- Email: [email protected]
Flexible AC transmission system devices: allocation and transmission pricing
Electrical Power and Energy Systems 21 (1999) 111–118
Flexible AC transmission system devices: allocation and transmission pricing E.J. de Oliveira a,*, J.W. Marangon Lima b, J.L.R. Pereira c a
Department of Electrical Engineering, Federal University at Juiz de Fora-UFJF, Juiz de Fora, MG, Brazil b Institute of Electrical Engineering, Federal Engineering School at Itajuba´-EFEI, Itajuba´, MG, Brazil c Department of Electrical Engineering, Federal University at Juiz de Fora-UFJF, Juiz de Fora, MG, Brazil
Abstract The transmission of electricity differs from transportation of any typical commodity by some inherent aspects such as: the production needs to match the consumption at the same time; system control is not an easy task; the electricity flows do not usually follow the economic law. The last aspect is normally observed when transmission systems are included in, for instance, an economic dispatch problem. One way to minimize the operational costs caused by an overloaded transmission system is through the installation of Flexible AC Transmission System (FACTS) devices in the system. They are able to change power flows by modifying the network parameters. This paper focuses on the ability of FACTS devices to change the overall costs of the system and their impact on transmission pricing. The allocation and the determination of the FACTS required are also discussed. Some examples using the IEEE-14 system and the Brazilian electrical system of the Southern region are given to illustrate the concepts introduced in this paper. 䉷 1998 Elsevier Science Ltd. All rights reserved. Keywords: Transmission pricing; FACTS devices; System operations; Optimal power flow
1. Introduction Transmission open access has been an important issue in the ongoing debate about deregulation and privatization of the power sector in many countries. Since the transmission business presents monopolistic characteristics, transmission open access means a vehicle for promoting competition in generation. But electricity flows do not usually follow the economic law when a transmission system is included in, for example, an economic dispatch problem. Transmission is, therefore, the main concern in the establishment of real competition in the electricity market. Advances in power electronics open a new way to deal with the inconveniences of fixed reactance and undesired flow directions. When devices based on power electronics known as Flexible AC Transmission Systems (FACTS) are introduced into power systems, a more flexible operation and control of transmission networks are obtained. Series compensations, phase shifters are examples of such devices. They can change the active flows of the transmission systems and are discussed in this paper. An increasing concern about environmental aspects and a search for optimal use of transmission capacities are moti-
vating many planners away from building new transmission lines. Therefore, the insertion of FACTS devices seems to be a promising strategy for decreasing transmission congestion and for increasing available transfer capability [1]. Moreover, if a line cannot be eliminated from the transmission plan, it is very likely to be postponed using such devices. These benefits can be obtained by properly allocating FACTS devices in the system. The ideal allocation of these devices is a subject tackled in this paper. Moreover, the impact of such allocation on the wheeling charges is also addressed. This is becoming an important issue when transmission open access schemes are introduced. These charges can really change the investments on both transmission and generation sides. Some examples with the IEEE-14 system and the Brazilian electrical system of the Southern region are used to emphasize the concepts introduced in this paper.
2. The allocation problem Neglecting the resistance, the active power flow, Fij, of a transmission line or a transformer is given by: Fij gij Vi Vj sin uij
* Corresponding author. Fax: ⫹ 55-32-229-3401; e-mail: [email protected].
where:
0142-0615/99/$ - see front matter 䉷 1998 Elsevier Science Ltd. All rights reserved. PII: S0142-061 5(98)00035-0
1
112
E.J. de Oliveira et al. / Electrical Power and Energy Systems 21 (1999) 111–118
Vi, Vj are bus voltages u ij is the angular difference between busses i and j g ij is the susceptance of branch i ⫺ j All equipment able to quickly change any of the above parameters by means of thyristor technology is considered as a FACTS device [2, 3]. Based on its fast response, it has been used to dampen system oscillation through the control of line compensation [4] and angle of phase shifters [5], etc. As the attention of this paper is on the ability of such devices to change the flow direction, a linear load flow model is used. Therefore, no voltage profiles are obtained. The economic dispatch problem which includes the transmission system can be simplified by using a DC load flow. In this case, the problem can be solved by using the following linear programming (LP) problem: Min CT × G st G ⫺ B × u L GⱕGⱕG
2
F ⱕ F ⱕ F where C is the vector of generation costs G is the vector of active generation B is the susceptance matrix u is the vector of voltage angles L is the vector of active load are the lower and upper generation limits G, G F, F are the lower and upper flow limits
Min C × G ⫹ Cfacts
• the line capacity limits, F vary when series capacitors are installed, because transmission line limits are usually based on stability problems and voltage profiles and not on thermal limits • the FACTS cost is a non-linear function of the susceptance variation. In the case of phase shifters, the cost is constant for any angle shift but the allowed limit may be changed. To overcome such difficulties, a technique based on successive LP is proposed in Appendix A. Once the FACTS device is located and dimensioned by solving problem Eq. (A3), it is important to identify the consequences for the transmission users. Since the trend towards unbundling services is becoming a reality in many countries, the introduction of FACTS devices into the transmission system changes the wheeling charges. These charges are dependent on the transmission pricing method adopted and they can affect the transmission-system users. In the next section some of these methods are described.
3. Main pricing methods
Since active power is under consideration, only two types of FACTS devices are considered: the series compensation (SC) and the phase shifter (PS). The phase shifter model can be included in Eq. (2) by a power injection technique. For each desired phase angle there is an injection pair at the end buses where the device is connected [6, 7]. The series compensation is represented by a change in the susceptance matrix. In this case, the LP given by Eq. (2) can be rewritten as: T
As the susceptances vary, Eq. (3) becomes non-linear because of the term B1 u . Other non-linearities also appear when a series compensation is included in the problem due to [8]:
3
st G ⫺ B1 u L ⫹ I
3:1
GⱕGⱕG
3:2
F ⱕ F ⱕ F
3:3
where B1 is the susceptance matrix which includes the susceptance variation due to the series compensation I are power injections representing the phase shifters Cfacts is the cost of the FACTS device (US$/year)
To assess the impact of FACTS devices on transmission charges, pricing methods are based on the incremental cost and embedded cost methods [9]. The incremental cost methods are based on the variation of system total costs when a wheeling transaction is accommodated. The marginal cost is an example of this technique. The embedded cost methods allocate the system total costs among the transmission users based on some ‘extent of use’ rule. These methods follow the cost based approach, i.e., the transmission costs are completely recovered by the wheeling charges. The MWmile based methods [10] are examples of this approach. 3.1. Marginal cost 3.1.1. Short-term marginal cost (STMC) Pricing transmission service by marginal cost has been the most accepted approach according to economic theory. Since the dual coefficients derived from Eq. (3) are the sensitivities of the objective function related to the constraints, they can be used as the marginal costs. In particular, the coefficients of constraints Eq. (3.1) are the bus short-term marginal cost (STMC). The wheeling charge of a transaction u which injects W(u) at bus i and retrieves the same amount at bus j can be obtained by [11]: R u W u STMCi ⫺ STMCj where:
4
E.J. de Oliveira et al. / Electrical Power and Energy Systems 21 (1999) 111–118
113
revenues are not generally enough to match the transmission fixed costs [12, 13]. On the other hand, the calculation of the LTMC which includes transmission investments, is very complex due to the inherent uncertainties associated with the planning horizon. Surrogate LTMC methods have been proposed to overcome such problems [14, 15]. One of them [16] will be used in this paper because it can easily be adapted to Eq. (3): X Min c k fk 5 all k
st G ⫺ B0 u L
5:1
where: ck is the unit cost of circuit k fk is the absolute net flow of circuit k
Fig. 1. IEEE-14 test system.
