Open-source Poly-phase Distribution SystemPower Flow Analysis Tool (DxFlow)

Hemanth Kumar Vemprala, Mohammad Asif Iqbal Khan, and Sumit PaudyalMichigan Technological University, Houghton, MI, USA

Emails: {hvempral, mkhan22, sumitp}@mtu.edu

Abstract—This paper details architecture, mathematicalmodeling, and use cases of a new distribution system powerflow analysis tool (called DxFlow). The DxFlow is developed inMATLAB environment and is made completely open source. Thefirst version of DxFlow can solve single-phase and poly-phasepower flow analyses using standard forward-backward sweep(FBS) method. Moreover, DxFlow allows easy integration withheuristic optimization algorithms for solving optimal settingsof equipment on distribution level (such as load tap changersand capacitor banks). DxFlow also allows time-series power flowanalysis. The mathematical model in DxFlow is validated usingthe IEEE test feeders, compared with OpenDSS and GridLAB-Dmodels, and additional new cases are presented to demonstrateits usefulness. The DxFlow is primarily intended for educationaland early-stage research purposes.

Index Terms—Distribution system, Radial networks, Timeseries analysis, Power flow, Distribution optimization.

I. INTRODUCTION

Distribution power flow (DPF) analysis serves as a coremodel in distribution planning and operational studies. DPFmethod is a well researched topic in the past, and severalattempts have been made to make the DPF model and solutionmethods robust, efficient, and comprehensive [1]–[9]. In thepast, the distribution systems were mainly passive circuitsand only a few controllable assets existed (e.g., voltageregulators, switches, capacitor banks) which were controlledat much slower time scale. With distribution feeders becomingactive circuits and with increased penetration of controllableresources at the grid edge, several of the grid functions (e.g.,voltage control, demand response, load following, frequencycontrol) can be achieved through aggregation of the spatiallydistributed resources on MV/LV feeders. Some of such gridfunctions require efficient and comprehensive modeling ofdistribution grids. Therefore, there is a renewed interest indeveloping novel DPF solution methods and tools. GridLAB-D [10] and OpenDSS [11] are excellent examples of such toolsdeveloped recently.

DPF solution methods are mainly three types: Implicit Z-bus methods, Newton-type methods, and Forward-Backwardsweep (FBS) methods. Implicit Z-bus methods are derivativefree, and hence, simple to implement [1]. Implicit Z-busmethods involve the solution of a linear set of equations forconstant current injections and an iterative method for constant

power loads. Recently, in [12], sufficient conditions for theconvergence of Implicit Z-bus method are derived for ZIPloads. However, its convergence could be slower and shown todiverge for distribution systems with voltage controlled buses[13].

Newton-Raphson (NR) method and its variants are commonin solving transmission level power flow problems [14].Due to radial or weakly meshed structure, and high R/Xratio, Newton-type methods may not be always preferredfor DPF [1], [12]. It is even mentioned that NR could failfor distribution systems [15]. However, Newton-type methodsare shown to be successful in solving DPF and generallyrequire less number of iterations [10], [16]. However, eachiteration of NR requires updating elements of Jacobian andinverting the Jacobian matrix, which are computationallyinvolving steps. NR method applied to current injection basedformulation, as in [7], [10], requires only a few of the Jacobianelements to be updated in the iteration; hence, this approachis faster compared to NR method applied to power injectionbased formulation. To achieve computational efficiency, afast decoupled version of DPF is developed in [3], but thisapproach is not the extension of the fast decoupled methodfor solving transmission level power flow which exploits thelow R/X ratio of transmission lines [17]. Several attempts weremade to improve performance of the fast decoupled methodwith high R/X ratio including the empirical adjustment madein the Jacobian matrix [18].

FBS methods exploit the radial or weakly meshed topologyof distribution systems and are based on ladder circuit analysis[19], [20]. Though FBS generally takes more iteration thanNR, it requires less memory and less computation complexityon each iteration as it does not need to store and invert fullY-bus or Jacobin matrices [21]. Several variants of FBS havebeen proposed to make the model more efficient [15], [22] andincorporate voltage controlled buses [23], [24]. FBS is shownas a robust approach for solving distribution grid power flowand its convergence is less dependent on R/X ratio of thefeeders [25].

The objective of this work is to develop an open-source,scalable, and comprehensive distribution system power flowanalysis tool; called DxFlow, using several state-of-the-artsolution methods discussed above. The first released version ofDxFlow uses FBS method; however, other methods includingImplicit Z-bus and the variants of the NR method will

be included in the future versions of the DxFlow. Open-source tools such as GridLAB-D [10] and OpenDSS [11]are becoming increasingly popular; however, we believe thata flexible MATLAB based open-source tool for DPF withan option of multiple solution approaches could still provideeducational and research values.

