linear iterative power flow approach based on the current

Received November 24, 2020, accepted December 18, 2020, date of publication December 29, 2020,date of current version January 21, 2021.

Digital Object Identifier 10.1109/ACCESS.2020.3047986

Linear Iterative Power Flow Approach Based onthe Current Injection Model of Load andGeneratorBEHZAD ZARGAR 1, (Member, IEEE), ANTONELLO MONTI 1, (Senior Member, IEEE),FERDINANDA PONCI 1, (Senior Member, IEEE), AND JOSÉ R. MARTí 2, (Life Fellow, IEEE)1Institute for Automation of Complex Power Systems, E.ON Energy Research Center, RWTH Aachen University, 52074 Aachen, Germany2Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, BC V6T 1Z4, Canada

Corresponding author: Behzad Zargar ([email protected])

This work was supported by the EU Horizon 2020 Project ‘‘IElectrix’’ under Grant 824392.

ABSTRACT Power flow (PF) is a fundamental tool for operation, automation and optimization of the powersystems. Due to the nonlinearity of the PF system equations, the classical PF solutions are computationallyvery demanding. As a common approach in solving the nonlinear equations, linearization is a potentialtechnique which can simplify and accelerate the PF calculations. In this context, this paper proposes alinear fast iterative method based on the fixed-point iteration technique in which a linearized model ofgenerator along with a ZI load model are integrated in a simplified system of linear equations (SLE) ofYv = i. The relaxation method is used during the deriving process of generator equivalent current in thisapproach. However, the already developed ZI load model based on the curve-fitting technique has beenexploited in this work. The accuracy of the proposed PF method has been compared with calculated resultsfrom DIgSILENT PowerFactory on the benchmark IEEE 33-bus test system and on a large medium voltagenetwork in Germany.

INDEX TERMS Linearization, fixed-point iteration technique, current injection model, relaxation method.

NOMENCLATUREThe main notation used in this paper is provided below; othersymbols are defined as required.

Y Admittance matrix of the power systemv Vector of bus voltagesi Vector of current injectionspg Active power of synchronous generatorE electromotive force (emf)δ Power angleig Synchronous generator injecting currentxg Generator synchronous reactanceγ Relaxation factor

I. INTRODUCTIONPower is supplied by conventional power plants and renew-able energy technologies and consumed by the loads in theelectrical power systems. To efficiently design and operate

The associate editor coordinating the review of this manuscript and

approving it for publication was Ruisheng Diao .

the power grids, the network operators analyze the impactof power flow (PF) on the power systems. Moreover, powersystem energy management tools include applications (suchas state estimation, volt-VAR optimization, network recon-figuration, etc.) which require computations based on powerflow solutions. In spite of existing several well-known PFmethods, the solutions designed for transmission grids [1]may not be optimal for distribution systems due to the differ-ent features of these networks (in terms of x/r ratio, topologyand configuration, length and type of lines). In this respect,the backward-forward sweep method [2], [3] and the cur-rent injection method (CIM) [4] are commonly exploitedto solve the PF problem in distribution grids. These meth-ods, however, have convergence problem when the sys-tem includes more than one P-V node (voltage-controllednodes) [3], [5]–[7]. To tackle this issue, this article proposesa new linearized model of synchronous generator which canbe used to integrate multiple P-V nodes in the grids (as elab-orated in the result section). Moreover, the conventional PFmethods have a low computational efficiency when solvingthe large networks encountered in distribution systems [8].

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https://orcid.org/0000-0003-2614-8559

https://orcid.org/0000-0003-1914-9801

https://orcid.org/0000-0003-0431-9169

https://orcid.org/0000-0003-3811-3687

https://orcid.org/0000-0002-8009-227X

B. Zargar et al.: Linear Iterative PF Approach Based on the CIM of Load and Generator

In this context, a lot of research has been undertaken toenhance the convergence rate and the computational cost ofthe PF solutions. As a potential technique, linearization hasbeen applied in some solutions to accelerate the PF calcula-tions. As one of the primary linear approaches, the classicaldirect-current (DC) PF provides a set of linear equationswhich yields only the active power injections and the phaseangle of the system bus voltages [9]. To derive the formulafor this method, the line resistances are ignored, the volt-age magnitudes are approximated to be 1 pu for all thebuses and the voltage angle difference across the lines areassumed to be small. The use of this method is limited mostlyto MW-oriented applications (such as contingency analysis,electricity market, scheduling, sensitivity analysis and so on)where the effects of network voltage and VAR conditions areminimal for the system’s operation. This approach, however,cannot be used in distribution systems since the resistiveand reactive parts of line impedances are comparable [10].Compared to [9], nevertheless, both line resistance and linereactance are considered in this article. To include the effectof reactive power transmission, Fatemi et al. [11] proposed anew formulation for DC PF in which the exact effect of netreactive loads on phase angles is considered. The accuracy ofthis approach is acceptable, even under cold-start conditions.Nevertheless, the voltage magnitude and the phase angle arenot completely decoupled, leading to a quadratic program-ming (QP) problem if this method is used for optimization.To advance a further step, Yang et al. [12] proposed a decou-pled linearized power flow (DLPF) model with respect to thevoltage magnitude and the phase angle. However, it has beenassumed that all lines have the same resistance-to-reactanceratios and the network is radial. This approach can be applieddirectly in the optimization problem. In addition to the decou-pling feature, a fast approximation of the matrices used in theDLPF model accelerates the computational speed, as fast asthe classical DC PF. In a novel work, a geometric approachis presented in [13] to derive a linear approximation of themanifold of feasible power flows. This linear approximationis sparse and can obtain a fast solution of a possibly unbal-anced three-phase power system, with either radial or meshedtopology. A linear balanced PF model for radial distributionsystemwas introduced in [14] assuming that the voltage dropsare much smaller than the nominal voltage. However, thevoltage-dependent characteristic of the loads is not taking intoaccount in this method. An interesting linear non-iterativeformulation of PF for distribution grids was proposed byMartí et al. in [15], which uses the curve-fitting technique toderive a voltage-dependent load model based on the ZI loadtype. This model has been exploited in this paper. Distributedgeneration units can be included as a standard constant P-Qmodel (negative load) in Martí’s approach. However, thisapproach is not able to model the DGs as voltage regulators(voltage-controlled nodes).

In this paper, a model-based linear iterative PF approachis introduced which exploiting a new linearized model ofgenerator (as the P-V node) along with the ZI load model

developed by Martí et al. [15]. The specific contribution ofthis paper is:

• To propose a linearized model of a synchronous genera-tor represented as a P-V node. To derive this model, thepower angle of the generator is relaxed by applying therelaxation method via a relaxation factor.

• To introduce a power flow formulation based on thefixed-point iteration technique in which the proposedlinearized model of a synchronous generator along withthe linear load model derived in [15] are exploited insuch a way that the generators equivalent currents areiteratively updated in parallel based on the power angleand the variations of the generator terminal voltage.To this aim, the node voltages are calculated by a sim-plified system of linear equations (SLE) of Yv = i.

• To obtain the optimal relaxation factor based on theMonte Carlo approach.

• To evaluate the computational burden of the proposedmethod based on the Floating Point Operations (FLOPs)and, in this respect, making a comparison with theNewton-Raphson approach.

• To assess the convergence rate of the problem based onthe variation of relaxation factor, system loading, gener-ator installation point, generator operating set points andthe predefined tolerance (ε).

To the best of our knowledge, nevertheless, the linearizedsynchronous generator model as a P-V node (the bus wherethe voltage is controlled either by synchronous generators orby grid forming converters [16]) has not been considered forpower flow studies in the related literatures. Considering thepoint that the Martí’s approach [15] cannot satisfy the P-Vnode in the PF formulation, the proposed current injectionmodel of synchronous generator could be a potential tech-nique to advance the Marti’s method.