R(u) is the charge of transaction u W(u) is the amount of transaction u STMCi is the short-term marginal cost at bus i The inclusion of FACTS devices changes the dual coefficients which, in turn, changes the transmission charges. 3.1.2. Long-term marginal cost (LTMC) As transmission costs are basically derived from the investments made in the transmission assets, it is important to take these investments into consideration when the transmission tariff is under design. The STMC, mentioned previously, depends only on generation costs and the Table 8 Additional data for IEEE-14 Cost (10 3 US$/year)
Limit (MW)
GEN-1 GEN-2 L 1–2 L 1–5 L 2–3 L 2–4 L 2–5 L 3–4 L 4–5 L 4–7 L 4–9 L 5–6 L 6–11 L 6–12 L 6–13 L 7–8 L 7–9 L 9–10 L 9–14 L 10–11 L 12–13 L 13–14
10 a 20 a 350.20 180.00 215.00 215.00 215.00 215.00 328.00 142.00 136.00 185.00 179.00 168.00 108.00 136.00 110.00 136.00 111.00 100.00 123.00 121.00
500.0 50.0 210.0 75.0 100.0 100.0 100.0 100.0 250.0 50.0 25.0 50.0 55.0 50.0 100.0 100.0 100.0 150.0 60.0 100.0 100.0 50.0
US$/MWh
3.2. Module method [17] This method derives from the original MW-mile method and is also based on load-flow calculations. The charge R(u) for a transaction u is determined by: R u
Equipment
a
The objective function represents the transmission revenues collected which are based on the capacity use of the transmission system. This term is similar to the one used in the original MW-mile [10]. The advantage is that in this case the dual variables of constraints in Eq. (5.1) may be interpreted as the LTMC at each bus. Therefore the wheeling charges can be determined using Eq. (4) by replacing the STMC by the LTMC.
X
f u Ck Xk fk s all k
6
all s
where: R(u) is the allocated cost to agent u Ck is the cost of circuit k flow caused by agent u fk(u) is the k-circuit P Total cost all k Ck If a FACTS device is installed at circuit k, the corresponding Ck will include the original circuit cost plus the FACTS cost. Therefore, the users of circuit k will pay for this new equipment according to the respective flows, fk(u). 3.3. Zero counter-flow method [18] This method is similar to the previous one but, in this case, the users that go in the opposite direction of the branch net flow do not pay any charge: f u for fk u ⬎ 0 Rk u Ck Xk fk s all sk ⫹
Rk u 0 for fk u ⱕ 0
7
114
E.J. de Oliveira et al. / Electrical Power and Energy Systems 21 (1999) 111–118
Table 1 VTC for the case without transaction. PC production cost; EC equipment cost; VTC PC ⫹ EC variable total cost Alternative
PC × 10 6 US$/year a
EC × 10 3 US$/year b
VTC × 10 6 US$/year c
Base Case SC at 1–5 PS at 1–5 NL at 1–5
24.23 22.73 22.73 22.73
0 28.74 1652.50 180.00
24.23 22.76 24.38 22.91
and R u
X
Rk u
all k
where k⫹
is the set of participants with positive flows on circuit k
This method tries to compensate the counter flow agent for alleviating the transmission system and improving system performance. This agent may also postpone future investments in some cases.
4. Examples 4.1. IEEE-14 4.1.1. Allocation problem The concepts presented are preliminary tested on the IEEE-14 system 1 which is shown in Fig. 1. Table 8 in Appendix B presents the limits and the costs for transmission lines and thermal generations adopted in this paper. A wheeling transaction of 15 MW from bus 1 to bus 4 is added to evaluate this transaction impact on the FACTS allocation cost. Table 1 shows the results of the optimization problem (A3), without considering any transactions, to allocate both SC and PS devices. The cost of an additional line 1– 5 is also included in Table 1, in order to compare the results obtained from the optimization. The SC at line 1–5 (addition of 1.3 MVAr, which represents a compensation of 8.67%) produces the most significant reduction in the Variable Total Cost (VTC). This cost represents the sum of the relevant cost components which vary according to the objective function. The PS alternative is not attractive because its cost is higher than the gain obtained through the variation of production costs (PC). The VTC for the
phase shifter is higher than the base case because the equipment cost (EC) is too high. Although not shown in Table 1, another simulation was carried out, by considering line 1–5 on its thermal limit. The allocation of SC has changed to line 1–2. In this case, the production cost also reduced to 22.73 million US$/year, but the FACTS cost increased to 53.34 thousand US$/year. This shows the importance of considering the line capacity variation when a SC is installed. However, in both cases (SC at line 1–5 or line 1–2) the decrease in fuel costs pays the SC cost. Table 2 shows the results of the optimization process Eq. (A3) when the wheeling transaction of 15 MW is considered. In this case, the SC cost increased to 92.8 thousand US$/year, but it is still the most attractive option. The PS alternative, in this case, is better than the one with no FACTS device (BC-15 MW). 4.1.2. Transmission charges Table 3 shows the annual transmission charges, using STMC, LTMC, MM and ZCF methods, for the 15 MW wheeling transaction, when the base case (BC) (without FACTS device), SC, PS and a new line (NL) from 1 to 5 are considered. For the STMC there is no transmission constraint when a new device is added. As Eq. (4) is used to calculate the transmission charges under the marginal cost approach, these charges vanish (see first row of Table 3). The same does not occur under LTMC where the investments in transmission facilities are considered, as shown in the second row of Table 3. For SC and PS there is an increase in the charges but for NL a decrease. This lower charge for the NL is caused by a significant variation in the circuit flows. For MM and ZCF charging methods the SC device produced the lowest tariff. The simultaneous analysis of Tables 2 and 3 shows that the SC device is the best option for the independent energy producer because it imposes a cost reduction of the order of
Table 2 VTC for the case with transaction. PC production cost; EC equipment cost; VTC PC ⫹ EC variable total cost Alternative
PC × 10 6 US$/year a
EC × 10 3 US$/year b
VTC × 10 6 US$/year c
BC–15MW SC at 1–5 PS at 1–5 NL at 1–5
26.93 22.73 22.73 22.73
0 92.80 1652.50 180.00
26.93 22.82 24.38 22.91
E.J. de Oliveira et al. / Electrical Power and Energy Systems 21 (1999) 111–118 Table 3 Total wheeling charges (10 3 US$/year). STMC short-term marginal cost; LTMC long-term marginal cost; MM module method; ZCF zero counter flow method Method a
STMC LTMC b MM c ZCF d
Base case
With SC
With PS
With NL
2697.36 56.16 160.09 130.93
0.0 58.62 162.59 134.93
0.0 155.9 259.37 228.38
0.0 55.4 165.57 139.65
4 million US$/year. However, the transmission-system user has to pay an additional amount of around 4 thousand US$/ year. These figures show that the tariff impact is rendered by the cost reduction in the transmission system investments. It is important to notice that the proposed method, for optimal FACTS allocation, places the FACTS on the system in such a way that there is a compromise in both system investment and energy production costs. This is a main contribution of this paper. 4.2. Brazilian system The same method is applied to the Brazilian power system of the Southern region. The system depicted in Fig. 2 is an equivalent which has 15 generators, 52 busses and 82 circuits. In this system, about 70% of the power source is from hydro generation (HG) and the rest from coal thermal generation (TG). In this paper it is assumed that the hydro generation cost has an average value which is determined for a medium hydrological condition. The cost of not providing energy to loads will also be considered in the objective function of the optimization problem (A3) by assuming that each bus has a thermal plant with a generation cost equal to the deficit cost.
