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Transmission loss allocation based on optimal power flow and sensitivity analysis
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Electrical Power and Energy Systems 30 (2008) 291–295 www.elsevier.com/locate/ijepes
Short Communication
Transmission loss allocation based on optimal power flow and sensitivity analysis E.A. Belati, G.R.M. da Costa
*
Department of Electrical Engineering of the Sa˜o Carlos Engineering School, Sa˜o Paulo University, 13.566-590 Sa˜o Carlos, SP, Brazil Received 27 July 2006; received in revised form 13 July 2007; accepted 26 July 2007
Abstract This paper presents a new approach to the transmission loss allocation problem in a deregulated system. This approach belongs to the set of incremental methods. It treats all the constraints of the network, i.e. control, state and functional constraints. The approach is based on the perturbation of optimum theorem. From a given optimal operating point obtained by the optimal power flow the loads are perturbed and a new optimal operating point that satisfies the constraints is determined by the sensibility analysis. This solution is used to obtain the allocation coefficients of the losses for the generators and loads of the network. Numerical results show the proposed approach in comparison to other methods obtained with well-known transmission networks, IEEE 14-bus. Other test emphasizes the importance of considering the operational constraints of the network. And finally the approach is applied to an actual Brazilian equivalent network composed of 787 buses, and it is compared with the technique used nowadays by the Brazilian Control Center. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Active transmission loss allocation; Optimal power flow; Sensitivity analysis
1. Introduction In the current world economic scenario, the electrical industry has been undergoing a restructure in many countries effectively with the aim of creating a more competitive industry and consequently many challenges have appeared. One challenge is in allocating losses to loads, generators or bilateral contracts, that is, regardless of the approach, the final allocation always contains a degree of arbitrariness. This allocation must be as fair as possible among the power system agents. Thus, the approaches have to treat all the constraints of the network. Active power losses in the transmission are about 4–8% of the total active power generated. In Brazil the cost of the loss is half a billion U.S. dollars a year [1]. The losses depend on the active and reactive power flows that are transmitted from a generator to a load through a number of alternative pathways, which in
*
Corresponding author. Tel.: +55 16 33738134; fax: +55 16 3373 9372. E-mail address: [email protected] (G.R.M. da Costa).
0142-0615/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2007.07.004
principle are unknown to the operator and on the system constraints and network topology [2]. The principle difficulty in allocating losses to system agents is the fact that the transmission loss function is non-separable and nonlinear, which makes it impossible to divide the system losses into the sum of terms, each one uniquely attributable to a generation or a load [3]. Several approaches were proposed to solve this problem, but so far there is no one that really satisfies the generation and the load, i.e., there is no approach that does not involve any arbitrary action. The approaches proposed for transmission-loss allocation are based on the solution of the power flow (PF) or optimal power flow (OPF) problems. They can be divided into several sets. The most simple loss allocation procedure is the so-called pro-rata technique [4], in which the losses are allocated to the generators and loads by considering the level of active-power injection, pro-rata (P), or current injection, pro-rata (I). In the set of the marginal procedures, the losses are assigned to generators and loads through the so-called incremental transmission loss (ITL) coefficients [5] or by using the decomposed marginal cost [6]. The
292
E.A. Belati, G.R.M. da Costa / Electrical Power and Energy Systems 30 (2008) 291–295
proportional sharing (PS) procedures use the results of the converged PF solution plus a linear proportional sharing principle [7–9]. Conejo et al. [3] proposed the Z-bus matrix in active transmission loss allocation. Recently, the integration of ITL coefficients [10] and alternatively marginal costs [5] have been proposed. In this paper we propose a new approach to the transmission loss allocation problem. This approach belongs to the set of marginal procedures. The idea is to solve the OPF problem with objective function active power loss transmission only once. From the solution of the optimal power flow the loads are perturbed and a new optimal operating point that satisfies the constraints is determined by the sensibility analysis. The ITL coefficients of each bus are obtained from the new optimal operating point. The sensitivity analysis takes account of the constraints of the problem, network topology and power injections into the buses.