The rest of the paper is organized as follows. Section IIdescribes the fixed-point iteration method as a backbone forthe proposed power flow calculation. The linearized mod-els of generator and load are formulated in section III andsection IV, respectively. Performance assessment of the pro-posed approach is compared to the classical Newton-Raphsonmethod based on the required FLOPs in section V. The simu-lation results are shown in section VI and finally a concludingpart is found in section VII.

II. FIXED-POINT POWER FLOW MODELConsidering the equivalent current injection model of sinkand source objects in the power system, the following systemformulation for a network with n nodes is derived by applyingthe Kirchhoff’s Current Law at each node:

Yv = i (v) (1)

in which Y ∈ Cn×n is the admittance matrix; i ∈ Cn isthe current vector (including load and generator equivalentcurrent injection model in this work) and v∈Cn is the vectorof bus voltages. To obtain the operating points of the network,

11544 VOLUME 9, 2021


system (1) can be solved by the fixed-point iteration (FPI)technique. The general fixed-point iteration is described herein Algorithm 1. As shown in [17], this method is fasterthan Newton and Dishonest Newton method. In addition toimplementing the simplified ZI loadmodel developed in [15],the applied concept of relaxation method in the proposedlinearized model of generator improves the convergence rateof the solver through selecting an optimal value for the relax-ation factor (γ ).

Algorithm 1 CIM Using Fixed-Point Iteration1: procedure Fixed-Point Iteration2: K = 03: while E ≥ ε do4: Update current injections i (v)5: Solve Yv = i (v) for vk+16: E = ‖vk+1 − vk‖7: k←k+ 18: end while9: end procedure

It is worth mentioning that the system including the onlyP-Q nodes converges without any iteration. However, itera-tions are required when there is a P-V node in the system.To put it differently, a kind of hybrid solution in terms ofthe required number of iterations to solve the PF problemis introduced in this paper. On the one hand, the systemincluding only P-Q nodes is solved in a single shot. On theother hand, a linearized P-V node model is integrated in aniterative simple formulation which can be computed even inparallel. In other words, from the computational point of view,parallel computing could enhance the system execution time.

FIGURE 1. Parallel computing of current injections.

During the iteration process, as shown in Fig 1, only theequivalent current injection nodes are updated as the restof the network’s admittance is constant. The independenceof all these current elements makes it possible for them tobe executed in parallel using a high-performance multi-corecomputer. This is one of the main advantages of the proposedPF formulation over other existing PF methods. It should benoted that, system (1) is a system of linear equations (SLE)in each iteration. The iteration process ends when the differ-ence of the two consecutive results satisfies the predefinedtolerance (ε). The closeness of the initial voltage guess to thefinal solution will improve the system convergence. Althoughthe bus voltages in the unloaded grid condition are chosen forsystem initial guess, the previous solution is used as a betterinitial guess for solving the next problem.

III. SYNCHRONOUS GENERATOR CURRENT INJECTIONMODELThe equivalent current injection model of the rotating syn-chronous generator used in this work, is derived in this sectionunder the assumption that the resistance of the armaturewinding can be ignored. The injected active power of this unitis obtained as [18]:

pg =Evgxg

sin (δ) (2)

where E is the electromotive force (emf) (i.e. the internalvoltage of the machine), xg is the generator’s synchronousreactance, vg is the terminal voltage of the generator and (δ) isthe angle between the terminal voltage and the generator emf,which is known as power angle. The Thévenin and Nortonequivalent circuits and the phasor diagram of the generatorare shown in Fig 2.

FIGURE 2. Single-line and phasor diagram of synchronous generator.

According to the Norton equivalent circuit, the equivalentcurrent of the generator electromotive force is calculated as:

ie =Ejxg

(3)

The generator is modeled here to behave as a P-V bus in thepower flow calculation. In this way, the generator injectedcurrent ig should be set to meet the generator reference oper-ating settings (Eref , pref ). Considering the generator phasordiagram, the internal voltage of the machine can be expressedas:

E = Eref exp(j[δ+]vg]) (4)

Thus, the machine emf can be adjusted by the power angleand the terminal voltage of the generator since the syn-chronous reactance is a constant value in (3). As the mag-nitude of the E is set to be Eref , the power angle and theangle of generator terminal voltage in (4) are varied duringthe iteration process, in such a way that the generator equiv-alent current injection ig can satisfy the pref . To obtain theupdated changing rate of the power angle at each iteration,a linearized form of equation (2) is derived by applying thepartial derivative rule in this work as follow:

1p =E .1vgxg

sin(δ)︸︷︷︸t1

+Evgxg

cos(δ)︸︷︷︸t2

.1δ (5)

According to the defined terms (t1, t2) in the formula (5), thechanging rate of the power angle is calculated as:

1δ =1p− t1

t2(6)

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In the above equation,1p is the difference of pref with respectto the generator’s active power. During the described fixed-point iteration procedure, this difference is made zero so thatthe generator can meet the reference power. To calculate theactive power of generator at each iteration, the real part of thecomplex power (sg) of the generator is taken into account:

sg = pg + jqg = vgi∗g = vg.

(E − vgjxg

)∗(7)

pg = Re[vg

(E − vgjxg

)∗](8)

The complex power (sg) of the generator is the multiplicationof generator terminal voltage and its current injection (ig)conjugation. So, during the iteration process both of thesevariables are varied to satisfy the active power reference point(pref ). As the initial power angle is set to zero in this work,the initial generator active power would be zero. By increas-ing the power angle, the generator active power would beclosed to pref . It should be noted that, the generator terminalvoltage is updated by solving the linear system (1) at eachiteration and the power angle is simultaneously varied basedon its changing rate. For updating the power angle, relaxationmethod (as explained in Appendix) is applied here as follow:

δ← δ + γ1δ (9)

γ is the relaxation factor in (9). The rate of change ofthe generator current injection, affecting the PF rate ofconvergence, is adjusted by this factor. In fact, γ = 1(Gauss-Seidel method) is the border between the under-andthe over-relaxation. Under over-relaxation condition, thesolution decays with oscillations. Compared to the over-relaxation condition, there is no oscillations in under-relaxation mode. The minimum number of iterations requiredto satisfy the solution can be obtained by selecting a rightvalue for the relaxation factor. Providing the grid parameters,the Monte Carlo simulation based on random operating casescenarios is considered to obtain the optimum value of therelaxation factor in this work (shown in the result section).To this end, for each random case scenario, the value of γis varied in a range starting from under-relaxation mode andended to a point in over-relaxation mode while the number ofrequired iterations satisfying the solutions is counted. Thus,the γ which results in the minimum number of requirediterations is selected as the optimal value of the relaxationfactor.

IV. A ZI LOAD MODELConventionally, the classical constant load model, in whichboth active and reactive power are assumed to be independentof the system voltage variations is used in load flow calcula-tions. This assumption works well at the transmission systemlevel since most of the nodes are regulated by the systemoperators. Considering the load’s complex power in (10), thecurrent injectionmodel of the load can be derived by applying

the relaxation method as (11),

sl = vi∗l = pl + jql (10)

il ← (1− γ ) il + γ.(pl + jqlvl

)∗(11)

In contrast to the transmission grid, in the distribution grid thevoltages vary widely along the system feeders as there are noor very few voltage control devices. Therefore, the voltage-dependent characteristic of the loads should be consideredin the power flow analysis of distribution grids. In this way,a more accurate system operating point is achieved. From apower quality point of view, the Distribution System Opera-tor (DSO) can reduce the cost of the delivered power to theend customers if more accurate PF results can be achieved.For example, if there is no need to use under- or over-compensation to regulate the voltage. In this context, then,both the constant active and reactive power in (11) should besubstituted with a right voltage-dependent equivalent charac-teristic of loads in distribution networks as:

il ← (1− γ ) il + γ .(pl (vl)+ jql(vl)

vl

)∗(12)

