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Real time loss sensitivity calculation in power systems operation
Electric Power Systems Research 73 (2005) 53–60
Real time loss sensitivity calculation in power systems operation Jizhong Zhu∗ , Davis Hwang, Ali Sadjadpour a AREVA
T&D Corporation, 11120 NE 33rd Place, Bellevue, WA 98004, USA
Received 24 September 2003; received in revised form 12 May 2004; accepted 13 May 2004 Available online 27 August 2004
Abstract Network losses are being considered in the standard market design. The calculation of loss sensitivity that is based on the distributed generator slack is commonly used for economic dispatch in energy control center. In order to obtain the accurate location-based marginal prices, an efficient and robust method to calculate loss sensitivity for an arbitrary slack bus within the real-time period (1–10 min) is needed in the real-time energy markets. This paper presents a fast and useful approach to calculate loss sensitivity for any slack bus. This method is based on the results taken from the distributed slack buses without repeating a traditional power flow analysis. The proposed approach is tested and verified. © 2004 Elsevier B.V. All rights reserved. Keywords: Standard market design; Economic dispatch; Loss sensitivity; Location-based marginal prices; Real-time energy markets
1. Introduction The electric power industry is being relentlessly pressured by governments, politicians, large industries, and investors to privatize, restructure, and deregulate. Despite changes with different structures, market rules, and uncertainties, an energy management system (EMS) control center must always be in place to maintain the security, reliability, and quality of electric service [1]. It means that EMS in the open energy market must respond quickly, reliably and efficiently to the market changes. In the early energy market, the transmission losses are neglected for reasons of computational simplicity, but are recently addressed in the standard market design (SMD). The loss calculation is considered for the dispatch functions of SMD such as location-based marginal prices (LMP). Loss allocation does not affect generation levels or power flows, however it does modify the value of LMP. The early and classic loss calculation approach is the loss formulas—B coefficient method [2], which is replaced by the more accurate inverse Jacobian transpose method [3]. ∗
Corresponding author. Fax: +1 425 889 1700. E-mail address: [email protected] (J. Zhu).
0378-7796/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2004.05.004
Numerous loss calculation methods have been proposed in the literature [4–11], and can be categorized into pro-rata [4], incremental [5], proportional-sharing [6], and Z-bus loss allocation [7]. The calculation of loss sensitivity that is based on the distributed slack buses is still interesting in the energy control center, because it is based on the well-established incremental transmission loss analysis, an approach known to power engineers for at least three decades [2,7,11]. The distributed slack buses refer to multiple generators or load buses that are selected to share the power mismatch during the power flow calculation. After the power flow problem has been solved, the difference (slack) between the total generation and the total load plus system losses are assigned to these distributed slack buses. In the real-time energy markets, it is frequently required to compute LMP based on market-based reference, which is a single slack bus, instead of the distributed slack buses in the traditional energy management system (EMS). Meanwhile, most of the existing loss calculation methods or traditional EMS products are generally based on the generator slack or reference buses, i.e., multiple generator buses are selected as power slack buses if these generators are AGC units. Since the patterns or status of AGC units are variable for the different time periods in the real time energy market,
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J. Zhu et al. / Electric Power Systems Research 73 (2005) 53–60
the loss sensitivity results will be changed even the network topology is unchanged. Thus, an efficient and robust method to calculate loss sensitivity for an arbitrary slack bus within the real-time period is needed. This paper presents a fast and useful method to calculate loss sensitivity for any single slack bus. This method uses the loss sensitivity results taken from the distributed slack buses and then transforms into single slack bus based results without computing a new set of sensitivity factors through the traditional power flow analysis. Especially, we can choose the load buses as the distributed slack buses rather than the usual generator slack buses. The loss sensitivity results will be consistent for the same network topology no matter how the status of the AGC units changes. The proposed method has been used in the practical market systems such as ISO New England (ISO-NE) and Midwest ISO (MISO), but AREVA T&D 60-bus test system is demonstrated here. The results show that the proposed method is very useful and efficient.
2. Loss sensitivity calculation
j
=
j
s.t.
2 (aj PGj + bj PGj + cj ),
PD + PL =
PGj ,
j ∈ NG
j ∈ NG
(1)
(2)
j
PGj min ≤ PGj ≤ PGj max ,
j ∈ NG
(3)
where PGj is the real power output at generator bus j. PGj min is the minimal real power output at generator j. PGj max is the maximal real power output at generator j. PL is the network losses. PD is the system load demands. fj is the generation cost function of unit j, and a, b, c are the coefficient of the generation cost function. NG is the number of units. All the equations in the paper are implemented in per-unit. Traditionally, the method based on generation slack buses is used in the calculation of loss allocation. In this method, all AGC units are selected to share the power mismatch during the power flow calculation. Since the AGC status or patterns of units are variable in the real time EMS or energy markets, the loss sensitivity values based on the distributed generator slack buses will keep changing due to the change of unit AGC status. Thus, the method based on the distributed load slack buses is used here. It means that the loads are allowed to vary uniformly during the calculation of loss sensitivity. In this way, the economic dispatch problem (1)–(3) can be rewritten as Min F = fj (PGj ), j ∈ NG (1) j
PDi + PL =
i
PGj ,
i ∈ ND, j ∈ NG
(4)
j
PDk + PLk = PGk ,
k∈n
(5)
PDi min ≤ PDi ≤ PDi max ,
i ∈ ND
(6)
PGj min ≤ PGj ≤ PGj max ,
j ∈ NG
(3)
where PDi is the real power load at load bus i. PDi min is the minimal demand at load bus i. PDi max is the maximum demand at load bus i. PGk is the real power output of unit related to bus k. PDk is the real power demand of load related to bus k. PLk is the real power losses at bus k. ND is the number of load buses. n is the total number of buses in the system. The Lagrangian function is obtained from Eqs. (1) and (4): (7) FL = fj (PGj ) + λ PDi + PL − PGj j
The problem of real power economic dispatch without line security constraints can be represented as follows [12]: Min F = fj (PGj )
s.t.
