Load Flow Solution U sing Simplified Newton-Raphson Method

Load Flow Solution U sing Simplified Newton-Raphson Method

S Hussain Mohisin Dr V. Ganesh

PG Scholar, JNTUAC Pulivendula Associate professor, JNTUAC Pulivendula

Abstract

The power flow analysis is of great importance in planning and designing for the future expansion of power systems as well as in determining the best operation of existing systems. There exist two widely-used numerical methods (the GaussSeidel: GS and the NewtonRaphson: NR) to solve this problem and therefore referred to as the GS and the NR power-flow solution methods, respectively. Although the standard Newton-Raphson (NR) method is the most powerful algorithm for the power flow analysis in electric power systems, the calculation of Jacobian matrix derivatives involves high computational time. The proposed method presents a simplified Newton-Raphson power flow solution method to simplify overall equation complexity and computation time. The simplified Newton-Raphson method employs nonlinear current mismatch equations instead of the commonly used power mismatch equations. Numerical results are presented with 5-bus test system and IEEE 30-bus test system and compared with standards NR method.

1. Introduction

   The main function of electric power systems is to deliver electric energy to its loads sufficiently, efficiently and economically. The steady-state performances of an interconnected power system during normal operation can be analyzed based on nonlinear nodal analysis to form power flow equations and must be solved by some efficient iterative methods [19]. Power flow analysis is commonly used as a part of power system operation and planning. Since AC power-flow solution methods were first developed over half a century ago, there exist two widely-used numerical methods (the

   GaussSeidel (GS) and the NewtonRaphson (NR) to solve this problem and therefore referred to as the GS and the NR power-flow solution methods, respectively. As broadly known, the NR method has been successfully developed and accepted as the most powerful algorithm for the power flow analysis in electric power systems. In large-scale power systems containing several

   hundred or up to thousand buses, the standard NR method gives a slow execution time due to a large updated Jacobian matrix that needs to be recalculated and factorized at each iteration [10, 11]. Consequently, the de-coupled and fast de- coupled power flow versions [12, 13] were released. Hence, the power-flow solution can be obtained faster. This method is very useful in practical power system analyses, e.g. contingency analysis, on-line power flow control, etc. [4, 14].

   Having a long history of development gives power flow algorithms a vast number and a various kind of applications. Enhancing the algorithm efficiency of power flow calculation has been carried out in many different approaches. Network partition technique can separate a whole power system into subsystems, therefore power flow solution of the complete system can be obtained by direct coupling of solutions from separate subsystems [15] based on the GS method. This concept is very useful to parallelize power flow algorithms in order to implement a parallel and sequential power flow performing on a computer cluster whether the GS method or some other numerical methods such as successive over- relaxation (SOR) method is used as the main solver [16, 17]. In some point of view, an initial guest solution of power flow calculation is one of key factors that cause slow computation.

   In [18, 19], an initial linear solution based on the decoupling principle of real and reactive power decomposition was utilized as the starting point to the power flow calculation. In addition, there are some modified versions of the NR power flow method to handle ill-conditioned power systems [20, 21]. The calculation algorithm has been continually developed by several researchers across the world. A complex form of the power flow calculation was introduced for a three phase unsymmetrical power-flow solution [22, 23]. Power-flow solutions based on a local search method were claimed [24] to be robust and be applicable to those cases in which conventional power flow method failed. Due to advancement of FACTS technology, the power flow equations were modified and rewritten into current-injected forms for the incorporation of FACTS devices and any kind of control strategy [25, 26]. Moreover, the

   study of power-flow solution methods for particular applications, e.g. economic dispatch [27], optimal power flow [28], FACTS devices [29, 30], and

   AC/DC power systems [31], was reported.

   The current balance equation at bus I is

   n

   n

   Ii Yik Vk

   k1

   n

   Over several decades, electrical power systems

   (Ig,i Id,i ) Yik Vk 0

   k1

   (1)

   have been characterized using the nodal analysis to solve for a set of voltage solutions. In general, electrical demands are defined in constant power. This leads to non-linearity of nodal voltage

   equations. To date, the standard NR power flow

   In practice, loads in electrical power systems are in form of powers, therefore it is convenient to rewrite eqn. (1) into a function of powers as follows.

