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Three-phase distribution network fast-decoupled power flow solutions
Whei-Min Lina, Jen-Hao Tengb,*aDepartment of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan, ROC
bDepartment of Electrical Engineering, I-Shou University, 1, Section 1, Hseuh-Cheng Road, Ta-Hsu Hsiang, Kaohsiung, Taiwan, 840, ROC
Abstract
An efficient power flow solution is introduced in this paper. This solution is a current-injection based Newton–Raphson method inrectangular coordinates. The Jacobian matrix of the proposed method could be decoupled into two identical sub-Jacobian matrices. TheG-matrix is used to execute load flow. The “Fast-Decoupled” idea is incorporated into the distribution network analysis the for first time. Usingthe rectangular-coordinate system, the matrix symmetry is retained and the memory requirement of the traditional fast-decoupled load flow isreduced to half. This method is significantly faster than any other method developed so far. Tests have shown that this method has greatpotential for on-line operations.q 2000 Elsevier Science Ltd. All rights reserved.
Keywords: Power flow; Gauss–Seidel method; Newton–Raphson method; Fast decoupled; Distribution automation
1. Introduction
Distribution Automation Systems (DAS) keep on evol-ving in both concept and implementation. As DAS grows toinclude management functions, a robust and efficient Distri-bution Power Flow (DPF) method for on-line applicationsbecomes desirable [1–9]. With the wide coverage of thedistribution network, any gain in executing the DPF couldsave tremendous resources. In addition, DPF will impactother applications, such as, network optimization, Var plan-ning, switching and so forth.
Gauss–Seidel and Newton–Raphson (NR) based algo-rithms are two common techniques used in the industryfor power-flow solutions. The Gauss–Seidel method needsmany iterations and is known to be slow. The NR algorithmhas been used predominantly in the Energy ManagementSystems (EMS), especially the Fast-Decoupled (FD)version which tends to surpass any other techniques inperformance [11].
The distribution network has characteristics differentfrom the transmission system. Distribution networks arecommonly:
1. three-phase unbalanced oriented;2. radial with sometimes weakly meshed topology;3. with high resistance to reactancer=x ratios.
Item (1) greatly increases the complexity of the network
model, since phase quantities have to be considered includ-ing mutual couplings. Items (2) and (3) are the well-knownobstacles which cause sophisticated power flow algorithmsused in EMS to fail [3], not to mention the issue of on-lineoperations. Considering the many uses in a DAS, a goodoperational DPF has to address the above factors properlyand has to be “fast”, like FD. Specialized algorithms havebeen developed to deal with the above issues [1,3,4,10]. Noalgorithm could rival the transmission FD NR in perfor-mance. In Ref. [1] a Gauss method is proposed whichuses a bi-factorized complexY admittance matrix (GBY).GBY is based on the equivalent current injection (ECI), andworks well as long as the power components can bemodeled in theY matrix or can be converted into ECI.
In this paper, a new DPF technique is presented. ThisDPF is also based on ECI. However, the proposed DPFuses the NR algorithm in rectangular coordinates. Aconstant Jacobian matrix can be obtained, which has thesame matrix dimension as the basic NR algorithm. Thisconstant Jacobian matrix can be further decoupled intotwo identical sub-Jacobian matrices. Similar to GBY, theECI-based DPF works for network components, which canbe either modeled in the Jacobian matrix or can beconverted into ECI.
2. ECI-based feeder model
A current-based feeder model was presented in Refs.[3,10], where theYadmittance matrix and the current injec-tion ECI are used.
Electrical Power and Energy Systems 22 (2000) 375–380
0142-0615/00/$ - see front matterq 2000 Elsevier Science Ltd. All rights reserved.PII: S0142-0615(00)00002-8
www.elsevier.com/locate/ijepes
* Corresponding author. Tel.:1 886-7-6563711, ext. 6613; fax:1 886-7-6563734.
E-mail address:[email protected] (J.-H. Teng).
2.1. Line section model
Fig. 1 shows a three-phase feeder line section. The lineconstants could be obtained by the method developed byCarson and Lewis. Fig. 2 is the equivalent circuit of Fig. 1with the ground wire removed. For Fig. 2, a 3× 3 admit-tance matrix,Y, could be generated.
2.2. Power equipment
Equipment could be modeled in phase quantities with orwithout mutual coupling terms. Examples can be found inRefs. [3,4,10] for capacitor, co-generator, and other compo-nents, where those components are modeled by ECI. Shuntadmittance can be either modeled as an admittance or as acurrent injection. However, distribution line charging isusually too small to be included. Loads are modeled asconstant power injections in this paper.
For convenience, phases are numbered as 1, 2, 3 insteadof a, b, c in the following sections.
