College Dublin. The authors600thank Dr. Nils Bornhorst, Elisabeth Drayer and FlorianSchäfer from University of Kassel for fruitful discussions.

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Introduction Modeling of Voltage Dependent Power Injection Smoothing function

Integration into the Power FlowPower Flow FormulationExternal AlgorithmInternal Algorithm

Test CasesTwo-Bus SystemConvergence PropertiesExperimental Validation

Performance Analysis

hlNumerical ExampleConclusion and Outlook