REPRESENTATION OF CHRONOLOGICAL HYDROTHERMAL OPERATION IN ANALYTICAL PRODUCTION COSTING

M.E.P .Maceira CEPEL

M . V.F .Pereira PSRI

Rio de Janeiro, Brazil

Abstract - This paper describes an extension of the Baleriaux methodology, able to handle the chronological aspects of reservoir operation. It is shown that the chronological hydrothermal production costing can be decomposed into a sequence of stochastic network flow problems with a special structure. These network problems are further decomposed into a sequence of analytical convolutions of reservoir storage levels, demand, inflows and generation capacity at each stage. The algorithm is very fast and easy to implement. The methodology is illustrated in a case study with a utility-derived system.

Keywords - analytical production costing, hydrothermal scheduling, chronological simulation

1. INTRODUCTION

Many power system planning and operation.problems can be represented as stochastic linear programming problems:

where:

z = Min ex subject to Ax';Z:b

z optimal objective function value c (row) vector of unit costs x vector of decision variables A coefficient matrix b vector of random variables (resource vector)

(1)

The objective is to determine the expected value of z over all possible outcomes of b:

where:

E(z) = L z(b) P(b) beB

(2)

z(b) optimal value ofLP problem (1) for outcome b of the random resource vector

P(b) probability of outcome b B set of possible values of vector b

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• For example, the need to represent uncertainties of future operating conditions in power system planning and operations is widely recognized by the utility industry. Probabilistic production costing (PPC) models, which evaluate the expected unit operating costs along the planning period, taking into account equipment outages and load variation, are especially relevant for utility studies. The probabilistic production costing (PPC) of a thermal generating system can be represented as a LP stochastic problem similar to ( 1):

where:

z= J

Min L c ·g· /=I J J

subject to J

:L gj';Z:d ; =1

gr~;;gj

forj = 1, ... , J

j indexes the generating units J number of units cj unit operation cost of the /h unit g · generation of the jth unit J system demand (random variable)

(3.1)

(3 .2)

(3 .3)

gj maximum generation capacity of the /h unit

(random variable)

The objective function (3 .1) is to mm1m1ze total generation cost. Constraint (3 .2) represents the system load supply. The set of constraints (3 .3) represents the generation limit of each unit. The PPC problem is to calculate the expected generation cost, E(z), taking into account load fluctuations and random equipment outages.

2. ANALYTICAL SOLUTION OF THE PPC

Stochastic LP problems are in general quite difficult to solve. However, the PPC problem (3) can be efficiently solved by a method proposed by Baleriaux [1,2] . The Baleriaux scheme decomposes the production costing

problem into J generation reliability evaluation subproblems. In turn, the reliability problems reduce to the comparison between the distribution of available capacity and system demand, which is carried out by discrete convolution. The Baleriaux scheme is thus accurate and very fast, especially when series expansion techniques are used to carry out the required convolutions [3-5].

The Baleriaux scheme is widely used by utilities in their production costing and expansion planning studies. It is also implemented in several popular computer packages such as PROMOD, w ASP and EGEAS.

One major limitation of the Baleriaux approach is related to the loss of chronological information. In order to carry out the analytical convolutions, it is assumed that the system states are independent of each other, i.e. system operation is represented as "snapshots". As a consequence, the operation of hydroelectric units, which is essentially chronological, has to be simplified. Hydro units are currently , represented as dummy thermal plants which are moved up in the loading order until their expected generation matches the preassigned hydro energy [6-7]. While this representation may be reasonably adequate for thermal­dominated systems, it is less acceptable for systems with a large component of hydro generation. This difficulty has been recognized by several utilities and international agencies such as the World Bank, IDB and IAEA.

This report describes an extension of the Baleriaux methodology, able to handle chronological aspects such as simulation of reservoir operation. It is shown that the Baleriaux scheme can be applied to stochastic optimization problems with a network flow structure. The problem is then decomposed into a sequence of analytical convolutions, proportional to the number of stages in the chronological simulation. Case studies with utility-derived systems are presented and discussed.

3. OVERVIEW OF THE BALERIAUX SCHEME

The Baleriaux algorithm is summarized in the following steps:

1. Assume that the generating units are ranked by increasing operating cost. Remove all units from problem (3), except the cheapest (g1, in this case), and solve the following stochastic subproblem:

w, =Min s subject to

s+ g 1 '?:.d (4)

2.

3.

4.

5.

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where s is a scalar variable, representing the load curtailment in the system due to insufficient generating capacity.

Problem (4) is a generation reliability evaluation problem, for which efficient solution algorithms are available [8] . In this particular case, the problem is easily solved by setting:

(5)

that is, the available generation is used up to its maximum capacity or up to the demand value. The expected value of w1 is calculated by the convolution of the probability distributions of d andg1 [8].