II. MODULES IN DxFlow

The developed DxFlow tool can analyze any poly-phasedistribution systems with a radial configuration. The DxFlowis made comprehensive in terms of component models andthere is no limit on the number of nodes, branches, and loadmodels. The general description of the DxFlow algorithm isshown in Fig. 1, and following are the main modules.

Node InformationOverhead lines

Underground cablesTransformers

Voltage Regulators..

LibraryConductor data

SpacingUser defined libraries

.

.

Input File Generator

ABCD/abcd

parameters

Impedance

Admittance

Calculations

Import function

Topology Builder

Power Flow Module

Load data & Models

Shunt elements: Caps

Voltage Regulators

Initialize

Forward Sweep

Load & capacitor currents

Backward Sweep

Converged?V, I, P+jQ

Off-peak

On-peak

DER

Storage

Snapshot DLF

Time varying load

Regulator switching

Capacitor switching

Time-series analysis

Volt/var optimization

Loss minimization

Balancing loads

User-defined obj.

Optimization

100

90

80

70

60

50

40

30

40

50

OUTPUTReport

B

C

DMATLAB script are used to access &

interface between modules DxFlow

Yes

No

Iter+1

Fig. 1. Overview of the DxFlow Functional Modules.

A. Input File Generator

It is a preprocessor that converts parameters of distributionnetwork element into data necessary for power flow analysis.The distribution system data and load information are importedfrom a spreadsheet or ‘.csv’ file. The key features of thismodules are:

• Importing network data and library from spreadsheet• Processing essential data necessary for topology builder

and power flow module• Computing ABCD/abcd parameters [26] for all series ele-

ments including transformers, overhead lines, undergroudcables, and voltage regulators

• Generate ‘.mat’ file for use in the next stageThe generic models for fundamental elements of thedistribution system such as overhead lines, undergroundcables, transformers, and voltage regulators are all developedusing the general information from the nameplate. Thispreprocessor stores data in ‘.mat’ file and is accessed byother modules. Due to the inherent advantage of handling dataefficiently, the multi-dimension arrays and structures are used.

B. Topology Builder

Topology builder module is used to generate a table withconnectivity information, nodes, serial branch type, and branchIDs that are adjacent. While solving radial or a weakly mesheddistribution network, structure and topology of the network

should be made available for the FBS algorithm. The solver,based on the hierarchy of the node and branches, computesvoltages and currents during the sweep. Various literaturediscusses methods for topology analysis starting with the nodeand branch ordering methods [22], [25], [27], where a specificcriterion has to be followed while numbering the nodes(sequentially numbering) and branches (pattern of sending andreceiving end nodes) [28]. Few other methods include use ofincidence matrices, a version of graph theory based approach[29], adjacency matrix [30], a topology analysis by undirectedgraph method [31]. The topology builder here produces a tableof connectivity from any ordering of nodes and branches,provided the reference/source node is specifically mentioned.The analysis starts with the reference bus and traverses fromnode to node to lookout for the junction nodes (a node withmore than two branches) till terminal node (a node with singlebranch) is reached.

C. Load Model Initialization

The DxFlow can accommodate various load types such asdelta, wye connections, and load models such as constantimpedance, constant power, and constant current. These loadmodels need to be initialized based on nominal voltage andspecified power, to compute the load currents needed for theFBS method. The time-varying load can also be initialized.

D. Power Flow Module

The processed input data file, network topology, and theload information are fed to this module. The present toolcan solve the poly-phase distribution network consisting ofsingle-, double- and three-phase elements. FBS [26] method isemployed for the first version of DxFlow. Section III providesthe mathematical model and steps involved while performingFBS.

E. Recording Module

The output solution is available in the form of magnitudesand angles for each node voltages and currents. The outputvariables are in the form of a multi-dimensional array, wherefirst subscript indicates phasors, second subscript is not usedin current version, third subscript is designated for elementid, and fourth subscript is for time-series representation. Theother system variables such as real, reactive powers, losses inthe system can also be derived.

F. Interfacing Modules

Using MATLAB script, various modules can be evoked fora single execution or can be iterated in a loop. Using thefundamental blocks of power flow analysis, various studiescan be performed. For instance, refer to section V-B, wheremodule C, D, and E are interfaced together in a loop to runa time-series power flow and in section V-C, modules D isintegrated with heuristic optimization code to perform volt-var optimization.