3. The basic NR algorithm
Starting from the power-based NR technique, themismatch functions can be written in the polar form or therectangular form as
DP
DQ
" #�
2P2u
2P2V
2Q2u
2Q2V
2666437775 Du
DV
" #;
DP
DQ
" #�
2P2e
2P2f
2Q2e
2Q2f
2666437775 De
Df
" # �1�
where the commonly used polar-form Jacobian matrix isgiven by
J �2P2u
2P2V
2Q2u
2Q2V
2666437775
J is a state-dependent non-symmetric matrix. The NR
algorithm will update the bus voltages at each iterationwith correction factors calculated from the powermismatches. For a practical HV transmission network,Jcould be greatly simplified by assuming that
1. the voltage magnitudesV ù 1 p:u:;2. the phase anglesu ù 0;3. ther p x:
After the simplifications,J becomes constant, symmetricand very sparse.J can be decoupled and needs to be factor-ized only once. When applied to a low voltage network,however, the method suffers from the following deficiencies:
1. Whenr=x ratios are increased, the assumptions of FD areno longer valid. FD NR could become unstable and evendiverge.
2. Without decoupling, the Jacobian matrix is non-symmetric, and a full matrix has to be stored.
3. J is state-dependent and has to be updated and factorizedat each iteration.
4. The ECI-based NR algorithm
An ECI-based NR uses current instead of power. For loadbuses, the specified constant powerPsp and Qsp can beconverted into ECI (I sp-eqv). For example, the bus-k ECI atthe ith iteration is given by
Isp-eqvk � �Pk 1 jQk�sp
Vik
!p
� Re�I sp-eqvk �1 jIm�I sp-eqv
k � �2�
From Fig. 2, the three-phase feeder section can be modeledwith the calculated current (I cal) as follows:
Ical1
Ical2
Ical3
2666437775 �
y11 y12 y13
y21 y22 y23
y31 y32 y33
26643775
v1 2 v10
v2 2 v20
v3 2 v30
26643775 �3�
whereykm� gkm 1 jbkm is the line admittance from nodekto m, and the calculated bus-k injection for a specified phase
W.-M. Lin, J.-H. Teng / Electrical Power and Energy Systems 22 (2000) 375–380376
Fig. 1. A three-phase line section.
Fig. 2. The equivalent circuit of a three-phase line section.
can be written as
I cal �Xl
j�1
Icalj
wherel is the number of intercepting line section at busk.Eq. (1) can be rewritten in the rectangular form based
upon current. The current mismatch function is
DI r
DI i
" #�
2I r
2e2I r
2f
2I i
2e2I i
2f
266664377775 De
Df
" #�4�
where
DI � I sp-eqv2 I cal � DI r 1 jDI i
DV � De1 jDf
are the real and imaginary components of the currents andvoltages, respectively.
Using Fig. 2 as an example, the Jacobian entries can beobtained as follows:
I r1 � �g11�e1 2 e10 �2 b11� f1 2 f10 �1 g12�e2 2 e20 �
2 b12� f2 2 f20 �1 g13�e3 2 e30 �2 b13� f3 2 f30 ��
I i1 � �g11� f1 2 f10 �1 b11�e1 2 e10 �1 g12� f2 2 f20 �
1 b12�e2 2 e20 �1 g13� f3 2 f30 �1 b13�e3 2 e30 ��
and
2I r1
2e1� g11;
2I i1
2e1� b11; etc:
Thus, the current mismatch function for the line section inFig. 2 can be modeled by a 6× 6 matrix in the 2× 2 blockedstructure. We get
�5�
The Jacobian matrix, given by Eq. (5), is a non-symmetricconstant matrix. Re-arranging the state variables, Eq. (5)becomes
�6�
This arrangement yields a symmetric matrix. Two- andsingle-phase lines can be processed similarly to get theJacobian matrix. The DPF problem can be solved itera-tively with the above current mismatch function. Thecurrent mismatches are updated by Eqs. (2)–(4). Withsymmetry, the memory requirements could be reducedto half. The time consuming factorization and updatehave to be executed only once. TheJ matrix can beobtained from the nodalY admittance matrix.J needsno approximation and provides a very robust model.The flow chart of executing the ECI-based NR isshown in Fig. 3.
W.-M. Lin, J.-H. Teng / Electrical Power and Energy Systems 22 (2000) 375–380 377
Fig. 3. Flow chart of the ECI-based NR method.