Fig. 2. System of the Southern Region of Brazil.
115
The following three wheeling transactions are considered independently: • transaction T1: 150 MW from bus 14 to bus 23, which uses the system in the direction of the main flow • transaction T2: 150 MW from bus 23 to bus 14, which uses the system in the counter-direction to the main flow • Transaction T3: 150 MW from bus 14 to bus 19, which is in the same direction as the main flow. In this analysis, three load levels for a typical working day are considered: light load for 12 h, medium load for 8 h and heavy load for 4 h. Only the inclusion of SC is analyzed.
4.2.1. Allocation problem Table 4 shows the FACTS location, production cost (PC), series compensation costs (SCC) and variable total cost (VTC) for the following cases: 1. BC–NT–NSC which means base case with no transactions and with no series compensation 2. BC–NT–SC which means base case with no transactions and with series compensation 3. T1–NSC which means wheeling transaction T1 with no series compensation 4. T1–SC which means wheeling transaction T1 with series compensation The remaining rows in Table 4, in pairs, refer to transactions T2 and T3. The results were obtained from the optimization problem Eq. (A3). The main objective of this table is to show the wheeling transaction impact on FACTS allocation and the associated costs. The cases T1–SC and BC–NT–SC allocate a SC at line 4–10, but the wheeling transaction T1 increases both PC and SCC. This happened because the wheeling transaction T1 is on the main flow direction. This does not occur for transaction T2 where the SC is not required. Remember that transaction T2 is in the counter direction of main flow and therefore alleviates the load of the 500 kV lines. Transaction T3 changes the location of the SC from line 4–10 to line 1– 9 because the energy added is now flowing through different circuits from transactions T1 and T2. In addition, the transaction T3–NSC produces a very high PC due to load curtailment. In contrast, with transaction T3–SC the VTC is drastically reduced but the SCC increases. It is important to note that the kind of transaction may affect the FACTS cost and its allocation. It was observed from the simulations performed that the SC is only effective for the heavy load period when the transmission system imposes some constraints, but nevertheless the gains are enough to justify the inclusion of this device. 1 The data of this system can be obtained by FTP anonymous from: wahoo.ee.washington.edu
116
E.J. de Oliveira et al. / Electrical Power and Energy Systems 21 (1999) 111–118
Table 4 Wheeling transaction effect Case
Location
PC × 10 6 US$/year
SCC × 10 3 US$/year
VTC × 10 6 US$/year
BC–NT–NSC BC–NT–SC T1–NSC T1–SC T2–NSC T2–SC T3–NSC T3–SC
– 4–10 – 4–10 – – – 1–9
52.896 51.735 54.907 52.245 51.277 51.277 98.011 51.848
– 538.88 – 1 047.30 – – – 2 066.80
52.896 52.274 54.907 53.292 51.227 51.277 98.011 53.916
4.2.2. Transmission charges The wheeling charges for transactions T1, T2 and T3 are shown in Table 5Table 6Table 7 respectively. The four pricing methods presented in item 3 are included. From the tables, it can be stated that the effect of FACTS devices on the transmission charges is dependent upon the pricing method. For transaction T1 (Table 5), the inclusion of the SC decreases the charges under STMC and LTMC and increases them under MM and ZCM. For STMC, it can be explained by the fact that the SC minimizes the transmission constraints and therefore tries to equalize the STMC of system busses. For the case of MM and ZCM, the addition of a new equipment cost at a certain circuit branch is shared by all wheeling agents who are using this circuit, which produces the increase in the transmission charges. For the LTMC the variation in the line flows decreases the charges. For transaction T2 (Table 6), the charges are negative under STMC and LTMC because this transaction alleviates the transmission circuits. Since no SC is necessary in this case, no values were provided in the third column. For transaction T3 (Table 7), the inclusion of the SC at circuit 1–9 decreases drastically the difference between the marginal costs of busses 1 and 9. This reflects in the variation of charges with and without the SC under STMC. Considering now the charges under LTMC, MM and ZCM, it can be seen that the results obtained for transaction T1 do not persist. In this case, the SC installed at the 230 kV line causes an increase in its flow changing the direction of the net flow at lines 9–5 and 5–19. This explains why the results from this transaction, under LTMC, MM and ZCF methods, do not follow the same logic of transaction T1. Tables 5 and 7 show that the inclusion of the SC device does not lead to a significant increase in the tariff, even for different transactions when LTMC, MM and ZCF methods are considered.
5. Conclusion
Table 5 Wheeling charges for T1 (US$/MWh)
Table 6 Wheeling charges for T2 (US$/MWh)
This paper focused on the allocation of FACTS devices and their impacts on transmission charges. From the results obtained from the allocation method proposed, some points can be emphasized: • the allocation method proposed is robust and easy to implement • based on the costs considered for the FACTS devices, series compensation proved to be the best alternative for controlling active flows • The location of FACTS devices is highly dependent on wheeling transactions. Therefore, transmission owners must have agreements with transmission users • The effect of FACTS devices on wheeling charges varies according to the pricing method. Moreover, the impact on one wheeling agent may be adverse whereas on others it may be beneficial • the optimal allocation of a SC device produces a reduction in the VTC which is sufficient to compensate for the tariff increase. In fact, the inclusion of a SC device in an optimal position does not cause a substantial impact on the tariff. All of these points need to be tackled by transmission owners, wheeling agents and regulators. The advance of FACTS technology tends to decrease its costs and, therefore, increases the number of such devices in the transmission system.
Acknowledgements The authors would like to thank CNPq (#522905/96-0)
Method
Without SC
With SC
Method
Without SC
With SC
STMC LTMC MM ZCM
1.57 2.19 8.13 7.36
0.53 2.17 8.38 7.93
STMC LTMC MM ZCM
⫺ 1.57 ⫺ 2.19 8.15 2.62
– – – –
E.J. de Oliveira et al. / Electrical Power and Energy Systems 21 (1999) 111–118
fk gok ⫺ fok Dgk fok gok
Table 7 Wheeling charges for T3 (US$/MWh) Method
Without SC
With SC
STMC LTMC MM ZCM
2.65 1.56 6.99 6.46
0.10 1.59 6.96 6.59
117
A2
Inserting Eq. (A1) and Eq. (A2) into Eq. (3) and taking into account the series compensation costs, Cfacts, the new optimization problem becomes: Min CTg G ⫹ Cfacts
A3
st G ⫺ B0 u ⫺ DB u L
A3:1
and FAPEMIG (TEC945/95), Brazil, for the financial support granted for the execution of this work.
fk gok ⫺ fok Dgk fok gok for all branch k
A3:2
Appendix A
GⱕGⱕG
A3:3
⫺F ⱕ F ⱕ F
A3:4
0 ⱕ DG ⱕ G
A3:5
An algorithm using LP is developed here to solve Eq. (3). The new susceptance matrix B1 is decomposed into two components in order to include the series compensation: B1 B ⫹ DB
A1
0
where: B 0 is the initial susceptance matrix DB is the susceptance matrix deviation due to series compensation
where: Cg is the vector of generation costs F is the vector of line capacity limits G is the vector of susceptance variation limits.