The optimal power flow problem can be formulated as: f ðxÞ;
subject to gi ðxÞi ¼ 0; i ¼ 1; . . . ; m < n; hj ðxÞ 6 0; j ¼ 1; . . . ; r þ n;
ð1Þ
where x 2 Rn . The control and state variable vector, x, represents the voltage magnitude (V), phase angles (h) and LTC taps (t). The objective function, f(x), represents the active power losses in transmission. The equality constraints, g(x) = 0, represent the power flow equations for scheduled load and generation. The inequality constraints, h(x) 6 0, represent the vector x and the functional constraints on the power flow. This is a typical nonlinear and non-convex problem, which can be solved by an augmented Lagrangian function approach [11]. 3. Sensitivity analysis The sensitivity analysis is based on a theorem proposed by Fiacco [12]. Its application in the OPF problem can be seen in detail in Ref. [13]. We intend to verify as the state of the network changes when a perturbation happens. The perturbation, that is, power increments on the buses, is associated with the equality constraints, g(x), of the problem (1). The perturbed problem can be represented as: Minimize
f ðxÞ;
subject to gi ðxÞ þ ei ¼ 0; i ¼ 1; . . . ; m < n; hj ðxÞ 6 0; j ¼ 1; . . . ; r þ n;
ð4Þ To evaluate Dx, Dl and Dx from Eq. (4), the optimal solution of the problem, (x*, l *, x*), is required. Therefore, the OPF problem must be solved by a Lagrangian method, in which the constraints are associated with Lagrange multipliers like in the example described in Ref. [11]. 4. Formulation of ITL procedure based on OPF and sensitivity analysis
2. Optimal power flow problem
Minimize
The necessary condition, of the Lagrangian function, yields a set of nonlinear equations that can be solved by Newton’s method at (x*, l*, x*), obtaining in the matrix form the expression 2 3 2 2 3 31 2 Dx 0 rxx Laðx ; x ; l Þ rx hðx Þ rx gðx Þ 6 7 6 7 7 6 6 Dx 7 ¼ 6 x rx hðx Þ 7 6 0 7e: hðx Þ 0 4 5 4 5 5 4 Dl 1 rx gðx Þ 0 0
ð2Þ
where e = (ei , . . . , em) is the perturbation vector. The Lagrangian function associated with the perturbed problem (2) is:
The proposed approach belongs to the set of marginal procedures. Firstly, the solution to the problem is determined by the OPF. After that, the power is perturbed in each bus of the network and the sensibility analysis is used to reach a new solution of the problem. Then the coefficients Ki are obtained. Ki is called the ITLi coefficient of bus i. In this paper we refer to coefficients Ki as SITLi (sensitivity to incremental transmission loss) due to the fact that they are obtained through the sensitivity technique. Ki can be calculated as: K i ¼ Li Lb ;
ð5Þ
where Lb is the active losses of the system to base case; Li is the active losses after of incremental variation of the power injection on bus i. The SITL of the slack bus is zero by definition, and in this case we use the pro-rata (P) method initially to obtain a coefficient Ki for this bus. Once the remaining coefficients K have been obtained via sensitivity, the losses are allocated to buses of the system, i.e.: LP i ¼ P i K i :
ð6Þ
However, due to the non linearity of the problem the sum of the allocated losses (L0b ) is different of the losses (Lb) of the network, that is, Lb 6¼
NB X
LP i ¼
i¼1
NB X
P i K i ¼ L0b :
ð7Þ
i¼1
Thus, a normalization procedure is used to match the allocation losses to the exact amount of losses (Lb) Lb ¼
NB1 X
P i K 0i ;
ð8Þ
i¼1
Laðx; l; xÞ ¼ f ðxÞ þ l½gðxÞ þ e þ xhðxÞ; where x P 0 and l are the Lagrange multipliers.
ð3Þ
where K 0i ¼ K i ðLb =L0b Þ, the normalized SITL coefficient for bus i.
E.A. Belati, G.R.M. da Costa / Electrical Power and Energy Systems 30 (2008) 291–295
Finally, the loss allocated to each bus is given by the expression, L0P i ¼ P i K 0i :
ð9Þ