i
j
The optimality criteria of the Lagrangian function (7) are written as follows: ∂FL dfj ∂PL = 0, i ∈ ND (8) = +λ 1+ ∂PDi dPDi ∂PDi dfj ∂PL ∂FL = +λ − 1 = 0, j ∈ NG (9) ∂PGj dPGj ∂PGj dfj LDi = λ, dPDi LDi = −
i ∈ ND
1 , 1 + (∂PL /∂PDi )
dfj LGj = λ, dPGj LGj =
(10) i ∈ ND
j ∈ NG
1 , 1 − (∂PL /∂PGj )
(11) (12)
j ∈ NG
(13)
where λ is the Lagrangian multiplier. ∂PL /∂PDi is the loss sensitivity with respect to load at bus i. ∂PL /∂PGj is the loss sensitivity with respect to unit at bus j. We use ∂PL /∂Pi , which is the loss sensitivity with respect to an injection at bus i, stand for both ∂PL /∂PDi and ∂PL /∂PGj . Since the distributed slack buses are used here, all loss sensitivity factors are non-zero. If bus k is selected as a slack bus, then Pk is the function of the other injections, i.e.: Pk = f (Pi ),
i ∈ n, i = k
(14)
where Pi is the power injection at bus i, which includes the load PDi and generation PGj . Actually, the load can be treated as a negative generation. Then Eqs. (11) and (13) can be
J. Zhu et al. / Electric Power Systems Research 73 (2005) 53–60
expressed as (15), and Eqs. (10) and (12) can be expressed as (16): Li =
1 , 1 − (∂PL /∂Pi )
dfi Li = λ, dPi
i∈n
(15)
i∈n
(16)
Eq. (4) will be rewritten as PL = Pk + Pi , i ∈ n
(17)
i=k
The new Lagrangian function can be obtained from (1) and (17): FL∗ = (18) fi (Pi ) + λ PL − Pk − Pi i
i=n
i ∈ n, i = k
(19)
From (17), we get ∂PL ∂Pk =1+ ∂Pi ∂Pi
(20)
From (19) and (20), we get dfi ∗ dfk L = dPi i dPk L∗i =
(21)
1 , 1 − (∂PL /∂Pi )
i ∈ n, i = k
(22)
It is noted that Li and L∗i are similar, but they have different meaning [13]. The former is based on the distributed slack buses, and the latter is based on the single slack bus k. Similarly, the loss sensitivity in Li is based on the distributed slack buses, i.e. ∂PL /∂Pi |DS (the subscript DS means the distributed slack buses); the loss sensitivity in L∗i is based on the single slack bus k, i.e. ∂PL /∂Pi |k . Note that the kth loss sensitivity, with bus k as the slack bus, is zero. From (14) and (19), we have the following equation: L∗i =
Li , Lk
L∗k = 1
(23)
From Eqs. (15), (22) and (23), we get 1 1 − (∂PL /∂Pk )|DS = 1 − (∂PL /∂Pi )|k 1 − (∂PL /∂Pi )|DS
a slack bus k can be calculated from the following equation: ∂PL /∂Pi |DS − ∂PL /∂Pk |DS ∂PL = (25) ∂Pi k 1 − (∂PL /∂Pk )|DS Loss sensitivity factors ∂PL /∂Pi |DS , which are based on the distributed slack buses, are determined from power flow calculation. The loss sensitivity calculation (25) is very simple, but is accurate and efficient for real-time energy markets. It will avoid computing a new set of the loss sensitivity factors whenever the slack bus k changes. Consequently, it means huge time savings. In addition, the loss factors based on the distributed load slack buses will not be changed no matter how the AGC statuses of units vary, as long as network topology is the same as before.
3. Test results and analysis
The optimality criteria can be obtained from the Lagrangian function (18): ∂FL∗ dfi dfk ∂Pk = + ∂Pi dPi dPk ∂Pi ∂PL ∂Pk +λ − − 1 = 0, ∂Pi ∂Pi
55
(24)
Hence, with one set of the incremental transmission loss coefficients for the distributed slack buses, the loss sensitivity for
The proposed method of loss sensitivity is implemented in our company’s EMS product, which is being used in the practical market systems. Here, the AREVA T&D 60-bus system is applied to illustrate the method. The one-line diagram of the system is shown in Appendix A. The system that has 3 areas consists of 24 generation units (15 units are available in the tests), 32 loads, 43 transmission lines and 54 transformers. The test results are shown in Tables 1–7. All loss sensitivity factors for units and loads are computed. In order to reduce the length of the paper, only loss sensitivities of generators are listed in Tables 1–7. Tables 1–4 are the test results and comparison of loss sensitivity calculation based on the distributed generation reference and distributed load reference, respectively. In Tables 1–4, column 1 is the name of station and units. Column 2 is the area number that the unit belongs to. Column 3 is the AGC status of the unit. The loss factors computed from the distributed unit reference are listed in column 4. The loss factors computed from the distributed load reference are listed in column 5. Generally, the values of loss sensitivities based on the generation references are different from those based on the load references, because the distribution of the units is not exactly the same as the distribution of loads in the power system. The loss factors will be close or equal if the units are close to the load locations. This can be observed from Table 1, where all units are on AGC status. For AREVA T&D 60-bus system, each load in area 3 has at least one unit connected, so the loss factors in area 3 are the same for both the distributed generation slack and distributed load slack. The distributed slack buses are determined as follows. For the distributed generator slack buses, the generators that are AGC units are selected as the distributed slack buses. For the distributed load slack buses, all loads are selected as the distributed slack buses. It is noted from Tables 1–4 that the loss sensitivity factors based on the distributed load slack are the same whether the
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J. Zhu et al. / Electric Power Systems Research 73 (2005) 53–60
Table 1 Test results and comparison of loss sensitivity calculation (case 1: all units on AGC) Station, generator
Area no.
AGC unit
Loss sensitivity distributed generation slack
Loss sensitivity distributed load slack
DOUGLAS, G2 DOUGLAS, G1 DOUGLAS, CT1 DOUGLAS, CT2 DOUGLAS, ST HEARN, G1 HEARN, G2 LAKEVIEW, G1 BVILLE, 1 WVILLE, 1 CHENAUX, 1 CHEALLS, 1 CHEALLS, 2 HOLDEN, 1 NANTCOKE, 1
1 1 1 1 1 1 1 1 2 2 3 3 3 3 3
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
0.015100 0.012100 0.009900 0.009900 0.009700 −0.016500 −0.016500 −0.018800 −0.001000 0.000700 −0.008900 0.021200 0.021200 0.001000 −0.012200
0.017000 0.014000 0.011800 0.011800 0.011600 −0.014600 −0.014600 −0.017000 −0.004200 −0.002500 −0.008900 0.021200 0.021200 0.001000 −0.012200
Table 2 Test results and comparison of loss sensitivity calculation (case 2: all units on AGC except the units under station Douglas in area 1) Station, generator
Area no.
AGC unit
Loss sensitivity distributed generation slack
Loss sensitivity distributed load slack
DOUGLAS, G2 DOUGLAS, G1 DOUGLAS, CT1 DOUGLAS, CT2 DOUGLAS, ST HEARN, G1 HEARN, G2 LAKEVIEW, G1 BVILLE, 1 WVILLE, 1 CHENAUX, 1 CHEALLS, 1 CHEALLS, 2 HOLDEN, 1 NANTCOKE, 1
1 1 1 1 1 1 1 1 2 2 3 3 3 3 3
No No No No No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
0.032800 0.029900 0.027800 0.027800 0.027600 0.001500 0.001500 −0.000800 −0.001000 0.000700 −0.008900 0.021200 0.021200 0.001000 −0.012200
0.017000 0.014000 0.011800 0.011800 0.011600 −0.014600 −0.014600 −0.017000 −0.004200 −0.002500 −0.008900 0.021200 0.021200 0.001000 −0.012200
Table 3 Test results and comparison of loss sensitivity calculation (case 4: all units on AGC except the units in area 2) Station, generator
Area no.