Next, we calculate the contribution of generating unit 1:

(6)

where E(d) is the mean system load. The

difference ~ 1 represents the expected generation

of unit 1 in the reliability problem ( 4 ).

The Baleriaux approach is based on the fact that ~ 1 is also the expected generation of unit 1 in the solution of the original PPC problem (3) . The intuitive reason is that unit 1, being the cheapest, is used at its maximum capacity to meet the demand of problem (3), before the more expensive units are loaded.

The above procedure is repeated for each unit in the system. The reliability evaluation problem, after the addition of the nth unit, becomes:

wn =Min s subject to

forj = 1, ... , n

(7)

The solution of problem (7) is similar to that of problem (4):

6.

n wn =Max { 0, d - L gj }

j=l (8)

As in (5), the expected value of wn, E(wn), is analytically obtained by convolution of the load d

and of then generation capacities {gj}· Note that E(wn) can be recursively obtained from the

convolutions carried out for unit wn-l• adding unit n.

The same reasoning applied . to the first unit shows us that the difference between E(wn) and E(wn_1) corresponds to the mean generation of the nlli unit in the PPC problem (3):

(9)

The expected value of the total generation cost in the PPC problem (3) is given by:

J E(z) = Lc· ll.·

j=I J J (10)

In summary, the Baleriaux approach decomposes the PPC problem (3) into J subproblems of calculating the mean load curtailment, which can be easily solved by analytical methods. As mentioned previously, this approach is accurate and computationally efficient, especially if techniques based on series expansion are used to carry out the convolutions.

4. EXTENSION OF THE BALERIAUX APPROACH

It is shown in (9] that Baleriaux's idea can be generalized for stochastic problems which can be represented as linear network flows. A network is composed of a set of nodes and a set of branches, which interconnect pairs of nodes. There are two classes of constraints in the network flow problem:

• flow conservation at each node • limits on the transportation capacity of each

branch

Figure 1 illustrates a network with three nodes and three branches.

Figure 1 • Network Model

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An important probabilistic power system problem which can be modeled in this way is chronological hydrothermal (HT) production costing, discussed next.

5. CHRONOLOGICAL HT PRODUCTION COSTING

The objective of hydrothermal dispatch is to minimize the operation cost of a system composed of J thermoelectric units and one hydroelectric plant:

where:

T J

Min L L cjgtj t=I j=l

subject to Vt+l = Vt + at - Ut

gtj $; gtj J

p Ut + L gtj ~ dt j=1

t indexes the stages T number of stages ct generation cost

(11)

gtj generation of thermal plant} in stage t gtj generation capacity ofplantj (random variable) vt stored volume in the hydro plant at the start of

stage t v maximum storage capacity of the hydro plant at inflow volume during stage t (random variable) ut turbined outflow during stage t Ut maximum outflow capacity of the hydro plant

(random variable) p hydro production coefficient put energy generated by the hydro plant in stage t dt energy demand in stage t (random variable) 01 stored volume at the beginning of the first stage OT+l target storage at the end of the last stage

The random variables in this problem are the loads {dt}, inflows {at}, thermal generation capacities {gtj} and turbine capacities {ut}· We assume that those variables are statistically independent at each stage.

Problem (11) can be represented as a network flow: the nodes are associated to water balance and load balance equations, and the arcs, to the storage volume, outflow, and thermal generation. Figure 2 illustrates a hydrothermal scheduling problem with three stages, one

thermal unit, and hydro production coefficient p equal to 1.

01 02 03 Figure 2 - Network Flow Representation of

Hydro-Thermal Dispatch

6. EXTENDED BALERIAUX SCHEME

6.1 Calculation of Expected Production Costs

Because hydro generation has no direct cost, it is added first in the extended Baleriaux scheme. The reliability problem, after the addition of the n cheapest thermal units, is formulated as:

T wn = Min LSt

t=I

subject to (12)

Vt+l = Vt + at - Ut

vt+l ~ v Ut ~ U

I\ Vl = Vl

I\

VT+l = VT+l

gtj ~gtj n

p Ut + L gtj + St ~ dt ;=1

The algorithm is implemented in the following steps.

1. assuming that the n cheapest thermal plants have been added, calculate the probability .distribution of the available energy, 'I' 1, at the first stage:

n '1' 1 =01 p + a 1 p + Min{0,Lg1j - d1} (13)

;=1

2. The probability distribution of 'I' 1 can be obtained by the convolution of the random variables in (13). Negative values indicate load curtailment. Positive values indicate excess energy, which can be transferred to the next time stage. The expected load curtailment in the first stage is given by:

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3.

4.

5.

6.

7.

The probability distribution of the energy arriving at the next stage is given by:

'1' 1 =Min [v p, Max { o, '1'1}] (15)

This distribution is obtained by truncating 'I' 1 at zero - indicating that the reservoir is empty, and that there is no energy available for the next stage - and at v p, indicating that the reservoir is full, and that any energy beyond that must be spilled.