This approach benefits the buses that operate closer of the optimal point, so stimulating the others to improve their performance. 5. Test results The tests were performed using the IEEE 14-bus [3] transmission system to verify the effectiveness of SITL approach. 5.1. Comparative test A few tests were performed to compare the proposed approach (STIL) to the following methods: Z-bus; pro-rata (I); pro-rata (P); ITL; ITLPOS and PS. The comparative tests of these methods were presented in [14], which served as a basis for the present test. The SITL starts with the OPF solution, unlike the other methods being tested, which used the PF solution. As the losses obtained through OPF are smaller than those obtained via PF, the results were expressed in percentages to allow comparison of all the results. In this test we did not consider the limits of reactive-power injection, the active flow in the lines and the LTC taps. The system used is the IEEE 14-bus, as described [3]. Table 1 shows the absolute active and reactive power injections into the buses, obtained by PF, and the percentages of the losses allocated to the buses, using the cited methods. The SITL, like the Z-Bus and ITL methods, can allocate negative losses to some buses, which can be interpreted as subsidies. The active power losses obtained for the system using PF were 13.6 MW. Using OPF, with the voltage magnitude
293
limited between 0.95 and 1.1 p.u., losses of 12.46 MW were obtained. The voltage magnitudes on buses 1 and 8 remained at the upper limit. 5.2. Variation of the participation factors of the buses considering operational limits Table 2 presents the test results using SITL that considers different operational limits of the system. In each test an operational constraint was added. In test 1, only the voltage magnitude limit was adopted; in test 2, the voltage magnitude and LTC taps limits; in test 3, the voltage magnitude, LTC taps and reactive power injection limits, and finally, in test 4, the voltage magnitude, LTC taps, reactive power injection and active power flow in line 5–6 were subjected to limits. In Test 1, the voltage magnitude on buses 1 and 8 remained at the upper limit. In Test 2, the voltage magnitude on buses 1 and 8 remained at the upper limit and the tap on line 4–9 at the lower limit. In Test 3 the voltage magnitude on buses 1 and 8 remained at the upper limit, the tap on line 4–9 at the lower limit and the reactive power injection of bus 6 at the upper limit, which was 31.0 MVAr. In Test 4, the voltage magnitude on buses 1 and 8 remained at the upper limit, the LTC taps on lines 4–9 and 7–4 at the lower limit and the active power flow in line 5–6 at 43.5 MW, but the reactive power injection limits were not reached. As expected, the total active power losses increased with the introduction of each operational limit. We can observe in columns 2–5 that these constraints impressively influence the loss allocation coefficients. 5.3. Brazilian equivalent network And finally the approach is applied to an actual Brazilian equivalent network and it is compared with the technique used nowadays by the Brazilian Control Center. It is composed of 787 buses as follows: 1 slack bus, 111 reac-
Table 1 Comparative test Bus
Absolute power
Methods Z-bus
1 2 3 4 5 6 7 8 9 10 11 12 13 14
MW
MVAr
233.65 18.30 94.20 47.80 7.60 11.20 0.00 1.00 29.50 9.00 3.50 6.10 13.50 14.90
22.31 20.29 1.37 3.90 1.80 42.56 0.00 28.65 16.60 5.80 1.80 1.60 5.80 5.00
56.36 1.17 20.58 6.26 0.52 3.52 0.00 0.17 3.80 1.28 0.37 0.72 1.88 3.31
Pro-rata (%) (I)
(P)
40.52 4.764 17.14 8.61 1.39 7.54 0.00 4.81 6.06 1.91 0.69 1.10 2.58 2.84
47.66 3.74 19.29 9.79 1.55 2.24 0.00 0.02 6.04 1.84 0.71 1.25 2.76 3.05
ITL (%)
ITLPOS (%)
PS (%)
SITL (%)
45.88 5.94 21.63 8.80 1.16 1.70 0.00 0.01 5.36 1.69 0.61 1.12 2.62 3.42
50.03 0.00 6.47 4.44 2.97 3.02 0.00 4.18 4.33 4.62 4.11 4.42 4.89 6.46
47.78 2.21 21.26 9.34 1.16 1.72 0.00 0.02 5.76 2.12 0.76 1.17 2.82 3.80
47.55 1.74 22.35 9.24 1.23 1.83 0.00 0.19 5.70 1.80 0.67 1.22 2.83 3.66
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E.A. Belati, G.R.M. da Costa / Electrical Power and Energy Systems 30 (2008) 291–295
Table 2 Operational constraints Bus
Constraints
1 2 3 4 5 6 7 8 9 10 11 12 13 14 Losses (MW)
0.95 6 V 6 1.10
0.95 6 V 6 1.10, 0.98 6 t 6 1.02
0.95 6 V 6 1.10, 0.98 6 t 6 1.02, 6 Q 6 Qupper Qlower n n
0.95 6 V 6 1.10, 0.98 6 t 6 1.02, Qlower 6 Q 6 Q upper, FlowP 56 6 43:50 MW
Test 1 (%)
Test 2 (%)
Test 3 (%)
Test 4 (%)
47.55 1.74 22.34 9.23 1.23 1.80 0.00 0.19 5.74 1.81 0.66 1.21 2.82 3.67 12.46
47.55 1.74 22.39 9.25 1.23 1.82 0.00 0.19 5.70 1.80 0.66 1.21 2.82 3.63 12.48
47.55 3.92 20.12 10.08 1.61 2.53 0.00 0.20 5.64 1.68 0.71 1.20 2.56 2.20 12.52
47.55 1.65 21.14 8.79 1.16 2.28 0.00 0.19 5.90 1.91 0.75 1.45 3.31 3.90 12.54