AGC unit
Loss sensitivity distributed generation slack
Loss sensitivity distributed load slack
DOUGLAS, G2 DOUGLAS, G1 DOUGLAS, CT1 DOUGLAS, CT2 DOUGLAS, ST HEARN, G1 HEARN, G2 LAKEVIEW, G1 BVILLE, 1 WVILLE, 1 CHENAUX, 1 CHEALLS, 1 CHEALLS, 2 HOLDEN, 1 NANTCOKE, 1
1 1 1 1 1 1 1 1 2 2 3 3 3 3 3
Yes Yes Yes Yes Yes Yes Yes Yes No No Yes Yes Yes Yes Yes
0.015200 0.012200 0.010000 0.010000 0.009900 −0.016700 −0.016700 −0.019100 −0.021000 −0.019300 −0.008900 0.021200 0.021200 0.001000 −0.012200
0.017000 0.014000 0.011800 0.011800 0.011600 −0.014600 −0.014600 −0.017000 −0.004200 −0.002500 −0.008900 0.021200 0.021200 0.001000 −0.012200
J. Zhu et al. / Electric Power Systems Research 73 (2005) 53–60
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Table 4 Test results and comparison of loss sensitivity calculation (case 5: all units on AGC except unit 3 under station HOLDEN in area 3) Station, generator
Area no.
AGC unit
Loss sensitivity distributed generation slack
Loss sensitivity distributed load slack
DOUGLAS, G2 DOUGLAS, G1 DOUGLAS, CT1 DOUGLAS, CT2 DOUGLAS, ST HEARN, G1 HEARN, G2 LAKEVIEW, G1 BVILLE, 1 WVILLE, 1 CHENAUX, 1 CHEALLS, 1 CHEALLS, 2 HOLDEN, 1 NANTCOKE, 1
1 1 1 1 1 1 1 1 2 2 3 3 3 3 3
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes
0.015100 0.012100 0.009900 0.009900 0.009700 −0.016500 −0.016500 −0.018800 −0.001000 0.000700 −0.008500 0.021600 0.021600 0.001400 −0.011800
0.017000 0.014000 0.011800 0.011800 0.011600 −0.014600 −0.014600 −0.017000 −0.004200 −0.002500 −0.008900 0.021200 0.021200 0.001000 −0.012200
Table 5 Test results of loss sensitivity calculation (distributed slack vs. single slack) Station, generator
AGC unit
Loss sensitivity distributed slack
Loss sensitivity single slack, HOLDEN 1
Loss sensitivity single slack, DOUGLAS ST
DOUGLAS, G2 DOUGLAS, G1 DOUGLAS, CT1 DOUGLAS, CT2 DOUGLAS, ST HEARN, G1 HEARN, G2 LAKEVIEW, G1 BVILLE, 1 WVILLE, 1 CHENAUX, 1 CHEALLS, 1 CHEALLS, 2 HOLDEN, 1 NANTCOKE, 1
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
0.017000 0.014000 0.011800 0.011800 0.011600 −0.014600 −0.014600 −0.017000 −0.004200 −0.002500 −0.008900 0.021200 0.021200 0.001000 −0.012200
0.016016 0.013013 0.010811 0.010811 0.010611 −0.015616 −0.015616 −0.018018 −0.005205 −0.003504 −0.009910 0.020220 0.020220 0.000000 −0.013213
0.005463 0.002428 0.000202 0.000202 0.000000 −0.026507 −0.026507 −0.028936 −0.015985 −0.014265 −0.020741 0.009713 0.009713 −0.010724 −0.024079
Table 6 Comparison of loss sensitivity calculation results for single slack bus at HOLDEN-1 (the proposed method vs. power flow method) Station, generator
AGC unit
Loss sensitivity, HOLDEN 1-PF method
Loss sensitivity, HOLDEN 1, Eq. (25)
|Error %|
DOUGLAS, G2 DOUGLAS, G1 DOUGLAS, CT1 DOUGLAS, CT2 DOUGLAS, ST HEARN, G1 HEARN, G2 LAKEVIEW, G1 BVILLE, 1 WVILLE, 1 CHENAUX, 1 CHEALLS, 1 CHEALLS, 2 HOLDEN, 1 NANTCOKE, 1
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
0.016029 0.013053 0.010817 0.010817 0.010621 −0.015630 −0.015630 −0.018110 −0.005220 −0.003500 −0.009920 0.020247 0.020247 0.000000 −0.013240
0.016016 0.013013 0.010811 0.010811 0.010611 −0.015616 −0.015616 −0.018018 −0.005205 −0.003504 −0.009910 0.020220 0.020220 0.000000 −0.013213
0.08110 0.30644 0.05547 0.05547 0.09415 0.08957 0.08957 0.50801 0.23002 0.02855 0.11088 0.13335 0.13335 0.00000 0.20393
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J. Zhu et al. / Electric Power Systems Research 73 (2005) 53–60
Table 7 Comparison of loss sensitivity calculation results for single slack bus at Douglas-ST (the proposed method vs. power flow method) Station, generator
AGC unit
Loss sensitivity, DOUGLAS ST-PF method
Loss sensitivity, DOUGLAS ST-Eq. (25)
|Error %|
DOUGLAS, G2 DOUGLAS, G1 DOUGLAS, CT1 DOUGLAS, CT2 DOUGLAS, ST HEARN, G1 HEARN, G2 LAKEVIEW, G1 BVILLE, 1 WVILLE, 1 CHENAUX, 1 CHEALLS, 1 CHEALLS, 2 HOLDEN, 1 NANTCOKE, 1
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
0.005467 0.002421 0.000202 0.000202 0.000000 −0.026530 −0.026530 −0.028950 −0.016000 −0.014280 −0.020770 0.009714 0.009714 −0.010730 −0.024090
0.005463 0.002428 0.000202 0.000202 0.000000 −0.026507 −0.026507 −0.028936 −0.015985 −0.014265 −0.020741 0.009713 0.009713 −0.010724 −0.024079
0.07317 0.28914 0.14829 0.14829 0.00000 0.08669 0.08669 0.04836 0.09999 0.10504 0.13962 0.01029 0.01029 0.07454 0.02491
status of the units is changed or not. But the loss factors based on the distributed generation references are changed according to the AGC status of the units. Generally, the change of AGC status of the units only affects the loss sensitivities in the same area that these units belong to. It can be seen from Table 2 that, when AGC status of the units in area 1 changes, only the loss factors in area 1 are affected. The loss factors in the other areas are unchanged. For Table 4, when AGC status of the units in area 3 changes, only the loss factors in area 3 are affected. The loss factors in the other areas are unchanged. But for Table 3, when AGC status of the units in area 2 changes, and all units in this area are not on AGC, it means that there is no unit reference in area 2. Then the units with AGC in the other areas will pick up the mismatch (i.e. area 1 in this case). Thus, the loss factors in area 1 and 2 are changed. The loss factors in the other areas are unchanged. Through the above comparisons, we can conclude that the method of the distributed load references for loss sensitivity calculation is superior to that of the distributed generation references in the real time energy markets, since the AGC status of the units are changeable in the real time system. The results of loss sensitivity calculation for a single slack, which are computed from Eq. (25), are shown in Table 5. Column 3 in Table 5 is the set of the loss sensitivity coefficients from the distributed slack buses. Column 4 in Table 5 is the set of loss sensitivity factors with a single slack bus at the location of HOLDEN 1. Column 5 in Table 5 is the set of loss sensitivity factors with a single slack bus at the location of Douglas. It is noted that all the loss sensitivities are nonzero if the distributed slack is selected. If the single slack is selected, the loss sensitivity of the slack equals zero. Since the loss sensitivity values based on the distributed load slacks are unchanged as long as the system topology is the same, we can easily and quickly get the loss factors for any single slack by use of the proposed loss sensitivity formula. Therefore, a large amount of the computations are avoided