   S S

   S S

   * n

   g,i d,i

   method is one of the most powerful algorithms,

   Fi

   V

   V

   i

   • Yik Vk 0

   k1

   (2)

   which has long history of development, and is widely used to develop commercial power-flow solution software. Although the standard NR power

   Define Fi = Gi + jHi be the current mismatch at bus

   i,

   V V ; V V

   flow method is very efficient and commonly used

   i i i

   k k k

   for the power flow calculation in several power system textbooks [19], to formulate iterative Jacobian updating matrix equations requires complicated formulae and long expressions. In this

   Yij Yij ij is

   Sg,i Sd,i Ssch,i Ssch,i i

   Expressing eqn. (2) in polar form,

   paper, the iterative NR method is still employed as *

   n

   n

   the main solution framework. The essential

   Ssch ,i i

   Y

   V 0

   (3)

   difference is that the proposed algorithm is to find roots of the current mismatch equations instead of

   Vi i

   ik

   k1

   ik k k

   those of the power mismatch equations. This

   Separating the real and imaginary parts,

   approach can simplify a very long and complicated mathematical formula to a very simplistic and short mathematical expression. With this simplification, reduction of the overall execution time is expected. To achieve this goal, expressions to obtain

   1. sch ,i cos n Y V cos(

      S

      S

      i

      i

      V

      V

      i i ik k

      i i ik k

      i k1

      S

      S

      V

      V

   2. sch ,i sin n Y V sin(

   i

   i

   ik k ) 0

   (4)

   ) 0

   elements of Jacobian updating matrix formulae must be derived.

   i i ik k

   i k1

   ik k

   (5)

   In this paper presents the formulation of the proposed NR power flow problem. Derivation of the Jacobian updating matrix elements is included and the floating-point operation counting to evaluate its computational effort. Numerical examples are selected to observe the effectiveness of the proposed method.

   Eqns. (4) and (5) constitute a set of nonlinear algebraic equations in terms of the inependent variables, voltage magnitude in per unit and phase angle in radians. Expanding eqns. (4) and (5) in Taylors series about the initial estimates and neglecting all higher order terms results a set of linear equations. In short form, it can be written as

   G

   G

   G

   V

2. Formulation of proposed simplified power flow solution

   H H

   H V

   V

   The power flow problem is a zero-finding problem

   G A A

   2

   2

   to determine voltage solutions of nonlinear power

   1

   (6)

   mismatch equations [1]. If alternative nonlinear current mismatch equations are selected and used as functions of estimating roots. Given that an n- bus power system, which bus number 1 is assigned to be a slack bus of constant voltage magnitude and

   zero phase angle. Considering the ith bus, current

   balance equations characterizing this bus can be expressed as follows. In this method, the set of nonlinear equations are formulated based on current mismatch equations. The mathematical equations for simplified Newton-Raphson method are as follows [34].

   H A3 A4 V

   The elements of sub matrices A1, A2, A3, and A4 can be derived in the similar manner as jacobian matrix of the standard NR method, which are the partial derivates of eqns. (4) and (5) with respect to and

   V .

   The equations are summarized as, the diagonal and the off-diagonal elements of A1 are

   Gi

   i

   sch ,i sin

   S

   S

   Vi

   (7)

   • V Y sin

   ii i

   k

   k

   ik

   ik

   Jacobian updating step dominates the overall execution time. In general, the time consumed to perform multiplication and division is about the same, but is larger than addition and subtraction. Hence, the operation counting of addition FLOPs is

   Gi

   k

   Vk

   Yik

   sin

   (8)

   negligible. Throughout this paper, FLOPs always

   i i i ii

   i i i ii

   means the multiplication FLOPs for short and it is employed to evaluate the computational effort of

   The diagonal and the off-diagonal elements of A2

   S

   S

   are

   the proposed algorithm. The amount of FLOPs required by each method to formulate Jacobian

   V

   V

   Gi

   sch ,i cos

   Y

   cos

   (9)

   matrices is summarized in Table 1 and where O(n)

   Vi

   Gi

   Vk

   2

   i

   Yik

   cos

   1. i ii

      k

      k

   2. i

   (10)

   means terms of order n.