tive power injection buses, 675 load buses, 1309 transmission lines, and 204 transformers. The OPF program takes solutions for base case after 18 iterations. Using the sensitivity technique the Ki coefficients were obtained taking all the operational constraints into account, with the voltage magnitude limits of 0.95–1.1 p.u. In this system, several buses had negative coefficients of allocation (Ki), caused Table 3 Comparative test Bus
Bus name
MW
SITL (%)
pro-rata (P) (%)
8 21 46 112 134 136 349 350 447 477 486 488 493 541 552 598 601 613 639 661 671 680 714 721 741 747 787
Bage (L) Condito Mota (G) Lajeado (L) Tres coroas (L) Trindade (L) Canoinha (L) Guarabi (G) Sosrio (G) Paranaiba (L) Angra do Reis (G) Itaipu (G) Marimbondo (G) Sa˜o Roque (G) Grajau (L) Jacarepagua´ (L) Iha Solteira (slack) Jupia (G) Araraquara (L) Tiete (L) Votuporanga (L) Altino (L) Leste (L) Jundiai (L) Pirituba (L) Miranda (L) S.Sima`o (G) Capivara
44.9 315 73.4 12 46.4 41.5 540 940 30.7 508.6 5500 1250 5053 2002 807 2195 1080 506 153 57.1 846 967 403 521 200 1400 0.38
0.115 0.852 0.018 0.025 0.060 0.037 1.669 1.507 0.058 2.105 2.957 0.002 9.298 10.165 4.223 3.747 0.098 0.779 0.537 0.117 2.327 2.231 1.003 1.665 0.407 0.032 0.0002
0.054 0.379 0.088 0.014 0.055 0.049 0.650 1.131 0.036 0.612 6.623 1.505 6.084 2.410 0.971 2.643 1.300 0.609 0.184 0.068 1.018 1.164 0.485 0.627 0.240 1.685 0.0004
by the feature of this system that it is essentially composed of hydropower generation far-removed from the load centers. Bus 598, hydroelectric plant of Ilha Solteira, represents the slack bus of the system. Table 3 shows the buses with larger positive and negative allocation coefficients. Column 2 gives the geographical name of each bus with the indication (G) for generation and (L) for load, column 3 shows the active power of the buses, column 4 refers to the percentage allocation by SITL, and the last column shows the percent allocation by pro-rata (P), which is the technique currently used in Brazil. The ‘‘generation’’ 493 bus, Sa˜o Roque (G), is near a load center. It gets its power through a direct current line that is not represented in the model of the equivalent system. The 477 bus, Angra dos Reis (G) has a negative allocation and is near an other load center. The load bus 8, Bage´ (L), with a negative allocation, is situated near abundant generation, far from the load centers. These negative losses can be interpreted as crossed subsidies. In Table 3 it can be seen that the allocation by SITL differs from the allocation by pro-rata (P), but the SITL technique is fairer than the pro-rata (P) as is shown in this paper because it considers all constraints and characteristics of the system. 6. Conclusions In this paper, a new approach is presented to transmission loss allocation, and the operational constraints of the system are taken into account. This approach, named SITL, is related to the incremental method of allocation. Starting with the primal and dual variables obtained via an OPF solution, the sensitivity technique is used to estimate the coefficients Ki of allocation of the losses among the generation and load buses of the system. In the comparative test, a discrepancy was observed among the
E.A. Belati, G.R.M. da Costa / Electrical Power and Energy Systems 30 (2008) 291–295
methods presented, the method of choice being the one that fairly allocates the responsibility of the companies for the losses. To show the relation of the binding constraints to the losses of the system, a study of the variation of the participation factors of the system agents with the operational limits was presented, using the IEEE 14-bus systems. As observed, for a fair allocation of losses, all the characteristics of the system must be taken into consideration. This approach benefits the buses that operate closer of the optimal point, so stimulating the others to improve their performance. Acknowledgements This work was partly supported by FAPESP, Fundac¸a˜o de Amparo a Pesquisa do Estado de Sa˜o Paulo and by CNPq–Conselho Nacional do Desenvolvimento Cientı´fico e Tecnolo´gico. References [1] Silva AML, Costa JGC. Transmission loss allocation: part I – single energy market. IEEE Trans Power Syst 2003;18:1389–94. [2] Salgado RS, Moyano CF, Medeiros ADR. Reviewing strategies for active power transmission loss allocation in power pools. Electr Power Energ Syst 2004;26:81–90.