whenever the loss factors are needed for a single slack in the real time energy markets. For example, a practical system with 25,000 buses, the CPU time of computing loss factors using the traditional power flow calculation is about 60 s, but less than 0.1 s if the proposed method is used. This is a huge time saving in the real time energy markets. In order to verify the correctness of the proposed loss sensitivity (25), the loss factors are computed and compared using the traditional power flow calculation. The results and comparison are shown in Tables 6 and 7, in which column 3 is the set of results from the power flow calculation, and column 4 is the set of results from (25). Table 6 shows the comparison of loss factor results for single slack bus at HOLDEN-1. Table 7 shows the comparison of loss factor results for single slack bus at DOUGLAS-ST. The difference or error of the results between the proposed method and power flow method is obtained from the following equation: LFPM (i) − LFPF (i) |Error%| = × 100% , i ∈ n (26) LFPF (i) where, Error% is the percentage of the computation error for the proposed formula. LFPM is the loss factor computed from the proposed method. LFPF is the loss factor obtained using the traditional power flow calculation. It can be known from Tables 6 and 7 that the loss sensitivity results from the two methods are very close. The maximum error is less than 0.6%. 4. Conclusion Network losses are being considered in the standard market design. The calculation of loss sensitivity that is based on the distributed slack buses is used in the energy control center. In order to obtain the accurate location-based marginal prices, it is frequently required that loss sensitivity is quickly computed based on market-based reference (i.e. a particular slack bus) in the real-time energy markets. This paper presents a
J. Zhu et al. / Electric Power Systems Research 73 (2005) 53–60
fast and useful approach to calculate loss sensitivity for any single slack bus. This method is based on the set of loss factors taken from the distributed slack buses. It is not necessary to compute a new set of loss coefficients through the traditional power flow calculation, whenever the slack bus changes. This results in huge time savings. The proposed approach is tested
59
on AREVA T&D 60-bus system. The test results show the proposed loss sensitivity calculation is very fast, useful and efficient. Meanwhile, the test results also show that the approach based on the distributed load slack buses is superior to the method based on the distributed generator slack buses in the real time system.
Fig. 1. One-line diagram of AREVA T&D system.
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J. Zhu et al. / Electric Power Systems Research 73 (2005) 53–60
Table A.1 Generator Data of AREVA T&D 60-bus system (p.u.) Station, generator DOUGLAS, G2 DOUGLAS, G1 DOUGLAS, CT1 DOUGLAS, CT2 DOUGLAS, ST HEARN, G1 HEARN, G2 LAKEVIEW, G1 BVILLE, 1 WVILLE, 1 CHENAUX, 1 CHEALLS, 1 CHEALLS, 2 HOLDEN, 1 NANTCOKE, 1
Area no. 1 1 1 1 1 1 1 1 2 2 3 3 3 3 3
Base MW 6.05 3.68 0.50 0.50 0.20 3.13 3.12 5.16 0.80 2.86 7.08 5.88 3.38 12.11 5.08
Minimum generation output 0.50 0.50 0.10 0.10 0.05 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
Maximum generation output 8.00 12.00 0.50 0.50 0.40 6.00 6.00 8.00 9.00 10.00 13.00 8.00 9.00 22.00 11.00
Table A.2 Load data of AREVA T&D 60-bus system (p.u.) Load in station DOUGLAS KINCARD LAKEVIEW BRIGHTON CEYLON HANOVER PARKHILL BVILLE GOLDEN JVILLE MTOWN WALDEN CHEALLS HOLDEN HUNTVILL MARTDALE NANTCOKE
Area no. 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3
Appendix A AREVA T&D 60-bus system is shown in Fig. 1. Area 1 is EAST, area 2 is WEST, and area 3 is ECAR. The generator and load data are listed in Tables A.1 and A.2. References [1] T.E. Dy-Liacco, Control centers are here to stay, IEEE Comput. Appl. Power 15 (4) (2002) 18–23. [2] L.K. Kirchmayer, Economic Operation of Power Systems, Wiley, New York, 1958. [3] H.W. Dommel, W.F. Tinney, Optimal power flow solutions, IEEE Trans. PAS PAS-87 (10) (1968) 1866–1876. [4] M. Ilic, F.D. Galiana, L. Fink, Power Systems Restructuring: Engineering and Economics, Kluwer Academic Publishers, Norwell, MA, 1998. [5] D. Kirschen, R. Allan, G. Strbac, Contributions of individual generators to loads and flows, IEEE Trans. Power Syst. 12 (1) (1997) 52–60.
Real power load 5.107 1.674 1.999 2.144 3.812 4.469 4.340 0.067 1.693 0.061 3.467 3.123 2.069 10.199 1.903 1.284 2.619
Reactive power load 0.7820 0.704 0.711 0.407 0.851 0.438 1.113 0.033 0.361 0.031 0.420 0.628 0.436 2.309 0.255 0.157 0.248
[6] F. Schweppe, M. Caramanis, R. Tabors, R. Bohn, Spot Pricing of Electricity, Kluwer Academic Publishers, Norwell, MA, 1988. [7] A.J. Conejo, F.D. Galiana, I. Kockar, Z-bus loss allocation, IEEE Trans. Power Syst. 16 (1) (2001) 105–110. [8] F.D. Galiana, A.J. Conjeo, I. Kockar, Incremental transmission loss allocation under pool dispatch, IEEE Trans. Power Syst. 17 (1) (2002) 26–33. [9] F.D. Galiana, M. Phelan, Allocation of transmission losses to bilateral contracts in a competitive environment, IEEE Trans. Power Syst. 15 (1) (2000) 143–150. [10] A.G. Exposito, J.M.R. Santos, T.G. Garcia, E.A.R. Velasco, Fair allocation of transmission power losses, IEEE Trans. Power Syst. 15 (1) (2000) 184–188. [11] O.I. Elgerd, Electric Energy Systems Theory: An Introduction, McGraw-Hill, New York, 1982. [12] J.Z. Zhu, G.Y. Xu, A new economic power dispatch method with security, Electric Power Syst. Res. 25 (1992) 9–15. [13] W.Y. Li, Power System Economic Operation with Security—Model and Method, Chongqing University Press, China, 1989.
