transmission access and pricing, most new electricity markets have introduced an ... charge covers the costs incurred in the electricity market due to the ..... 12 READ, E.G.: 'Transmission pricing in New Zealand', Utilities Policy,. 13 BUSHNELL ...
Comparison of two methods for long-run marginal cost-based transmission use-of-system pricing A.Bakirtzis, PBiskas, A.Maissis, A.Coronides, J.Kabouris and M.Efstathiou Abstract: Two methods that provide geographically differentiated transmission usage tariffs, based on the long-run marginal cost of transmission, are presented and compared: investment cost-related pricing and the DC load flow pricing. The basic assumptions and approximations behind their design are analysed. Very efficient implementations of both methods, based on sensitivity analysis, are presented. Both methods are applied to the computation of transmission usage tariffs in the Greek power system, and the resulting tariffs are compared.
List of symbols
SB = system power base (MVA)
p.
ei
= =
E
=
e..!I
=
PDi
cy =
xij = f. = !I
0
=
P = B’
=
eij
=
net real power injection at node i (MW) (peak demand conditions) power demand at node i (MW) voltage phase angle at node i (rad) expansion constant (f/MW/km/yr) length of line ij (km) annual cost of line ij per MW (€/MWlyr) (eo = ? . CIJ) reactance of line i j @.u.) power flow of line ij (MW) voltage phase angle vector (phase angle of reference node is excluded) real power injections vector (reference node is excluded) negative of network admittance matrix: [BG]= -l/xo, [Bii]= C,l/x,, (row and column that correspond to reference node are excluded) vector with all its elements zero, except for the element that corresponds to node i, which equals +1, and the element that corresponds to nodej, which equals -1 e23
1
=
i
if 8,
2 0,
- e t J , if OZ < 0,
Introduction
The restructuring of the power industry is based on the separation of electricity generation, transmission, distribu0IEE, 200 1 IEE Proceedwigs o n h e no. 20010408 DOL 10.1W9/ip-gtd:200IONX? Paper first received 13th July 2000 and in revised form 18th January 2001 A. Bakutns and P. Biskas are with the Department of Electrical & Computer Engineering,Aristotle University of Thessaloniki,540 06 Thessaloniki,Greece A. Maissis, A. Coronides, J. Kabouris and M. Efstathiou are with the Department of Generation and Transmission Studies, Public Power Corporation, 89 Deirachiou St., 104 43 Athens, Greece IEE Proc.-Gener. Trurism. DisIrih., Vol. 148, No. 4, July 2001
tion and retailing. Electricity transmission and distribution are considered natural monopolies, whereas generation and retailing are open to competition. Open access to the transmission system and fair, cost reflective pricing of transmission services are very important for healthy competition in the power sector. To ensure fair and non-discriminatory transmission access and pricing, most new electricity markets have introduced an independent transmission system operator (TSO), who is responsible for the operation and pricing of the transmission system. Owing to the economies of scale in electricity transmission [I], it is problematic to price transmission services solely on the basis of short run marginal costs (SRMC) since collected revenues cannot cover the revenue requirements of transmission owners [ 2 4 ] .The charges for transmission services introduced in countries that have restructured their power sectors are usually separated into three components [7-IS]. (i) Connection charge: this charge covers the cost of network reinforcements required to provide service to a transmission customer. It is characterised as ‘deep’ or ‘shallow’, depending on how far from the customer site the customer’s liability extends. (ii) Transmission use-of-system charge (capacity charge): this charge compensates the transmission owner for the sunk costs of the existing transmission system assets, as well as the transmission system operating and maintenance costs. (iii) Transmission operating charge (energy charge). This charge covers the costs incurred in the electricity market due to the existence of a ‘non-perfect’ transmission system. These are the costs of transmission losses and transmission limitations (congestion). The revenues collected from energy charges are used to compensate the providers of the corresponding services (generation adjustment to cover losses, generation or demand adjustment to relieve congestion) [9, 15-18]. If based on SRMC, energy charges leave the TSO with a profit, which can be used to either reduce the transmission use-of-system charges, or compensate holders of capacity rights [16, 191. Our focus in this paper is on the transmission use-ofsystem (TUoS) charge, which constitutes the largest part of transmission service charges. There are two broad categories of TUoS charging method, 411