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[3] Conejo AJ, Galiana FD, KocKar I. Z-bus loss allocation. IEEE Trans Power Syst 2001;16(1):105–10. [4] Ilic M, Galiana FD, Fink L. Power systems restructuring: engineering and economics. Norwell, MA: Kluwer; 1998. [5] Galiana FD, Conejo AJ, Kockar I . Incremental transmission loss allocation under pool dispatch. IEEE Trans Power Syst 2002;17:26–33. [6] Finney J, Othman HA, Rutz WL. Evaluating transmission congestion constraints in system planning. IEEE Trans Power Syst 1997;12(3):1143–50. [7] Bialek JW. Tracing the flow of electricity. IEE Proc Gener Transm and Distrib 1996;143:313–20. [8] Kirschen D, Strbac G. Tracing active and reactive power between generators and loads using real and imaginary currents. IEEE Trans Power Syst 1999;14:1312–9. [9] Strbac G, Kirschen D, Ahmed S. Allocating transmission system usage on the basis of traceable contributions of generators and loads to flows. IEEE Trans Power Syst 1998;13:527–34. [10] Gonzalez JJ, Basagoiti P. Spanish power exchange market and information system. Design concepts, and operating experience. In: Proceeding of the 1999 IEEE power industry computer applications conference, Santa Clara, USA. p. 245–52. [11] da Costa GRM. Modified newton method for reactive dispatching. Int J Electr Power Energ Syst 2002;24(10):815–9. [12] Fiacco AV. Sensitivity analysis for nonlinear programming using penalty methods. Mathematics Programming 1976;10:278–311. [13] Belati EA, Baptista EC, da Costa GRM. Optimal operation studies of the power via sensitivity analysis. Electr Power Syst Res 2005;75:79–84. [14] Lima DA, Padilha-Feltrin A. Allocation of the costs of transmission losses. Electr Power Syst Res 2004;72:13–20.
Electrical Power and Energy Systems 30 (2008) 291–295 www.elsevier.com/locate/ijepes
Short Communication
Transmission loss allocation based on optimal power flow and sensitivity analysis E.A. Belati, G.R.M. da Costa
*
Department of Electrical Engineering of the Sa˜o Carlos Engineering School, Sa˜o Paulo University, 13.566-590 Sa˜o Carlos, SP, Brazil Received 27 July 2006; received in revised form 13 July 2007; accepted 26 July 2007
Abstract This paper presents a new approach to the transmission loss allocation problem in a deregulated system. This approach belongs to the set of incremental methods. It treats all the constraints of the network, i.e. control, state and functional constraints. The approach is based on the perturbation of optimum theorem. From a given optimal operating point obtained by the optimal power flow the loads are perturbed and a new optimal operating point that satisfies the constraints is determined by the sensibility analysis. This solution is used to obtain the allocation coefficients of the losses for the generators and loads of the network. Numerical results show the proposed approach in comparison to other methods obtained with well-known transmission networks, IEEE 14-bus. Other test emphasizes the importance of considering the operational constraints of the network. And finally the approach is applied to an actual Brazilian equivalent network composed of 787 buses, and it is compared with the technique used nowadays by the Brazilian Control Center. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Active transmission loss allocation; Optimal power flow; Sensitivity analysis
1. Introduction In the current world economic scenario, the electrical industry has been undergoing a restructure in many countries effectively with the aim of creating a more competitive industry and consequently many challenges have appeared. One challenge is in allocating losses to loads, generators or bilateral contracts, that is, regardless of the approach, the final allocation always contains a degree of arbitrariness. This allocation must be as fair as possible among the power system agents. Thus, the approaches have to treat all the constraints of the network. Active power losses in the transmission are about 4–8% of the total active power generated. In Brazil the cost of the loss is half a billion U.S. dollars a year [1]. The losses depend on the active and reactive power flows that are transmitted from a generator to a load through a number of alternative pathways, which in
*
Corresponding author. Tel.: +55 16 33738134; fax: +55 16 3373 9372. E-mail address: [email protected] (G.R.M. da Costa).
0142-0615/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2007.07.004
principle are unknown to the operator and on the system constraints and network topology [2]. The principle difficulty in allocating losses to system agents is the fact that the transmission loss function is non-separable and nonlinear, which makes it impossible to divide the system losses into the sum of terms, each one uniquely attributable to a generation or a load [3]. Several approaches were proposed to solve this problem, but so far there is no one that really satisfies the generation and the load, i.e., there is no approach that does not involve any arbitrary action. The approaches proposed for transmission-loss allocation are based on the solution of the power flow (PF) or optimal power flow (OPF) problems. They can be divided into several sets. The most simple loss allocation procedure is the so-called pro-rata technique [4], in which the losses are allocated to the generators and loads by considering the level of active-power injection, pro-rata (P), or current injection, pro-rata (I). In the set of the marginal procedures, the losses are assigned to generators and loads through the so-called incremental transmission loss (ITL) coefficients [5] or by using the decomposed marginal cost [6]. The
292
E.A. Belati, G.R.M. da Costa / Electrical Power and Energy Systems 30 (2008) 291–295
proportional sharing (PS) procedures use the results of the converged PF solution plus a linear proportional sharing principle [7–9]. Conejo et al. [3] proposed the Z-bus matrix in active transmission loss allocation. Recently, the integration of ITL coefficients [10] and alternatively marginal costs [5] have been proposed. In this paper we propose a new approach to the transmission loss allocation problem. This approach belongs to the set of marginal procedures. The idea is to solve the OPF problem with objective function active power loss transmission only once. From the solution of the optimal power flow the loads are perturbed and a new optimal operating point that satisfies the constraints is determined by the sensibility analysis. The ITL coefficients of each bus are obtained from the new optimal operating point. The sensitivity analysis takes account of the constraints of the problem, network topology and power injections into the buses.