Real time loss sensitivity calculation in power systems operation Jizhong Zhu∗ , Davis Hwang, Ali Sadjadpour a AREVA
T&D Corporation, 11120 NE 33rd Place, Bellevue, WA 98004, USA
Received 24 September 2003; received in revised form 12 May 2004; accepted 13 May 2004 Available online 27 August 2004
Abstract Network losses are being considered in the standard market design. The calculation of loss sensitivity that is based on the distributed generator slack is commonly used for economic dispatch in energy control center. In order to obtain the accurate location-based marginal prices, an efficient and robust method to calculate loss sensitivity for an arbitrary slack bus within the real-time period (1–10 min) is needed in the real-time energy markets. This paper presents a fast and useful approach to calculate loss sensitivity for any slack bus. This method is based on the results taken from the distributed slack buses without repeating a traditional power flow analysis. The proposed approach is tested and verified. © 2004 Elsevier B.V. All rights reserved. Keywords: Standard market design; Economic dispatch; Loss sensitivity; Location-based marginal prices; Real-time energy markets
1. Introduction The electric power industry is being relentlessly pressured by governments, politicians, large industries, and investors to privatize, restructure, and deregulate. Despite changes with different structures, market rules, and uncertainties, an energy management system (EMS) control center must always be in place to maintain the security, reliability, and quality of electric service [1]. It means that EMS in the open energy market must respond quickly, reliably and efficiently to the market changes. In the early energy market, the transmission losses are neglected for reasons of computational simplicity, but are recently addressed in the standard market design (SMD). The loss calculation is considered for the dispatch functions of SMD such as location-based marginal prices (LMP). Loss allocation does not affect generation levels or power flows, however it does modify the value of LMP. The early and classic loss calculation approach is the loss formulas—B coefficient method [2], which is replaced by the more accurate inverse Jacobian transpose method [3]. ∗
Corresponding author. Fax: +1 425 889 1700. E-mail address: [email protected] (J. Zhu).
0378-7796/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2004.05.004
Numerous loss calculation methods have been proposed in the literature [4–11], and can be categorized into pro-rata [4], incremental [5], proportional-sharing [6], and Z-bus loss allocation [7]. The calculation of loss sensitivity that is based on the distributed slack buses is still interesting in the energy control center, because it is based on the well-established incremental transmission loss analysis, an approach known to power engineers for at least three decades [2,7,11]. The distributed slack buses refer to multiple generators or load buses that are selected to share the power mismatch during the power flow calculation. After the power flow problem has been solved, the difference (slack) between the total generation and the total load plus system losses are assigned to these distributed slack buses. In the real-time energy markets, it is frequently required to compute LMP based on market-based reference, which is a single slack bus, instead of the distributed slack buses in the traditional energy management system (EMS). Meanwhile, most of the existing loss calculation methods or traditional EMS products are generally based on the generator slack or reference buses, i.e., multiple generator buses are selected as power slack buses if these generators are AGC units. Since the patterns or status of AGC units are variable for the different time periods in the real time energy market,
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J. Zhu et al. / Electric Power Systems Research 73 (2005) 53–60
the loss sensitivity results will be changed even the network topology is unchanged. Thus, an efficient and robust method to calculate loss sensitivity for an arbitrary slack bus within the real-time period is needed. This paper presents a fast and useful method to calculate loss sensitivity for any single slack bus. This method uses the loss sensitivity results taken from the distributed slack buses and then transforms into single slack bus based results without computing a new set of sensitivity factors through the traditional power flow analysis. Especially, we can choose the load buses as the distributed slack buses rather than the usual generator slack buses. The loss sensitivity results will be consistent for the same network topology no matter how the status of the AGC units changes. The proposed method has been used in the practical market systems such as ISO New England (ISO-NE) and Midwest ISO (MISO), but AREVA T&D 60-bus test system is demonstrated here. The results show that the proposed method is very useful and efficient.
2. Loss sensitivity calculation
j
=
j
s.t.
2 (aj PGj + bj PGj + cj ),
PD + PL =
PGj ,
j ∈ NG
j ∈ NG
(1)
(2)
j
PGj min ≤ PGj ≤ PGj max ,
j ∈ NG
(3)
where PGj is the real power output at generator bus j. PGj min is the minimal real power output at generator j. PGj max is the maximal real power output at generator j. PL is the network losses. PD is the system load demands. fj is the generation cost function of unit j, and a, b, c are the coefficient of the generation cost function. NG is the number of units. All the equations in the paper are implemented in per-unit. Traditionally, the method based on generation slack buses is used in the calculation of loss allocation. In this method, all AGC units are selected to share the power mismatch during the power flow calculation. Since the AGC status or patterns of units are variable in the real time EMS or energy markets, the loss sensitivity values based on the distributed generator slack buses will keep changing due to the change of unit AGC status. Thus, the method based on the distributed load slack buses is used here. It means that the loads are allowed to vary uniformly during the calculation of loss sensitivity. In this way, the economic dispatch problem (1)–(3) can be rewritten as Min F = fj (PGj ), j ∈ NG (1) j
PDi + PL =
i
PGj ,
i ∈ ND, j ∈ NG
(4)
j
PDk + PLk = PGk ,
k∈n
(5)
PDi min ≤ PDi ≤ PDi max ,
i ∈ ND
(6)
PGj min ≤ PGj ≤ PGj max ,
j ∈ NG
(3)
where PDi is the real power load at load bus i. PDi min is the minimal demand at load bus i. PDi max is the maximum demand at load bus i. PGk is the real power output of unit related to bus k. PDk is the real power demand of load related to bus k. PLk is the real power losses at bus k. ND is the number of load buses. n is the total number of buses in the system. The Lagrangian function is obtained from Eqs. (1) and (4): (7) FL = fj (PGj ) + λ PDi + PL − PGj j
The problem of real power economic dispatch without line security constraints can be represented as follows [12]: Min F = fj (PGj )
s.t.