   Table 1 Number of FLOPs

   ik

   ik

   Sub-matrix		Number of FLOPs
   		Standard NR	Proposed NR
   J1	Diagonal Off-diagonal Total	3(n-1)	4(n-2)
   		3n2+O(n)	2(n-2)
   		3n2+O(n)	6(n-2)
   J2	Diagonal Off-diagonal Total	2(n-1)+3	4(n-2)
   		2n2+O(n)	(n-2)
   		2n2+O(n)	5(n-2)
   J3	Diagonal Off-diagonal Total	3(n-1)	4(n-2)
   		3n2+O(n)	2(n-2)
   		3n2+O(n)	6(n-2)
   J4	Diagonal Off-diagonal Total	2 (n-1)+3	4(n-2)
   		2n2+O(n)	(n-2)
   		2n2+O(n)	5(n-2)
   Overall		10n2+O(n)	22(n-2)

   Sub-matrix		Number of FLOPs
   		Standard NR	Proposed NR
   J1	Diagonal Off-diagonal Total	3(n-1)	4(n-2)
   		3n2+O(n)	2(n-2)
   		3n2+O(n)	6(n-2)
   J2	Diagonal Off-diagonal Total	2(n-1)+3	4(n-2)
   		2n2+O(n)	(n-2)
   		2n2+O(n)	5(n-2)
   J3	Diagonal Off-diagonal Total	3(n-1)	4(n-2)
   		3n2+O(n)	2(n-2)
   		3n2+O(n)	6(n-2)
   J4	Diagonal Off-diagonal Total	2 (n-1)+3	4(n-2)
   		2n2+O(n)	(n-2)
   		2n2+O(n)	5(n-2)
   Overall		10n2+O(n)	22(n-2)

   The diagonal and the off-diagonal elements of A3

   are

   Hi Ssch ,i

   i Vi

   cos i i Vi Yii cos ii i

   (11)

   Hi V Y cos

   (12)

   k

   k ik

   ik k

   The diagonal and the off-diagonal elements of A4

   S

   S

   are

   V

   V

   Hi

   sch ,i sin

   Y

   sin

   (13)

   Vi

   Hi

   Vk

   2

   i

   Yik

   sin

   1. i ii

      k

      k

      ik

      ik

   2. i

   (14)

   As a total number of buses n gets larger, the number of FLOPs grows quadratically in the standard NR method. Interestingly, the FLOP number required by the proposed NR method is

   The new estimates for bus voltages are

   ( t1) ( t) ( t)

   (15)

   linearly proportional to the total number of buses n. Fig. 1 shows the amount of FLOPs required by the

   i i i

   V( t1) V( t) V( t)

   (16)

   two methods.

   i i i

   N

   N

   The process is continued until the current mismatch

   9 Number of FLOPs per iteration to update the Jacobian matrix

   M

   M

   ( t ) i

   and

   ( t ) i

   are less than the specified accuracy,

   10

   Standard NR

   i.e.,

   G( t) & H( t)

   (17)

   10 Proposed NR

   8

   8

   7

   i i 10

   To compare the effectiveness of the proposed NR method against the standard NR method, expressions of the Jacobian matrix elements of A1, A2, A3 and A4, the calculated real and imaginary current matrix elements of G and H, and the calculated real and reactive power matrix elements of Pcal and Qcal need to be evaluated using the floating point operation.