Commonly, the voltage dependency of the active and reactivepower is expressed via an exponential model [18]:

pl (vl)pl0

=

(vlvl0

)α(13)

ql (vl)ql0

=

(vlvl0

)β(14)

where the nominal values are indicated by a zero subscript.The active and reactive power exponents are defined by αand β, respectively, and they can be extracted from measure-ments. The polynomial model is another common way ofthe load model representation in power systems [18]. Thismodel consists of three major parts: a constant-impedance(Z ), a constant-current (I ) and a constant-power (P). Thevoltage dependency of this model is formulated here:

pl (vl)pl0

= FZ .(vlvl0

)2

+ FI .(vlvl0

)+ FP (15)

ql (vl)ql0

= F ′Z .(vlvl0

)2

+ F ′I .(vlvl0

)+ F ′P (16)

where constants F and F ′ are fractions which can beobtained from measurements; the subscripts Z , I and P standfor constant-impedance, constant-current and constant-powercontributions, respectively. Note that there are only two inde-pendent parameters in (15) and (16) because FZ + FI +FP = 1 and F ′Z + F ′I + F ′P = 1. By providing the voltagecharacteristics for either the power exponents in (13) and(14), or polynomial coefficients in (15) and (16), from mea-surements, the equivalent currents of the loads are updatedat each iteration so that the system converges to the rightstates in distribution grids. In addition to the mentioned load

11546 VOLUME 9, 2021


modelling techniques, Martí et al. [15] consider the followingmodified polynomial load model:

pl (vl)pl0

= CZ .(vlvl0

)2

+ CI .(vlvl0

)(17)

ql (vl)ql0

= C ′Z .(vlvl0

)2

+ C ′I .(vlvl0

)(18)

The vertical intercept is assumed to be zero in Martí’sproposed polynomial load model. This in turn allows fordecreasing the number of load independent parameters by one(CZ + CI = 1 and C ′Z + C ′I = 1) while the accuracy of thePF calculated results is not affected. In general, the describedvoltage-dependent characteristics ((13)-(18)) impose nonlin-earities in the right-hand side of the formula (1). Consideringthe modifiedMartí’s load model and assuming that the imagi-nary part of the voltage is smaller than the real part by severalorders of magnitude, a ZI load model (an impedance (Z ) inparallel with a current source (I ) shown in Fig 3) is obtainedwhich simplifies the PF problem to a linear non-iterative PFsolution for the systems consisting of only P-Q nodes [15].To this end, the modified Martí’s load model ((17) - (18)) isintegrated in the load complex power as (19), shown at thebottom of the page.

Representing the terms vrevl

and vimvl

as follow:

vlrevl=

vlre√v2lre + v

2lim

(20a)

vlimvl=

vlim√v2lre + v

2lim

(20b)

And, applying the assumption vim ≈ 0 in (20), formula (19)can be simplified as:

ilre = <{il} ≈ql0C ′Zv2l0

vlim +pl0CZv2l0

vlre +pl0CIvl0

(21a)

ilim = = {il} ≈pl0CZv2l0

vlim −ql0C ′Zv2l0

vlre −ql0C ′Ivl0

(21b)

Then, the formula (21) can be written as a ZI load model.Considering the current drawn by a constant-impedance as:

il = Ylvl = (Gl + jBl) . (vlre + jvlim)

= (Glvlre − Blvlim)+ j(Blvlre + Glvlim) (22)

the synthesized impedance load model (as shown in Fig 3) isextracted from (21) as follow:

GL + jBL =P0CZV 20

− jQ0C ′ZV 20

(23)

while the rest of the terms in (21a)–(21b) can be representedas a constant-current source.

ip + jiq =P0CIV0− j

Q0C ′IV0

(24)

The applied assumptions simplify the PF problem to a linearnon-iterative PF solution [15] for the systems consisting ofonly P-Q nodes. To this aim, the load coefficients are calcu-lated based on the curve fitting technique. In fact, a suitablecurve should fit the P-V and the Q-V characteristics basedon the collected measured data. The form of Martí’s linearPF (LPF) system equation is the same as (1). Therefore,the impedance of the loads and the load current sourcesare added to the admittance matrix diagonal elements andthe current vector, respectively. Thus, the combination ofthe ZI load model developed by Martí et al. [15] and theproposed linearized generator model yields a linear iterativepower-flow formulation that can incorporate both P-Q andP-V node types. It should be noted that the proposed PF solveris accurate enough to be used for the operation of distributiongrids (as shown in the section VI). Due to the assumptions inthe derivation of the ZI load model, however, the calculatedresults cannot exactly match those obtained with the non-linear load model (NLPF). To compensate for these assump-tions, the non-linear load model based on (17) and (18) canalso be implemented as (25), shown at the bottom of the nextpage. Formula (25) is another ZI representation of the loadmodel imposing non-linearity to the PF system of equationsand requires iterations to satisfy the solution. However, thismodel has not been included in the final system of equationsin this work. In fact, the load voltage and consequently theload equivalent current injection, are updated per iteration sothat the solver can meet the predefined tolerance (ε).

Algorithm 2 describes the implementation of the proposedlinear iterative power flow procedure, including the currentinjection model of generator and load.

il =(pl + jqlvl

)∗

=

pl0.(CZ .(vlvl0

)2+ CI .

(vlvl0

))+ jql0.(C

′Z .(vlvl0

)2+ C ′I .

(vlvl0

))

vl

∗

=

(pl0CZ vlre + ql0C ′Z vlim

v2l0+pl0CI vlre + ql0C ′I vlim

vl0vl

)

+j

(pl0CZ vlim − ql0C ′Z vlre

v2l0+pl0CI vlim − ql0C ′I vlre

vl0vl

)(19)

VOLUME 9, 2021 11547


FIGURE 3. ZI load model.

V. COMPUTATIONAL BURDEN OF THE PROPOSEDAPPROACH COMPARED TO CLASSICALNEWTON-RAPHSON METHODConventional PF methods have a low computational effi-ciency when solving the large networks encountered in distri-bution systems [8]. In this respect, the computational burdenof the proposed technique in comparison with the classicalNewton-Raphson method has been assessed in terms of arough estimation of the Floating Point Operations (FLOPs)for the main required mathematical operation functions andthe solution convergence in this section:

A. FLOPs OF THE MAIN MATHEMATICAL OPERATORS INTHE PF ALGORITHMSThe mathematical operators are the main elements of the PFalgorithm. To evaluate the computational burden of the PFmethods, the FLOPs and the computational cost of the mainmathematical operators, used in the system of the equationof the proposed method, are compared here with those inthe Newton-Raphson method. To obtain a general picture ofalgorithm, the system of equation for both approaches arerepresented here:

1) NEWTON-RAPHSON SOLUTION CALCULATION

1vk = −[J k]−1 [1P (vk)

1Q(vk) ] (26)

where [J ] is the Jacobian matrix:

(2N−2−M )×(2N−2−M )︷︸︸︷[J k ] =

(N−1)×(N−1)︷︸︸︷

[J kPθ ]

(N−1)×(N−M )︷︸︸︷[J kPV ]

(N−M )×(N−1)︷︸︸︷[J kQθ ]

(N−M )×(N−M )︷︸︸︷[J kQV ]

(27)

where N and M, respectively, are the number of buses andthe number of P-V nodes in the grid. The Jacobian subma-trices and the power mismatch formulation are found in theAppendix C.

2) THE PROPOSED APPROACH SOLUTION CALCULATION

1vk =

N×N︷︸︸︷[Z ]

[ikg (v)il

](28)

where [Z ] is the impedance matrix.To assess the computational burden of the PF methods, the

number of FLOPs in (26), for the Newton-Raphson method,should be compared with the number of FLOPs in (28) of theproposed method. The number of FLOPs for mathematicaloperators altogether used in the Jacobian matrix and in thepower mismatch matrices in (26) is significantly larger thanthe FLOPs for the operators used in the current injectionmatrix in (28). However, this number is neglected in theassessment here as the main operational functions which havethe adverse impact on the execution time of the equation(26) are the inversion function of the Jacobian matrix andthe matrix multiplication of the Jacobian matrix with thepower mismatches. In fact, the computational complexityof a matrix inversion in terms of the required number ofFLOPs is cubically O(N 3) dependent on the size of thematrix (N×N ) and the required number of FLOPs for matrixmultiplication is quadratically dependent O

(N 2)[19], [20].