The optimal power flow problem can be formulated as: f ðxÞ;
subject to gi ðxÞi ¼ 0; i ¼ 1; . . . ; m < n; hj ðxÞ 6 0; j ¼ 1; . . . ; r þ n;
ð1Þ
where x 2 Rn . The control and state variable vector, x, represents the voltage magnitude (V), phase angles (h) and LTC taps (t). The objective function, f(x), represents the active power losses in transmission. The equality constraints, g(x) = 0, represent the power flow equations for scheduled load and generation. The inequality constraints, h(x) 6 0, represent the vector x and the functional constraints on the power flow. This is a typical nonlinear and non-convex problem, which can be solved by an augmented Lagrangian function approach [11]. 3. Sensitivity analysis The sensitivity analysis is based on a theorem proposed by Fiacco [12]. Its application in the OPF problem can be seen in detail in Ref. [13]. We intend to verify as the state of the network changes when a perturbation happens. The perturbation, that is, power increments on the buses, is associated with the equality constraints, g(x), of the problem (1). The perturbed problem can be represented as: Minimize
f ðxÞ;
subject to gi ðxÞ þ ei ¼ 0; i ¼ 1; . . . ; m < n; hj ðxÞ 6 0; j ¼ 1; . . . ; r þ n;
ð4Þ To evaluate Dx, Dl and Dx from Eq. (4), the optimal solution of the problem, (x*, l *, x*), is required. Therefore, the OPF problem must be solved by a Lagrangian method, in which the constraints are associated with Lagrange multipliers like in the example described in Ref. [11]. 4. Formulation of ITL procedure based on OPF and sensitivity analysis
2. Optimal power flow problem
Minimize
The necessary condition, of the Lagrangian function, yields a set of nonlinear equations that can be solved by Newton’s method at (x*, l*, x*), obtaining in the matrix form the expression 2 3 2 2 3 31 2 Dx 0 rxx Laðx ; x ; l Þ rx hðx Þ rx gðx Þ 6 7 6 7 7 6 6 Dx 7 ¼ 6 x rx hðx Þ 7 6 0 7e: hðx Þ 0 4 5 4 5 5 4 Dl 1 rx gðx Þ 0 0
ð2Þ
where e = (ei , . . . , em) is the perturbation vector. The Lagrangian function associated with the perturbed problem (2) is:
The proposed approach belongs to the set of marginal procedures. Firstly, the solution to the problem is determined by the OPF. After that, the power is perturbed in each bus of the network and the sensibility analysis is used to reach a new solution of the problem. Then the coefficients Ki are obtained. Ki is called the ITLi coefficient of bus i. In this paper we refer to coefficients Ki as SITLi (sensitivity to incremental transmission loss) due to the fact that they are obtained through the sensitivity technique. Ki can be calculated as: K i ¼ Li Lb ;
ð5Þ
where Lb is the active losses of the system to base case; Li is the active losses after of incremental variation of the power injection on bus i. The SITL of the slack bus is zero by definition, and in this case we use the pro-rata (P) method initially to obtain a coefficient Ki for this bus. Once the remaining coefficients K have been obtained via sensitivity, the losses are allocated to buses of the system, i.e.: LP i ¼ P i K i :
ð6Þ
However, due to the non linearity of the problem the sum of the allocated losses (L0b ) is different of the losses (Lb) of the network, that is, Lb 6¼
NB X
LP i ¼
i¼1
NB X
P i K i ¼ L0b :
ð7Þ
i¼1
Thus, a normalization procedure is used to match the allocation losses to the exact amount of losses (Lb) Lb ¼
NB1 X
P i K 0i ;
ð8Þ
i¼1
Laðx; l; xÞ ¼ f ðxÞ þ l½gðxÞ þ e þ xhðxÞ; where x P 0 and l are the Lagrange multipliers.
ð3Þ
where K 0i ¼ K i ðLb =L0b Þ, the normalized SITL coefficient for bus i.