0
0
0
…
0
0
0
Dg
0
…
⫺Dg
0
0
0
0
…
0
0
The problem Eq. (A3) is a non-linear optimization problem which is solved here by using a successive LP. This can be done by supposing that the angles of the term DB u are constant and equal to the last iteration.The series compensation cost in line k is given by: ÿ 2 ft A4 Cfacts k cXc k k BP
DB …
…
…
…
…
…
where:
0
⫺Dg
0
…
Dg
0
0
0
0
…
0
0
For a series compensation in just one circuit ij, the susceptance matrix deviation becomes:
col ⫺ i
row ⫺ i
row ⫺ j
col ⫺ j
Notice that in the case where the installation of series compensation in every circuit of the system is allowed, DB represents the sum of all the above combinations.When the susceptance of a line varies, the stability and the voltage limit associated with this line changes. In general, the transmission capacity limits are associated with stability or voltage limits and not with the thermal limit. One simple method of including such a limit change in a line k is by using the assumption that the differences between the voltage angles of adjacent busses i and j, uij , should not change. Therefore the new limit, fk can be expressed in terms of the initial limit, fok , and the susceptance variation Dg k, as follows: fok gok sin uij fk gok ⫹ Dgk sin uij Eliminating sin uij in the above equations:
c is the FACTS cost (US$/MVAr-year) ft k is the thermal limits of line k BP is the base power The series capacitor reactance Xc(k) is given by: Xc k
1 Dg g0k g0k ⫹ Dgk k
A5
From Eq. (A4) and Eq. (A5) follow: Cfacts k ak Dgk ⫹ bk
A6
where:
ÿ 2 ft 1 ak c k 0 0 BP gk gk ⫹ Dgk b k is the installation cost.
As Eq. (A6) is a non-linear function of Dgk, the coefficient ak needs to be re-evaluated at each iteration of Eq. (A3). At the initial iteration the values of Dgk may be set equal to zero. The installation cost is only introduced into the algorithm after the first iteration where the location is roughly determined.The same structure as in Eq. (A3) can be used, with slight modification, to include the phase shifter. The flow of
118
E.J. de Oliveira et al. / Electrical Power and Energy Systems 21 (1999) 111–118
a line k between nodes i and j is given by: fk gk uij ⫹ ck Dgk ⫹ gk ·uij
A7
where c k is the phase angle of the phase shifter.In this paper, the phase shifter is represented by a modification at the admittance matrix B, i.e., the flow variation caused by a shift in c k is represented by the flow variation caused by Dg k. Eq. (A7) shows this equivalence. The results are similar to the ones obtained by power injections [19]. Appendix B Table 8 provides the additional data used to evaluate the series capacitor allocation and the wheeling charges for the IEEE-14 system.The FACTS cost (c) adopted in the examples presented in the paper was 135 US$/kVA [8]. Note that it will mean an annual installment of 22 thousand US$/ MVA-year if it is considered an income tax of 10% and a mortgage period of 10 years. References [1] Galiana K, Almeida M, Toussaint J, Griffin D, Atanackovic BT, Ooi D, McGrillis T. Assessment and control of the impact of FACTS devices on power system performance. IEEE Trans on PWRS 1996;11(4):1931–1936. [2] Hingorani NG. FACTS: flexible AC transmission systems. EPRI workshop. Cincinnati, OH, November 1990. [3] Gronquist JL, Sethares WA, Alvarado FL, Lasseter RH. Power oscillation damping control strategies for FACTS devices using locally measurable quantities. IEEE Trans on PWRS 1995;10(3):1598–1605. [4] Taranto GN, Chow JH. A robust frequency domain optimization technique for tuning series compensation damping controllers. IEEE Trans on PWRS 1995;10(3):1219–11225. [5] de Oliveira EJ, Oliveira JC, Moraes AJ, Guimara˜es GC. AC systems stabilization via phase shift transformer with thyristor commutation. Japan Industry Applications Society Conference JIASC-IEEE, EIME UNIV. Matsuyana (Japan), 24–26 August 1994.
[6] Taranto GN, Pinto LMV, Pereira MVF. Representation of FACTS devices in power system economic dispatch. IEEE Trans on PWRS 1992;7(2):572–576. [7] Lie TT, Deng W. Optimal flexible AC transmission systems (FACTS) devices allocation. Electrical Power and Energy Systems 1997;19(2):125–134. [8] Griffin J, Atanackovic D, Galiana FD. A study of the impact of FACTS on the secure-economic operation of power systems. 12th Power Systems Computation Conference. Dresden, August 19–23 1996:1077–1082. [9] Shirmohammadi D, Filho XV, Goresntin B, Pereira MVP. Some fundamental technical concepts about cost based transmission pricing. IEEE Trans on PWRS 1996;11(2):1002–1008. [10] Shirmohammadi D, Gribik PR, Law ETK, Malinowski JH, O’Donnel RE. Evaluation of transmission network capacity use for wheeling transactions. IEEE Trans on PWRS 1989;4:1405–1413. [11] Schweppe FC, Caraminis MC, Tabors RD, Bohn RE. Spot pricing of electricity. Boston, MA: Kluwer Academic Publishers, 1988. [12] Marangon Lima JW, Pereira MVF, Pereira JLR. An integrated framework for cost allocation in a multi-owned transmission system. IEEE Trans on PWRS 1995;10(2):971–977. [13] Pe´rez-Arriaga IJ, Rubio FJ, Puerta JF, Arceluz J, Marı´n J. Marginal pricing of transmission services: an analysis of cost recovery. IEEE Trans on PWRS 1995;10(1):546–553. [14] Jonard F, Smeers Y, Bruel P, Heilbronn, B. Wheeling costs: an economic analysis illustrated by short term and long term simulation. Stockholm Power Tech Conference, paper SPT PS 23-02-0287. Sweden, June 1995. [15] Marangon Lima JW, de Oliveira EJ. The long-term impact of transmission pricing. Paper PE-248-PWRS-0-12-1997, presented at 1998 IEEE Winter Meeting. Tampa, FL, USA. [16] Calviou MC, Dunnett RM, Plumptre PH. Charging for use of a transmission system by marginal cost methods. 11th PSCC. Avignon France, September 1993. [17] Kovacs R, Leverett A. A load flow based method for calculating embedded, incremental and marginal cost of transmission capacity. IEEE Trans on PWRS 1994;9(1):272–278. [18] Marangon Lima JW. Transmission fixed charge allocation: an overview. IEEE Trans on PWRS 1996;11(3):1409–1418. [19] de Oliveira EJ, Marangon Lima JW, de Almeida KC. Optimal allocation of FACTS devices. VI SEPOPE. Salvador, Brazil, May 1998.
