Contract Selection Framework Using Decision Analysis Including Unknown Correlation Between Price Drivers

 

Gerald B. Sheblé &  Daniel Berleant

Iowa State University

Ames, IA 50011

 

Abstract: This paper presents a decision analysis model for contract selection.  The power industry in the United States is presently under a changing business environment.  While planning to meet future peak demand is still a concern, the efficient utilization of existing generation and transmission resources and markets is fast becoming of primary interest.  The electric market is being deregulated to promote competition.  In such a situation, market players will have many choices for generation and transmission contracts.  In this new environment, the contract selection problem is complicated to solve because of the risk involved in economic and reliable operation arising from the uncertainty in many factors.  This work provides a framework, which integrates the economy and reliability aspects of power system in the new business environment even when the correlations between price factors are unknown.  The approach is based on a decision analysis methodology, which involves the use of a set of mathematical techniques to explicitly address risk, uncertainty, and complexity in the contract selection problem combined with interval mathematics.  The methodology successfully finds the best tradeoff between low risk and low cost.

 

 

I.  INTRODUCTION

 

The basis of power system operation and economics in the United States is undergoing major structural changes.  Economists believe that the electric pricing must be regulated by free market forces rather than by state and federal commissions.  To facilitate the growth of free market electricity, US Senate passed a comprehensive National Energy Policy Act (NEPA) in 1992.  The act defines exempt wholesale generators (EWGs) as any company owning or operating all or part of an eligible facility and selling electricity at wholesale cost.  Utilities are permitted to purchase from an affiliated EWG under the jurisdiction of a state commission.  FERC may issue a transmission order if such an order meets certain requirements and would be in the public interest.  A utility has 60 days to respond to a transmission request before an applicant can file for a wheeling order with FERC.  This requires utilities to have efficient decision tools in order to perform contract selection.  This paper presents the kernel of such a tool.

 

A contract selection problem is difficult to solve because of the involved risks and uncertainties.  In the new framework of the electric market, major sources of uncertainties are availability of transmission capacity, market prices, load forecasts, and availability of a generating unit.  Many of this uncertainty sources are dependent or strongly correlated. Moreover, the idea of a deregulated environment is based on the efficient and full utilization of existing generation and transmission sources.  Increased transmission loading has occurred and is expected to continue.  The probability of loosing transmission transfer capability is higher.  These uncertainties adversely affect the power system reliability, resulting in risk of less secure power system, of unserved energy, and of loss of opportunities. All parameters need to be integrated in a unified framework to address risk and uncertainty in a contract selection problem.

 

The potential deregulation of the industry is based on the idea of a free market.  In a free market, consumers will have many choices for purchasing electricity.  Therefore, fear of loosing customers will prevail in the market and utilities will consider the option of buying insurance.  Usually, the cost of insurance is very high because of the high cost associated with the commitment of spinning units.  Thus, in the new electric market, the cost of insurance would be one of the most significant components to be considered for the contract selection problem.  Each contract should be evaluated in conjunction with associated insurance cost.

 

 The aforementioned economic problems with other engineering constraints complicate the problem of contract selection. This problem needs a unified framework that can integrate the production costing model and reliability model to address the cost and associated risk.  All uncertainties associated with the new market, such as probability of price coupling between competing units needs to be included.  The costs of contingent contracts have to be included in the framework so that a comprehensive economic analysis can be made. 

 

Decision analysis can be viewed as a methodology for making decisions with uncertain outcomes[1,2].  Comprehensive treatments can be found in many volumes [3].  Decision analysis is not a competitor to the other modeling methodologies.  Rather, it is complementary in that it integrates the results of various models and applies them to decision making.  This approach finds the strategy using decision analysis that provides the best trade-off between low risk and low cost. 