5. Fast-decoupledG-matrix method
For Eq. (5), the state variables could also be arranged asfollows:
�7�
The current mismatch function can be written in a moregeneral form as
G 2B
B G
" #De
Df
" #� DI r
DI i
" #
Here, the Jacobian matrix is asymmetric, and this formulacan also be used for the ECI-based NR approach. The NRmethod is a gradient minimization problem in solving non-
linear equations, where the Jacobian matrix provides theoptimal direction for the root finding. Simplifications inJacobian tend to alter the direction, which generally increasethe number of iterations. If the simplification can be doneproperly, the computation could be compensated for by animproved overall performance. Whatever the simplifica-tions did, the final solution should remain unchanged. Thetransmission FD NR [11] is an example to show thecomputational advantages. Using constant Jacobian matrixapproximations, FD NR reduces the computing time per itera-tion, and storage. Withg . b in the lower-voltage distribu-tion system, it is interesting to note that if the decoupled ideais applicable. A decoupled formulation is thus investigated byneglecting the off-diagonal blocks. We get
GDe� DI r GDf � DI i �8�where the Jacobian matrix
G� 2I r=2e� 2I i
=2f
Note that the distributionr=x ratio may not be as high asthe transmissionx=r ratio. However, when compared withthe transmission FD NR, the proposed method does notrequire any assumptions on the voltage magnitudes orphase angles, which might lend the proposed methodsome advantages in robustness.
Fast-decoupledG-matrix (FDG) method uses the sameNR algorithm as shown in Fig. 3. The compactG matrixis used for the Jacobian matrix in solving Eq. (8) instead ofEq. (6).
6. Test results
A UNIX workstation was used to conduct various tests.All methods mentioned above were tested except the power-based FD NR which is known to diverge for highr=x ratios.
Method 1 (PNR): Power-based NR method [4].Method 2 (GBY): Gauss–Seidel method with bi-factor-ized Y matrix [3].Method 3 (CI): ECI-based NR algorithm.Method 4 (FDG): ECI-based FDG algorithm.
W.-M. Lin, J.-H. Teng / Electrical Power and Energy Systems 22 (2000) 375–380378
Table 1The test feeder
Feeder No. No. of buses Length (km)
1 45 1.52 90 2.53 135 3.24 180 4.05 270 7.4
Fig. 4. Performance test with normalized execution time.
Fig. 5. r=x Ratio test.
A main feeder trunk with 3× 90 phase buses was usedwith unbalanced loads. The averager=x ratio of the lines isaround 1.88. Convergence tolerance is 1023 for De andDf.This trunk will be chopped into various sizes for testing asshown in Table 1. The substation is modeled as the slackbus.
6.1. Performance test
Fig. 4 shows the performance test with the normalizedexecution time. It can be seen that while FDG and CI areefficient, FDG is even more impressive. FDG shows a betterperformance as the network increases in size. For example,if the FDG execution time for the 45-bus feeder is 1, CI is2.6, GBY is 1.95 and PNR is 5. For the 270-bus feeder, CI is6.1–1 against FDG, GBY is 4.5 and PNR is 17.8. Withcomplex values used in a smaller matrix, GBY shows abetter performance than CI.
6.2. Robustness—r=x ratio test
Fig. 5 shows ar=x ratio test for FDG. Ther =x ratio isadjusted by multiplying a factorK that ranges between 0.5and 1.5. Various tests were conducted for FDG to show itsrobustness. CI is insensitive to ther=x ratio, which is asteady three-iteration for all test cases, and is not shownhere. On the contrary, FDG needs a higherr=x ratio (around1.1) to converge effectively. That is, FDG is designed todeal with low-voltage highr=x ratios circuits. Higherr=xratio yields better performance, which is different from the
conventional FD NR needing a lowr=x ratio to converge.For networks with lowr=x ratios, CI could be used instead.More researches are still conducted by the authors to makeFDG applicable to all circumstances.
6.3. Storage
Fig. 6 shows the storage requirement for each method.Occupying one half of a quadrant, the FDG Jacobian matrixis also symmetric and sparse. Only an upper 12.5% of thematrix is used. Fig. 7 shows the actual computer memoryused by Feeder 1 with 45 buses. There is one branch inFeeder 1 as shown in the figure. One dot indicates a non-zero term in this Jacobian matrix. Consequently, thememory requirement is minimal for FDG.
6.4. Meshed network test
One more test was run for a feeder with many branches,four meshes and 225 buses, which is mixed with singlephase, two-phase and three-phase conductors. Test showsthat this feeder needs five iterations and 26.55 normalizedexecution time to converge. This test shows that FDG is alsoinsensitive to the network structure including the meshes.
7. Discussion and conclusion
This paper presents an ECI-based full-matrix CI and anECI-based FDG method for distribution power networks.The ECI-based algorithm uses a constant symmetric Jaco-bian matrix, which needs to be factorized only once. Theconstant Jacobian matrix is obtained without any assump-tions and the algorithm is very robust. By the use of aconstant JacobianG-matrix, the execution time can begreatly saved.