E.A. Belati, G.R.M. da Costa / Electrical Power and Energy Systems 30 (2008) 291–295
Finally, the loss allocated to each bus is given by the expression, L0P i ¼ P i K 0i :
ð9Þ
This approach benefits the buses that operate closer of the optimal point, so stimulating the others to improve their performance. 5. Test results The tests were performed using the IEEE 14-bus [3] transmission system to verify the effectiveness of SITL approach. 5.1. Comparative test A few tests were performed to compare the proposed approach (STIL) to the following methods: Z-bus; pro-rata (I); pro-rata (P); ITL; ITLPOS and PS. The comparative tests of these methods were presented in [14], which served as a basis for the present test. The SITL starts with the OPF solution, unlike the other methods being tested, which used the PF solution. As the losses obtained through OPF are smaller than those obtained via PF, the results were expressed in percentages to allow comparison of all the results. In this test we did not consider the limits of reactive-power injection, the active flow in the lines and the LTC taps. The system used is the IEEE 14-bus, as described [3]. Table 1 shows the absolute active and reactive power injections into the buses, obtained by PF, and the percentages of the losses allocated to the buses, using the cited methods. The SITL, like the Z-Bus and ITL methods, can allocate negative losses to some buses, which can be interpreted as subsidies. The active power losses obtained for the system using PF were 13.6 MW. Using OPF, with the voltage magnitude
293
limited between 0.95 and 1.1 p.u., losses of 12.46 MW were obtained. The voltage magnitudes on buses 1 and 8 remained at the upper limit. 5.2. Variation of the participation factors of the buses considering operational limits Table 2 presents the test results using SITL that considers different operational limits of the system. In each test an operational constraint was added. In test 1, only the voltage magnitude limit was adopted; in test 2, the voltage magnitude and LTC taps limits; in test 3, the voltage magnitude, LTC taps and reactive power injection limits, and finally, in test 4, the voltage magnitude, LTC taps, reactive power injection and active power flow in line 5–6 were subjected to limits. In Test 1, the voltage magnitude on buses 1 and 8 remained at the upper limit. In Test 2, the voltage magnitude on buses 1 and 8 remained at the upper limit and the tap on line 4–9 at the lower limit. In Test 3 the voltage magnitude on buses 1 and 8 remained at the upper limit, the tap on line 4–9 at the lower limit and the reactive power injection of bus 6 at the upper limit, which was 31.0 MVAr. In Test 4, the voltage magnitude on buses 1 and 8 remained at the upper limit, the LTC taps on lines 4–9 and 7–4 at the lower limit and the active power flow in line 5–6 at 43.5 MW, but the reactive power injection limits were not reached. As expected, the total active power losses increased with the introduction of each operational limit. We can observe in columns 2–5 that these constraints impressively influence the loss allocation coefficients. 5.3. Brazilian equivalent network And finally the approach is applied to an actual Brazilian equivalent network and it is compared with the technique used nowadays by the Brazilian Control Center. It is composed of 787 buses as follows: 1 slack bus, 111 reac-
Table 1 Comparative test Bus
Absolute power
Methods Z-bus
1 2 3 4 5 6 7 8 9 10 11 12 13 14
MW
MVAr
233.65 18.30 94.20 47.80 7.60 11.20 0.00 1.00 29.50 9.00 3.50 6.10 13.50 14.90
22.31 20.29 1.37 3.90 1.80 42.56 0.00 28.65 16.60 5.80 1.80 1.60 5.80 5.00
56.36 1.17 20.58 6.26 0.52 3.52 0.00 0.17 3.80 1.28 0.37 0.72 1.88 3.31
Pro-rata (%) (I)
(P)
40.52 4.764 17.14 8.61 1.39 7.54 0.00 4.81 6.06 1.91 0.69 1.10 2.58 2.84
47.66 3.74 19.29 9.79 1.55 2.24 0.00 0.02 6.04 1.84 0.71 1.25 2.76 3.05
ITL (%)
ITLPOS (%)
PS (%)
SITL (%)
45.88 5.94 21.63 8.80 1.16 1.70 0.00 0.01 5.36 1.69 0.61 1.12 2.62 3.42
50.03 0.00 6.47 4.44 2.97 3.02 0.00 4.18 4.33 4.62 4.11 4.42 4.89 6.46
47.78 2.21 21.26 9.34 1.16 1.72 0.00 0.02 5.76 2.12 0.76 1.17 2.82 3.80
47.55 1.74 22.35 9.24 1.23 1.83 0.00 0.19 5.70 1.80 0.67 1.22 2.83 3.66
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Table 2 Operational constraints Bus
Constraints
1 2 3 4 5 6 7 8 9 10 11 12 13 14 Losses (MW)
0.95 6 V 6 1.10
0.95 6 V 6 1.10, 0.98 6 t 6 1.02
0.95 6 V 6 1.10, 0.98 6 t 6 1.02, 6 Q 6 Qupper Qlower n n
0.95 6 V 6 1.10, 0.98 6 t 6 1.02, Qlower 6 Q 6 Q upper, FlowP 56 6 43:50 MW
Test 1 (%)
Test 2 (%)
Test 3 (%)
Test 4 (%)
47.55 1.74 22.34 9.23 1.23 1.80 0.00 0.19 5.74 1.81 0.66 1.21 2.82 3.67 12.46