Flexible AC transmission system devices: allocation and transmission pricing E.J. de Oliveira a,*, J.W. Marangon Lima b, J.L.R. Pereira c a
Department of Electrical Engineering, Federal University at Juiz de Fora-UFJF, Juiz de Fora, MG, Brazil b Institute of Electrical Engineering, Federal Engineering School at Itajuba´-EFEI, Itajuba´, MG, Brazil c Department of Electrical Engineering, Federal University at Juiz de Fora-UFJF, Juiz de Fora, MG, Brazil
Abstract The transmission of electricity differs from transportation of any typical commodity by some inherent aspects such as: the production needs to match the consumption at the same time; system control is not an easy task; the electricity flows do not usually follow the economic law. The last aspect is normally observed when transmission systems are included in, for instance, an economic dispatch problem. One way to minimize the operational costs caused by an overloaded transmission system is through the installation of Flexible AC Transmission System (FACTS) devices in the system. They are able to change power flows by modifying the network parameters. This paper focuses on the ability of FACTS devices to change the overall costs of the system and their impact on transmission pricing. The allocation and the determination of the FACTS required are also discussed. Some examples using the IEEE-14 system and the Brazilian electrical system of the Southern region are given to illustrate the concepts introduced in this paper. 䉷 1998 Elsevier Science Ltd. All rights reserved. Keywords: Transmission pricing; FACTS devices; System operations; Optimal power flow
1. Introduction Transmission open access has been an important issue in the ongoing debate about deregulation and privatization of the power sector in many countries. Since the transmission business presents monopolistic characteristics, transmission open access means a vehicle for promoting competition in generation. But electricity flows do not usually follow the economic law when a transmission system is included in, for example, an economic dispatch problem. Transmission is, therefore, the main concern in the establishment of real competition in the electricity market. Advances in power electronics open a new way to deal with the inconveniences of fixed reactance and undesired flow directions. When devices based on power electronics known as Flexible AC Transmission Systems (FACTS) are introduced into power systems, a more flexible operation and control of transmission networks are obtained. Series compensations, phase shifters are examples of such devices. They can change the active flows of the transmission systems and are discussed in this paper. An increasing concern about environmental aspects and a search for optimal use of transmission capacities are moti-
vating many planners away from building new transmission lines. Therefore, the insertion of FACTS devices seems to be a promising strategy for decreasing transmission congestion and for increasing available transfer capability [1]. Moreover, if a line cannot be eliminated from the transmission plan, it is very likely to be postponed using such devices. These benefits can be obtained by properly allocating FACTS devices in the system. The ideal allocation of these devices is a subject tackled in this paper. Moreover, the impact of such allocation on the wheeling charges is also addressed. This is becoming an important issue when transmission open access schemes are introduced. These charges can really change the investments on both transmission and generation sides. Some examples with the IEEE-14 system and the Brazilian electrical system of the Southern region are used to emphasize the concepts introduced in this paper.
2. The allocation problem Neglecting the resistance, the active power flow, Fij, of a transmission line or a transformer is given by: Fij gij Vi Vj sin uij
* Corresponding author. Fax: ⫹ 55-32-229-3401; e-mail: [email protected].
where:
0142-0615/99/$ - see front matter 䉷 1998 Elsevier Science Ltd. All rights reserved. PII: S0142-061 5(98)00035-0
1
112
E.J. de Oliveira et al. / Electrical Power and Energy Systems 21 (1999) 111–118
Vi, Vj are bus voltages u ij is the angular difference between busses i and j g ij is the susceptance of branch i ⫺ j All equipment able to quickly change any of the above parameters by means of thyristor technology is considered as a FACTS device [2, 3]. Based on its fast response, it has been used to dampen system oscillation through the control of line compensation [4] and angle of phase shifters [5], etc. As the attention of this paper is on the ability of such devices to change the flow direction, a linear load flow model is used. Therefore, no voltage profiles are obtained. The economic dispatch problem which includes the transmission system can be simplified by using a DC load flow. In this case, the problem can be solved by using the following linear programming (LP) problem: Min CT × G st G ⫺ B × u L GⱕGⱕG
2
F ⱕ F ⱕ F where C is the vector of generation costs G is the vector of active generation B is the susceptance matrix u is the vector of voltage angles L is the vector of active load are the lower and upper generation limits G, G F, F are the lower and upper flow limits
Min C × G ⫹ Cfacts
• the line capacity limits, F vary when series capacitors are installed, because transmission line limits are usually based on stability problems and voltage profiles and not on thermal limits • the FACTS cost is a non-linear function of the susceptance variation. In the case of phase shifters, the cost is constant for any angle shift but the allowed limit may be changed. To overcome such difficulties, a technique based on successive LP is proposed in Appendix A. Once the FACTS device is located and dimensioned by solving problem Eq. (A3), it is important to identify the consequences for the transmission users. Since the trend towards unbundling services is becoming a reality in many countries, the introduction of FACTS devices into the transmission system changes the wheeling charges. These charges are dependent on the transmission pricing method adopted and they can affect the transmission-system users. In the next section some of these methods are described.
3. Main pricing methods
Since active power is under consideration, only two types of FACTS devices are considered: the series compensation (SC) and the phase shifter (PS). The phase shifter model can be included in Eq. (2) by a power injection technique. For each desired phase angle there is an injection pair at the end buses where the device is connected [6, 7]. The series compensation is represented by a change in the susceptance matrix. In this case, the LP given by Eq. (2) can be rewritten as: T
As the susceptances vary, Eq. (3) becomes non-linear because of the term B1 u . Other non-linearities also appear when a series compensation is included in the problem due to [8]:
3
st G ⫺ B1 u L ⫹ I
3:1
GⱕGⱕG
3:2
F ⱕ F ⱕ F
3:3
where B1 is the susceptance matrix which includes the susceptance variation due to the series compensation I are power injections representing the phase shifters Cfacts is the cost of the FACTS device (US$/year)
To assess the impact of FACTS devices on transmission charges, pricing methods are based on the incremental cost and embedded cost methods [9]. The incremental cost methods are based on the variation of system total costs when a wheeling transaction is accommodated. The marginal cost is an example of this technique. The embedded cost methods allocate the system total costs among the transmission users based on some ‘extent of use’ rule. These methods follow the cost based approach, i.e., the transmission costs are completely recovered by the wheeling charges. The MWmile based methods [10] are examples of this approach. 3.1. Marginal cost 3.1.1. Short-term marginal cost (STMC) Pricing transmission service by marginal cost has been the most accepted approach according to economic theory. Since the dual coefficients derived from Eq. (3) are the sensitivities of the objective function related to the constraints, they can be used as the marginal costs. In particular, the coefficients of constraints Eq. (3.1) are the bus short-term marginal cost (STMC). The wheeling charge of a transaction u which injects W(u) at bus i and retrieves the same amount at bus j can be obtained by [11]: R u W u STMCi ⫺ STMCj where:
4
E.J. de Oliveira et al. / Electrical Power and Energy Systems 21 (1999) 111–118
113
revenues are not generally enough to match the transmission fixed costs [12, 13]. On the other hand, the calculation of the LTMC which includes transmission investments, is very complex due to the inherent uncertainties associated with the planning horizon. Surrogate LTMC methods have been proposed to overcome such problems [14, 15]. One of them [16] will be used in this paper because it can easily be adapted to Eq. (3): X Min c k fk 5 all k
st G ⫺ B0 u L
5:1
where: ck is the unit cost of circuit k fk is the absolute net flow of circuit k
Fig. 1. IEEE-14 test system.
