Review of Power Flow Studies on Distribution Network with Distributed Generation

K. Balamurugan and Dipti Srinivasan

National University of Singapore kbalahari@yahoo.com.sg, dipti@nus.edu.sg

Abstract- With the perspective of the emerging Smart grid concept, tomorrows distribution network will require repeated and fast load flow solution that must be resolved as efficiently as possible in some applications particularly in distribution planning, automation, optimization of power system etc. This necessitates the continued search for accurate and fast power flow algorithms for distribution network. This paper presents a review and summary of research developments in the field of distribution network power flow which is an essential part of development of effective smart distribution system analysis tools. Different solution strategies including the modeling of distributed generation sources involved in the distribution network power flow are presented. The solution techniques of distribution network power flow problem are classified under two major reference groups namely phase frame approach and sequence frame approach. The Forward and Backward sweep method, Compensation method, Implicit Gauss method, modified Newton or Newton like methods or any other miscellaneous power flow methods are the different algorithms used under each reference frame. Attention is given to the techniques to deal with balanced/unbalanced, radial/weakly meshed/mesh configuration, with or without Distributed Generation (DG) and convergence criteria.

I. INTRODUCTION

Recently modernization of electric power system in developed countries is driven by the smart grid initiative. Implications of Smart Grid for distribution systems will be the integrated functions of Advanced Metering Infrastructure (AMI), Distribution Automation (DA) and Distributed Energy Resources (DER)[1]. In the near future most radial distribution systems would become networked or looped with more points of power injection (through managed integration of three phase and single phase renewable resources) [2].

Distribution Management System (DMS) is a fundamental tool for managing the smart grid. Distribution power flow [DPF] is one of the core modules in a DMS and the results are used by other DMS applications, such as FDIR (Fault detection, isolation and service restoration) and IVVC (Integrated voltage/var control) for analyses. In advanced DMS, DPF will be used on a more frequent basis [3].

Hence in future DPF is required to solve the three phase unbalanced load flow for both meshed and radial operation scenarios of the distribution network integrated with distributed and active resources (i.e. renewable power generation, loads, storages and electric vehicles etc.).This

creates interest on the review of research developments in the power flow studies on the distribution network.

Load flow is the procedure used for obtaining the steady state voltages of electric power systems at fundamental frequency [4]. An efficient power flow solution looks for fast convergence, minimum usage of memory (computationally efficient) and numerically robust solution for all the scenarios. Load Flow studies on transmission networks are well developed using Gauss-Seidal, Newton-Raphson and its decoupled versions [5]. The distribution networks because of the some of the following special features fall in the category of ill-conditioned power systems for these conventional load flow methods [6]:

Radial or weakly meshed networks High R/X ratios Multi phase, unbalanced operation Unbalanced distributed load Distributed generation Single phase representation of three phase system is used

for power flow studies on transmission system which is assumed as a balanced network in most cases. But due to the unbalanced loads, radial structure of the network and untransposed conductors makes the distribution system as an unbalanced system. Hence three phase power flow analysis need to be used for distribution systems. The three phase power flow analysis can be carried out in two different reference frames namely phase frame and sequence frame. Phase frame deals directly with unbalanced quantities. Sequence frame deals with three separate phasor systems which, when superposed, give the unbalanced conditions in the circuit.

We have structured our review as follows: (1) Phase frame power flow approach and its solution techniques using different algorithms (2) Sequence frame power flow approach and its coupled and decoupled algorithms (3) Discussion and brief summary of algorithms used in both phase and reference frames.

II. PHASE FRAME POWER FLOW

Computations in Phase frame power flow analysis are carried out in 3 phase a-b-c quantities. The various algorithms used under this category are forward and backward sweep method, compensation methods, Implicit Z Bus method,

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Modified Newton or Newton like methods and other miscellaneous power flow methods.