47.55 1.74 22.39 9.25 1.23 1.82 0.00 0.19 5.70 1.80 0.66 1.21 2.82 3.63 12.48
47.55 3.92 20.12 10.08 1.61 2.53 0.00 0.20 5.64 1.68 0.71 1.20 2.56 2.20 12.52
47.55 1.65 21.14 8.79 1.16 2.28 0.00 0.19 5.90 1.91 0.75 1.45 3.31 3.90 12.54
tive power injection buses, 675 load buses, 1309 transmission lines, and 204 transformers. The OPF program takes solutions for base case after 18 iterations. Using the sensitivity technique the Ki coefficients were obtained taking all the operational constraints into account, with the voltage magnitude limits of 0.95–1.1 p.u. In this system, several buses had negative coefficients of allocation (Ki), caused Table 3 Comparative test Bus
Bus name
MW
SITL (%)
pro-rata (P) (%)
8 21 46 112 134 136 349 350 447 477 486 488 493 541 552 598 601 613 639 661 671 680 714 721 741 747 787
Bage (L) Condito Mota (G) Lajeado (L) Tres coroas (L) Trindade (L) Canoinha (L) Guarabi (G) Sosrio (G) Paranaiba (L) Angra do Reis (G) Itaipu (G) Marimbondo (G) Sa˜o Roque (G) Grajau (L) Jacarepagua´ (L) Iha Solteira (slack) Jupia (G) Araraquara (L) Tiete (L) Votuporanga (L) Altino (L) Leste (L) Jundiai (L) Pirituba (L) Miranda (L) S.Sima`o (G) Capivara
44.9 315 73.4 12 46.4 41.5 540 940 30.7 508.6 5500 1250 5053 2002 807 2195 1080 506 153 57.1 846 967 403 521 200 1400 0.38
0.115 0.852 0.018 0.025 0.060 0.037 1.669 1.507 0.058 2.105 2.957 0.002 9.298 10.165 4.223 3.747 0.098 0.779 0.537 0.117 2.327 2.231 1.003 1.665 0.407 0.032 0.0002
0.054 0.379 0.088 0.014 0.055 0.049 0.650 1.131 0.036 0.612 6.623 1.505 6.084 2.410 0.971 2.643 1.300 0.609 0.184 0.068 1.018 1.164 0.485 0.627 0.240 1.685 0.0004
by the feature of this system that it is essentially composed of hydropower generation far-removed from the load centers. Bus 598, hydroelectric plant of Ilha Solteira, represents the slack bus of the system. Table 3 shows the buses with larger positive and negative allocation coefficients. Column 2 gives the geographical name of each bus with the indication (G) for generation and (L) for load, column 3 shows the active power of the buses, column 4 refers to the percentage allocation by SITL, and the last column shows the percent allocation by pro-rata (P), which is the technique currently used in Brazil. The ‘‘generation’’ 493 bus, Sa˜o Roque (G), is near a load center. It gets its power through a direct current line that is not represented in the model of the equivalent system. The 477 bus, Angra dos Reis (G) has a negative allocation and is near an other load center. The load bus 8, Bage´ (L), with a negative allocation, is situated near abundant generation, far from the load centers. These negative losses can be interpreted as crossed subsidies. In Table 3 it can be seen that the allocation by SITL differs from the allocation by pro-rata (P), but the SITL technique is fairer than the pro-rata (P) as is shown in this paper because it considers all constraints and characteristics of the system. 6. Conclusions In this paper, a new approach is presented to transmission loss allocation, and the operational constraints of the system are taken into account. This approach, named SITL, is related to the incremental method of allocation. Starting with the primal and dual variables obtained via an OPF solution, the sensitivity technique is used to estimate the coefficients Ki of allocation of the losses among the generation and load buses of the system. In the comparative test, a discrepancy was observed among the
E.A. Belati, G.R.M. da Costa / Electrical Power and Energy Systems 30 (2008) 291–295
methods presented, the method of choice being the one that fairly allocates the responsibility of the companies for the losses. To show the relation of the binding constraints to the losses of the system, a study of the variation of the participation factors of the system agents with the operational limits was presented, using the IEEE 14-bus systems. As observed, for a fair allocation of losses, all the characteristics of the system must be taken into consideration. This approach benefits the buses that operate closer of the optimal point, so stimulating the others to improve their performance. Acknowledgements This work was partly supported by FAPESP, Fundac¸a˜o de Amparo a Pesquisa do Estado de Sa˜o Paulo and by CNPq–Conselho Nacional do Desenvolvimento Cientı´fico e Tecnolo´gico. References [1] Silva AML, Costa JGC. Transmission loss allocation: part I – single energy market. IEEE Trans Power Syst 2003;18:1389–94. [2] Salgado RS, Moyano CF, Medeiros ADR. Reviewing strategies for active power transmission loss allocation in power pools. Electr Power Energ Syst 2004;26:81–90.
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