   6

   10

   FLOPs

   FLOPs

   5

   10

   4

   10

   3

   10

   2

   10

   1

   10

   0 1 2 3 4

   10 10

   10 10 10

   Total Number of Buses

3. FLOPs Evolution

   The execution time of the power flow calculation depends on the amount of floating-point operations (FLOPs) [34, 35]. Assume that other steps of the two NR methods are exactly the same, therefore the

   Fig. 1 Number of FLOPS per iteration to update the Jacobian Matrix

4. Results and Analysis

   The effectiveness of the simplified Newton Raphson power flow method was tested against 5- bus [2] and 30-bus [1] IEEE test systems. Each individual test was performed by using Intel i5 Processer in which the power flow programs were coded in MATLAB [35. From the computer simulation, the voltage solution of each test case was calculated. Both NR power flow methods used here took 110-6 per-unit as the termination criteria for the maximum allowable voltage tolerance.

   The 5-bus power systems voltages and line losses are calculated using proposed simplified NR method. The obtained results are compared with the solution of existing standard NR method, has shown in the Table 2 and Table 3 respectively and observed that the results are nearly matched. The 30-bus systems voltages of standard NR method and simplified NR method are given in Table 4.

   Table 2 Voltages for the 5-Bus System

   Bus No	Standard NR		Proposed NR
   	Voltage		Voltage
   	|V| p.u.	Angle Deg.	|V| p.u.	Angle Deg.
   1	1.0600	0.0000	1.0600	0.0000
   2	1.0000	-2.0612	1.0000	-2.0502
   3	0.9872	-4.6367	0.9872	-4.6286
   4	0.9841	-4.9570	0.9841	-4.9483
   5	0.9717	-5.7649	0.9717	-5.7547

   Table 3 Line flows and line losses details for the 5- Bus System

   From Bus	To Bus	Standard NR		Proposed NR
   		Line Losses		Line Losses
   		MW	MVAr	MW	MVAr
   1	2	2.486	1.087	2.479	1.065
   1	3	1.518	-0.692	1.515	-0.701
   2	3	0.360	-2.871	0.360	-2.869
   2	4	0.461	-2.554	0.462	-2.552
   2	5	1.215	0.729	1.215	0.730
   3	4	0.040	-1.823	0.040	-1.823
   4	5	0.043	-4.652	0.043	-4.653
   Total		6.122	-10.777	6.114	-10.803

   For 5-bus system the power mismatch is 9.82099e-010 and number of iterations is 4 in standard NR method where as in current mismatch is 4.17729e-007 and number of iterations is 5 in proposed NR method. For 30-bus system the power mismatch is 4.6806e-008 and number of iterations is 4 in standard NR method whereas in current mismatch is 1.16066e-007 and number of iterations is 8 in proposed NR method. The fig. 2 and fig. 3 shows the power and current mismatches with respect to standard NR method and proposed simplified NR method

   Table 4 Voltages for the 30-Bus System

   Bus No	Standard NR		Proposed NR
   	Voltage		Voltage
   	|V| p.u.	Angle Deg.	|V| p.u.	Angle Deg.
   1	1.0600	0.0000	1.0600	0.0000
   2	1.0430	-5.3504	1.0430	-5.0522
   3	1.0205	-7.5309	1.0210	-7.1807
   4	1.0115	-9.2830	1.0120	-8.8467
   5	1.0100	-14.1684	1.0100	-13.4622
   6	1.0100	-11.0625	1.0103	-10.5093
   7	1.0022	-12.8651	1.0024	-12.2493
   8	1.0100	-11.8154	1.0100	-11.1283
   9	1.0499	-14.1031	1.0501	-13.5661
   10	1.0432	-15.6944	1.0434	-15.1660
   11	1.0820	-14.1031	1.0820	-13.5661
   12	1.0565	-14.9577	1.0566	-14.4731
   13	1.0710	-14.9577	1.0710	-14.4731
   14	1.0415	-15.8500	1.0416	-15.3593
   15	1.0367	-15.9373	1.0369	-15.4407
   16	1.0432	-15.5301	1.0434	-15.0270
   17	1.0382	-15.8606	1.0384	-15.3399
   18	1.0268	-16.5480	1.0270	-16.0400
   19	1.0241	-16.7193	1.0243	-16.2044
   20	1.0281	-16.5205	1.0283	-16.0022
   21	1.0308	-16.1386	1.0310	-15.6101
   22	1.0313	-16.1243	1.0316	-15.5959
   23	1.0259	-16.3233	1.0261	-15.8134
   25	1.0199	-16.4931	1.0201	-15.9651
   26	1.0162	-16.0730	1.0164	-15.5237
   27	0.9985	-16.4936	0.9987	-15.9441
   28	1.0066	-11.6871	1.0067	-11.1059
   29	1.0025	-16.7844	1.0029	-16.2214
   30	0.9911	-17.6688	0.9914	-17.1052