1. Forward and Backward sweep Algorithm

The solution techniques using forward and backward

sweep algorithm may be classified as Current summation methods, Power summation methods and Admittance summation methods [7].

a. Current summation methods Efficient power flow algorithm for solving single and three

phase balanced or unbalanced radial distribution networks was developed by W.H.Kersting et al [8,9] and R.Berg et.al[10]. The approach, known by Kersting as the modified Ladder iterative technique, involves forward and backward sweeps through the network using Kirchoffs voltage and current laws. The forward sweep (starts from the last node and ends with source node) is primarily a current summation with possible voltage updates. The backward sweep (starts from the source node and ends with last node) is primarily a voltage drop calculation with possible current updates.

The Forward and Backward sweep algorithm is very pioneer and most commonly used method for the power flow calculation of balanced and unbalanced radial feeders. This is often used as a bench mark for comparison with other algorithms. But this algorithm however was not designed to solve meshed networks.

G.W.Chang et.al.,[11] proposed an improved Backward / Forward sweep load flow algorithm for radial distribution systems which includes the backward sweep and the decomposed forward sweep. Backward sweep uses KVL and KCL to obtain the calculated voltage at each upstream bus. The ratio of specified voltage to the calculated voltage at source bus (first node) is calculated at the end of backward sweep. This ratio is multiplied by calculated voltage at each bus based on the linear proportional principle to update the voltage at each bus in the forward sweep. This algorithm demonstrated its better convergence compared to the commonly used backward /forward sweep method in terms of computational efficiency.

Forward and backward sweep method using quadratic equation was given by U.Eminoglu et.al [12] for balanced radial networks. The difference is that forward sweep node voltage calculations are carried out using quadratic equation and backward sweep using Kirchoffs law like the earlier methods. Maximum node voltage mismatch is checked and node voltages are adjusted using voltage ratio as proposed by Jianwei et al [13]. It is demonstrated that iteration number of quadratic equations based power flow methods is quite small compared to KVL based sweep algorithms, except for the system with more voltage dependent loads [14].

b. Power summation method Haque[15] used the robust forward and backward sweep

algorithm for the distribution system with multiple sources

and/or mesh configurations. The solution technique involves converting the multiple sources and mesh network in to single source radial network by introducing Generator Break Points (GBP) and Loop Break Points (LBP) respectively. The generator break points are created by isolating the generator from the network and the generator effect is created by injecting negative real power and positive reactive power. The loop break points are created by breaking the loops and injecting equal and opposite amount of complex power at the break points. In backward sequence the active and reactive power of sending end of each branch is found by algebraic sum of load powers, branch flow and injections [16].

Earlier Das et al. [17, 18] also proposed the power summation method for balanced three phase radial network using a unique node, branch and lateral numbering scheme to solve the radial load flow problem. But this method evaluates only the node voltage magnitudes without any trigonometric terms.

S.M.Moghaddas et.al [19] discussed the three phase power flow algorithm for radial distribution networks based on power summation method in backward and forward sweep algorithm including the modeling various DG sources. It is demonstrated that power flow summation method can handle DG units in both PQ and PV modes.

c. Admittance Summation method Dragoslav et al.,[20] introduced admittance summation

method for forward and backward sweep analysis for radial and weakly meshed distribution network. This is designated for the cases where all active and reactive node loads are of constant admittance type. If all loads are of a constant admittance type, it is a non-iterative and much faster than the other methods. Its advantage increase with increase of network loads.

2. Compensation methods

D. Shirmohammadi et al.[21] developed a new iterative

compensation power flow method for solving balanced weakly meshed distribution networks, using a multiport compensation technique and basic formulations of Kirchoffs laws. Weakly meshed network is a radial network with few simple loops. By choosing appropriate break points weakly meshed network can be converted to radial configuration. The resulting radial network can now be solved by the forward-backward iterative technique. The voltages of the two nodes of each break point are found from the backward and forward sweep method. Each of the break point voltage can then be obtained by subtracting the voltages at the two end nodes of the break point. The branch currents interrupted by the creation of every break point can be replaced by current injections at its two end nodes, without affecting the operating condition. Thereby the loops are simulated.