III. MATHEMATICAL MODEL

A comprehensive set of distribution system componentsare considered in DxFlow that includes single-phase, two-phase, three-wire three-phase, four-wire three-phase cables andconductors, various transformer connections for three-phasetransformers, single-phase transformers, voltage regulators, Y-and Δ– connected ZIP loads, single-phase loads, switchedcapacitors. In DPF analysis, modeling series elements such aslines, transformers, and regulators are represented using abcdparameters [26], which in a compact form can be representedas in Fig. 2. As mentioned in the previous section, the modifiedladder iterative technique based FBS method is shown to beefficient and computationally robust for DPF analysis andis chosen for building the mathematical model in DxFlow.Among three main variants of the FBS method; namely, thecurrent summation method, the power summation method,and the admittance summation method; the DxFlow tool usesthe current summation method due to its simplicity in theformulation and lesser computational effort [32]. The FBSperforms the load flow analysis through two computationaltasks at every iteration: forward sweep and backward sweep.The formulation of the FBS can be expressed mathematicallyas following:

𝒂𝟏 𝒃𝟏𝒄𝟏 𝒅𝟏

𝒂𝟐 𝒃𝟐𝒄𝟐 𝒅𝟐

[𝑽𝒂𝒃𝒄]𝟏 [𝑽𝒂𝒃𝒄]𝟐 [𝑽𝒂𝒃𝒄]𝟑

[𝑰𝒂𝒃𝒄 ]𝟏 [𝑰𝒂𝒃𝒄 ]𝟐 [𝑰𝒂𝒃𝒄 ]𝟑𝒂𝒏-𝟏 𝒃𝒏-𝟏𝒄𝒏-𝟏 𝒅𝒏-𝟏

[𝑰𝒂𝒃𝒄 ]𝒏[𝑰𝒂𝒃𝒄 ]𝒏-𝟏

[𝑽𝒂𝒃𝒄]𝒏-𝟏 [𝑽𝒂𝒃𝒄]𝒏

Fig. 2. Generalized abcd parameter based approach for series elements.

Forward Sweep: The purpose of the forward sweep is todetermine the voltages at each node, starting from the sourcenode till the end node with the following equation,

[VLNabc]m = [A].[VLNabc]n – [B].[Iabc]m (1)

Backward Sweep: The backward sweep provides a solutionof current flowing through series components, starting from theend node till the source node using the following equation,

[Iabc]n = [c].[VLNabc]m + [d].[Iabc]m (2)

where, [A] = [a]–1, [B] = [a]–1.[b] and [a], [b], [c] and [d] arethe generalized abcd matrices of series components [26].

The load currents are updated based on load models, nodalvoltages are obtained from (1), and the current summations areused in conjunction with (2) to update the branch currents.

IV. MODEL VALIDATION

In order to validate the component models, the IEEE testfeeders are solved and the results are compared with the reportin [33]. Table I summarizes the component models validatedusing the IEEE 4-node and 13-node test feeders.

TABLE ILIST OF VERIFIED DISTRIBUTION SYSTEM COMPONENTS USING THE IEEE

4-NODE AND 13-NODE FEEDERS.

Component 4-node 13-nodeYg-Yg Transformer D DYg-Δ Transformer DY-Δ Transformer DΔ-Yg Transformer D DΔ-Δ Transformer D

1-φ wire D2-φ wire D

3-φ, 3-wire D3-φ, 4-wire D D

Underground Cable DConstant Power Load D D

Constant Current Load DConstant Impedance Load DΔ - connected Load D DY - connected Load D DVoltage Regulator DShunt Capacitor D

A. IEEE 4-node Test Feeder

IEEE 4-node system consists of several transformerconnection types as well as balanced and unbalanced loadingconditions. When solved using DxFlow, the maximumpercentage difference on IEEE 4-node system for anunbalanced loading condition was observed to be 0.045% forvoltage and 0.039% for current magnitudes. Fig. 3 shows thecomparative performance (% voltage deviation) of the DxFlowand the results in the IEEE report with different transformerconnections for the 4-node system.

GrY - GrY

Node 2 Node 3 Node 40

0.02

0.04

% d

evia

tion ø-a

ø-bø-c

GrY -

Node 2 Node 3 Node 40

0.02

0.04

% d

evia

tion

-

Node 2 Node 3 Node 40

0.02

0.04

% d

evia

tion

Y -

Node 2 Node 3 Node 40

0.02

0.04

% d

evia

tion

- GrY

Node 2 Node 3 Node 40

0.02

0.04

% d

evia

tion

Fig. 3. % voltage deviation on the solutions with different transformerconnections for the IEEE 4-node test feeder.