   Fig. 2 Power and Current Mismatches for 5-Bus Power System

   Table 4.7 is the summary of the effectiveness of the proposed method by giving the required iteration and calculation time in comparison with those of the standard NR method. It notes that, in Table 5, SNR and PNR denote the standard NR method and the proposed NR method, respectively.

   Fig. 3 Power and Current Mismatches for 30-Bus Power System

   Table 5 Simulation result for required iteration and time of computation

   Test system	Method	Required Iterations	Execution time (s)	Calculating time ratio
   5-Bus	SNR	4	0.0231	1.5197
   	PNR	5	0.0152	–
   30-Bus	SNR	4	0.1686	1.3255
   	PNR	8	0.1272	–

   From the Table 5, the PNR method spends shorter calculation times for all test cases even the though the test cases iteration high in PNR method compared with SNR. Undoubtedly, the PNR method is faster for these two test cases. Since the PNR method takes less requirement of re- calculation in its Jacobian matrix per iteration, the calculation time ratios for these three test cases are remarkably larger with a factor of 1.5197 and 1.3255 respectively.

5. Conclusions

   Power flow calculation is one of the most essential parts in electric power system operation in order to analyze, simulate, design and control the steady- state system performances properly. Although there exist several powerful power flow solvers based on the standard NR method, their problem formulation gives complication due to the need to calculate derivatives in the Jacobian matrix. The proposed method uses nonlinear current mismatch equations instead of the commonly-used power mismatches to simplify overall equation complexity. With performance evaluation found in session 3, a total number of operations required by the proposed NR method is linearly proportional to the size of the Jacobian matrix, while that of the standard NR method is quadratic. This means that the

   calculation time of the standard NR method increases more rapidly as a total bus number increases than that of the proposed NR method does. From this advantage, the calculation time consumed by the proposed NR method is expected to be less than that of the standard one. This can leads to improvement of power-flow software development in fast computational speed and less memory usage.

6. References

   1. Saadat H. Power system analysis. McGraw-Hill; 2004.

   2. Stagg GW, El-Abiad AH. Computer methods in power system analysis. McGraw-Hill; 1968.

   3. Grainger JJ, Grainger JJ, Stevenson WD. Power system analysis. McGraw-Hill; 1994.

   4. Wood AJ, Wollenberg BF. Power generation, operation, and control. John Wiley & Sons; 1996.

   5. Bergen AR, Vittal V. Power systems analysis.

      Prentice-Hall; 2000.

   6. Glover JD, Sarma M. Power system analysis and design. PWS Publishing; 1994.

   7. Kothari DP, Nagrath IJ. Modern power system analysis. McGraw-Hill; 2004.

   8. Natarajan R. Computer-aided power system analysis. Marcel Dekker; 2002.

   9. Weedy BM, Cory BJ. Electric power systems. Johns Wiley & Sons; 1999.

   10. Sambarapu KM, Halpin SM. Sparse matrix techniques in power systems. In: The 39th southeastern symposium on system theory, Macon, Georgia; 46 March 2007. p. 212216.

   11. Chan KW. Parallel algorithms for direct solution of large sparse power system matrix equation. IEE Proc Gener Transm Distr 2001;148:615622.

   12. Stott B. Decoupled Newton load flow. IEEE Trans Power Apparatus Syst 1972;91:19551959.

   13. Stott B. Fast decoupled load flow. IEEE Trans Power Apparatus Syst 1974;93:859869.

   14. Mori H, Tanaka H, Kanno J. A preconditioned fast decoupled power flow method for contingency screening. IEEE Trans Power Syst 1996;11:357 363.