It appears from a computational viewpoint, decoupledversions are generally preferable to the full matrix version.So the FDG algorithm was investigated and extensive testswere further conducted for FDG. Test results have shownthat FDG is very effective for higherr=x ratio. The advan-tages of the FDG algorithm can be summarized as follows:
1. FDG uses a constant sparse and symmetric Jacobianmatrix, with the factorization having to be done onlyonce.
2. FDG has a lower memory requirement.3. FDG has a lower execution time, especially for large
networks.4. FDG is very effective for highr=x ratio.5. FDG exhibits a steady iteration number.
Compared with the transmission FD NR, FDG does notrequire any voltage magnitude or angle assumptions. Inaddition, when using ECI, state variables are already avail-able in the rectangular form without changing coordinates.V andI could be updated easily on each FDG iteration whilepower-based NR or FD need additional effort and cost in
W.-M. Lin, J.-H. Teng / Electrical Power and Energy Systems 22 (2000) 375–380 379
Fig. 6. The memory requirement of each method.
Fig. 7. The FDG Jacobian of feeder 1.
computing power mismatches, and sine and cosine func-tions.
The standard sparse matrix techniques could be used forFDG. Optimal ordering for radial networks will create nofill-in terms and will result in a very clean factorization.Also, a weakly meshed structure will create very few fill-in terms. The FDG is mainly developed for the distributionnetwork. Additional models have been investigated by theauthors for a wider use of the technique and will bediscussed in future papers.
References
[1] Sun DI, Abe S, Shoults RR, Chen MS, Eichenberger P, Farris D.Calculation of energy losses in a distribution system. IEEE Transac-tions on Power Apparatus and Systems 1980;PAS-99(4):1347–56.
[2] Vempati N, Shoults RR, Chen MS, Schwobel L. Simplified feedermodeling for load flow calculations. IEEE Transactions on PowerSystems 1987;PWRS-2(1):168–74.
[3] Chen TH, Chen M, Hwang KJ, Kotas P, Chebli EA. Distributionsystem power flow analysis—a rigid approach. IEEE Transactionson Power Delivery 1981;6(3):1146–52.
[4] Birt KA, Graffy JJ, McDonald JD, El-Abiad AH. Three phase loadflow program. IEEE Transactions on Power Apparatus and Systems1976;PAS-95(1):59–65.
[5] Baran ME, Wu FF. Network reconfiguration in distribution system forloss reduction and load balancing. IEEE Transactions on PowerDelivery 1989;4(2):1401–7.
[6] Liu CC, Lee SJ, Vu K. Loss minimization of distribution feeder:optimality and algorithm. IEEE Transactions on Power Delivery1989;4(2):1281–9.
[7] Chiang HD, Jumeau RJ. Optimal network reconfigurations in distri-bution systems: Part 1: a new formulation and a solution methodol-ogy. IEEE Transactions on Power Delivery 1990;5(4):1902–9.
[8] Chen CS, Cho MY. Determination of critical switches in distributionsystem. IEEE Transactions on Power Delivery 1992;7(3):1443–9.
[9] Distribution automation: a practical tool for shaping a more profitablefuture. Special Report, Electrical World, December 1986. p. 43–50.
[10] Chen TS, Chen MS, Inoue T, Chebli EA. Three-phase cogeneratorand transformer models for distribution system analysis. IEEE Trans-actions on Power Delivery 1991;6(4):1671–81.
[11] Stott B, Alsac O. Fast decoupled load flow. IEEE Transactions onPower Apparatus and Systems 1974;PAS-93(3):859–67.
W.-M. Lin, J.-H. Teng / Electrical Power and Energy Systems 22 (2000) 375–380380
Whei-Min Lin was born on 3 October 1954. He received his BS-EE fromthe National Chao-Tung University, MS-EE from the University ofConnecticut and his PhD EE from the University of Texas in 1985.He worked at Chung-Hwa Institute for Economic Research, Taiwan,as a visiting researcher after his graduation. He joined Control DataCorp. in 1986 and worked with Control Data Asia in 1989. He has beenwith the National Sun Yat-Sen university, Taiwan, since 1989. Dr Lin’sinterests are GIS, Distribution Automation System, SCADA and Auto-matic Control System.
Jen-Hao Teng was born in 1969 in Tainan, Taiwan. He received his BS,MS and PhD degree in electrical engineering from the National SunYat-Sen University in 1991, 1993 and 1996, respectively. He has beenwith the I-Shou University, Taiwan, since 1998. His current researchinterests are Energy Management System, Distribution AutomationSystem and Power System Quality.