R(u) is the charge of transaction u W(u) is the amount of transaction u STMCi is the short-term marginal cost at bus i The inclusion of FACTS devices changes the dual coefficients which, in turn, changes the transmission charges. 3.1.2. Long-term marginal cost (LTMC) As transmission costs are basically derived from the investments made in the transmission assets, it is important to take these investments into consideration when the transmission tariff is under design. The STMC, mentioned previously, depends only on generation costs and the Table 8 Additional data for IEEE-14 Cost (10 3 US$/year)
Limit (MW)
GEN-1 GEN-2 L 1–2 L 1–5 L 2–3 L 2–4 L 2–5 L 3–4 L 4–5 L 4–7 L 4–9 L 5–6 L 6–11 L 6–12 L 6–13 L 7–8 L 7–9 L 9–10 L 9–14 L 10–11 L 12–13 L 13–14
10 a 20 a 350.20 180.00 215.00 215.00 215.00 215.00 328.00 142.00 136.00 185.00 179.00 168.00 108.00 136.00 110.00 136.00 111.00 100.00 123.00 121.00
500.0 50.0 210.0 75.0 100.0 100.0 100.0 100.0 250.0 50.0 25.0 50.0 55.0 50.0 100.0 100.0 100.0 150.0 60.0 100.0 100.0 50.0
US$/MWh
3.2. Module method [17] This method derives from the original MW-mile method and is also based on load-flow calculations. The charge R(u) for a transaction u is determined by: R u
Equipment
a
The objective function represents the transmission revenues collected which are based on the capacity use of the transmission system. This term is similar to the one used in the original MW-mile [10]. The advantage is that in this case the dual variables of constraints in Eq. (5.1) may be interpreted as the LTMC at each bus. Therefore the wheeling charges can be determined using Eq. (4) by replacing the STMC by the LTMC.
X
f u Ck Xk fk s all k
6
all s
where: R(u) is the allocated cost to agent u Ck is the cost of circuit k flow caused by agent u fk(u) is the k-circuit P Total cost all k Ck If a FACTS device is installed at circuit k, the corresponding Ck will include the original circuit cost plus the FACTS cost. Therefore, the users of circuit k will pay for this new equipment according to the respective flows, fk(u). 3.3. Zero counter-flow method [18] This method is similar to the previous one but, in this case, the users that go in the opposite direction of the branch net flow do not pay any charge: f u for fk u ⬎ 0 Rk u Ck Xk fk s all sk ⫹
Rk u 0 for fk u ⱕ 0
7
114
E.J. de Oliveira et al. / Electrical Power and Energy Systems 21 (1999) 111–118
Table 1 VTC for the case without transaction. PC production cost; EC equipment cost; VTC PC ⫹ EC variable total cost Alternative
PC × 10 6 US$/year a
EC × 10 3 US$/year b
VTC × 10 6 US$/year c
Base Case SC at 1–5 PS at 1–5 NL at 1–5
24.23 22.73 22.73 22.73
0 28.74 1652.50 180.00
24.23 22.76 24.38 22.91
and R u
X
Rk u
all k
where k⫹
is the set of participants with positive flows on circuit k
This method tries to compensate the counter flow agent for alleviating the transmission system and improving system performance. This agent may also postpone future investments in some cases.
4. Examples 4.1. IEEE-14 4.1.1. Allocation problem The concepts presented are preliminary tested on the IEEE-14 system 1 which is shown in Fig. 1. Table 8 in Appendix B presents the limits and the costs for transmission lines and thermal generations adopted in this paper. A wheeling transaction of 15 MW from bus 1 to bus 4 is added to evaluate this transaction impact on the FACTS allocation cost. Table 1 shows the results of the optimization problem (A3), without considering any transactions, to allocate both SC and PS devices. The cost of an additional line 1– 5 is also included in Table 1, in order to compare the results obtained from the optimization. The SC at line 1–5 (addition of 1.3 MVAr, which represents a compensation of 8.67%) produces the most significant reduction in the Variable Total Cost (VTC). This cost represents the sum of the relevant cost components which vary according to the objective function. The PS alternative is not attractive because its cost is higher than the gain obtained through the variation of production costs (PC). The VTC for the
phase shifter is higher than the base case because the equipment cost (EC) is too high. Although not shown in Table 1, another simulation was carried out, by considering line 1–5 on its thermal limit. The allocation of SC has changed to line 1–2. In this case, the production cost also reduced to 22.73 million US$/year, but the FACTS cost increased to 53.34 thousand US$/year. This shows the importance of considering the line capacity variation when a SC is installed. However, in both cases (SC at line 1–5 or line 1–2) the decrease in fuel costs pays the SC cost. Table 2 shows the results of the optimization process Eq. (A3) when the wheeling transaction of 15 MW is considered. In this case, the SC cost increased to 92.8 thousand US$/year, but it is still the most attractive option. The PS alternative, in this case, is better than the one with no FACTS device (BC-15 MW). 4.1.2. Transmission charges Table 3 shows the annual transmission charges, using STMC, LTMC, MM and ZCF methods, for the 15 MW wheeling transaction, when the base case (BC) (without FACTS device), SC, PS and a new line (NL) from 1 to 5 are considered. For the STMC there is no transmission constraint when a new device is added. As Eq. (4) is used to calculate the transmission charges under the marginal cost approach, these charges vanish (see first row of Table 3). The same does not occur under LTMC where the investments in transmission facilities are considered, as shown in the second row of Table 3. For SC and PS there is an increase in the charges but for NL a decrease. This lower charge for the NL is caused by a significant variation in the circuit flows. For MM and ZCF charging methods the SC device produced the lowest tariff. The simultaneous analysis of Tables 2 and 3 shows that the SC device is the best option for the independent energy producer because it imposes a cost reduction of the order of
Table 2 VTC for the case with transaction. PC production cost; EC equipment cost; VTC PC ⫹ EC variable total cost Alternative
PC × 10 6 US$/year a
EC × 10 3 US$/year b
VTC × 10 6 US$/year c
BC–15MW SC at 1–5 PS at 1–5 NL at 1–5
26.93 22.73 22.73 22.73
0 92.80 1652.50 180.00
26.93 22.82 24.38 22.91
E.J. de Oliveira et al. / Electrical Power and Energy Systems 21 (1999) 111–118 Table 3 Total wheeling charges (10 3 US$/year). STMC short-term marginal cost; LTMC long-term marginal cost; MM module method; ZCF zero counter flow method Method a
STMC LTMC b MM c ZCF d
Base case
With SC
With PS
With NL
2697.36 56.16 160.09 130.93
0.0 58.62 162.59 134.93
0.0 155.9 259.37 228.38
0.0 55.4 165.57 139.65
4 million US$/year. However, the transmission-system user has to pay an additional amount of around 4 thousand US$/ year. These figures show that the tariff impact is rendered by the cost reduction in the transmission system investments. It is important to notice that the proposed method, for optimal FACTS allocation, places the FACTS on the system in such a way that there is a compromise in both system investment and energy production costs. This is a main contribution of this paper. 4.2. Brazilian system The same method is applied to the Brazilian power system of the Southern region. The system depicted in Fig. 2 is an equivalent which has 15 generators, 52 busses and 82 circuits. In this system, about 70% of the power source is from hydro generation (HG) and the rest from coal thermal generation (TG). In this paper it is assumed that the hydro generation cost has an average value which is determined for a medium hydrological condition. The cost of not providing energy to loads will also be considered in the objective function of the optimization problem (A3) by assuming that each bus has a thermal plant with a generation cost equal to the deficit cost.