To observe this fact, the required computational time to invertmatrices [Matrix]−1N×N in MATLAB by increasing the size ofthe matrices along with the required computational time formatrix multiplication [Matrix]N×N∗[Matrix]N×1 is shown inFig 4. As can be seen, the execution time ofmatrix inversion iscubically increasing with the size of the matrix. Instead, thereis a quadratic trend for the matrix multiplication. It shouldbe noted that, 5000 tests were done in which the size of thematrix to be inverted increases from (1×1) to (5000×5000).And, the same matrix used to perform the matrix multiplica-tion. The matrix elements were randomly selected. As mak-ing the inversion takes a lot of time, only 5000 tests wereperformed. However, a rough estimation was obtained by the

il ← (1− γ ) il + γ .(pl (vl)+ jql(vl)

vl

)∗

= (1− γ ) il + γ.

pl0.(CZ .(vlvl0

)2+ CI .

(vlvl0

))+ jql0.(C

′Z .(vlvl0

)2+ C ′I .

(vlvl0

))

vl

∗

→ il = (1− γ ) il + γ.

(pl0CZv2l0− j

ql0C ′Zv2l0

)︸︷︷︸

Admittance part

vl +(pl0CIvl0− j

ql0C ′Ivl0

)︸︷︷︸

Current part

(25)

11548 VOLUME 9, 2021


FIGURE 4. Computational cost of matrix inversion and multiplication.

curve fitting tool for both the inversion and the multiplicationtasks which are shown in the Fig 4.

To validate this estimation, a matrix inversion was per-formed when the size of the matrix was (9000×9000). It wasobserved that it takes around 23 sec to invert this matrix.Comparing (26) and (28), thus, the proposed method doesnot involve the time-consuming calculations of the Jacobianmatrix (as the Z matrix is constant in proposed method)and the matrix inversion (with the O((2N − 2−M)3) com-plexity) in each iteration as in Newton-Raphson algo-rithm. Moreover, the matrix multiplication operation in theNewton-Raphson technique has O((2N − 2−M)2) com-plexity per iteration, while the complexity of same operationin the proposed method is O(MN ) per iteration, when the

number of P-V nodes in the system is M . The followingformulations ((28 – a) – (28 – d)), shown at the bottom of thepage and next page, respectively, show how this complexityis derived in the proposed method. To this end, two generalscenarios have been considered:

a: WHEN GRID INCLUDES ONLY P-Q NODESIn this case, the system is solved without iteration.As explained before, a linearized ZI load model has beenexploited so as the equivalent constant value of the loadimpedance and load current injection is used in the system ofequation (28−a). It should be noted that the constant valuesare indicated by the black color in the following formulations.In this respect, the complexity of the proposed method isO(N 2) and the complexity of the NRmethod isO((2N − 2)2)per iteration. Compared to the Newton-Raphson algorithm,thus, less complexity is required to reach the solution by theproposed method.

b: WHEN GRID INCLUDES BOTH P-V AND P-Q NODESThe power system includes both P-Q and P-V nodes in thisscenario. In this case, the proposed linearized model of theP-V node requires some iterations to reach the solution.And, the complexity of the matrix multiplication operation in(28) is O(MN ) as per iteration. This number is significantly

Z11 Z12 . . . Z1NZ21 Z22 . . . Z2NZ31 Z32 . . . Z3N...

.... . .

...

ZN1 ZN2 . . . ZNN

×iL1iL2iL3iL4. . .

iLN

=

v1v2v3v4...

vN

=

v1 =

V1︷︸︸︷Z11iL1 + Z12iL2 + Z13iL3 + . . .+ Z1N iLN

v2 =


v3 =


...

vN =

VN︷︸︸︷ZN1iL1 + ZN2iL2 + ZN3iL3 + . . .+ ZNN iLN

=

v1 = V1v2 = V2v3 = V3...

vN = VN

(28-a)

Z11 Z12 . . . Z1NZ21 Z22 . . . Z2NZ31 Z32 . . . Z3N...

.... . .

...

ZN1 ZN2 . . . ZNN

×

iL1ikG2+iL2iL3iL4...

iLN

=

vk1

vk2

vk3

vk4...

vkN

=

vk1 =

V11︷︸︸︷[Z11iL1 + Z12iL2]+Z12ikG2 +

V12︷︸︸︷[Z13iL3 + . . .+ Z1N iLN ]

vk2 =

V21︷︸︸︷[Z21iL1 + Z22iL2]+Z22ikG2 +

V22︷︸︸︷[Z23iL3 + . . .+ Z2N iLN ]

vk3 =

V31︷︸︸︷[Z31iL1 + Z32iL2]+Z32ikG2 +

V32︷︸︸︷[Z33iL3 + . . .+ Z3N iLN ]

...

vkN =

VN1︷︸︸︷[ZN1iL1 + ZN2iL2]+ZN2ikG2 +

VN2︷︸︸︷[ZN3iL3 + . . .+ ZNN iLN ]

=

vk1 = Z12ikG1 + V11 + V12

vk2 = Z22ikG1 + V21 + V22

vk3 = Z32ikG1 + V31 + V32...

vkN = ZN2ikG1 + VN1 + VN2

=

vk1 = Z12ikG1 + V1

vk2 = Z22ikG1 + V2

vk3 = Z32ikG1 + V3...

vkN = ZN2ikG1 + VN

=

Z12Z22Z32...

ZN2

×[ikG1]=

vk1 − V1

vk2 − V2

vk3 − V3...

vkN − VN

(28-b)

VOLUME 9, 2021 11549


less than the corresponding FLOPs, O((2N − 2−M)2),in Newton-Raphson algorithm (26), as there are a few P-Vnodes in distribution grids (M < N ). To drive this complexityin the proposed method, the first assumption is that there isonly one P-V node (at node 2) in the system where each nodeconnected to a load. Therefore, the corresponding system ofequation, as per iteration, is described in (28−b). It shouldbe noted that the black parts are constant values and arecomputed before the iteration process. However, only blueparts are iteratively updated. In the final derived formulation,

then, the size of the Zmatrix is N×1which has the complexityof O(N × 1) FLOPs as per iteration.Considering two P-V nodes (at nodes 2 and 4) in the grid,

the system (28−c) requiresO(N ×2) FLOPs as per iteration.Finally,O(N×M ) FLOPs are required, as per iteration, whenthe number of P-V nodes isM in the system. In a nutshell, therequired FLOPs as per iteration for the proposed method andthe Newton-Raphson technique is compared in the table 1.

As can be seen the required FLOPs, O(NM ), for theproposed method is significantly less than required FLOPs,

Z11 Z12 . . . Z1NZ21 Z22 . . . Z2NZ31 Z32 . . . Z3N...

.... . .

...

