Electricity planning under uncertainty Risks, margins and the uncertain planner

Electricity planning under uncertainty

Risks, margins and the uncertain planner

Nigel Lucas and Dimitrios Papaconstantinou

The uncertainty of plant performance and unpredictable demand for electricity has always been a problem for power system planners. This consideration has come to dominate all others and this article presents a methodology which incorporates many of the uncertainties of power system planning in one model. This model is then used to show that proposals to greatly lower planning margins entail high risks. The effect of uncertainty on plant mix is shown to lead to greater installation of middle merit plant. Price regulation as a means of equilibrating supply and demand under uncertainty is shown to be of limited, but distinct, application.

Keywords: Electricity supply; Planning; Uncertainty

Coping with the uncertain performance of plant and with unpredictable demand has long been a principal priority of power system planners. Now the uncertainties attached to the planning of the electricity supply industry have come to dominate all other considerations. Economies in scale of plant have meant lower availabilities and this combination of larger units with less predictable performance has increased uncertainties on the supply side. More important is that the extent of demand in the planning year is subject to such powerful and unpredictable influences that it cannot be estimated with any precision.

The intrinsic intractability of the problem is exacerbated by the long lead times associated with modem plant and the costs of dealing with this problem are enhanced by the trend to capital intensive technologies of power generation. Reasonably satisfactory methods exist for calculating the consequences of these various uncertainties, generally in a piecemeal fashion.

In this paper the authors present a methodology which incorporates all the relevant aspects of the problem in a single lucid model. This model is then used to show that proposals from outside the industry to greatly lower planning margins entail high risks. The high planning margins presently used by utilities may be justified by the current uncertainties.

The effect of uncertainties on the plant mix is examined and shown to lead to a greater installation of middle merit type plant; coal in many circumstances. Price regulation as a method of equilibrating supply and demand under uncertainty is examined and shown to be of limited, but distinct, application.

The authors are with the Energy Policy Unit, Department of Mechanical Engineering, Imperial College of Science and Technology, Exhibition Road, London SW7 2BX.

Planning margins Electricity utilities throughout the world have always installed extra plant as a c~r'tingency against uncertainties in supply and demand. Initially, when plant was unreliable, systems were small and lead times (and therefore planning horizons) were short, then the principal uncertainties

0301-4215/82/020143-10503.00 © 1982 Butterworth & Co (Publishers) Ltd 143


lSee for example the Monopolies and Mergers Commission, Report on the Central Electricity Generating Board, Report HC 315, HMSO, May 1981. 2Probabilistic simulation was first applied to Power System Planning by H. Baleriaux and E. Jamoulle, 'Simulation de rexploita- tion d'un parc des machines thermiques de production d'electricit(~ coupld & des stations de pompage', Revue E. edition SRBE, Vol V, No 7, 1967. A good review is given by D. S. Joy and R. T. Jenkins, A Probabilistic Model for Estimating the Operating Cost of an Electric Power Generating System, ORNL-TM-3549, Oak Ridge National Laboratory, USA, 1971.

came from the supply side. With the development of more complex plant (requiring long construction times) the uncertainties on the demand side began to influence the volume of generating capacity necessary to achieve specified reliability levels.

At the present time uncertainty about the evolution of demand is especially high because of the difficulties of predicting the outcome of the opposing influences of the economic recession (depth and duration), energy conservation, the probable improved position of electricity within energy price relativities and the extent of the public and government perception of the need and desirability of promoting nuclear electricity as a substitute for oil.

Three principal factors influence the calculation of planning margins:

• the uncertainties associated with generating plant; • the uncertainties in the timing and extent of demand; and • the costs of outages.

For our purposes we consider only plant uncertainties associated with the availability of commissioned plant, although for some utilities uncertainties associated with licencing and construction periods are also important.

On the demand side it is important to distinguish between long- and short-term uncertainties. Short-term uncertainties about the exact timing of demand are implicit in the conventional presentation of demand as a load duration curve (LDC), which is ex ante a probabilistic conception. Long-term uncertainties in demand could in principle be considered as implicit in the LDC in the same way without any leap of the imagination. As far as we know utilities have never adopted this approach.