(a) Uniform tariff methods: these methods (also referred to as postage stamp methods) uniformly charge all transmission customers, irrespective of their location within the grid, according to a measure of their usage of the transmission network [2]; this is usually their system-peak-coincident demand (kW) or annual energy demand (kWh). The postage stamp methods define a uniform tariff (in f k W or E/kWh) for all grid users. They are simple, but fail to provide locational s ignals to the transmission customers. They are used in many countries, due to their simplicity, especially when combined with energy charges that provide the necessary locational signals to the customers. In this case, they can be theoretically justified as additive revenue reconciliation adjustment to SRMC-based transmission prices [24]. (b) Locational tariff methods: these methods differentiate the TUoS tariff according to the customers’ location within the grid. Usually, they separate the entire network into different zones of charge, with a uniform tariff within each zone. The intention behind their design is to send long-term signals for the positioning of new generators and new loads in the grid, so as to avoid network reinforcements as far as possible. In this paper we present two locational TUoS tariff methods based on the long-run marginal cost (LRMC) of electricity transport: investment cost-related pricing (ICRP) and DC load flow Firicing (DCLFP). In fact, both pricing methods are investment cost-related, but we reserve the name ICRP for the transport model method originally developed by the EIritish National Grid Company [20]. Both methods lead to zonal TUoS tariffs. The assumptions behind their design and the design methodology are analysed. Fast LRMC computations by both methods, based on efficient seiisitivity analysis, are presented. Transmission tariffs prodiiced by both methods for the Greek transmission system are presented and compared. 2 Transmission pricing using long-run marginal cost
The calculation of the LRMC of transmission is related to the construction of an optimum transmission network. The optimum network is the minimum cost network that serves the users under specified reliability standards. The LRMC of transporling electric power to (from) a user is the reinforcement cost of the optimum network, required for servicing a unit increase in the demand (production) of the particular user. The LRMC of transmission is calculated by sensitivity analysis around the solution of the problem of constructing the optimum transmission network. The precise calculation of the LRMC of transmission is very difficult and must be based on a variety of assumptions about the future. An approximate calculation of the LRMC of transmission is usually performed in practice, based on several simplifying assumptions concerning the optimum network. ‘Two methods for the approximate calculation of LRMC of transmission, are analysed below: ICRP and DCLFP.
2.I Investment cost-related pricing 2.I. I Basic assumptions: The following simplifying assumptions, used for the construction of the optimum network by the ICRP method, were developed by the British National Grid Company (NGC) [20]. (i) The optimum network is constructed using only the existing routes of the current network. 478
(ii) The optimum network is constructed assuming peak demand conditions. The system load is allocated to the generators in proportion to their registered capacity (pro rata), with no regard to economic criteria. The underlying assumption is that maximum transmission system stress occurs under peak demand conditions, which is not always correct [2]. (iii) All the lines of the optimum network are of the same type. Their cost is proportional to their length. The length of the cables is scaled up, to reflect their higher price relative to that of the overhead lines. (iv) The transmission capacity of the lines of the optimum network is taken to be precisely equal to the power transported from generators to loads at peak demand conditions. The construction of the optimum network does not take into account the required network redundancy ( N - 1 or hgher) and ignores the fact that network expansions take place in discrete quantities. (v) When constructing the optimum network, it is assumed that electric power can be routed at will on the existing routes. The flow of electric power in the optimum network conforms to the real power balance equations, but ignores Kirchhoff s voltage law. Consequently, many routes of the existing network become useless, and the optimum network is very sparse; in fact, it is radial. The shorter routes are being used, whereas the longer routes are not used and do not contribute to the optimum network construction cost.
2.1.2 Construction of optimum network: The optimum network is constructed by solving the optimisation problem: [MW.km] 23
S.t.
Cft3 = Pt,
for every node i ( l b )
3
The annualised network construction cost is subsequently computed by multiplying the total MW.km with the network expansion constant C (average annual transmission cost in fIMWIkmlyr.). The optimisation problem (eqn. 1) is easily transformed into a linear programming (LP) problem of the linear network flow type. In our implementation of ICRP, the linear network flow problem is efficiently solved using the RELAX code [21].