B. IEEE 13-node Test Feeder

The IEEE 13-node test feeder is used to validate the modelssuch as single-, two-, and three-phase lines, overhead lines,underground cables, regulators, capacitor banks, Delta/Wye

load connection, and ZIP load. Table II and III summarize thepower flow results obtained from DxFlow and the IEEE report[33]. From the Tables, it can be observed that the voltages,current and power flow obtained from DxFlow match closelywith the IEEE reports.

TABLE IIPOWER AND CURRENT AT SUBSTATION LEVEL AND SYSTEM LOSSES

COMPUTED FROM DxFlow COMPARED WITH THE IEEE REPORT.

Variables IEEE Report DxFlowP+jQ (φ-a), kVA 1251.39+681.57i 1251.42+681.77iP+jQ (φ-b), kVA 977.33+373.42i 977.02+373.66iP+jQ (φ-c), kVA 1348.46+669.78i 1348.80+669.60iCurrent (φ-a), A 558.40 6 -28.58◦ 558.41 6 -28.58◦Current (φ-b), A 414.87 6 -140.91◦ 415.00 6 -140.93◦Current (φ-c), A 586.60 6 93.59◦ 586.38 6 93.59◦

Total Losses, kW 111.06 111.02

TABLE IIIVOLTAGE AT SELECTED NODES OBTAINED FROM DxFlow COMPARED WITH

THE IEEE REPORT.

Node IEEE Report [pu, deg] DxFlow[pu, deg]632 (φ-a) 1.0210 6 -2.49◦ 1.0210 6 -2.49◦632 (φ-b) 1.0420 6 -121.72◦ 1.0414 6 -121.72◦632 (φ-c) 1.0174 6 117.83◦ 1.0180 6 117.83◦

634 (φ-a) 0.9940 6 -3.23◦ 0.9941 6 -3.23◦634 (φ-b) 1.0218 6 -122.22◦ 1.0211 6 -122.23◦634 (φ-c) 0.9960 6 117.34◦ 0.9966 6 117.34◦

646 (φ-b) 1.0311 6 -121.98◦ 1.0305 6 -121.98◦646 (φ-c) 1.0134 6 117.9◦ 1.0140 6 117.90◦

671 (φ-a) 0.9900 6 -5.3◦ 0.9900 6 -5.30◦671 (φ-b) 1.0529 6 -122.34◦ 1.0523 6 -122.35◦671 (φ-c) 0.9778 6 116.02◦ 0.9784 6 116.02◦

611 (φ-c) 0.9738 6 115.78◦ 0.9744 6 115.78◦

V. USE CASES

The DxFlow is an open-source tool; thus, all the codesand simulation results presented in this work are available onhttps://github.com/hvempral/DxFlow

A. Snapshot Power Flow

A large scale single-phase circuit is simulated using DxFlowto show its performance for a snapshot power flow analysis.A modified Baran and Wu system is considered [34]. The12.66 kV MV circuit in [34] is modified by adding 43 LVlaterals (0.24 kV) as shown in Fig. 4, which resulted in 730-node MV/LV system. The load flow problem was solved withan error tolerance of 1e–4 on voltage magnitudes. The DxFlowtook 5 iterations to converge with a computation time of 0.862seconds. Fig. 5 shows the voltage profile for the 730-nodesystem.

B. Time-Series Power Flow

The time-series functionality of DxFlow is demonstratedusing the European Low voltage test feeder [33]. Thisdistribution network has various features that are worth testingthe scalability and flexibility of DxFlow. Time-series loadinformation is available on a minute resolution for 24 hours

Substation12.66 kV0.24 kV

Fig. 4. A 730-node MV/LV Single Phase Feeder.

0 100 200 300 400 500 600 700Node Number

0.9

0.95

1

1.05

Volta

ge, p

u

Fig. 5. Voltage profile on 730-node system obtained from DxFlow.

(1440 minutes). Fig. 6 shows the 50 Hz European test feederwith 11 kV MV source and 416 V laterals supplying the load.

For performing time-series simulation, the Power FlowModule in the DxFlow is invoked in a loop while constantlyupdating the time-series load data and monitoring the nodesand branches of interest. Convergence tolerance is specified to1e–4 on voltages. The simulation results are compared with thesolutions obtained from OpenDSS and GridLAB-D providedin [33]. Fig. 7 presents the time-series plot for total active andreactive power observed at the substation and Fig. 8 presentsthe magnitude of load voltages at select load nodes. It canbe observed from the Table IV and % active/reactive powerdifference plot in Fig. 9 that the active and reactive power onall phases observed at the substation bus match closely withthe solutions in the IEEE report.