The Single phase representation of three phase weakly meshed radial system is directly extended to three phase

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unbalanced weakly meshed system including dispersed generation by C.S.Cheng and Shirmohammadi [22] and later followed by Sarika Khuslani et al.[23] for radial terrestrial distribution system with DGs. DGs are modeled as PQ and PV nodes. For weakly meshed networks along with break point voltage compensation PV Node compensation also carried out by calculating PV node sensitivity matrix to eliminate the voltage magnitude mismatch of PV node. But for radial networks PV node compensation only carried out. Uniformly distributed load on each phase of a section is lumped half-half at the line sections two end nodes.

Zhu and Tomsovic [24] analyzed the Convergence property of break point voltage and PV node compensation algorithm and concluded that convergence rate is insensitive to breakpoint selection, dispersed generation installation position and the no. of meshed loops. Convergence rate is more sensitive to large single phase loads than the load unbalance. Increased number of PV nodes increases the no. of iterations.

3. Implicit ZBus Gauss Method

Tsai-Hsiang Chen et al.,[25] proposed implicit ZBus

algorithm which uses optimally ordered triangular factorization YBus solution method. The complete solution of network equations involves the sparse bifactored YBus matrix ([L] and [U]) and equivalent current injections. This algorithm can handle balanced or unbalanced systems which can be radial, network or mixed type distribution systems.

The voltage of each bus is considered to arise from two different contributions, the specified source voltage and equivalent current injection. The loads, cogenerators, capacitors and reactors are modeled as current injection sources/sinks at their respective buses. According to the superposition principle only one type of source will be considered at a time when calculating the bus voltages. The component of each bus voltage is obtained by activating only the swing bus (substation) voltage source represents the no-load system voltage. This component can be determined directly as equal to the swing bus voltage for every bus in the system. However the other components affected by load currents and cogenerator currents cannot be determined directly. Since load and cogenerator currents are affected by bus voltages and vice-versa, these quantities must be determined in an iterative manner. Each cogenerator bus is handled as a P-Q specified bus and substation is bus is handled as a P-V bus. This method is suitable for a distribution network with many PQ specified buses and only one P-V bus.

4. Modified Newton/Newton like methods

Fan Zhang et.al.,[26] proposed a modified Newton method

for radial distribution systems under the assumptions of no

shunt branches and small voltage difference between two adjacent nodes. The network radial topological structure is explored to express the approximate Jacobian matrix as an product of UDUT where U is a constant triangular superior matrix depends solely on system topology and D is a diagonal matrix resulting from the radial structure of the distribution systems, the elements of which are updated at every iteration. Under the above assumptions this method demonstrated its robustness and efficiency as the backward/forward sweep method of radial distribution systems.

J.H.Heng et al.[27], proposed an algorithm for unbalanced radial distribution system which uses conventional bus-branch oriented data as the input data and branch voltage as the state variable for the power flow solution. This method takes the advantages of topological characteristics of the distribution network to form the Jacobian matrix and traditional Newton Raphson technique is used to find the solution.

Due to radial structure the Jacobian matrix formed is a upper triangular matrix. Traditionally upper triangular matrix are solved by triangular factorization. But here upper triangular jacobian matrix is decomposed into identity matrix and diagonal matrix. Branch voltages are easily obtained by solving the identity matrix by a backward substitution algorithm. Convergence is checked by the specified tolerance for current mismatch. This method put its advantage as LU factorization (which makes ill conditioned numerical problem) is not necessary and saves computer resources by proposing a solution technique to solve upper triangular matrix (due to radial nature of the distribution circuit). This method is robust and fast for large distribution networks particularly of radial structure. But for weakly meshed network or meshed network said advantages may not be effective.

Paulo et.al., [28]proposed a new solution method based on current injection technique using the Newton-Raphson method for unbalanced three phase networks. The current injection equations are expressed in rectangular coordinates and computations are carried out independently on real and imaginary parts. Each element of admittance matrix is an 6*6 sub matrix to accommodate three phase rectangular coordinates. The Jacobian matrix retains its same structure as nodal admittance matrix. The off-diagonal elements of the Jacobian matrix are equal to those of the nodal admittance matrix and it is constant, excluding PV buses. Only the diagonal elements are updated during the iteration, thereby the number of elements to be recalculated during the iteration process is very small. Also for radial distribution system without any DG sources, the Jacobian matrix is almost constant.