i
j
The optimality criteria of the Lagrangian function (7) are written as follows: ∂FL dfj ∂PL = 0, i ∈ ND (8) = +λ 1+ ∂PDi dPDi ∂PDi dfj ∂PL ∂FL = +λ − 1 = 0, j ∈ NG (9) ∂PGj dPGj ∂PGj dfj LDi = λ, dPDi LDi = −
i ∈ ND
1 , 1 + (∂PL /∂PDi )
dfj LGj = λ, dPGj LGj =
(10) i ∈ ND
j ∈ NG
1 , 1 − (∂PL /∂PGj )
(11) (12)
j ∈ NG
(13)
where λ is the Lagrangian multiplier. ∂PL /∂PDi is the loss sensitivity with respect to load at bus i. ∂PL /∂PGj is the loss sensitivity with respect to unit at bus j. We use ∂PL /∂Pi , which is the loss sensitivity with respect to an injection at bus i, stand for both ∂PL /∂PDi and ∂PL /∂PGj . Since the distributed slack buses are used here, all loss sensitivity factors are non-zero. If bus k is selected as a slack bus, then Pk is the function of the other injections, i.e.: Pk = f (Pi ),
i ∈ n, i = k
(14)
where Pi is the power injection at bus i, which includes the load PDi and generation PGj . Actually, the load can be treated as a negative generation. Then Eqs. (11) and (13) can be
J. Zhu et al. / Electric Power Systems Research 73 (2005) 53–60
expressed as (15), and Eqs. (10) and (12) can be expressed as (16): Li =
1 , 1 − (∂PL /∂Pi )
dfi Li = λ, dPi
i∈n
(15)
i∈n
(16)
Eq. (4) will be rewritten as PL = Pk + Pi , i ∈ n
(17)
i=k
The new Lagrangian function can be obtained from (1) and (17): FL∗ = (18) fi (Pi ) + λ PL − Pk − Pi i
i=n
i ∈ n, i = k
(19)
From (17), we get ∂PL ∂Pk =1+ ∂Pi ∂Pi
(20)
From (19) and (20), we get dfi ∗ dfk L = dPi i dPk L∗i =
(21)
1 , 1 − (∂PL /∂Pi )
i ∈ n, i = k
(22)
It is noted that Li and L∗i are similar, but they have different meaning [13]. The former is based on the distributed slack buses, and the latter is based on the single slack bus k. Similarly, the loss sensitivity in Li is based on the distributed slack buses, i.e. ∂PL /∂Pi |DS (the subscript DS means the distributed slack buses); the loss sensitivity in L∗i is based on the single slack bus k, i.e. ∂PL /∂Pi |k . Note that the kth loss sensitivity, with bus k as the slack bus, is zero. From (14) and (19), we have the following equation: L∗i =
Li , Lk
L∗k = 1
(23)
From Eqs. (15), (22) and (23), we get 1 1 − (∂PL /∂Pk )|DS = 1 − (∂PL /∂Pi )|k 1 − (∂PL /∂Pi )|DS
a slack bus k can be calculated from the following equation: ∂PL /∂Pi |DS − ∂PL /∂Pk |DS ∂PL = (25) ∂Pi k 1 − (∂PL /∂Pk )|DS Loss sensitivity factors ∂PL /∂Pi |DS , which are based on the distributed slack buses, are determined from power flow calculation. The loss sensitivity calculation (25) is very simple, but is accurate and efficient for real-time energy markets. It will avoid computing a new set of the loss sensitivity factors whenever the slack bus k changes. Consequently, it means huge time savings. In addition, the loss factors based on the distributed load slack buses will not be changed no matter how the AGC statuses of units vary, as long as network topology is the same as before.
3. Test results and analysis
The optimality criteria can be obtained from the Lagrangian function (18): ∂FL∗ dfi dfk ∂Pk = + ∂Pi dPi dPk ∂Pi ∂PL ∂Pk +λ − − 1 = 0, ∂Pi ∂Pi
55
(24)
Hence, with one set of the incremental transmission loss coefficients for the distributed slack buses, the loss sensitivity for
The proposed method of loss sensitivity is implemented in our company’s EMS product, which is being used in the practical market systems. Here, the AREVA T&D 60-bus system is applied to illustrate the method. The one-line diagram of the system is shown in Appendix A. The system that has 3 areas consists of 24 generation units (15 units are available in the tests), 32 loads, 43 transmission lines and 54 transformers. The test results are shown in Tables 1–7. All loss sensitivity factors for units and loads are computed. In order to reduce the length of the paper, only loss sensitivities of generators are listed in Tables 1–7. Tables 1–4 are the test results and comparison of loss sensitivity calculation based on the distributed generation reference and distributed load reference, respectively. In Tables 1–4, column 1 is the name of station and units. Column 2 is the area number that the unit belongs to. Column 3 is the AGC status of the unit. The loss factors computed from the distributed unit reference are listed in column 4. The loss factors computed from the distributed load reference are listed in column 5. Generally, the values of loss sensitivities based on the generation references are different from those based on the load references, because the distribution of the units is not exactly the same as the distribution of loads in the power system. The loss factors will be close or equal if the units are close to the load locations. This can be observed from Table 1, where all units are on AGC status. For AREVA T&D 60-bus system, each load in area 3 has at least one unit connected, so the loss factors in area 3 are the same for both the distributed generation slack and distributed load slack. The distributed slack buses are determined as follows. For the distributed generator slack buses, the generators that are AGC units are selected as the distributed slack buses. For the distributed load slack buses, all loads are selected as the distributed slack buses. It is noted from Tables 1–4 that the loss sensitivity factors based on the distributed load slack are the same whether the
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J. Zhu et al. / Electric Power Systems Research 73 (2005) 53–60
Table 1 Test results and comparison of loss sensitivity calculation (case 1: all units on AGC) Station, generator
Area no.
AGC unit
Loss sensitivity distributed generation slack
Loss sensitivity distributed load slack
DOUGLAS, G2 DOUGLAS, G1 DOUGLAS, CT1 DOUGLAS, CT2 DOUGLAS, ST HEARN, G1 HEARN, G2 LAKEVIEW, G1 BVILLE, 1 WVILLE, 1 CHENAUX, 1 CHEALLS, 1 CHEALLS, 2 HOLDEN, 1 NANTCOKE, 1
1 1 1 1 1 1 1 1 2 2 3 3 3 3 3
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
0.015100 0.012100 0.009900 0.009900 0.009700 −0.016500 −0.016500 −0.018800 −0.001000 0.000700 −0.008900 0.021200 0.021200 0.001000 −0.012200
0.017000 0.014000 0.011800 0.011800 0.011600 −0.014600 −0.014600 −0.017000 −0.004200 −0.002500 −0.008900 0.021200 0.021200 0.001000 −0.012200
Table 2 Test results and comparison of loss sensitivity calculation (case 2: all units on AGC except the units under station Douglas in area 1) Station, generator
Area no.
AGC unit
Loss sensitivity distributed generation slack
Loss sensitivity distributed load slack
DOUGLAS, G2 DOUGLAS, G1 DOUGLAS, CT1 DOUGLAS, CT2 DOUGLAS, ST HEARN, G1 HEARN, G2 LAKEVIEW, G1 BVILLE, 1 WVILLE, 1 CHENAUX, 1 CHEALLS, 1 CHEALLS, 2 HOLDEN, 1 NANTCOKE, 1
1 1 1 1 1 1 1 1 2 2 3 3 3 3 3
No No No No No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
0.032800 0.029900 0.027800 0.027800 0.027600 0.001500 0.001500 −0.000800 −0.001000 0.000700 −0.008900 0.021200 0.021200 0.001000 −0.012200
0.017000 0.014000 0.011800 0.011800 0.011600 −0.014600 −0.014600 −0.017000 −0.004200 −0.002500 −0.008900 0.021200 0.021200 0.001000 −0.012200
Table 3 Test results and comparison of loss sensitivity calculation (case 4: all units on AGC except the units in area 2) Station, generator
Area no.