2.1.3 Computation of LRMC of transmission: The LRMC of transmission is computed by sensitivity analysis around the optimal solution of the problem in eqn. 1. According to optimisation theory [2], the LRMC of transporting electric power to node i is the Lagrange multiplier A,, associated with the power balance equation at node i (eqn. 16); this gives the sensitivity of the optimum network construction cost with respect to the power demand at node i:
The LRMC of power transport to node i is calculated multiplying Aiwith the expansion constant c: L R M C , = E . A,
[e/MW/yr]
(3)
2.1.4 Physical interpretation of LRMC of transportation: reference node: Since the optimum network is radial, there is a unique route from a given node to another. A generation node is arbitrarily chosen as a reference node. I E E Proc.-Genrr. Transm. Disfrib., Vol. 148. No. 4. July 2001
The LRMC of transporting electric power to node i (a,) expresses the total MW.km of reinforcement of the optimum network required to serve a 1MW increase in the demand of node i. it is implied that the demand increase at node i will be supplied by an equal increase of the reference node generation. The required reinforcement of the optimum network is the addition of 1 MW of transport capacity to all the lines along the unique path from the reference node to node i. Therefore, d,is the distance (in km) from the reference node to node i, traversing the optimum network. The concept of distance is used for the efficient computation of the nodal LRMCs.
s.t. B' . e = -S1pB To solve the optimisation problem in eqn. 5, the DC load flow equations (eqn. 5b) are first solved for 6, and the resulting vector 8 is then substituted in eqn. 5a, to compute the total M W . h of the optimum network. The annualised cost of the optimum network can then be computed by multiplying the total MW.km with the expansion constant -
c.
2.2 DC load flow pricing
2.2.7 Basic assumptions: The following simplifying assumptions are used for the construction of the optimum network by the DCLFP approach. (i) The optimum network is identical to the existing network as far as the topology and the electrical characteristics of transmission lines are concerned. (ii) The optimum network is constructed assuming peak demand conditions. The system peak load is allocated to generators either pro rutu or based on economic criteria. (iii) The cost of the optimum network is computed based on either of the following: ( U ) the equipment cost of the optimum network is the replacement cost of the equipment of the existing network. (b) all the lines of the optimum network are of the same type; their cost is proportional to their length; the length of the cables is scaled up, to reflect their higher price relative to that of the overhead lines. (iv) The transmission capacity of the lines of the optimum network is taken to be precisely equal to the power transported from generators to loads at peak demand conditions. The construction of the optimum network does not take into account the required network redundancy ( N - 1 or higher) and ignores the fact that network expansions take place in discrete quantities. (v) The optimum network satisfies Kirchhoffs laws. DC load flow simplifications are assumed.
2.2.2 Construction of optimum network: The optimum network is constructed by solving the optimisation problem (in the following analysis it is assumed that all lines are of the same type):
2.2.3 Computation of LRMC of transmission: The LRMC of transmission is computed by sensitivity analysis around the optimal solution of the problem in eqn. 5. However, since the optimisation problem in eqn. 5 is trivial (it has no degrees of freedom), the transportation LRMC can be computed by a sensitivity analysis around the solution of the base-case load flow. The LRMC of power transportation at node i of the network represents the change in construction cost of the optimum network (in €'yr or MW.km) required for servicing an incremental change of 1MW in the demand at node i. it is considered that an increase in demand at node i is covered by an equal increase in the reference node generation. The calculation of the LRMC of power transport to node i involves the solution of an additional DC load flow problem (change-case load flow). This case differs from the base-case load flow, in that the generated power at the reference node and the power demand at node i are increased by 1 MW. The LRMC of node i is computed as the difference between the MW.km of the change-case and the basecase load flows. The solution of N change-case load flows is required for the computation of the LRMCs of all nodes of the network. As a consequence, the computation of all LRMCs requires N forward-back substitutions with the table of factors of B'. Using sensitivity analysis around the solution of the basecase load flow, the LRMCs of all nodes of the system can be calculated more efficiently, requiring only one fonvardback substitution. Differentiating eqn. 5u with respect to the power demand vector, Po we get:
Using eqn. 5b
S.t.