 

2.  FORMULATION OF THE PROBLEM

 

From the buyers’ perspective, the contract selection problem can be defined as follows.  Given a set of possible contracts, select a contract plan that minimizes the net expected cost of production and cost of insurance, which are subject to risks due to uncertainty in power system reliability.  Mathematically, the problem can be expressed as follows:

 

Minimize:  (Expected production and insurance cost)

Subject to:    (Load constraint)

 

 

  (Transmission capacity constraint)  (1)

 

where,  H = nonlinear relationship  between power produced and purchased

            F(p)                 = production cost of the utility company

            I                       = cost of insurance

            Tp                     = contract power

            Ga                    = summation of maximum capacity of available generating units

            Ip                      = power purchased from insurance company

            L                      = total load of the utility company

            Tf                     = loading of transmission line for contract

            Tcmax     = capacity of transmission for contract

 

Probabilistic modeling of the random variables has been done using power system reliability data.  Probability of loading transmission line exceeding its capacity can be achieved using probabilistic power flow formulation.  Identifying a suitable production-costing model can do the cost calculation.

 

3.  MODELING ISSUES

 

There are a number of factors to be considered when analytical models of reliability, and production costing need to be developed.  We have focused on factors such as composition of the system, system failure criteria, and assumptions in modeling.  Note that all the parameters under consideration are continuous state variables.  The probability distribution for all the parameters have been converted to a discrete distribution, but this is not necessary in the decision analysis framework.  The discretizations are constructed in a way that provides a good approximation to the first few moments of the original distribution.

 

RELIABILITY MODEL

 

The electric utility industry in North America, in the words of its North American Electric Reliability Council (NERC), uses reliability in a bulk electric system to indicate “ the degree to which the performance of the elements of that system results in electricity being delivered to customers within accepted standards and in the amount desired.  The degree of reliability may be measured by the frequency, duration, and magnitude of adverse effects on the electric supply”.  The council also suggests that reliability can be addressed by considering the two basic and functional aspects of the bulk electric system - adequacy and security.

 

Adequacy

The static evaluation of the system’s ability to satisfy the system load requirements can be designated as adequacy evaluation.  For each possible system state,  we define:

 

            f(Pg)                             = probability of generation adequacy

            f(Pl)                             = probability of forecasted load

            f(Tf ≥ Tcmax)      = probability function of transmission adequacy,

.                                                  that is, probability of line flow exceeding the capability of

                                                   transmission line resulting into loss of contract.

 

Probability of generation adequacy [f(Pg)]: This is expressed by the generating unit availability distribution.  Our approach includes multi-state unit, i.e., a unit that can exist in one or more derated or partial output states as well as in the fully up and fully down states.  Typical generation unit availability is shown in Figure 1.  The data for resource availability may be convolved with the load data set to create capacity outage probability. 

 

            Probability of  forecasted load [f(Pl)]: This is usually expressed  by the probability density function.  We divide the load forecast probability distribution into class intervals, the number of which depends upon the accuracy desired.  The area of each class interval represents the probability that the load is the class interval mid-value.  Figure 2 presents a sample load duration curve divided into class intervals.

Figure 1.  Probability of unit availability

Figure 2.  Time probability of load

 

            Probability of transmission adequacy [f(Tf ≥ Tcmax)]: Probability distribution function of network parameters can be achieved by probabilistic load flow (PLF).  The algorithm for determining probability of transmission adequacy using PLF is given as follows:

 

Figure 3.  Algorithm for determining transmission adequacy

 

 

Security

When a contract occurs, the system security margin is decreased due to increased transmission  loading.  There are many engineering tools to provide the static [4] and dynamic security  regions [5].  However, we intend to treat security as a probabilistic problem [7].

 

Alvarado [8] has attempted to integrate the aforementioned concept as a probabilistic measure of likelihood of system failure, which is defined as probability of normal operating status .  Distance to system operational limits has been suggested  as a measure of system security margin.  The work defines an operational limit boundary (OLB) in load demand space.  Distances to this boundary are then translated into probabilistic measures of likelihood of system failure,  which is quantified by (PN)  and is defined in reference [8].  We note that the definition of PN in reference [8] is based on two dimensional OLB.  The choice of two parameters defining OLB can be made by considering the worst case contingency.  However, if OLBS (operating limit boundary surface) is of interest, PN  can be replaced by the distance measure of security [9].