TABLE IVMAX. REAL AND REACTIVE POWER AT THE SUBSTATION IN KW/ KVAR.

Simulator P+Q i (φ-a) P+Q i (φ-b) P+Q i (φ-c)OpenDSS 23.817+7.775i 37.900+12.005i 20.421+6.595i

GridLAB-D 23.817+7.775i 37.903+12.006i 20.422+6.595iDxFlow 23.815+7.779i 37.907+12.066i 20.423+6.608i

Performance of the DxFlow is presented next in terms ofnumber of iterations and computation time for the Europeantest system. Since, the load is time-varying, converged nodalvoltages at t–1 is used as the initial guess of the node voltagesfor t = 1, which avoids flat start. This has improved the

Substation

Fig. 6. Three-phase European Low Voltage Test Feeder [33].

0 200 400 600 800 1000 1200 1400Time(minute)

0

1

2

3

4

5

6

kW/k

VAr

104

Real PowerReactive Power

Fig. 7. Time-series simulation results: active and reactive power observed atthe substation.

convergence rate and reduced the computation burden whichis of the essence in real-time operation of distribution systems.Fig. 10 presents computation time taken for each of the 1440time-series load points. At t = 0, maximum computation timeis observed, which is around 0.626s with 4 FBS iterations. Forsubsequent intervals, the solution time is less and the solutionsconverged within 1 to 5 FBS iterations.

0 500 1000 1500Time (minute)235

240

245

250

255

Load

Vol

tage

, V

Load 1(ph A)Load 53(ph B)Load 32(ph C)

Fig. 8. Load profile obtained from time-series simulation.

C. Integration with Heuristic Optimization Algorithm

DxFlow is interfaced with a Genetic Algorithm (GA) codefor the optimal setting of voltage regulators and switchedcapacitor banks (similar to Volt/var optimization in [35]). Forthis, IEEE 13-node test feeder is modified by assuming acontrollable voltage regulator at node 650 with tap positionsfrom -16 to +16 (modeled as an integer variable) and a

0 500 1000 1500Time (minute)0

0.005

0.01

0.015

0.02

% d

iffer

ence

OpenDSSGridLAB-D

Fig. 9. Difference (%) of active power at substation between DxFlow vsOpenDSS and GridLAB-D.

0 200 400 600 800 1000 1200 1400Time (minute)

0

0.2

0.4

0.6

0.8

Solv

e tim

e, s

econ

ds

Fig. 10. Computation time for convergence for time-series simulation.

controllable capacitor (modeled as a continuous variable) atnode 675, and the code is run with loss minimization as anobjective. Here, a GA code is wrapped around DxFlow andcalled multiple times whenever the GA needs to computepower flow for loss calculation. The settings of the voltageregulator and switched capacitor banks are updated using theGA code [36]. Fig. 11 shows the performance of the GAmethod over several GA generations.

0 5 10 15 20 25Genetic Algorithm Generations

100

105

110

Loss

es, k

W

Fig. 11. Performance of GA for Volt/var Optimization in the modified IEEE13-node System.

VI. FEATURES PLANNED IN NEXT VERSION

Next version of DxFlow will include modeling of center-tapped transformer secondaries, Y-bus extraction, inclusion ofvoltage controlled buses, DER and energy storage models,probabilistic load flow analysis modules, short-circuit analysismodule, feeder re-configuration module, modeling of weaklymeshed networks, inclusion of more IEEE test systemsin the library, format conversions to/from GridLAB-D,

ePHASORSIM, and new solvers based on variants of Newton-type methods.

VII. CONCLUSION

A new FBS-based poly-phase power flow analysis tool,DxFlow, is developed in MATLAB and is made fully open-source. The main modules released in the first version of thetool are briefly discussed. The accuracy of DxFlow is validatedby comparing with the power flow results of the IEEE 4-node and 13-node test feeders. Further, to demonstrate themodular ability of the developed code, application examplesof snapshot power flow, time-series power flow, and volt-var optimization using a heuristic approach were carried out.Looking at the performance of the load flow solution forvarious cases explained, the execution time, iteration andaccuracy of the test results closely match with the existingtools such as OpenDSS and GridLAB-D.

ACKNOWLEDGMENT

This work was in part supported by Michigan Tech.Research Excellence Funds and National Science FoundationGrant ECCS-1751460.

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