1 -(Oz 223
Pi
- 0,) = - for every node i
SB
(4b)
The constraints of the optimisation problem in eqn. 4 have no degrees of freedom. Therefore, the solution of the optimisation problem coincides with the solution of a DC load flow problem. Owing to the linear dependence of the power flow equations (eqn. 4b), a generation node is selected as the reference node and its voltage phase angle is arbitrarily set to zero. The power balance equation of the reference node is omitted when writing the load flow equations (eqn. 4b). The problem in eqn. 4 can be written using vector notation: IEE Proc.-Gener. Tninsm. Df.s.trib., Vol. 148. No. 4, July 2001
Using eqn. 6, the LRMCs of all nodes A; are computed, except for the LRMC of the reference node, which equals zero. The calculation involves only one forward-back substitution of the vector inside the large parentheses in eqn. 6 with the triangular factors of matrix B'. The nodal LRMCs in [€/MW/yr] are computed using eqn. 3.
2.3 Properties of LRMC of transmission The nodal LRMCs computed by both methods have the following properties. (U) The LRMC of the reference node is zero. 419
(6) Changing the choice of reference node results in a constant shift of all nodal LRMCs, thus preserving the geographical differentiation of nodal LRMCs. Additive adjustment of the nodal LRMCs to satisfy the transmission owner’s revenue requirements results in unique transmission tariffs, indepen’dentof the reference node selection (as discussed later). (c) The LRMCs for generation and for demand at the same node are equal and opposite. 2.4 Computation of zonal TUoS tariffs Charging every node in the system according to its LRMC would be complex to implement in practice. For simplicity, the system is separated into zones of TUoS charge. Each zone of charge is a collection of neighbouring nodes with similar LRMCs. Consumers within a zone are charged according to their peak coincident demand. Producers are charged according to their registered capacity. The tariff within a zone is computed as the weighted average of the LRMCs of the nodes within the zone, with the weights being the corresponding nodal generator registered capacities or consumer peak coincident demands. If the above zonal tariffs were applied to all generators and loads, the revenues collected would represent only a small fraction (e.g. 20Y0) of the transmission system sunk costs. This is because during the optimum network construction the network reliability (which increases considerably the network construction cost) was neglected. Therefore, if TUoS tariffs are solely based on LRMCs, the revenues collected fi-om TUoS charges are not sufficient to cover the transmission owner’s revenue requirements. An additive adjustment of the LRMC-based TUoS tariffs is required in order to cover the revenue requirements. The zonal transmission tariffs are increased by a uniform amount (calculated separately-for generators and loads). The different uplift applied to the zonal transmission tariffs for producers and consumers gives an additional degree of freedom, so that the allocation of the revenue requirements to producers and consumers can be arbifkily selected. Imposing the revenue requirement constraints also results in the derivation ol‘ unique transmission tariffs, independent of the initial refkrence node selection. 3 Calculation oif transmission use-of-system tariffs for Greek power system
In this Section, we present and compare the TUoS tanffs calculated by the ICRP and the DCLFP methods for the Greek power systern. We also summarise the main findings of a study undertaken by the Public Power Corporation for the design of cost-reflective TUoS tariffs. The Greek power system is characterised by bulk power transfers from the lignite-rich North to the South, where the densely populated Athens Metropolitan area is located. Several critical operating conditions, mostly related to voltage instability, may lead to North-South transport limitations, with considerable generation rescheduling costs. In order to send the appropriate locational signals to the transmission users, a LRMC based locational TUoS tariff was designed. For this purpose, the Greek power system was separated into six TUoS charge zones, after careful study of the system geography and the nodal LRMCs [22]. Fig. 1 shows the registered capacity and the peak coincident demand w i t h each zone, forecast for year 2001. Total registered generating capacity is 9 61 3 MW. and peak demand is 7819M’W. After examining the tariffs for different levels of allocation, a 500/0-50% allocation of the 480
charges between generators and loads (G-L) was selected and all results presented in this Section are based on this G-L allocation [22].