A hybrid algorithm based on Ratio-Flow method which uses the voltage ratio as convergence control in forward and backward method and conventional Newton-Raphson method was developed by Jianwei et al.[13] for complex distribution systems (radial networks are connected to a strong connected loop or have multiple sources). It is demonstrated that Ratio-

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Flow method has fast convergent ability and less sensitive to the distribution system parameters.

Baran and Wu[29] have obtained the load flow solution in a distribution network by the iterative solution of three non-linear equations for each branch representing real power, reactive power and voltage magnitude. The number of equations is subsequently reduced by using terminal conditions associated with the main feeder and its laterals, and the Newton-Raphson method is applied to this reduced set. The computational efficiency is improved by making some simplifications in the Jacobian using a chain rule. The speed of convergence and hence the computational efficiency of the method would probably suffer as the number of loops increases.

H.L.Nguyen [30] proposed a three phase power flow formulation for balanced or unbalanced meshed transmission or distribution systems in which Jacobian matrix is expressed in complex form, but some simplifications are introduced by neglecting the component of the mismatch arising from voltage changes. This provides the solution in whole phasor format.

5. Miscellaneous Power Flow Methods The algorithms developed in [31] and [32] not using any of

the conventional methods and proposed new methods using the advantage of the topological characteristics of the distribution networks.

a. Direct method (BIBC/BCBV matrix method) J.H Teng proposed another modified algorithm of his

earlier work [27]. He developed two matrices by taking advantage of the topological characteristics of the distribution system and solves the distribution load flow directly without using Newton Raphson technique [31]. For any power system equipment, if its equivalent current injection or admittance matrix can be obtained, it can be integrated into the proposed method. The two developed matrices are Bus injection to branch current (BIBC) and Branch current to bus voltage (BCBV) matrices. BIBC matrix represents the relationship between bus current injections and branch currents. The BCBV matrix represents the relationship between branch currents and bus voltages. The two matrices provide a novel viewpoint in observing the relations between bus voltages, branch currents and bus current injections. This algorithm uses the maximum load current mismatch as a convergence criterion.

Branch current [B] is related with Bus current [I] injection as (1)

[B]=[BIBC][I] (1) BIBC is the upper triangular matrix represented by ones.

Then the relation between branch current [B] to branch voltage [V] which is the difference of bus voltages is expressed as (2)

[V]=[BCBV][B] (2)

BCBV is the lower triangular matrix represented by branch impedences.

This method eliminates the time consuming procedures, such as LU factorization and forward/backward substitution of the Jacobian matrix or Y admittance matrix. This new three phase load flow algorithm was compared with Gauss implicit Z-matrix method and shows the same accuracy. Also this method demonstrated its robustness and efficiency for Radial network and weakly-meshed network just by modifying BIBC and BCBV matrices. Since BIBC and BCBV matrices are upper and lower triangular matrices, respectively, and therefore the computational time can be reduced. The disadvantage of this algorithm is that two matrices BIBC and BCBV are built by direct observation method. This makes difficult for the programmers to develop the program and also extension.

Since the method fully exploit the network structure to build the matrices, the integration of DGs into the proposed load flow does not degrade the performance of the original power flow algorithm. Different types of DG sources are included in this direct method by the same author [33]. DGs are integrated as constant power factor model (power and power factor specified), variable reactive power model and constant voltage model (power and voltage are specified).

b. Loop impedence matrix method T.H.Chen and N.C.Yang in 2010 [32] proposed a three

phase power flow solution for unbalanced radial distribution system based on loop frame of reference while all the other methods are based on bus frame of reference. This method exploits the topological characteristics of the radial distribution systems and combines the concepts of loop impedence matrix [34] and current injection techniques proposed in [28].

This method can be extended to weakly meshed networks using the concept of break point impedence matrix. The main advantage of this method is that the solution procedure dealing with bus admittance matrix and other algorithms discussed in all above algorithms are eliminated. This method has demonstrated its robust and fast convergence for large unbalanced radial distribution systems. Since this method exploits the topological structure of the radial and weakly meshed network, it may not give the same advantage for meshed networks.