ZN1 ZN2 . . . ZNN

×

iL1ikG1+iL2iL3ikG2+iL4...

iLN

=

vk1

vk2

vk3

vk4...

vkN

=

vk1 =

V11︷︸︸︷[Z11iL1 + Z12iL2]+Z12ikG1 +

V12︷︸︸︷[Z13iL3]+Z14ikG2 +

V13︷︸︸︷[Z14iL4 . . .+ Z1N iLN ]

vk2 =

V21︷︸︸︷[Z21iL1 + Z22iL2]+Z22ikG1 +

V22︷︸︸︷[Z23iL3]+Z24ikG2 +

V23︷︸︸︷[Z24iL4 . . .+ Z2N iLN ]

vk3 =

V31︷︸︸︷[Z31iL1 + Z32iL2]+Z32ikG1 +

V32︷︸︸︷[Z33iL3]+Z34ikG2 +

V33︷︸︸︷[Z34iL4 . . .+ Z3N iLN ]

...

vkN =

VN1︷︸︸︷[ZN1iL1 + ZN2iL2]+ZN2ikG1 +

VN2︷︸︸︷[ZN3iL3]ZN4ikG2 +

VN3︷︸︸︷[ZN4iL4 . . .+ ZNN iLN ]

=

vk1 = Z12ikG1 + Z14ikG2 + V11 + V12 + V13

vk2 = Z22ikG1 + Z24ikG2 + V21 + V22 + V23

vk3 = Z32ikG1 + Z34ikG2 + V 31 + V32 + V33

...

vkN = ZN2ikG1 + ZN4ikG2 + VN1 + VN2 + VN3

=

vk1 = Z12ikG1 + Z14ikG2 + V1

vk2 = Z22ikG1 + Z24ikG2 + V2

vk3 = Z32ikG1 + Z34ikG2 + V 3

...

vkN = ZN2ikG1 + ZN4ikG2 + VN

=

Z12 Z14Z22 Z24Z32 Z34...

...

ZN2 ZN4

×

[ikG1ikG2

]=

vk1 − V1

vk2 − V2

vk3 − V3...

vkN − VN

(28-c)

vk1 = Z12ikG1 + Z14ikG2 + . . .+ Z1N i

kGM + V1

vk2 = Z22ikG1 + Z24ikG2 + . . .+ Z1N i

kGM + V2

vk3 = Z32ikG1 + Z34ikG2 + . . .+ Z1N i

kGM + V 3

...

vkN = ZN2ikG1 + ZN4ikG2 + . . .+ Z1N ikGM + VN

=

Z12 Z14 . . . Z1MZ22 Z24 . . . Z2MZ32 Z34 . . . Z3M...

.... . .

...

ZN2 ZN4 . . . ZNM

×ikG1ikG2...

ikGM

=

vk1 − V1

vk2 − V2

vk3 − V3...

vkN − VN

(28-d)

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Algorithm 2 Linear Iterative PF AlgorithmInputs: Grid parameters, Power flow data, Iteration,Tolerances (ε, ε), Power plant settings (Eref ,pref ),Relaxation factor (γ ), Load Fitting coefficients(Cz,C

′z,CI and C

′I )

Outputs: Power flow solution (v)Per Unit Values ConversionBuilding Admittance Matrix (including theequivalent load impedance defined in (23)):YCalculate Impedance Matrix: Z = Y−1

Making Current Vector (including equivalentcurrent of load defined (24) in and generator)Initialization: v = v0K = 0

while e ≥ ε doUpdate generators current injection model ig:

ik∗g =(Ek−vkgxg

)∗pkg = Re

[vkg.i

k∗g

]1pk = pref − pkg

tk1 =E .(vkg−v

k−1g

)xG

sin(δk)

tk2 =Evkgxgcos

(δk)

1δk =pkg−t

k1

tk2

if t2 > ε→ relaxing the power angleδk+1 = δk + γ1δk

endEk+1 = Eref exp(j[δk+1+vkg])

ik+1e =Ek+1jxg

solve v = Z .i (v) for vk+1

e = ‖vk+1 − vk‖2k ← k + 1

Return

TABLE 1. Number of FLOPs of the solution as per iteration.

(O((2N − 2−M)3

)+ O((2N − 2−M)2)), for Newton-

Raphson method in each solution iteration.

B. SOLUTION CONVERGENCEBoth methods require some iterations so that the differenceof the two consecutive results satisfies the predefined toler-ance (ε). However, Newton’s method is quadratically con-vergent, while the fixed point iteration converges linearlyto a fixed point [21]. And, this is the main advantage of

the Newton-Raphson method over the fixed point iterationtechnique. However, in the proposed fixed point method, thelinearized model of the load and generator is exploited whichsignificantly simplifies the system of equation (as shown in(28-d)) and boosts the rate of the convergence. As mentionedearlier, there is a kind of hybrid solution. On the one hand,the system is solved in a single shot (without iteration) whenit includes only P-Q nodes. On the other hand, the iteration isneeded when the proposed linearized model of the P-V nodeis integrated in the more simplified system of equations (28-d). In this case, moreover, the required number of iterations isdecreased when the optimal relaxation factor is applied [22].

In this context, however, the exact comparison of thecomputational cost between the solvers is not possible asthe way that the PF algorithms have been programmed andimplemented may not be optimal. Moreover, the iterativesolver computational cost depends on the required numberof iterations to reach the solution. However, the number ofrequired iterations varies in different problems due to thevarious operating conditions (node number, system loading,line parameters, the number of P-V nodes and so on) and thesolver initial set points. Then, there is no way to determine,in advance, the exact number of the required iterations tosolve a PF problem. Nevertheless, considering the requiredFLOPs for each iteration and the convergence rate of thesolvers, a rough estimation for comparing the computationalburden of the solvers can be obtained and the systems forwhich a solver is expected to be faster can be found.

To this aim, in this work, the execution time (elapsed byMATLAB) of the required FLOPs (the matrix inversion andmultiplication) for NR method (as shown in Fig (5-b)) isdivided by the execution time of the required FLOPs (themultiplication) for the PM method (as shown in Fig (5-a)).The outcome of this division (as shown in Fig (5-c)) showsthat the PM method is much faster than the NR method (asper iteration) when the size of the network is large. In thecase of 15 P-V nodes in a grid with 2500 nodes, for exam-ple, the proposed method is approximately 2000 times fasterthan the NR method for each iteration. To put it differently,if the NR method solves a PF problem in one iteration, theproposed solution has 2000 iteration chances to reach thesolution so as the computational cost of NR method is met(Accordingly, the PM method has 4000 and 6000 iterationchances to solve the problem when NR method reach thesolution in 2 and 3 iterations, respectively). As elaboratedin the scientific literature [21], however, Newton’s methodis quadratically convergent, while the fixed-point iterationconverges linearly to a fixed point [21]. Moreover, [23] haspointed that the fixed-point iterative solution converges quitewell formost distribution systems that have adequate capacityto serve the load. The key is to have a dominant bulk powersource, which is the case formost distribution systems. There-fore, the mentioned number of iteration chances (2000, 4000,and 6000 and so on) are much more than the required numberof iterations to solve the PF problem by the PM method.Then, the proposes method is faster than the NR method for

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FIGURE 5. MATLAB execution time of the required FLOPs for the matrixinversion and multiplication. (a) shows the elapsed time of[M1 ≡ Z ]N×M × [M2 ≡ iG]1×M . (b) shows the elapsed time of[M3 ≡ J]−1

(2N−2−M)×(2N−2−M) × [M4 ≡ [P;Q]]1×2N−2−M . (c) shows howmany times the PM is faster than NR method per iteration.

large grids. As the convergence rate of the Newton-Raphsonalgorithm is quadratic, the following assumption (shown inTable 2) describes the expected computational cost of theNewton-Raphson and the proposed method. To show theimpact of relaxation factor (γ ) on the rate of convergence,a factor (α) is multiplied by the expected required numberof iterations of the proposed method (it2). And, accordingly,the required number of iterations to solve a PF problem isdecreased in the proposedmethodwhen an optimal relaxationfactor (γ ) is selected.

It should be noted that the described formulas in table 2,are considered here just as the indications for estimating thecomputational cost of the solvers. And, there is no theoreticalassessment on defining the required number of iterations (it)

TABLE 2. Expected computational cost of PM and NR methods.

and the value of α in this work. However, as shown in theresult section, the optimal relaxation factor decreases therequired number of iterations to solve a PF problem in theproposed method. In general, as shown in Fig (5-c), the PMmethod is faster than NR method for large grids. And, it isworth noting that, the fact that the system structure of theproposed method is faster than Newton-Raphson techniqueis also observed in a similar evaluation has been done in [24]when the number of elements in the Jacobian matrix is large.