Power system planning, affected as it is by the performance of many plants over many years in supplying a commodity with especially exigent conditions of demand, is an extremely complicated business. The models of electricity systems on which plans are based all make simplifications; the validity of the simplifications depends on what are the crucial influences on the particular aspect of planning with which the model is intended to deal.

The Central Electricity Generating Board (CEGB) establishes a background system mix from a programme which models the performance of the system over many years and contains no representation of the probabilistic nature of plant availability; net effective system costs of given options are then established against this background. The probabilistic nature of long-term demand is incorporated separately into planning through a calculation of the volume of plant necessary to provide a specified level of reliability. 1 This exogenous level of reliability is determined by yet another calculation which trades off the social costs of outage against the costs of extra capacity. This procedure, excellent and practical in many respects, has weaknesses. It is not clear how the planning margin is distributed among plant types, nor, as a result, is it clear what are the costs of the planning margin to be set against the social costs of outage. A methodology which emphasizes uncertainty at the expense of other factors, might also be of interest.

Other utilities proceed differently. Especially appealing in any dis- cussion of uncertainties is the technique of probabilistic simulation.2 This elegant procedure permits calculation of the economic consequences of the interaction between the supply and demand side uncertainties of any specified system. There is for example a finite probability that high merit plant will be unavailable in unexpected quantities when demand is high,

144 ENERGY POLICY June 1982

~ 1.0 o

~p

t'~ x

D e m a n d

Figure 1. Load duration curve.

3p. R. Booth, 'Power system simulation model based on probability analysis', IEEE Transactions, Vol PAS-91, 1972, pp 62-69. 4Electricit6 de France, Methods and Models used by EDF in the Choice of its Investments for its Power Generation System, M. Garlet, E. L'Hermitage and D. Levy, XXIII Meeting of the Institute of Management Sciences, Athens, July 1977.


resulting in an unexpected call upon low merit plant and therefore higher costs; the reverse, higher than expected availability at times of low demand does not have a happy, inverse economic consequence. Prob- abilistic simulation was used by an Australian utility to explain why it ran its gas turbines so often and was most successful, a This technique therefore provides part of the necessary equipment for planning under uncertainty.

A second technique which helps in our attempt to synthesize the three elements of the problem identified, is to incorporate the social cost of an outage directly into the objective function of the optimization, thereby avoiding the need to pass through an intermediate reliability level. This is easily done using probabilistic simulation because the technique naturally calculates the probability that electricity demand is not met and the volume of demand that is shed. This demand shortfall can be multiplied by an estimate of the social cost of outage per unit and the result intro- duced into the objective function as a penalty; Electricit6 de France have adopted this method.4

There is no reason why the gross uncertainties in future demand should not be combined with the information of a conventional load duration curve to provide a probability distribution of future demand. This specifi- cation of future demand can be incorporated with a penalty function describing the social cost of outage into a probabilistic simulation model to provide a powerful means of describing and discussing the broad issues surrounding the consequences of uncertainty of power system planning.

The load duration curve

As it is central to this paper it is worth labouring the distinction between the various meanings of a load duration curve. The curve is drawn as in Figure 1. It has three meanings according to the context. For a past year the curve relates to a demand that is known in every respect; it has a finite intercept on the horizontal axis. It is a probability distribution in the sense that if one were to pick a random time in that past year then the probability of demand having exceeded the level X at that time is p; but this is an incidental property of no particular practical consequence, the curve is essentially a convenient way of representing a completely determined situation (ie that in the past year demand exceeded the level X for a fraction p of the year).

A load duration curve for a future year has a genuine probabilistic nature; it appears to us that there is an important but imprecise distinction to be drawn between probabilistic effects arising from within a broadly determined overall level of activity and those which arise from uncertainties in the overall level. For the current year, or next year, the overall level of activity is reasonably closely established; the stock of electricity using equipment, national income and its distribution are known within a narrow range.

Much of the shape of the load duration curve will be determined by the rather predictable distribution of demand between seasons and between night and day or by probabilistic events of limited effect and known distribution, mainly weather. When load duration curves for the distant future are used in system planning models it is this information that they are generally intended to represent ie they represent a distribution of demand within a broadly defined level of activity and not the overall uncertainty in activity. This is the second interpretation.