III. SEQUENCE FRAME POWER FLOW ANALYSIS

The phase frame approach for the solution of whole distribution network which comprises of three phase, two phase and single phase sections is tedious due to complicated data structure required for the solution process. Sequence frame approach uses decoupled positive, negative and zero sequence networks of unbalanced three phase system to solve the unbalanced three phase load flow.

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Two problems were faced in the implementation of sequence frame approach compared to phase frame approach for power flow studies on unbalanced system [35]. The mutual inductance between different phases of untransposed distribution lines is not equal to each other. Consequently, unsymmetrical line models established in positive-negative-zero sequence reference frame cannot be broken into independent positive-negative and zero sequence models. So the first is the fact that the coupling in the untransposed line in sequence component model still exists. Second, the phase shifts introduced by special transformer connections are difficult to be represented. But in phase frame, the coupling between lines and the phase shifts are included in the phase components model.

By injecting certain compensation currents into both terminal buses of the line, the three sequence coupled network model can be broken into three decoupled models in different sequence reference frames [36]. Using this decoupled sequence components distribution line model and the sequence component transformer model [37] sequence frame power solution was implemented.

Mamdouh Abdel-Akher et al. [35] split the unbalanced three phase distribution network into main three phase network with three phase line segments and unbalanced laterals with two and single phase line segments. Thereby the unbalanced laterals are decoupled from the main three phase circuit and replaced by equivalent current injections. The sequence frame power flow analysis includes Newton Raphson or fast decoupled to the solution of main unbalanced three phase network and forward/backward sweep method to the solution of unbalanced laterals. This hybrid iterative power flow method involves the following three steps: 1. Calculate the equivalent current injection for each unbalanced lateral using backward sweep in phase components 2. Perform power flow on main three phase network (including the lateral current injection) in sequence reference frame. The symmetrical component method is exploited to decouple the unbalanced three phase system of main three phase network into positive, negative and zero sequence networks. The standard single phase Newton-Raphson or fast decoupled methods are used for solving the positive sequence network whereas the negative and zero sequence networks are represented by two nodal voltage equations. 3. After solving the sequence network in step 2, the root voltages of the unbalanced laterals are known. Update the voltages of the nodes in the unbalanced lateral using forward sweep in phase components. The above three steps are repeated until convergence is reached by using phase voltages mismatch criterion.

The above steps 1 and 3 deals with lateral networks in phase components and step 2 deals with main three phase network in sequence components. The solution in phase and sequence components is coupled together. Hence both phase and sequential frame analysis is performed in every iteration. But by decoupling the main three phase and lateral network

the size of the problem is reduced and hence the execution time and computational burden also reduced.

In order to further reduce the execution time and computational burden Mohamed Zakaria Kamh et al.,[38,39] decoupled the phase and sequence component solutions. Also synchronous generator based and inverter based distributed energy sources are modeled in sequential frame.

The decoupled sequence frame algorithm is as follows: 1. Decompose the microgrid into a three phase section and single phase laterals. It is assumed that the considered single phase lateral is not equipped with single phase voltage controlled DG units or voltage regulating devices. 2. Using the concept of coincidence factor, each single phase lateral and its associated loads are lumped and represented as a single phase load at the node (termed as head node or root node) where the laterals are connected to the three phase system. 3. Perform 3phase sequential frame power flow as discussed earlier by Mamdouh Abdel-Akher et al. [35]. Evaluate the voltage mismatch termination criterion instead of power mismatch. Upon satisfaction of termination criterion, extract the phase voltages of head nodes of all the single phase laterals. 4. Perform the single phase backward/forward sweep power flow for each single phase lateral.

This approach perform the backward and forward sweep power flow only after the three phase sequential frame power flow satisfies the termination criteria. Since both frame computations are decoupled it reduces the execution time and computational burden.

IV. SUMMARY AND DISCUSSION

As discussed earlier due to the concept of smart grid, the distribution system will have more than one feeding node or voltage controlled bus. Also in future the structure of a distribution system may not necessarily be radial. A mesh or mixed type network can be found in modern distribution systems.