VI. TEST CASES AND SIMULATION RESULTSIn the following case scenarios, the calculated results of theproposedmethod (PM) are comparedwith those of theDIgSI-LENT PowerFactory on the benchmark IEEE 33-bus testsystem and on a large medium voltage (MV) network (20 kV)in Germany. The MV network consists of 555 nodes. Thisgrid hosts three distributed generators which are operatedin P-Q mode. To assess the effectiveness of the proposedlinearized generator model, a bus is assumed to be operatedas a P-V node in these systems. Moreover, it has been shownthat a multiple number of the proposed generator modelcan be integrated into the grid (on the benchmark IEEE 33-bus test system) without encountering the convergence issue.It should be noted that DIgSILENT PowerFactory solves thePF problem based on the Newton-Raphson iteration tech-nique. The results of the proposed solver accurately trackthe results of the DIgSILENT PowerFactory software show-ing that the proposed method is accurate. In terms of thecomputational cost, however, the proposed method has notbeen compared here with the DIgSILENT as the proposedmethod is coded by MATLAB but the DIgSILENT has itsown interface. Moreover, the MATLAB code is not optimallyprogrammed. The previous section, nevertheless, shows thatthe proposed method is faster than NR method for the largegrids. To quantify the accuracy of the calculated results, themaximum percentage error in the voltage magnitude and themaximum absolute error in the voltage phase angle over allbuses are provided. Moreover, the influence of the relaxationmethod on the computational cost and the convergence rateof the proposed solver are compared for different relaxationfactors. It should be noted that the proposed solver algorithmwas implemented in the MATLAB platform on an Intel Corei5-6500 CPU@ 3.2 GHz with 8GB of RAM.

A. CASE A, IEEE 33-BUS INCLUDING ONLY LOADSThe layout of this grid is shown in Fig 6. This system includesloads on all buses (except the node 1).

The line and the load data are available in the Appendix.The calculated voltage magnitude and voltage angle bythe DIgSILENT PowerFactory, the Martí’s linear load

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FIGURE 6. IEEE 33-bus test system (including only loads).

FIGURE 7. Voltage magnitude of the buses.

FIGURE 8. Voltage angle of the buses.

model technique (LPF) and the proposed non-linear loadmodel (NLPF) are shown in Fig 7 and Fig 8, respectively.It should be noted that the considered load fitting coefficientsare: Cz = −1,C ′z = −1,CI = 2, and C ′I = 2. Althoughthe calculated values of the LPF accurately track the corre-sponding values solved by the DIgSILENT PowerFacroty,the PF results from non-linear load model are more accurate(especially, for the calculated phase angles). According tothe standard EN 50160 (or EN 61000), the voltage deviationshould be confined in a range of ±10 % of nominal voltagein distribution grids. In this case scenario, the voltage drop isabout 10 %. And, the tabulated (in Table 3) maximum volt-age magnitude percentage error and the maximum voltageangle absolute error over all nodes indicate that the proposedsolution is accurate in such a high voltage drop. And, theelapsed time for solving the system taken by the MATLAB,was 2.0269e-04 sec and 3.0890e-04 sec for the LPF and theNLPF, respectively.

In addition to this, the accuracy of the proposed methodfor some extra high load conditions (loading of 150 % and200%) are evaluated here. The corresponding PF results are

TABLE 3. Error of voltage magnitude and phase angle.

FIGURE 9. Voltage magnitude of the buses in high loading conditions.

FIGURE 10. Voltage angle of the buses in high loading conditions.

shown in Fig 9 and Fig 10 for the voltage magnitude and thevoltage angle, respectively. The maximum voltage magnitudepercentage error for 150 % and 200 % loading conditionsare 0.3140% and 0.9051%, respectively. Moreover, the cor-responding maximum voltage angle absolute errors, respec-tively, are 0.0874 deg and 0.17 deg in these high loadingconditions. It should be noted that the voltage of slack busis assumed to be 1 pu in this paper. In general, the highervoltage set point of the slack bus gives the more accurate PFresults.

B. CASE B, IEEE 33-BUS INCLUDING LOADS AND AGENERATORIn this scenario, a generator, as voltage controller, is con-nected to the end of the main branch at bus 18. The internalvoltage of the synchronous generator is adjusted to 0.98 pu.To further assess the correctness of the proposed linearizedgenerator model, two different machine settings in terms ofthe generator reference power and the generator synchronousreactance are considered in this part. Fig 11 and Fig 12 showthe impact of different generator power references (0.5 MWand 1MW) on the voltage magnitude and the voltage phase

VOLUME 9, 2021 11553





TABLE 5. Solver execution time.

angle of all the buses, respectively. In this scenario, themaximum voltage magnitude percentage error and the max-imum voltage angle absolute error over all nodes along withthe solver’s execution times are summarized in Table 4 andTable 5, respectively.

As can be seen, the proposed solutions are accurate asDIgSILENT PowerFactory. Compared to the LPF, however,the NLPF yields more accurate results at the expense of moreexecution time.

In another case scenario, the internal voltage of the syn-chronous generator is set to 0.95 pu and the PF calculationsare compared for two different synchronous reactances, withvalues of 1 pu and 1.7 pu. The calculated errors for powerflow calculations are provided in Table 6.


FIGURE 13. Impact of relaxation factor on the required iterations.

In the following the convergence rate of problem isassessed with respect to the relaxation factor, system loading,generator installation node, generator operation set points andthe tolerance (ε).

1) OPTIMAL RELAXATION FACTORTo obtain the optimal relaxation factor yielding the minimumnumber of required iterations to converge the system, 5000Monte Carlo trials were applied. To this end, the active andreactive power of loads was varied based on normal distri-bution in each trial. And, the value of γ was varied with thestep of 0.01 from 0.2 to 1.9 for each trial. Fig 13 shows thechanging of iteration numbers with respect to the relaxationfactor for 5000 Monte Carlo simulations. It should be notedthat the internal voltage and the active power set point ofgenerator were 0.98 pu and 1MW, respectively, for all thetrials.

As can be seen, a relaxation factor of 1.76 yields theminimum number of required iterations (27 iterations) for allthe scenarios. To show the impact of the relaxation factor onthe power angle, applied in equation (9), and, consequently,on the generator active power, different relaxation factorswere considered. The corresponding results are shown inFig 14 and Fig 15, respectively. Since the linearized formof the generator active power is used to update the changingrate of the power angle, the variations of both power angleand generator active power are the same. However, the powerangle changes so that the generator feeds the reference setvalue (1 MW) to the system. As explained before, underover- and under-relaxation modes, the solver converges to thesolution with and without oscillations, respectively. However,the required number of iterations is decreased when the relax-ation factor is set to around 1.76.

It should be noted that the selected optimal γ has a veryminor impact on the accuracy of the results which can beneglected.

To show this fact, the calculated voltage magnitude and thevoltage angle of node 18 where the generator is connected

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FIGURE 14. Power angle variation for different relaxation factors.

FIGURE 15. Generator active power variation for different relaxationfactors.

FIGURE 16. The impact of γ on the accuracy of the calculated voltagemagnitude.

is plotted over the relaxation factor in Fig 16 and Fig 17,respectively. As can be seen, the accuracy of the calculatedresults has changed significantly low in the expense of fewernumber of iterations.

2) SYSTEM LOADINGTo be more precise on the sensitivity of γ with respect tothe system loading, all the loads in this grid were multipliedby a factor increasing in stepwise way (with the step of 0.1)from 0.025 to 1. And, the relaxation factor was varied withthe step of 0.01 from 0.2 to 1.9. As can be seen in Fig 18 theoptimal value of γ yielding the minimum number of itera-tions has changed in very narrow band from 1.74 to 1.76.To put it differently, γ is robust with respect to the loadingcondition.