A similar procedure is applied to each time stage:

(a) The energy available at stage t is given by:

- n '1'1 = '1'1_1 +at p + Min{O, :Lg1j - d1 (16)

j=1

(b) The expected load curtailment at stage t is given by:

(c) The energy available to be transferred to the next stage is:

'l't =Min [ v p, Max { 0, 'l't}1 (18)

The total expected unserved energy is the sum of the load curtailments along the stages:

T E(wn) = L E(st)

t=l (19)

As in the purely thermal PPC, the difference between E(wn) and E(wn- ) corresponds to the mean generation of the n(h unit in the hydro­thermal production problem:

(20)

The calculation of the mean operation cost in the hydro-thermal problem (11) is also similar to the purely thermal PPC:

J E(z) = L c· !l·

j=l J 1 (21)

The above procedure transforms the stochastic hydrothermal optimization problem into T problems of convolutions of random variables.

6.2 Calculation of Marginal Costs

The extended Baleriaux scheme can also produce marginal cost information, i.e. the variation of system operation costs with respect to incremental variations in generation capacity and load level. This information is useful for planners, and can also be used as part of an automated expansion planning procedure, based on decomposition [10). The expected total system operating cost is given by:

J

E(z) = L cj dj + cd E(wJ) ;=1

(22)

where cd is the cost ofunserved energy. Substituting (6) and (20) into (22), and rearranging terms, we obtain:

J

E(z) = L (er cj.) E(wj_1) + (cd- cJ) E(wJ) (23) ;=1

We take c0 as zero and E(w0) as E(Gh). Taking the derivative of (23) with respect to the demand d, we have:

J

8E(z)/ad= L (er cj_1) LOLPj.1 + (cd- cJ) LOLP1 (24) ;=1

where LOLI) is the risk of deficit in any of the stages after the addition of the 1111 thermal unit. For LOLP0 we take the risk of deficit after the addition of the hydro unit. The risk in each stage corresponds to the probability of having a negative energy available at the stage. Note that, in order to avoid double counting, those negative states are not carried over to the next stage in the convolution.

Using a similar reasoning, we calculate the marginal costs with respect to generation capacities as:

J

ngk = . L (er cj-1) LOLPj-1 + (cd- CJ) LOLP1 (25) ;=k+l

fork= 1, ... , J.

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6.3 Representation of Terminal Conditions

The algorithm allows the representation of a target storage at the end of the study period. It is only necessary to add an additional stage with a "demand" equal to the target storage. This "demand" can be "supplied" either by the remaining volume at the last stage, or by a "thermal" unit with very high operating cost. Figure 3 illustrates the approach.

AT-1

DT-1

AT

DT

"Thermal" Plant with High Cost

Figure 3 - Representation of Target Storage

A similar approach can be used to represent future-cost functions at the end of study period.

6.4 Limitations of the Approach

The efficiency of the extended Baleriaux approach is due to the fact that the chronological hydrothermal scheduling problems for each scenario are not solved explicitly. For that reason, it is not possible to calculate the individual reservoir trajectories, or the turbined outflows.

7. CASE STUDIES

7.1 Test System

The application of the methodology will be illustrated with a utility-derived generation system, composed of a 600-MW hydro plant and 15 thermal plants, totaling 584 MW. Table 1 presents the equipment data : plant number, installed capacity, failure probability, and unit operation cost.

Plant

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15

Table 1 Equipment Data

Cap. Prob. of (MW) Failure

60 0.10 10 0.09 4 0.08

10 0.07 10 0.06 15 0.05 32 0.04

130 0.10 125 0.09 60 0.08 11 0.10 12 0.09 24 0.10 30 0.10 51 0.10

Op.Cost ($/MWh)

0.0 0.0 0.0 0.0 0.0 0.0 0.0

24.5 24.5 24.5 27.6 27.6 28.7 32.0 41.2

The hydro plant data (maximum storage, production coefficient and maximum outflow) are shown in Table 2.

Table 2 Hydro Plant Data

Hvdro Plant Data Value maximum storage 100Hm3

production coefficient 5 MW/m3/s maximum outflow 120 m3/s

Inflows in each stage were represented by a three­parameter lognormal distribution. Table 3 shows the mean, standard deviation, and skewness coefficients.

Table 3 - Hydrological Data

Month Mean Std.Dev. Skewn. {m3/s) (m3/s) Coef.

1 65 5 0.05 2 73 6 0.05 3 188 12 0.08 4 97 8 0.06 5 151 10 0.07 6 146 10 0.07 7 121 9 0.06 8 141 10 0.06 9 109 8 0.05

10 169 11 0.07 11 68 5 0.04 12 65 5 0.04

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Demand variation in each stage was represented by three equiprobable load levels, shown in Table 4.