Table 1 and 2 gives a brief summary of different algorithms used in phase frame and sequence frame power flow approaches.

Forward/Backward algorithm by Kersting and others is the best method to solve the radial part of the distribution network and used as a bench mark comparison method. But this was not designed to solve the meshed networks.

Compensation based method by Shirmohammadi and others of handling weakly meshed systems are based on choosing break points to convert the system to a radial structure. The network is then solved using a radial load flow algorithm forward and backward sweep method. This can solve only radial or weakly meshed systems.

Gauss implicit z bus method convergence rate is comparable to Newton - Raphson approach if the only voltage specified bus is the swing bus. The convergence characteristics of this method are highly dependent on the

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Table 1 Summary of Phase frame power flow

Phase frame

Algorithm R/WM/M UB DGs CG

Forward/ Backward sweep R 9 Max. Bus voltage mismatch Compensation Based method R/WM 9 9 Max. Bus Real and reactive

Power mismatch Gauss Implicit ZBus R/WM/M 9 9 Max. bus voltage mismatch

Modified Newton/Newton like methods R/WM/M 9 9 Max. Bus Real and reactive Power mismatch / current mismatch/ loop voltage mismatch

Miscellaneous power flow methods

BIBC/BCBV matrices method

R/WM 9 9 Max. Load current mismatch

Loop impedence Matrix method

R/WM 9 Max. bus voltage mismatch

Table 2 Summary of Sequence frame power flow Sequence frame R/WM/M

UB DG CG

Forward /Backward sweep Radial Part of Mixed network

9 Max. Bus voltage mismatch

Compensation Based method Not discussed Gauss Implicit ZBus Not discussed Newton Based methods Main 3 phase

part of mixed network

9 9 Positive sequence voltage mismatch or power mismatch

R-Radial network; WM-Weakly meshed network; M- Mesh or Mixed type network UB-Unbalanced; DG-Distributed Generation; CG-Convergence criteria

number of specified voltage buses in electrical network. Now a day the penetration of distributed generation is day by day increasing. In this scenario this Gauss implicit method may not give good convergence characteristics.

Modified Newton/Newton like methods deals without LU factorization and its forward/backward substitution to avoid the ill conditioned problem. When the mismatch equations are solved by using the jacobian matrix concept the researchers refers this as Newton like methods.

The miscellaneous methods exploit the topological structure of the distribution networks which reduces the number of equations and hence reduce the computational burden. But these methods demonstrated their performance only for radial systems.

Phase frame approach computations are carried out in three phase a-b-c representation. In sequence frame approach computations are carried out in positive-negative and zero sequence components which are decoupled from each other. In phase coordinates conventional Newton Raphson method falls into category of ill-conditioned problem. However in sequence frame single phase Newton Raphson and its diagonal dominance of the Jacobian are utilized because of the balance nature of Positive sequence network. But for the

networks with higher R/X ratios both frame of approaches will provide same results because of the lack of diagonal dominance of the Jacobian. Sequence frame approach reduces the problem size and the computational burden compared to phase frame approach. The different software programmes which can handle distribution network power flows are DIgSILENT power factory [40], PSCAD, CYMDIST[41], RDAP[42], Matlab SimPower Systems[43] and GridLAB-D[44]. All these software except RDAP can handle DGs.

V. CONCLUSIONS

This review has tried to summarize the research developments in distribution power flow including the modeling of DG sources from the research papers which the authors came across. We first classified the techniques of distribution system power flow solution under two different frames called phase and sequence frame approaches. Secondly various algorithms used under each reference frame were discussed in detail with respect to balanced/unbalanced system, radial/weakly meshed/meshed network, with or without DG and mismatch criterion. Lastly the pros and cons

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are summarized for the algorithms used in both frames of approach with their ability to deal different systems. Researchers and practitioners can benefit from this detailed review on the distribution power flow studies in order to develop enhanced tools for the vision of smart distribution system.

VI. ACKNOWLEDGMENT

This work was supported by National Research Foundation (NRF), Singapore under CERP research grant R-263-000-522-272

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