Fig. 2. System of the Southern Region of Brazil.
115
The following three wheeling transactions are considered independently: • transaction T1: 150 MW from bus 14 to bus 23, which uses the system in the direction of the main flow • transaction T2: 150 MW from bus 23 to bus 14, which uses the system in the counter-direction to the main flow • Transaction T3: 150 MW from bus 14 to bus 19, which is in the same direction as the main flow. In this analysis, three load levels for a typical working day are considered: light load for 12 h, medium load for 8 h and heavy load for 4 h. Only the inclusion of SC is analyzed.
4.2.1. Allocation problem Table 4 shows the FACTS location, production cost (PC), series compensation costs (SCC) and variable total cost (VTC) for the following cases: 1. BC–NT–NSC which means base case with no transactions and with no series compensation 2. BC–NT–SC which means base case with no transactions and with series compensation 3. T1–NSC which means wheeling transaction T1 with no series compensation 4. T1–SC which means wheeling transaction T1 with series compensation The remaining rows in Table 4, in pairs, refer to transactions T2 and T3. The results were obtained from the optimization problem Eq. (A3). The main objective of this table is to show the wheeling transaction impact on FACTS allocation and the associated costs. The cases T1–SC and BC–NT–SC allocate a SC at line 4–10, but the wheeling transaction T1 increases both PC and SCC. This happened because the wheeling transaction T1 is on the main flow direction. This does not occur for transaction T2 where the SC is not required. Remember that transaction T2 is in the counter direction of main flow and therefore alleviates the load of the 500 kV lines. Transaction T3 changes the location of the SC from line 4–10 to line 1– 9 because the energy added is now flowing through different circuits from transactions T1 and T2. In addition, the transaction T3–NSC produces a very high PC due to load curtailment. In contrast, with transaction T3–SC the VTC is drastically reduced but the SCC increases. It is important to note that the kind of transaction may affect the FACTS cost and its allocation. It was observed from the simulations performed that the SC is only effective for the heavy load period when the transmission system imposes some constraints, but nevertheless the gains are enough to justify the inclusion of this device. 1 The data of this system can be obtained by FTP anonymous from: wahoo.ee.washington.edu
116
E.J. de Oliveira et al. / Electrical Power and Energy Systems 21 (1999) 111–118
Table 4 Wheeling transaction effect Case
Location
PC × 10 6 US$/year
SCC × 10 3 US$/year
VTC × 10 6 US$/year
BC–NT–NSC BC–NT–SC T1–NSC T1–SC T2–NSC T2–SC T3–NSC T3–SC
– 4–10 – 4–10 – – – 1–9
52.896 51.735 54.907 52.245 51.277 51.277 98.011 51.848
– 538.88 – 1 047.30 – – – 2 066.80
52.896 52.274 54.907 53.292 51.227 51.277 98.011 53.916
4.2.2. Transmission charges The wheeling charges for transactions T1, T2 and T3 are shown in Table 5Table 6Table 7 respectively. The four pricing methods presented in item 3 are included. From the tables, it can be stated that the effect of FACTS devices on the transmission charges is dependent upon the pricing method. For transaction T1 (Table 5), the inclusion of the SC decreases the charges under STMC and LTMC and increases them under MM and ZCM. For STMC, it can be explained by the fact that the SC minimizes the transmission constraints and therefore tries to equalize the STMC of system busses. For the case of MM and ZCM, the addition of a new equipment cost at a certain circuit branch is shared by all wheeling agents who are using this circuit, which produces the increase in the transmission charges. For the LTMC the variation in the line flows decreases the charges. For transaction T2 (Table 6), the charges are negative under STMC and LTMC because this transaction alleviates the transmission circuits. Since no SC is necessary in this case, no values were provided in the third column. For transaction T3 (Table 7), the inclusion of the SC at circuit 1–9 decreases drastically the difference between the marginal costs of busses 1 and 9. This reflects in the variation of charges with and without the SC under STMC. Considering now the charges under LTMC, MM and ZCM, it can be seen that the results obtained for transaction T1 do not persist. In this case, the SC installed at the 230 kV line causes an increase in its flow changing the direction of the net flow at lines 9–5 and 5–19. This explains why the results from this transaction, under LTMC, MM and ZCF methods, do not follow the same logic of transaction T1. Tables 5 and 7 show that the inclusion of the SC device does not lead to a significant increase in the tariff, even for different transactions when LTMC, MM and ZCF methods are considered.
5. Conclusion
Table 5 Wheeling charges for T1 (US$/MWh)
Table 6 Wheeling charges for T2 (US$/MWh)
This paper focused on the allocation of FACTS devices and their impacts on transmission charges. From the results obtained from the allocation method proposed, some points can be emphasized: • the allocation method proposed is robust and easy to implement • based on the costs considered for the FACTS devices, series compensation proved to be the best alternative for controlling active flows • The location of FACTS devices is highly dependent on wheeling transactions. Therefore, transmission owners must have agreements with transmission users • The effect of FACTS devices on wheeling charges varies according to the pricing method. Moreover, the impact on one wheeling agent may be adverse whereas on others it may be beneficial • the optimal allocation of a SC device produces a reduction in the VTC which is sufficient to compensate for the tariff increase. In fact, the inclusion of a SC device in an optimal position does not cause a substantial impact on the tariff. All of these points need to be tackled by transmission owners, wheeling agents and regulators. The advance of FACTS technology tends to decrease its costs and, therefore, increases the number of such devices in the transmission system.
Acknowledgements The authors would like to thank CNPq (#522905/96-0)
Method
Without SC
With SC
Method
Without SC
With SC
STMC LTMC MM ZCM
1.57 2.19 8.13 7.36
0.53 2.17 8.38 7.93
STMC LTMC MM ZCM
⫺ 1.57 ⫺ 2.19 8.15 2.62
– – – –
E.J. de Oliveira et al. / Electrical Power and Energy Systems 21 (1999) 111–118
fk gok ⫺ fok Dgk fok gok
Table 7 Wheeling charges for T3 (US$/MWh) Method
Without SC
With SC
STMC LTMC MM ZCM
2.65 1.56 6.99 6.46
0.10 1.59 6.96 6.59
117
A2
Inserting Eq. (A1) and Eq. (A2) into Eq. (3) and taking into account the series compensation costs, Cfacts, the new optimization problem becomes: Min CTg G ⫹ Cfacts
A3
st G ⫺ B0 u ⫺ DB u L
A3:1
and FAPEMIG (TEC945/95), Brazil, for the financial support granted for the execution of this work.
fk gok ⫺ fok Dgk fok gok for all branch k
A3:2
Appendix A
GⱕGⱕG
A3:3
⫺F ⱕ F ⱕ F
A3:4
0 ⱕ DG ⱕ G
A3:5
An algorithm using LP is developed here to solve Eq. (3). The new susceptance matrix B1 is decomposed into two components in order to include the series compensation: B1 B ⫹ DB
A1
0
where: B 0 is the initial susceptance matrix DB is the susceptance matrix deviation due to series compensation
where: Cg is the vector of generation costs F is the vector of line capacity limits G is the vector of susceptance variation limits.