4 D
r 0
x 3
z
2 1
n Athens
Central
South West North-East carpacity andpeuk couzcldeMt denwd u2 d$’rent
North
:ones of
13 registered cdpdcity W peak coincident demand
The transmission system revenue requirements for the year 2001 were projected to be fl80Miyr. The system expansion constant (average annual p.u. line cost) was calculated to be f34.2lMWlkmlyr. Assuming 50%G-50%L allocation, the postage stamp tariff would be €9 362iMWl yr for generators and €1 15lliMWlyr for the loads. The network model used to compute the nodal LRMCs consisted of 832 nodes and 1049 branches. Hydro plant Plastiras, in Central Greece was selected as the reference node. Both NGC’s [20] and our RELAX-based ICRP code produced the same nodal LRMCs, but our code executed 45 times faster (134s compared to 3s on a P-11-350MHz PC). The DCLFP approach produced slightly different results when based on assumptions similar to ICRP (pro vutu dispatch and uniform line type). The execution time of the DCLFP approach was 2s on a P-11-350MHz PC. Figs. 2 and 3 give the zonal TUoS tariffs for generators and loads calculated by ICRP and DCLFP for 500/oG5O%L allocation. The zone sequence in all the Figures is such that generator tariffs (Fig. 2) are in ascending order. The following conclusions can be drawn from the results. (i) Both the ICRP and DCLFP methods lead to heavy charges for generators located in the North and consumers in the metropolitan area of Athens. These charges provide economic incentives for the location of new generators in the South and of new loads in the North. (ii) In most of the cases, the ICRP method leads to greater differentiation of charges between zones. This is a major feature of the ICRP method; in some cases, this is desirable as it produces stronger economic signals. (iii) Both the ICRP and DCLFP methods provide the same ranking of the zones with respect to generation tariffs (Fig. 2). The South and West systems are interchanged in the ranking of the zones with respect to load tariffs (Fig. 3). Differentiation of the charges calculated by the ICRP and DCLFP methods is mainly attributed to the following reasons: (U) The Greek transmission system consists of two parallel networks of 400kV and 150kV voltage levels. The absence of a 400kV network in the North-East leads to lower marginal prices for those areas when calculated by the ICRP method (Fig. 3). On the contrary, for areas where a 400kV network dominates (i.e. the North), the marginal prices calculated by both methods are sirmlar. (b) The Western part of the country is an area with high production and low load (exporting area). Despite this, IEE P i o c -Gener T r u n m Dtrrrrb
Vol 148, N o 4, July 2001
the load charging (especially by DCLFP) is quite high (Fig. 3). This is because the load, although not high, is dispersed over a wide area through long transmission lines. *O1-
P
$ 10 W
:5
Athens
Central
South
West
North-East North und DCLFP
Fig.2 Zonul trcuzsniisxun tLirfls for generutors by ICRP
upproid (5iMG-jO‘YL.ullocution) 0 ICRP
4
Conclusions
Two locational transmission use-of-system tariff methods, based on long run marginal cost of transmission, have been presented and compared: investment cost related pricing and DC load flow pricing. Transmission usage tariffs have been computed using both methods for the Greek power system. These tariffs provide adequate incentive to increase generation in the vicinity of high consumption via a tariff diversification of 5:l between high and low tariff zones for generators. Reasonable incentive for loads to move near the production areas is also given by a 1:3 tariff diversification for load charging zones. The designed TUoS tariffs are capacity tariffs and charge the users according to their peak demand usage (in kW). Work is under way for the development of energy usage tariffs to account for losses and congestion in the transmission network, as well as connection tariffs. 5
DCLFP
References