ENERGY POLICY June 1982 145


sUK Department of Energy, Energy Policy -A Consultative Document, Cmnd 7101, HMSO, London, February 1978, paragraph 14.20, p 75. 6Department of Energy, Energy Commis- sion Paper 6, DOE, UK, January 1978.

The third interpretation which seems to us valid and useful, is quite simply as a probability distribution of demand for a distant year including diurnal and seasonal contributions, but also, and more importantly, uncertainties in the overall level of activity.

Methodology The technique of probabilistic simulation is well established. We believe that we have developed a more concise and direct exposition of the method than has previously been published (the current method is summarized in Appendix 1).

We have represented uncertainty in overall demand as a normal distribution; that is to say we have taken a probability distribution such as that shown in Figure 1 characterized by a finite intercept on the horizontal axis. We have then considered the family of distributions which arise from scaling the curve along the horizontal axis by say + X, we have then weighted the members of this family with probabilities corresponding to normal distributions of Xwith standard deviations of 5, 10, 15, 20% of the maximum demand in the base case.

The compound probability distributions obtained are those that should be used by planners who are 5, 10, 15, 20% etc uncertain about the future. Uncertainty is located in the planner, rather than in the future itself to emphasize the direct relationship between optimal policies and the extent of the uncertainty with which individuals view the future, ie to stress the subjectivity of optimal policies.

The decision making criterion used is maximum expected utility, Another criterion that could be used is the 'minimax loss', which has received some attention recently, s We do not think it a suitable criterion as it is difficult to establish the worst possible outcome of each decision (power system expansion strategy). There are also theoretically based objections; minimax strategies are usually applied in two player games where both players follow the same principle; despite the occasional doubts we cannot really believe for long that future demand is hostile and is adopting a minimax strategy; the minimax criterion is probably not therefore appropriate. The methods described here could be adapted easily to a minimax criterion if so desired.

Data The illustrative calculations reported here relate to an imaginary system. The relevant characteristics of new and existing plant are summarized in Table 1. The economic characteristics of new plant are obtained from the Department of Energy. 6 We emphasize that the calculations are illustrative of the principles; they do not set out to be precise and up to date estimates of particular aspects. Here and elsewhere costs relate to 1977 levels.

Table 1. System data,

Capacity Operating Number Capital Life (MW) FOR cost (p/kWh) per type cost (£/kW) (yr)

New nuclear 1000 0.20 0.64 - 600 25 Old nuclear 800 0.20 0.66 4 New coal 700 0.10 1.20 - 334 30 Old coal 500 0.15 1.30 18 New gas 100 0.05 4.00 - 225 30 Old gas 100 0.05 4.50 10



The central estimate of the LDC in the planning year is defined by a maximum demand of 14 GW, a minimum demand of 4 GW, a load factor of 57% and a shape similar to that recently experienced by the CEGB. A unit outage cost has been calculated from the ratio of wages and salaries to the consumption of electricity in commerce and industry; the value found is 55p/kWh. Identification of this parameter is extremely con- tentious and this figure lies within the range of others' estimates (see Appendix 2).

Outages can also arise from distribution failure and the social costs of such failure are similar, although geographically limited. In an optimized generation/distribution system the marginal costs of increasing reliability by investment either in extra generation or distribution should be equal. The analysis presented here deals only with generation capacity, which is politically the more significant area.

c 5 0

o 30 K

E

0 I0 20 30 Demand uncertointy (SD) (%)

Figure 2. Effect of demand uncertainty on the planning margin.

A

~14

~to

~ 6 ._=

~z

I0 20 50 Demand uncertointy (SD) (%)

Figure 3. Effect of demand uncertainty on expected system costs.

Application of and results from the model

Using these data the system cost of meeting demand in the planning year ie operating cost plus amortization of new plant, plus outage costs can be assessed by a probabilistic simulation and the optimal expansion strategy selected. The following sections assess the consequences for margins, system costs and system mix.

Margins and system costs

The effect that the planner's perception of uncertainty has on the planning margin is shown in Figure 2. A certain planner (who attaches zero uncertainty to his forecasts) will get by with a planning margin of just under 20%; the planning margin increases rapidly and non-linearly as the uncertainty of the planner increases.