FIGURE 17. The impact of γ on the accuracy of the calculated voltageangle.

FIGURE 18. Impact of system loading on the optimal value of γ .

FIGURE 19. The convergence minimum number of iterations as pergenerator installation point and its active power set point.

3) GENERATOR INSTALLATION NODE AND THE POWER SETPOINTIn this part, the convergence rate of the PF problem is assessedwith respect to the installation point of the generator and itspower set point while the relaxation factor varies with thestep of 0.01 from 0.2 to 1.9. To this aim, for three differentpower set points (1, 1.5 and 2 MW), the required number ofiterations to solve the problem is counted for different valuesof γ while the generator installation node is changed overthe grid nodes. It should be noted that the required numberof iterations increases when the applied γ deviates from itsoptimal value (as shown in Fig 18).

The minimum and average number of iterations as pergenerator installation point and its active power set point

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FIGURE 20. The convergence average number of iterations as pergenerator installation point and its active power set point.

FIGURE 21. The convergence minimum number of iterations as pergenerator installation point and its internal voltage set point.

FIGURE 22. The convergence average number of iterations as pergenerator installation point and its internal voltage set point.

are shown in Fig 19 and Fig 20, respectively. The obtainedresults reveal that a fewer number of iterations is countedwhen• the electrical distance of the generator with respect to theslack bus is lower

• the lower power is set for the generator

4) GENERATOR INSTALLATION NODE AND THE VOLTAGESET POINTTo show the impact of the generator voltage set point on thenumber of required iterations to solve the PF problem, forvarious value of γ (from 0.2 to 1.9 with a step of 0.01), theEref is increased from 0.94 pu to 1 pu with the step of 0.02 pu

FIGURE 23. The required iteration to solve the problem in the presence ofmultiple P-V nodes.

FIGURE 24. A medium voltage network in Germany.


while the installation node of generator is changed over all thenodes. As can be seen from Fig 21 and Fig 22, a fewer numberof iterations is required for the larger voltage set points andwhen the electrical distance of the generator with respect tothe slack bus is lower.

5) IEEE 33-BUS INCLUDING LOADS AND MULTIPLEGENERATORSIn this subsection, the convergence rate of the problem isassessed when multiple generators exist in the system. In this

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FIGURE 27. MW grid including DGs operating in P-Q mode.

way, a generator (Eref = 0.98pu, pref = 1MW) is grad-ually added to all the grid nodes starting from node 1 tonode 33 while the number of iterations to solve the problemis counted. As can be seen in Fig 23, the lowest numberof iterations (two iterations) is obtained when a generator isinstalled on node 1 (the shortest electrical distance).

By gradually adding the other generators from node1to node 18, the number of iterations sharply increased(from 2 to 190 iterations), as the electrical distance of thenewly added generators is increased. However, there is nosignificant increase on the number of iterations when theother generators are installed from node 19 to node 33. Apartfrom the convergence rate, these tests show that the proposedmodel is robust when there is more than one generatorin the grid.

C. CASE C, A MEDIUM VOLTAGE NETWORK IN GERMANYTo evaluate the performance of the proposed PF solver forlarge distribution grids, a large MV system operated by theBayernwerk Netz GmbH (a DSO in Germany) is considered.The layout of this network is shown in Fig 24, which consistsof both radial and mesh feeders. Then, this system can show



the capability of the proposed PF method in solving thesystem with complicated structures.

The PF results for different operating conditions are shownin the following case scenarios:

1) PASSIVE NETWORKTheMV system consists of only passive loads. The calculatedvoltage magnitude and the voltage angle for a section of thegrid including 10 nodes connected radially to each other areshown in Fig 25 and Fig 26, respectively.

The line length of the first three line segments (L533-536,L536-539, L539-542) is significantly smaller than the otherlines. As can be seen, thus, there is a sharp voltage dropfrom node 542 to node 544. As there is no generation unitin this scenario, the voltage does not exceed 1 pu. More-over, the iteration is not required to satisfy the solution andthe execution time for solving the system is 9.7494e-04 s.The maximum voltage magnitude percentage error and themaximum voltage angle absolute error over all nodes are3.7065e-05 % and 1.8745e-06 deg, respectively.

2) ACTIVE NETWORKIn addition to the passive loads, three distributed generators(operating in P-Q mode) are added to three buses (343, 424and 503) in the MV grid as shown in Fig 27.

Consequently, due to the bidirectional power flow, thevoltage profile exceeds from 1 pu at the feeders containingthose generators. For instance, Fig 28 and Fig 29 show,respectively, the calculated voltagemagnitude and the voltage

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FIGURE 30. MW grid including four DGs. Three DGs operating in P-Qmode and one (node 153) operating in P-V mode.



angle for a grid section where nodes are radially connectedand generation unit (G343, P-Q control mode) is installed onnode 343.

Due to the generator injection at bus 343, the voltage of thetwo nodes (node 316 and node 343) at the end of the sectionhas boostedmore than the other nodes. Themaximum voltagemagnitude percentage error and the maximum voltage angleabsolute error over all nodes are 9.4360e-05 % and 4.7331e-04 deg, respectively. The execution time of the system is9.7214e-04 s in this scenario.

FIGURE 33. MW grid including four DGs. Three DGs operating in P-Qmode and one (node 554) operating in P-V mode.



3) BUS 153 AS A P-V NODEIn this scenario, a generator operating in P-V mode (0.98 pu,10 MW) is connected to the bus 153 in the previous activesystem as shown in Fig 30.

The maximum voltage magnitude percentage error and themaximum voltage angle absolute error over all nodes are6.8265e-04 % and 0.0046 deg, respectively. To show the volt-age controlling impact of the generator, a radial path includ-ing 10 nodes is selected in such a way that generator is locatedat the end of the selected section. And, the calculated voltagemagnitude and the voltage angle for those nodes are shown in

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FIGURE 36. Generator Power angle variation for different relaxationfactors.

FIGURE 37. Generator active power variation for different relaxationfactors.

FIGURE 38. Generator power angle variation for relaxation factor of 1.8.

FIGURE 39. Generator active power variation for relaxation factor of 1.8.

Fig 31 and Fig 32, respectively. Although the internal voltageof the machine is set to 0.98 pu, the calculated voltage at thegenerator terminal is more than the Eref . It should be notedthat, moreover, the line length between node 137 and 153

is significantly smaller than other segments. The system isexecuted slower (9.8374e-04 s) than the previous scenariosas some iterations are required to reach the solution in thisscenario.

4) BUS 554 AS A P-V NODEThe previous generator is connected to bus 554 in this sce-nario (shown in Fig 33).

The voltage profiles including the voltage magnitude andthe phase angle for radially connected 10 nodes are shownin Fig 34 and Fig 35, respectively. The maximum voltagemagnitude percentage error and the maximum voltage angleabsolute error over all nodes are 0.0041 % and 0.0291 deg,respectively. The system is executed in 0.0011 s.

To show the impact of the relaxation factor on the powerangle and the generator active power, different relaxation fac-tors are applied in this scenario and the corresponding resultsare shown in the following figures (Fig 36, 37, 38 and 39).

This impact is the same as the scenario explained in theCase B in this section. However, as shown in Fig 38 andFig 39, for the γ= 1.8, the solver is unstable in the first 60thiterations and then it converges to the right solution.