AGC unit
Loss sensitivity distributed generation slack
Loss sensitivity distributed load slack
DOUGLAS, G2 DOUGLAS, G1 DOUGLAS, CT1 DOUGLAS, CT2 DOUGLAS, ST HEARN, G1 HEARN, G2 LAKEVIEW, G1 BVILLE, 1 WVILLE, 1 CHENAUX, 1 CHEALLS, 1 CHEALLS, 2 HOLDEN, 1 NANTCOKE, 1
1 1 1 1 1 1 1 1 2 2 3 3 3 3 3
Yes Yes Yes Yes Yes Yes Yes Yes No No Yes Yes Yes Yes Yes
0.015200 0.012200 0.010000 0.010000 0.009900 −0.016700 −0.016700 −0.019100 −0.021000 −0.019300 −0.008900 0.021200 0.021200 0.001000 −0.012200
0.017000 0.014000 0.011800 0.011800 0.011600 −0.014600 −0.014600 −0.017000 −0.004200 −0.002500 −0.008900 0.021200 0.021200 0.001000 −0.012200
J. Zhu et al. / Electric Power Systems Research 73 (2005) 53–60
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Table 4 Test results and comparison of loss sensitivity calculation (case 5: all units on AGC except unit 3 under station HOLDEN in area 3) Station, generator
Area no.
AGC unit
Loss sensitivity distributed generation slack
Loss sensitivity distributed load slack
DOUGLAS, G2 DOUGLAS, G1 DOUGLAS, CT1 DOUGLAS, CT2 DOUGLAS, ST HEARN, G1 HEARN, G2 LAKEVIEW, G1 BVILLE, 1 WVILLE, 1 CHENAUX, 1 CHEALLS, 1 CHEALLS, 2 HOLDEN, 1 NANTCOKE, 1
1 1 1 1 1 1 1 1 2 2 3 3 3 3 3
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes
0.015100 0.012100 0.009900 0.009900 0.009700 −0.016500 −0.016500 −0.018800 −0.001000 0.000700 −0.008500 0.021600 0.021600 0.001400 −0.011800
0.017000 0.014000 0.011800 0.011800 0.011600 −0.014600 −0.014600 −0.017000 −0.004200 −0.002500 −0.008900 0.021200 0.021200 0.001000 −0.012200
Table 5 Test results of loss sensitivity calculation (distributed slack vs. single slack) Station, generator
AGC unit
Loss sensitivity distributed slack
Loss sensitivity single slack, HOLDEN 1
Loss sensitivity single slack, DOUGLAS ST
DOUGLAS, G2 DOUGLAS, G1 DOUGLAS, CT1 DOUGLAS, CT2 DOUGLAS, ST HEARN, G1 HEARN, G2 LAKEVIEW, G1 BVILLE, 1 WVILLE, 1 CHENAUX, 1 CHEALLS, 1 CHEALLS, 2 HOLDEN, 1 NANTCOKE, 1
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
0.017000 0.014000 0.011800 0.011800 0.011600 −0.014600 −0.014600 −0.017000 −0.004200 −0.002500 −0.008900 0.021200 0.021200 0.001000 −0.012200
0.016016 0.013013 0.010811 0.010811 0.010611 −0.015616 −0.015616 −0.018018 −0.005205 −0.003504 −0.009910 0.020220 0.020220 0.000000 −0.013213
0.005463 0.002428 0.000202 0.000202 0.000000 −0.026507 −0.026507 −0.028936 −0.015985 −0.014265 −0.020741 0.009713 0.009713 −0.010724 −0.024079
Table 6 Comparison of loss sensitivity calculation results for single slack bus at HOLDEN-1 (the proposed method vs. power flow method) Station, generator
AGC unit
Loss sensitivity, HOLDEN 1-PF method
Loss sensitivity, HOLDEN 1, Eq. (25)
|Error %|
DOUGLAS, G2 DOUGLAS, G1 DOUGLAS, CT1 DOUGLAS, CT2 DOUGLAS, ST HEARN, G1 HEARN, G2 LAKEVIEW, G1 BVILLE, 1 WVILLE, 1 CHENAUX, 1 CHEALLS, 1 CHEALLS, 2 HOLDEN, 1 NANTCOKE, 1
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
0.016029 0.013053 0.010817 0.010817 0.010621 −0.015630 −0.015630 −0.018110 −0.005220 −0.003500 −0.009920 0.020247 0.020247 0.000000 −0.013240
0.016016 0.013013 0.010811 0.010811 0.010611 −0.015616 −0.015616 −0.018018 −0.005205 −0.003504 −0.009910 0.020220 0.020220 0.000000 −0.013213
0.08110 0.30644 0.05547 0.05547 0.09415 0.08957 0.08957 0.50801 0.23002 0.02855 0.11088 0.13335 0.13335 0.00000 0.20393
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J. Zhu et al. / Electric Power Systems Research 73 (2005) 53–60
Table 7 Comparison of loss sensitivity calculation results for single slack bus at Douglas-ST (the proposed method vs. power flow method) Station, generator
AGC unit
Loss sensitivity, DOUGLAS ST-PF method
Loss sensitivity, DOUGLAS ST-Eq. (25)
|Error %|
DOUGLAS, G2 DOUGLAS, G1 DOUGLAS, CT1 DOUGLAS, CT2 DOUGLAS, ST HEARN, G1 HEARN, G2 LAKEVIEW, G1 BVILLE, 1 WVILLE, 1 CHENAUX, 1 CHEALLS, 1 CHEALLS, 2 HOLDEN, 1 NANTCOKE, 1
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
0.005467 0.002421 0.000202 0.000202 0.000000 −0.026530 −0.026530 −0.028950 −0.016000 −0.014280 −0.020770 0.009714 0.009714 −0.010730 −0.024090
0.005463 0.002428 0.000202 0.000202 0.000000 −0.026507 −0.026507 −0.028936 −0.015985 −0.014265 −0.020741 0.009713 0.009713 −0.010724 −0.024079
0.07317 0.28914 0.14829 0.14829 0.00000 0.08669 0.08669 0.04836 0.09999 0.10504 0.13962 0.01029 0.01029 0.07454 0.02491
status of the units is changed or not. But the loss factors based on the distributed generation references are changed according to the AGC status of the units. Generally, the change of AGC status of the units only affects the loss sensitivities in the same area that these units belong to. It can be seen from Table 2 that, when AGC status of the units in area 1 changes, only the loss factors in area 1 are affected. The loss factors in the other areas are unchanged. For Table 4, when AGC status of the units in area 3 changes, only the loss factors in area 3 are affected. The loss factors in the other areas are unchanged. But for Table 3, when AGC status of the units in area 2 changes, and all units in this area are not on AGC, it means that there is no unit reference in area 2. Then the units with AGC in the other areas will pick up the mismatch (i.e. area 1 in this case). Thus, the loss factors in area 1 and 2 are changed. The loss factors in the other areas are unchanged. Through the above comparisons, we can conclude that the method of the distributed load references for loss sensitivity calculation is superior to that of the distributed generation references in the real time energy markets, since the AGC status of the units are changeable in the real time system. The results of loss sensitivity calculation for a single slack, which are computed from Eq. (25), are shown in Table 5. Column 3 in Table 5 is the set of the loss sensitivity coefficients from the distributed slack buses. Column 4 in Table 5 is the set of loss sensitivity factors with a single slack bus at the location of HOLDEN 1. Column 5 in Table 5 is the set of loss sensitivity factors with a single slack bus at the location of Douglas. It is noted that all the loss sensitivities are nonzero if the distributed slack is selected. If the single slack is selected, the loss sensitivity of the slack equals zero. Since the loss sensitivity values based on the distributed load slacks are unchanged as long as the system topology is the same, we can easily and quickly get the loss factors for any single slack by use of the proposed loss sensitivity formula. Therefore, a large amount of the computations are avoided