This calculation is not based on a specific, real system, but nevertheless note that the planning margin of 28% used by the CEGB appears very reasonable in the circumstances. The expected system costs of uncertainty are summarized in Figure 3. A planner with 10% uncertainty would expect to pay 2-3% more in system costs than would the completely certain planner and the 20% uncertain planner would expect to pay 8% more. Better to be certain one might deduce. But of course that is not the case. What is important is the relationship between the certainty of the planner and probability of him being correct. Figure 4 makes the point clearer. The figure shows the total system cost (TSC) for the optimal strategies of planners of various uncertainties as a funtion of the actual outcome. The highly uncertain planner produces the strategy which shows the least variation in TSC; the certain planner produces the strategy with the greatest variation, but with the lowest absolute value, providing the outcome is less than forecast demand.

These graphs can be adapted to a minimax criterion. If the worst possible outcome is considered to be a demand 15% higher than forecast then the best strategy among those shown is that produced by the 20% uncertain planner; a worst outcome 5% higher than forecast and a minimax criterion selects the strategy of the 5% uncertain planner. We are aware that the results will depend on the probability distribution chosen to describe future demand. We have taken normal distributions; it is likely that these are not the best but we have chosen them to simplify the initial stages of a complicated debate.



F i g u r e 4. Effect of demand outcome on system costs for the optimal strategies of uncertain planners.

Note: O - 0% SD; [] -- 10% SD; A --- 20% SD; and Q -- 30% SD.

c ._o

E

0

1600

800

600

80 t00 120 SMD outcome as o % of central forecast SMD (%)

Uncertainty and plant mix We have also examined the different mixes of plant in the expansion strategies of planners of varying certainty. The numbers of nuclear and coal additions and their difference for our hypothetical example are shown in Figure 5, the results are shown for various values of the capital cost for nuclear capacity. Naturally the greater the cost of nuclear capacity, the less it is preferred. But the point of interest is the interaction with uncertainty. For low nuclear costs both the certain and the uncertain planner eschew coal; the uncertain planner builds more.

Over a rather wide range in the middle they agree on future nuclear capacity but the uncertain planner gets his margin from coal. This range seems to include the cost ratios of practical significance. The consequences are substantial. At a nuclear cost of $570/kW (1977 cost levels), both planners select three nuclear units, but the uncertain planner also builds three coal units to provide the margin; this is an important result because it shows a place for coal generating plant in a risk bearing

1 4 8 E N E R G Y P O L I C Y J u n e 1 9 8 2

Figure 5. Effect of demand uncertainty on the plant mix.

Note: Symbols as for Figure 4.

"q'here are interesting relations between uncertainty and scale of plant; larger plants make for less reliable system because there are fewer units and possibly also because their availabilities are lower. This classic problem was analysed by Casazza and Hoffman (see J. A. Casazza and C. H. Hoffman, Relationship between Pool Size, Unit Size and Transmission Requirements, Paper 32-09, CIGRE, 1968). It is amenable to treatment by the techniques described here. There is a further interesting relationship with uncertainty in future demand if the leadtimas of the plants vary with size; results on this problem will be presented in a later paper. sit may also be varied by policy induced conservation at an unquantifiable cost, but for which the price effects in the following analysis may be an adequate proxy. 9UK Department of Energy, The Report of the Working Group on Energy Elasticities, Energy Paper 17, HMSO, London, 1977, puts price elasticities for electricity in the UK between - 0.2 and - 0.3.

6


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I 1 1 I I I L 520 540 560 580 600 67'0 640

Capitol cost of nuclear plants (Lf/kW)

r61e under economic conditions where it would not be competitive in a deterministic comparison.7

Effects of price regulation It has been assumed so far that electricity demand is completely beyond control . However demand could be varied by price, s We have assumed that in the planning year demand can instantaneously be varied by price with an elasticity of--0.2. Clearly this is a crude simplification, but it serves the present purpose.9

The change in social welfare arising from this change in price can be calculated. The new demand:

el -0.2

leads to a consumers benefit of

W = pdD = - 0 . 2 Ddp Do Po

o r

W = - 4 LXPo/ -



The base price Po which we have used in subsequent calculations is equal to the long-run marginal cost of expanding supply at the expected demand level, whilst preserving the shape of the LDC and therefore the load factor.