   15. Carre BA. Solution of load flow problems by partitioning systems into trees. IEEE Trans Power Apparatus Syst 1968;87:19311938.

   16. Huang G, Ongsakul W. Managing the bottlenecks in parallel GaussSeidel type algorithms for power flow analysis. IEEE Trans Power Syst 1994;9:677 684.

   17. Huang G, Ongsakul W. An adaptive SOR algorithm and its parallel implementation for power system applications. In: The 6th IEEE symposium on parallel and distributed processing, Dallas, USA; 2629 October 1994. p. 8491.

   18. Leoniopoulos G. Efficient starting point of load- flow equations. Int J Electric Power Energy Syst 1994;16:419422.

   19. Hubbi W, Refsum A. Starting algorithm and modification for NewtonRaphson load-flow

       method. Int J Electric Power Energy Syst

       1983;5:166172.

   20. Prasad GD, Jana AK, Tripathy SC. Modifications to NewtonRaphson load flow for ill-conditioned power systems. Int J Electric Power Energy Syst 1990;12:192196.

   21. Slochanal SMR, Mohanram KR. A novel approach to large scale system load flows NewtonRaphson method using hybrid bus. Electric Power Syst Res 1997;41:219223.

   22. Nguyen HL. NewtonRaphson method in complex form [power system load flow analysis]. IEEE Trans Power Syst 1997;12:13551359.

   23. Strezoski VC, Trpezanovski LD. Three-phase asymmetrical load-flow. Int J Electric Power Energy Syst 2000;22:511520.

   24. Acharjee A, Goswami SK. Robust load flow based on local search. Expert Syst Appl 2008;35:1400 1407.

   25. Vinkovic A, Mihalic R. A current-based model of an IPFC for NewtonRaphson power flow. Electric Power Syst Res 2009;79:12471254.

   26. da Costa VM, Pereira JLR, Martins N. An augmented NewtonRaphson power flow formulation based on current injections. Int J Electric Power Energy Syst 2001;23:305312.

   27. Chen S, Chen J. A direct NewtonRaphson economic emission dispatch. Int J Electric Power Energy Syst 2003;25:411417.

   28. Hur D, Park J, Kim BH. On the convergence rate improvement of mathematical decomposition technique on distributed optimal power flow. Int J Electric Power Energy Syst 2003;25:3139.

   29. Fuerte-Esquivel CR, Acha E. NewtonRaphson algorithm for the reliable solution of large power networks with embedded FACTS devices. IEE Proc Gener Transm Distr 1996;143:447454.

   30. Wanliang F, Ngan HW. Extension of Newton Raphson load flow techniques to cover multi unified power flow controllers. In: The 4th international conference on advances in power system control, operation and management (APSCOM-97); 1114 November 1997. p. 383388.

   31. Tylavski DJ, Trutt FC. The NewtonRaphson load flow applied to AC/DC systems with commutation impedance. IEEE Trans Ind Appl 1983;19: 940 948.

   32. El-Hawary ME. Electric power applications of fuzzy systems. IEEE Press; 1998.

   33. Wan HB, Song YH. Hybrid supervised and unsupervised neural network approach to voltage stability analysis. Electric Power Syst Res 1998;47:115122.

   34. Thanatchai Kulworawanichpong, Simplified Newton-Raphson power flow solution method, Elect. Power and Energy Syst., 201:32:551 558.

   35. Fausett LV. Applied numerical analysis using MATLAB. Prentice-Hall; 1999.

7. Bibliography

1. S.Hussain Mohisin, PG scholar at JNTUAC, pulivendula.

2. Dr.V.Ganesh, presently working as a associate professor at JNTUAC, pulivendula. Previously he worked as a head of the department for electrical and electronics engineering at JNTUAC, pulivendula for about 4 years. Presently he is guiding 7 p.hd scholars. His areas of interest are electrical distribution system, FACTs devices, renewable energy sources etc.