0
0
0
…
0
0
0
Dg
0
…
⫺Dg
0
0
0
0
…
0
0
The problem Eq. (A3) is a non-linear optimization problem which is solved here by using a successive LP. This can be done by supposing that the angles of the term DB u are constant and equal to the last iteration.The series compensation cost in line k is given by: ÿ 2 ft A4 Cfacts k cXc k k BP
DB …
…
…
…
…
…
where:
0
⫺Dg
0
…
Dg
0
0
0
0
…
0
0
For a series compensation in just one circuit ij, the susceptance matrix deviation becomes:
col ⫺ i
row ⫺ i
row ⫺ j
col ⫺ j
Notice that in the case where the installation of series compensation in every circuit of the system is allowed, DB represents the sum of all the above combinations.When the susceptance of a line varies, the stability and the voltage limit associated with this line changes. In general, the transmission capacity limits are associated with stability or voltage limits and not with the thermal limit. One simple method of including such a limit change in a line k is by using the assumption that the differences between the voltage angles of adjacent busses i and j, uij , should not change. Therefore the new limit, fk can be expressed in terms of the initial limit, fok , and the susceptance variation Dg k, as follows: fok gok sin uij fk gok ⫹ Dgk sin uij Eliminating sin uij in the above equations:
c is the FACTS cost (US$/MVAr-year) ft k is the thermal limits of line k BP is the base power The series capacitor reactance Xc(k) is given by: Xc k
1 Dg g0k g0k ⫹ Dgk k
A5
From Eq. (A4) and Eq. (A5) follow: Cfacts k ak Dgk ⫹ bk
A6
where:
ÿ 2 ft 1 ak c k 0 0 BP gk gk ⫹ Dgk b k is the installation cost.
As Eq. (A6) is a non-linear function of Dgk, the coefficient ak needs to be re-evaluated at each iteration of Eq. (A3). At the initial iteration the values of Dgk may be set equal to zero. The installation cost is only introduced into the algorithm after the first iteration where the location is roughly determined.The same structure as in Eq. (A3) can be used, with slight modification, to include the phase shifter. The flow of
118
E.J. de Oliveira et al. / Electrical Power and Energy Systems 21 (1999) 111–118
a line k between nodes i and j is given by: fk gk uij ⫹ ck Dgk ⫹ gk ·uij
A7
where c k is the phase angle of the phase shifter.In this paper, the phase shifter is represented by a modification at the admittance matrix B, i.e., the flow variation caused by a shift in c k is represented by the flow variation caused by Dg k. Eq. (A7) shows this equivalence. The results are similar to the ones obtained by power injections [19]. Appendix B Table 8 provides the additional data used to evaluate the series capacitor allocation and the wheeling charges for the IEEE-14 system.The FACTS cost (c) adopted in the examples presented in the paper was 135 US$/kVA [8]. Note that it will mean an annual installment of 22 thousand US$/ MVA-year if it is considered an income tax of 10% and a mortgage period of 10 years. References [1] Galiana K, Almeida M, Toussaint J, Griffin D, Atanackovic BT, Ooi D, McGrillis T. Assessment and control of the impact of FACTS devices on power system performance. IEEE Trans on PWRS 1996;11(4):1931–1936. [2] Hingorani NG. FACTS: flexible AC transmission systems. EPRI workshop. Cincinnati, OH, November 1990. [3] Gronquist JL, Sethares WA, Alvarado FL, Lasseter RH. Power oscillation damping control strategies for FACTS devices using locally measurable quantities. IEEE Trans on PWRS 1995;10(3):1598–1605. [4] Taranto GN, Chow JH. A robust frequency domain optimization technique for tuning series compensation damping controllers. IEEE Trans on PWRS 1995;10(3):1219–11225. [5] de Oliveira EJ, Oliveira JC, Moraes AJ, Guimara˜es GC. AC systems stabilization via phase shift transformer with thyristor commutation. Japan Industry Applications Society Conference JIASC-IEEE, EIME UNIV. Matsuyana (Japan), 24–26 August 1994.
[6] Taranto GN, Pinto LMV, Pereira MVF. Representation of FACTS devices in power system economic dispatch. IEEE Trans on PWRS 1992;7(2):572–576. [7] Lie TT, Deng W. Optimal flexible AC transmission systems (FACTS) devices allocation. Electrical Power and Energy Systems 1997;19(2):125–134. [8] Griffin J, Atanackovic D, Galiana FD. A study of the impact of FACTS on the secure-economic operation of power systems. 12th Power Systems Computation Conference. Dresden, August 19–23 1996:1077–1082. [9] Shirmohammadi D, Filho XV, Goresntin B, Pereira MVP. Some fundamental technical concepts about cost based transmission pricing. IEEE Trans on PWRS 1996;11(2):1002–1008. [10] Shirmohammadi D, Gribik PR, Law ETK, Malinowski JH, O’Donnel RE. Evaluation of transmission network capacity use for wheeling transactions. IEEE Trans on PWRS 1989;4:1405–1413. [11] Schweppe FC, Caraminis MC, Tabors RD, Bohn RE. Spot pricing of electricity. Boston, MA: Kluwer Academic Publishers, 1988. [12] Marangon Lima JW, Pereira MVF, Pereira JLR. An integrated framework for cost allocation in a multi-owned transmission system. IEEE Trans on PWRS 1995;10(2):971–977. [13] Pe´rez-Arriaga IJ, Rubio FJ, Puerta JF, Arceluz J, Marı´n J. Marginal pricing of transmission services: an analysis of cost recovery. IEEE Trans on PWRS 1995;10(1):546–553. [14] Jonard F, Smeers Y, Bruel P, Heilbronn, B. Wheeling costs: an economic analysis illustrated by short term and long term simulation. Stockholm Power Tech Conference, paper SPT PS 23-02-0287. Sweden, June 1995. [15] Marangon Lima JW, de Oliveira EJ. The long-term impact of transmission pricing. Paper PE-248-PWRS-0-12-1997, presented at 1998 IEEE Winter Meeting. Tampa, FL, USA. [16] Calviou MC, Dunnett RM, Plumptre PH. Charging for use of a transmission system by marginal cost methods. 11th PSCC. Avignon France, September 1993. [17] Kovacs R, Leverett A. A load flow based method for calculating embedded, incremental and marginal cost of transmission capacity. IEEE Trans on PWRS 1994;9(1):272–278. [18] Marangon Lima JW. Transmission fixed charge allocation: an overview. IEEE Trans on PWRS 1996;11(3):1409–1418. [19] de Oliveira EJ, Marangon Lima JW, de Almeida KC. Optimal allocation of FACTS devices. VI SEPOPE. Salvador, Brazil, May 1998.