On this basis we have then calculated the optimal expansion programmes of planners of varying degrees of certainty who know they can control deviations from the central forecast by price changes. The objective is to minimize final social cost, (FSC) defined as

system cost + outage cost + cost of reduced consumption from price control

The consequences for a particular planner (the certain planner as it happens) are summarized in Table 2. If SMD is less than forecast, then the social benefit of reducing price is rather small and the price changes required to balance are disproportionately large. For example to balance a shortfall of 4% in demand requires a reduction in price of 10% and improves the FSC quite negligibly. The theoretical social benefits do not justify the financial problems which would be created by the loss of revenue to the utility. At the other extreme, when the certain planner has under-estimated demand, large improvements in FSC are theoretically possible, because of the large costs of outage avoided, but only with price changes so large as to be impractical.

Figure 6 summarizes the consequences for the optimal expansion strategies of our various planners. Shown in the figure are the com- ponents (total additions and coal additions) of the strategies of each planner according to whether he can vary price completely, within + 20%, or not at all.

It is found that price variation has no effect (for the particular circumstances represented here) on the nuclear additions and bites little into the coal margin. The certain planner is unaffected, as he is certain of the outcome anyway he has no interest in other means of equilibrating supply and demand; the other planners benefit by one or two plants the effect becoming increasingly small proportionally as uncertainty increases.

9!

~ 7 o 6

= 51

I I I 0 I0 20 30

Demand uncertainty (SD) (%)

Figure 6. Effect of price regulation.

Note: • =- Nuclear plant additions; O -~ Opt imum price variation; [ ] -= Limited price variation; and X =- No price variation.

Conclusions

Planning margins in power systems are a very strong function of the certainty with which the planner views the future. The low margins recommended by many outside the industry imply a certainty that is inconsistent with any reasonable degree of doubt about our ability to predict the future. A relatively modest uncertainty about the future and

7 e 5 o Table 2. Effects of price changes on TSC and FSC for various outcomes to the certain planners 4 ~ expansion strategy. 3 o

2 _ ~o SMD/ TSC FSC dplp FSC [ ~ (SMD)o (£mill ion) optimum optimum (clp/p <~ 0.2) O o ~ 0.80 636,5 625.0 -0.3 -

0.84 671.4 661.9 -0,3 - 0.88 707.5 700.3 -0.2 - 0.92 746.1 742.0 -0.2 - 0.96 787.8 786.5 -0.1 - 1.00 835.8 835.0 0.0 - 1.04 892.2 889.7 0.1 - 1.08 963.5 948.6 0.2 - 1.12 1053.0 1014.6 0,4 1019.4 1.16 1165.9 1085.7 0.5 1107.3 1.20 1315.0 1163.0 0.6 1217.0



r ecogn i t ion o f the high costs o f not having electricity when it is wanted can easi ly gene ra t e the large planning margins used by utilities.

T h e sys tem mix can be strongly influenced by the uncertainty of the p lanner . Unce r t a in d e m a n d tends to be 'me t ' by middle merit plant and t he r e fo re uncer ta in ty can lead to extra plant o f this nature, increasing especia l ly the installation o f coal burning plant.

Pr ice could be used as a means o f balancing supply and demand. If d e m a n d falls shor t o f forecast then the practical benefits o f varying price are minimal , but there is some scope for using price to ration demand in the reverse case. The knowledge that price control would be possible in the p lann ing yea r has a small but distinct influence on optimal strategies for uncer ta in planners.

Appendix 1

Methodology

Given probabilistic representations of demand and supply, their relative costs and the costs of demand not served; the optimization problem is simply to minimize:

E T C = E S C + E O C

where: E T C = expected total costs; E S C = expected supply cost, including capital, operating and fuel costs; and E O C = expected outage cost arising from a reduction of economic activity because of energy shortage.

To solve this problem we arrange the energy supplies in order of their vari- able cost, analagous to the merit order employed in power station despatch- ing. We then calculate the cumulative probability distribution of the quantity.

D - ~ . S i i=l,n

where: D = demand; and S i = supply from source i.

This cumulative probability distribution we denote Fn(X); it has the form in Figure 7. The area of the hatched por- tion is the expected amount of energy not provided by the n sources; the cost of that shortage is the area times the unit cost of outage, which quantity we denote EOC.