Table 4 - Load Data

Month Load# 1 Load# 2 Load# 3 (MW) (MW) (MWJ

1 592 830 901 2 592 826 897 3 592 830 901 4 605 842 917 5 605 847 917 6 600 838 913 7 592 830 905 8 588 821 892 9 592 826 897

10 609 851 926 11 617 863 938 12 638 892 972

7.2 Calculation of Expected Operation Cost

Each convolution step of the Baleriaux procedure can be applied either to individual plants or to groups of plants with the same operating cost. The thermal plants were then aggregated into 6 cost classes, as shown below.

Table 5 - Aggregate Cost Classes

Class Op. Cost Inst. Cap. ($/MWh) (MW)

l 0.0 141 2 24.5 315 3 27.6 23 4 28.7 24 5 32.0 30 6 41.2 51

The probability distribution of generating capacity for each thermal class was calculated by discrete convolution.

The extended Baleriaux procedure was then applied to the system. Table 6 shows the mean load curtailment, E(wn), after the addition of the hydro plant and of each thermal class. The expected plant generation is the difference between ex-pected load curtailments before and after the addition of each plant. The expected operation cost is the product of fuel cost and expected generation.

Table6 Extended Baleriaux Scheme

It Mean Load Mean Mean Op. # Curtailed Plant Cost (106$)

Gener. - 9431.33 - -H 3004.25 6427.09 0.0 1 1780.16 1224.09 0.0 2 147.61 1632.54 29.20 3 107.62 40.00 0.81 4 74.31 33.30 0.70 5 44.56 29.75 0.69 6 16.86 27.70 0.83

7.3 Calculation of Marginal Costs

Table 7 presents the expected marginal costs for the demand and for each cost class, calculated through the procedure of Section 6.2.

Table 7 Expected Marginal Cost

Sytem Marginal Cost Parameters ($/MWh)

demand 274.51 Hvdro 274.51 Class 1 274.51 Class 2 250.00 Class 3 247.37 Class 4 246.54 Class 5 244.41 Class 6 240.00

8. CONCLUSIONS

This paper described an extension of the Baleriaux methodology, able to handle chronological aspects such as simulation of reservoir operation. It was shown that the Baleriaux scheme can be applied to stochastic optimization problems with a network flow structure. The problem was then decomposed into a sequence of analytical convolutions, proportional to the number of stages in the chronological simulation. The application of the algorithm was illustrated for a utility-derived system.

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9. REFERENCES

1) H.Baleriaux,E.Jamoulle,F.Guertechin,"Simulation de L'exploitation d'un pare de machines thermiques de production d'electricite couple a des stations de pompage", Revue E, Vol. 5, No. 7, pp. 225-245, 1967.

2) RR.Booth, "Power System Simulation Model Based on Probability Analysis", IEEE Transactions on Power Apparatus and Systems, Vol. PAS-91, pp.62-69, 1972.

3) N.S.Rau, P.Toy, K.F.Schenk, "Expected Energy Production Costs by The Method of Moments", IEEE Transactions on PAS, Vol. PAS-99, No. 5, pp. 1908-1917, 1980.

4) J.P.Stremel,R.T.Jenkins, RA.Babb, W.D.Bayless, "Production Costing Using the Cumulant Method of Representing the Equivalent Load Curve", IEEE Transactions on PAS, Vol. PAS-99, No. 5, pp. 1947-1956, 1980.

5) G.Gross, N.V.Garapic, B.McNutt, "The mixture of Normals Approximation Techniques for Equivalent Load Duration Curves", IEEE Transactions on Power Systems, Vol. PWRS-3, No. 2, pp. 368-374, 1988.

6) J.A.Bloom,L.Charny, "Long Range Generation Planning with Limited Energy and Storage Plants, Part I: Production Costing", IEEE Transactions on Power Apparatus and Systems, Vol. PAS-102, No. 9, pp. 2861-2870, 1983.

7) B.Manhire, "Probabilistic Simulation of Multiple Energy Storage Devices for Production Cost Calculations", Electric Power Research Institute Project Report EA-1411, May 1980.

8) R.Billinton, RN.Allan, Reliability Evaluation of Power Systems, Pitman Advanced Publishing Program, New York, 1984.

9) M.E.P.Maceira, M.V.F.Pereira, "Extension of Baleriaux-Booth Analytical Production Costing to Multi-Area Chronological Hydrothermal Scheduling", Cepel!PSRI Technical Report, 1993.

10) B.Gorenstin,N.Campod6nico,J.Costa, M.Pereira, "Power System Expansion Planning Under Uncertainty", IEEE Transactions on Power Systems, Vol. PWRS-7, No. 2, pp. 791-797, 1992.