VII. CONCLUSIONA new model-based power-flow formulation including thelinearized models of both P-Q and P-V nodes is integratedin a simplified system of linear equations in this paper. Theproposed approach suits for distribution grids and considersboth resistive and reactive parts of the lines. Compared to theclassical Newton-Raphson method, the proposed formulationrequires a much less Floating Point Operations (FLOPs) tosolve the problem as per iteration. A ZI load model has beendeployed that can accurately model the voltage dependencyof the loads in distribution grids (including as the P-Q nodes).The use of this model leads to a linear formulation in whichthe system of power flow equations is solved without any iter-ation. For P-V nodes, a linearized model of synchronous gen-erator has been derived and integrated in a relaxation-methodfixed-point iteration procedure which can be treated in paral-lel. And, as shown in the result section, a multiple number ofthis model can be integrated in the proposed PF formulation.The convergence rate and the stability of the solution areaffected by the applying different relaxation factors in theiterations. In this respect, the results for various tests havebeen assessed to show how the convergence is obtained underover- and under-relaxation modes, with and without oscilla-tions, respectively. The optimal relaxation factor is achievedby the Monte Carlo method which is robust with respect tothe loading of the system. To solve the proposed PF problem,moreover, the required number of iterations is increased byincreasing the electrical distance of the installed generators,increasing the generator operating active power set pointand decreasing the generator operating internal voltage setpoint. The obtained results indicate that the accuracy of theproposed solver (even in the high voltage drop conditions) iscomparable with DIgSILENT PowerFactory.

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TABLE 7. Line and load data.

APPENDIXA. RELAXATION METHODThe relaxation method is one of the iterative methods isused for solving the system Ax = b. The relaxation methodgeneralizes the Gauss-Seidel method by introducing a relax-ation factor, γ > 0. If γ is optimized for the system thiscan increase the rate of convergence of the solution xk by

modifying the size of the correction:

xk+1i =xki +γ

Aii

bi− i−1∑j=1

Aijxk+1j −

n∑j=i

Aijxkj

, 1 ≤ i ≤ nThis is called the successive relaxation (SR) method and for:• 0 <γ< 1→ under-relaxation• γ= 1→ Gauss-Seidel method• γ> 1→ over-relaxation

The form of the above formulation is used in this work to relaxthe power angle (9) of the synchronous generator. To exploitthe load equivalent currentmodel (11), however, another formof this equation can be written as follow to:

xk+1i = (1− γ )xik +γ

Aii

bi − i−1∑j=1

Aijxk+1j −

n∑j=i

Aijxkj

,1 ≤ i ≤ n

B. THE JACOBIAN SUBMATRICES AND THE POWERMISMATCH FORMULATION• Jacobian submatrices [22]:

JPθjk =∂Pj (v)∂θk

=∣∣Vj∣∣ |Vk | (Gjk sin (θj − θk)

−Bjk cos(θj − θk

))(29-a)

JPθjj =∂Pj (v)∂θj

= −Qj (v)− Bjj∣∣Vj∣∣2 (29-b)

JQθjk =∂Qj (v)∂θk

= −∣∣Vj∣∣ |Vk | (Gjk cos (θj − θk)

+Bjk sin(θj − θk

))(29-c)

JQθjj =∂Qj (v)∂θk

= Pj (v)− Gjj∣∣Vj∣∣2 (29-d)

JPVjk =∂Pj (v)∂ |Vk |

=∣∣Vj∣∣ (Gjk cos (θj − θk)

+Bjk sin(θj − θk

))(29-e)

JPVjj =∂Pj (v)

∂∣∣Vj∣∣ = Pj (v)∣∣Vj∣∣ + Gjj ∣∣Vj∣∣ (29-f)

P2 (v)− P2...

PN (v)− PN−−−−

QNG+1 (v)− QNG+1...

QN (v)− QN

=

N∑j=1|V2|

∣∣Vj∣∣ (G2jcos(θ2 − θj

)+ B2jsin

(θ2 − θj

))− P2

...N∑j=1|VN |

∣∣Vj∣∣ (GNjcos (θN − θj)+ B2jsin (θN − θj))− PN−−−−

N∑j=1

∣∣VNG+1∣∣ ∣∣Vj∣∣ (GNG+1,jcos (θNG+1 − θj)+ BNG+1,jsin (θNG+1 − θj))− PNG+1...

N∑j=1|VN |

∣∣Vj∣∣ (GNjcos (θN − θj)+ B2jsin (θN − θj))− PN

11560 VOLUME 9, 2021


JQVjk =∂Qj (v)∂ |Vk |

=∣∣Vj∣∣ (Gjksin (θj − θk)

−Bjkcos(θj − θk

))(29-g)

JQVjj =∂Qj (v)

∂∣∣Vj∣∣ = Qj (v)∣∣Vj∣∣ − Bjj ∣∣Vj∣∣ (29-h)

• Power mismatch formulation as shown at the bottom ofthe previous page [22].

C. GRID DATA OF THE IEEE 33-BUS SYSTEMSee Table 7.

ACKNOWLEDGMENTThe authors would like to thank the Bayernwerk NetzGmbH [25] (the largest regional distribution system operatorin Germany) for providing them with the grid data of themedium voltage system.

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[25] Bayernwerk AG. Accessed: 2020. [Online]. Available: https://www.bayernwerk.de/de.html

BEHZAD ZARGAR (Member, IEEE) received theM.Sc. degree in electrical power engineering fromRWTH Aachen University, Aachen, Germany,in 2015. He is currently a Research Assistant withthe Institute for Automation of Complex PowerSystems, E.ON Energy Research Center, RWTHAachen University. His current research interestsinclude distribution network automation functionswith a focus on monitoring systems, state estima-tion, and power flow calculation.

ANTONELLO MONTI (Senior Member, IEEE)received the M.Sc. degree (summa cum laude) andthe Ph.D. degree in electrical engineering from thePolitecnico di Milano, Italy, in 1989 and 1994,respectively. He started his career with AnsaldoIndustria and then moved to the Politecnico diMilano as an Assistant Professor, in 1995. In 2000,he joined the Department of Electrical Engineer-ing, University of South Carolina, Columbia, SC,USA, as an Associate Professor, and then a Full

Professor. Since 2008, he has been the Director of the Institute for Automa-tion of Complex Power System with the E.ON Energy Research Cen-ter, RWTH Aachen University. He is the author or coauthor of morethan 300 peer-reviewed papers published in international Journals and inthe proceedings of International conferences. He is an Associate Editor ofthe IEEE SYSTEM JOURNAL and IEEE Electrification Magazine, a Member ofthe Editorial Board of the JOURNAL SUSTAINABLE ENERGY, GRIDS AND NETWORKS

(Elsevier), and a member of the Founding Board of the Journal ‘‘ENERGY

INFORMATICS (Springer). He was a recipient of the 2017 IEEE Innovation inSocietal Infrastructure Award.

VOLUME 9, 2021 11561


FERDINANDA PONCI (Senior Member, IEEE)received the Ph.D. degree in electrical engineer-ing from the Politecnico di Milan, Milan, Italy,in 2002. In 2003, she joined the Departmentof Electrical Engineering, University of SouthCarolina, Columbia, SC, USA, as an AssistantProfessor, where she became an Associate Pro-fessor, in 2008. In 2009, she joined the E.ONResearch Center, Institute for Automation of Com-plex Power Systems, RWTH Aachen University,

Aachen, Germany, where she is currently a Professor of Monitoring andDistributed Control for Power Systems. Her current research interests includeautomation and advanced monitoring of active distribution systems.

JOSÉ R. MARTí (Life Fellow, IEEE) was bornin Lleida, Spain. He received the degree in elec-trical engineering from the Central Universityof Venezuela, Caracas, Venezuela, in 1971, theMEEPE degree from the Rensselaer PolytechnicInstitute, Troy, NY, USA, in 1974, and the Ph.D.degree from The University of British Columbia,Vancouver, BC, Canada, in 1981. He has been withThe University of British Columbia, since 1988,where he is currently a Professor of Electrical and

Computer Engineering. His research interests include the ElectromagneticTransients Program, real time, parallel processing, system-of-systems simu-lation, critical infrastructures, and smart grid technologies.

11562 VOLUME 9, 2021

linear iterative power flow approach based on the current

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