whenever the loss factors are needed for a single slack in the real time energy markets. For example, a practical system with 25,000 buses, the CPU time of computing loss factors using the traditional power flow calculation is about 60 s, but less than 0.1 s if the proposed method is used. This is a huge time saving in the real time energy markets. In order to verify the correctness of the proposed loss sensitivity (25), the loss factors are computed and compared using the traditional power flow calculation. The results and comparison are shown in Tables 6 and 7, in which column 3 is the set of results from the power flow calculation, and column 4 is the set of results from (25). Table 6 shows the comparison of loss factor results for single slack bus at HOLDEN-1. Table 7 shows the comparison of loss factor results for single slack bus at DOUGLAS-ST. The difference or error of the results between the proposed method and power flow method is obtained from the following equation: LFPM (i) − LFPF (i) |Error%| = × 100% , i ∈ n (26) LFPF (i) where, Error% is the percentage of the computation error for the proposed formula. LFPM is the loss factor computed from the proposed method. LFPF is the loss factor obtained using the traditional power flow calculation. It can be known from Tables 6 and 7 that the loss sensitivity results from the two methods are very close. The maximum error is less than 0.6%. 4. Conclusion Network losses are being considered in the standard market design. The calculation of loss sensitivity that is based on the distributed slack buses is used in the energy control center. In order to obtain the accurate location-based marginal prices, it is frequently required that loss sensitivity is quickly computed based on market-based reference (i.e. a particular slack bus) in the real-time energy markets. This paper presents a
J. Zhu et al. / Electric Power Systems Research 73 (2005) 53–60
fast and useful approach to calculate loss sensitivity for any single slack bus. This method is based on the set of loss factors taken from the distributed slack buses. It is not necessary to compute a new set of loss coefficients through the traditional power flow calculation, whenever the slack bus changes. This results in huge time savings. The proposed approach is tested
59
on AREVA T&D 60-bus system. The test results show the proposed loss sensitivity calculation is very fast, useful and efficient. Meanwhile, the test results also show that the approach based on the distributed load slack buses is superior to the method based on the distributed generator slack buses in the real time system.
Fig. 1. One-line diagram of AREVA T&D system.
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J. Zhu et al. / Electric Power Systems Research 73 (2005) 53–60
Table A.1 Generator Data of AREVA T&D 60-bus system (p.u.) Station, generator DOUGLAS, G2 DOUGLAS, G1 DOUGLAS, CT1 DOUGLAS, CT2 DOUGLAS, ST HEARN, G1 HEARN, G2 LAKEVIEW, G1 BVILLE, 1 WVILLE, 1 CHENAUX, 1 CHEALLS, 1 CHEALLS, 2 HOLDEN, 1 NANTCOKE, 1
Area no. 1 1 1 1 1 1 1 1 2 2 3 3 3 3 3
Base MW 6.05 3.68 0.50 0.50 0.20 3.13 3.12 5.16 0.80 2.86 7.08 5.88 3.38 12.11 5.08
Minimum generation output 0.50 0.50 0.10 0.10 0.05 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
Maximum generation output 8.00 12.00 0.50 0.50 0.40 6.00 6.00 8.00 9.00 10.00 13.00 8.00 9.00 22.00 11.00
Table A.2 Load data of AREVA T&D 60-bus system (p.u.) Load in station DOUGLAS KINCARD LAKEVIEW BRIGHTON CEYLON HANOVER PARKHILL BVILLE GOLDEN JVILLE MTOWN WALDEN CHEALLS HOLDEN HUNTVILL MARTDALE NANTCOKE
Area no. 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3
Appendix A AREVA T&D 60-bus system is shown in Fig. 1. Area 1 is EAST, area 2 is WEST, and area 3 is ECAR. The generator and load data are listed in Tables A.1 and A.2. References [1] T.E. Dy-Liacco, Control centers are here to stay, IEEE Comput. Appl. Power 15 (4) (2002) 18–23. [2] L.K. Kirchmayer, Economic Operation of Power Systems, Wiley, New York, 1958. [3] H.W. Dommel, W.F. Tinney, Optimal power flow solutions, IEEE Trans. PAS PAS-87 (10) (1968) 1866–1876. [4] M. Ilic, F.D. Galiana, L. Fink, Power Systems Restructuring: Engineering and Economics, Kluwer Academic Publishers, Norwell, MA, 1998. [5] D. Kirschen, R. Allan, G. Strbac, Contributions of individual generators to loads and flows, IEEE Trans. Power Syst. 12 (1) (1997) 52–60.
Real power load 5.107 1.674 1.999 2.144 3.812 4.469 4.340 0.067 1.693 0.061 3.467 3.123 2.069 10.199 1.903 1.284 2.619
Reactive power load 0.7820 0.704 0.711 0.407 0.851 0.438 1.113 0.033 0.361 0.031 0.420 0.628 0.436 2.309 0.255 0.157 0.248
[6] F. Schweppe, M. Caramanis, R. Tabors, R. Bohn, Spot Pricing of Electricity, Kluwer Academic Publishers, Norwell, MA, 1988. [7] A.J. Conejo, F.D. Galiana, I. Kockar, Z-bus loss allocation, IEEE Trans. Power Syst. 16 (1) (2001) 105–110. [8] F.D. Galiana, A.J. Conjeo, I. Kockar, Incremental transmission loss allocation under pool dispatch, IEEE Trans. Power Syst. 17 (1) (2002) 26–33. [9] F.D. Galiana, M. Phelan, Allocation of transmission losses to bilateral contracts in a competitive environment, IEEE Trans. Power Syst. 15 (1) (2000) 143–150. [10] A.G. Exposito, J.M.R. Santos, T.G. Garcia, E.A.R. Velasco, Fair allocation of transmission power losses, IEEE Trans. Power Syst. 15 (1) (2000) 184–188. [11] O.I. Elgerd, Electric Energy Systems Theory: An Introduction, McGraw-Hill, New York, 1982. [12] J.Z. Zhu, G.Y. Xu, A new economic power dispatch method with security, Electric Power Syst. Res. 25 (1992) 9–15. [13] W.Y. Li, Power System Economic Operation with Security—Model and Method, Chongqing University Press, China, 1989.