If the calculation is repeated omit- ting the most expensive (n th) energy source we obtain a cumulative prob-

ability distribution for

D-ZS, i=l,n-I

which we denote F n_ I(X); similarly the distribution for

D - ~_, S i i= l , j

we denote Fj(X). These distributions are shown in Figure 8.

The area enclosed by the X axis, the line X = 0 and the curve is in each case the expectation value of the energy not served by the first j sources, which quantity we denote EOSj; it follows that the expectation value of the energy served by the jth source is

ESj = EOS _,- eOS

Knowing the costs of the various sources of supply the total cost is easily obtained. The result is an optimum in the sense that the available supplies have been optimally allocated; the calculation can be repeated for other supply structures corresponding to different investment policies for the purpose of finding an optimum in a wider sense.

Our programmes use a different formulation, much more economical in computer time, but this we think is the most transparent. For a description of

the exact methodology used, see Papaconstantinou. 1

tD. V. Papaconstantinou, Power System Planning Under Uncertainty Using Proba- bilistic Simulation, Internal Report EPU/DP/1, Energy Policy Unit, Depart- ment of Mechanical Engineering, Imperial College of Science and Technology, London, May 1980.

I 0 - - - -

0

o x Demond

Figure 7. Cumulative probability distribution (Fn) and energy not served.

i.ol

0

o Dero~nd

Figure 8. Cumulative probability distributions (F).



Appendix 2

Outage costs

The cost of an outage to the utility, at least a monopoly utility, is the loss of revenue. The cost to be used in calculations such as these must be consider- ably higher, to reflect the loss of social welfare. Differing views have been expressed about how to calculate this parameter ; estimates of its magnitude are summarized in Table 3. See also Kaufman 2 and Telson. 3

2Alvin Kaufman, Reliability Criterion - a cost-benef i t analysis, Office of Research, New York State Public Services Commis- sion, August 1975. aM. L. Telson, 'The economics of alter- nat ive levels of reliability for electric power generat ion systems', The Be//Journal of Economics, Vol 6, No 2, Autumn 1975.

Table 3. Outage costs (S/kWh).

Type of consumer Report (year) Industrial Commercial Domestic Average Fully developed country (1972) a 0.25-0.33 1.06-1.16 0.58 - Much developed country (1972) a 0.31 0.16 0.10 - Much developed country (1972) a 0.54 - - - Developing country (1972) a 1.56 0,72 0.25 - Developing country (1972) a 0.15 - - - Very undeveloped country (1972) a 0.28 - - - Developing country cit~. (1976) u 1.00-6.00 - 1.30-1.70 1.12-129 Great Britain (1972) c, o - - 0.50-1.50 - California (1976) c, o - - 0.10 - Ontario Hydro (1976) c, e, f 1.00-2.69 1.00 0.10 - USA and Canada (1973) c, g 2.68 - - - France (1975) h - - - 0.45 USSR (1978) ~ - - - 0.60

Sources: aT.W. Berrie, 'How to work out what quality of electricity supply we can afford', Electrical Review, Vol b2~L2, No 3, 20 January 1978.

M Munasinghe, A New Approach to Power System Planning, IEEE, PES Winter Meeting, New York, February 1979. CD.D. Koval, and R. Billinton, Statistical and Analytical Evaluation of the Duration and Cost of Consumers Interruptions, IEEE, PES Winter Meeting, New York, February 1979. dAnalysis of Electric Power System Reliability, Prepared for Energy Assessments Division of the California Energy Resources Conservation and Development Commission, by System Control Inc, Palo Alto, CA, USA, 1979. eL. H. Berk and E. M. MacKay, Ontario Hydro Survey on Supply Reliability: Viewpoint of Large Users, CEA, 21 March 1977. fOntano Hydro Survey on Power System Reliability: Viewpoint of Large Users, Report No PMA 76-5, 1976. gReport on Reliability Survey of Industrial Plants, Part 2, Reliability Subcommittee of Industrial and Commercial Power System Committee of IEEE, May 1973. h'Risque de defaillance et decision d'investissement', EdF, France. ICybernetics in Electric Power Systems, Mir Publishers, Moscow, USSR, 1978.

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Electricity planning under uncertainty Risks, margins and the uncertain planner