1 DISMUKES, D.E., COPE, R.F., and MESYANZHINOV, D.: Capacity and economies of scale in electric power transmission’, Utilities Policy, 1998, 7, pp. 155-162 2 EPRl TR-105121: ‘Transmission services costing framework’. Electric Power Research Institute, 3412 Hillview Ave., Palo Alto, CA, 94303, 1995,l & 2 3 CARAMANIS, M.C., BOHN, R.E., and SCHWEPPE, F.C.: ‘The costs of wheeling and optimal wheeling rates’, IEEE Truns., Power Syst., 1986, 1, (l), pp. 63-73 4 RUDNICK, H., PALMA. R., and FERNANDEZ, J.E.: ‘Marginal pricing and supplement cost allocation in transmission open access’, IEEE Truns., Power Syst., 1995, 10, (2), pp. 1125-1 142 5 FARMER, E.D., PERERA, B.L.P.P., and CORY, B.J.: ‘Optimal pricing of transmission services application to large power-systems’, IEE Proc., Cener. Trunsm. Distrib., 1995, 142, (3), pp. 263-268 6 PERERA, B.L.P.P., FARMER, E.D., and CORY, B.J.: ‘Revenue reconciled-optimum pricing of transmission services’, IEEE Truns., Power Syst., 1996, 11, (3), pp. 1419-1426 7 SHIRMOHAMMADI, D., RAJAGOPALAN, C., ALWARD, E.R., and THOMAS, C.L.: ‘Cost of transmission transactions: an introduction’, IEEE Truns., Power Syst., 1991, 6, (4), pp. 1546-1560 8 GREEN, R.: ‘Electricity transmission pricing: an intemational comparison’, Utilities Policy, 1997, 6, (3), pp. 177-184 9 GREEN, R.: ‘Transmission pricing in England and Wales’, Utilities Policy,, 1997, 6, (3), pp. 185-194 10 RUDNICK, H., and RAINERI, R.: ‘Transmission pricing practices in South America’, Utilities Policy, 1997, 6, (3), pp. 21 1-218 1I BRATON, J.: ‘Transmission pricing in Norway’, Utilities Policy, 1997, 6. (3). DD. 219-226 12 READ, E.G.: ‘Transmission pricing in New Zealand‘, Utilities Policy, 1997, 6, (3), pp. 227-236 13 BUSHNELL, J., and OREN, S.: ‘Transmission pricing in California’s proposed electricity market’, UrilifiesPolicy, 1997, 6, (3), pp. 237-244 14 Transmission License Condition 10. ‘Statements of Charges for Use of the Transmission System and Connection to the Transmission System for the year 1997198’. The National Grid Company plc, UK, 1997 I5 ‘Transmission tariffs. Computational tools and practical application’. SINTEF Energy Research, 1998 16 SlNGH, H., HAO, S.Y., and PAPALEXOPOULOS, A.: ‘Transmission congestion management in competitive electricity markets’, IEEE Truns., Power Syst., 1998, 13, (2), pp, 672-680 17 SHIRMOHAMMADI, D., WOLLENBERG, B., VOJDANT, A., SANDRN, P., PEREIRA, M., RAHIMI, F., SCHNEIDER, T., and STOTT, B.: ‘Transmission dispatch and congestion management in the emerging energy market structures’, IEEE Truns., Power Syst., 1998, 13, (4), pp. 14661474 18 GRIBIK, P.R., ANGELIDIS, G.A., and KOVACS, R.R.: ‘Transmission access and pricing with multiple separate fonvard markets’, IEEE Truns., Power Syst., 1999, 14, (3), pp. 865-876 19 HOGAN, W.W.: ‘Contract networks for electric power transmission’, J. Reguhfory Econ., 1992,4, (3), pp. 211-242 20 FALVIOU, M.C., DUNNETT, R.M., and PLUMPTRE, P.H.: Charging for use of a transmission system by marginal cost methods’. Proc. Eleventh Power Systems Computation Conference, Avignon, France, August 30 - September 4 1993, pp. 385-391 21 BERTSEKAS, D.P., and TSENG, P.: ‘RELAX: a computer code for minimum cost network flow programs’, Ann. Oper. Res., 1988, 13, pp. 127-190 22 KORONIDES, A., KABOURIS, J., EFSTATHIOU, S., MAISSIS, A., and BAKIRTZIS, A.: ‘Comparative analysis of transmission charging policies for the Greek electric power system’. Cigre, 38th Session, Paris, France, 27 August - I September 2000 -
Athens
Central
South
West
North-East
North
Fi . 3 Zonul trun~mlssionturgs for lods by ICRP curd DCLFP upprouch
(j8vd-joxLulionrtion)
0 ICRP W DCLFP
The DCLFP method was finally selected for the TUoS tariff calculation in the Greek power system, since it provides more realistic power flows. The disadvantage of the method is that it requires more input information than the ICRP method. Of particular interest to market participants engaged in bilateral contracts is the TuoS-related cost per transported MWh. Table 1 gives the TUoS charge per MWh of a bilateral contract from one zone to another. A 80% load factor of the contract is assumed. The contract TUoS charge ranges from a low of €l/MWh for Athens-to-North transport to a high of €3.6/MWh for North-to-Athens transport. Table 1: Bilateral contract transmission charge in E per MWh (80% loadfactor) ~-
-
From generator North- North Central Athens South West east Toload
Northeast
2.4
2.8
1.7
1.5
1.9
2.2
North
1.9
2.3
1.2
1.0
1.4
1.7
Central
2.7
3.1
2.0
1.8
2.2
2.5
Athens
3.2
3.6
2.4
2.2
2.7
3.0
South
2.8
3.2
2.0
1.8
2.3
2.6
West
2.9
3.3
2.2
2.0
2.5
2.7
IEE Prm-Geiier. Trrmsni. Distrih., Vol. 148, No. 4, July 2001
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