 

PRODUCTION COSTING MODEL

Production costs comprise a significant portion of the operating costs for electric utility companies.  Traditionally, production costing has been formulated as a probabilistic problem.  The three most common methods used for production costing are as follows:

 

(1) Load duration curve[10]: This approach is fast but it gives incorrect information about peaking units.

(2) Chronological simulations[11]  This approach is computationally intensive and hence is not suitable for probabilistic simulation.

(3) Probabilistic Simulation[12]:  The probabilistic simulation is the most popular algorithm because of fast computation.  Probability methods correctly predict utilization of units.  They produce reliability measures as an extra benefit.

 

In this work, we have discretized the reliability model.  If we consider each discrete scenario separately, production costing can be formulated as a deterministic optimization problem.  We have proposed to use a linear programming (LP) model for deterministic production costing since it provides an uncomplicated means of computation that is fast, versatile and efficient. 

 

LP- based production cost model

Our approach is based on LP based production costing algorithm[13] for multiple time periods.  For each path, we form a linear program to forecast the production costs.  The probabilistic production costs can be computed by calculating expected production cost using the decision tree.

 

            Problem formulation: The costs associated with satisfying the power production requirements expressed by a load duration curve are :

 

            (1) Start-up cost                                  

            (2) Unit running costs

                       

            The constraints can be listed as:

 

(1) Enough units should be running at any given time to satisfy the spinning requirements

(2) Enough power should be produced at any given time to meet the load requirements.

(3) Only units available can  participate in meeting the spinning or load requirements.

(4) Only those units which are participating in meeting the load requirement at any give time can be used to meet the total spinning requirements.

 

Mathematically, the problem can be expressed as follows:

 

Minimize

 

Subject to

    (spin constraints)

  (load constraints)  (3)

          (start constraints)

         (availability constraints)

 

where, xrun(s)             = number of units running through segment s,

            xprod(u,s)        = level of output above minimum of unit u in segment s,

            capmin(u)        = minimum capacity of u,

            capmax(u)       = maximum capacity of u,

            crun(u,s)          = cost of runing unit u through segment s at minimum output,

            cprod(u,s)        = costs above crun of u at full load in segment s,

            spin(s)              = spinning MW` requirement for segment s,

            load(s)             = generation MW requirement in segment s.

 

 

4.  DECISION ANALYSIS IN CONTRACT SELECTION

 

The first step of decision analysis is to construct an influence diagram to represent the contract selection problem.  Influence diagrams are a means of representing the same decision problem much more compactly.  For each chance and decision variables in the decision tree, there is a single object in the graph.  Arrows connecting these objects represent probabilistic and informational relationships.  An influence diagram is a theoretical construct and an evaluation device with all the power of a decision tree.  We present the influence diagram of contract selection problem and the corresponding decision tree in Figure 4 and 5 respectively.  Note that uncertainties are portrayed using chance nodes (hollow round nodes) with possible events shown on top.  Decision nodes are represented by squares.

 

 

 

Figure 4.  Influence diagram for contract selection

 

 

Decision analysis has been used widely in engineering economic modeling.  Comprehensive treatments can be found in many volumes [14].  We use the influence diagram to restate the contract selection problem in DA framework as follows:

 

 

Figure 5.  Decision tree for contract selection

 

Problem formulation in DA framework

            Alternative:       •Do nothing      •Contract 1 (T1 MW)  •Contract 2 (T2 MW)

            Information:      (a) Generator data

            (b) load data

            (c) transmission system data

                                    (d) Expected value of price (EVP) in ($/MW)

                                    (e) Expected value of cost of insurance (I) in $/(MW)

            Preference:

                                    Select contract with highest payoff considering system security

 

Risk Preference Function

 

The developed decision tree does not address the risk of security discussed in section 3.  We propose to embed the security issue using a risk preference function given below.

 

                     (4)

where u(v) is the certain monetary equivalent(CME) for the pay off value v . These two are related through a risk preference parameter y.  We propose to formulate the risk preference parameter y as the system security index PN.  Thus, the risk preference function of the utility company is given by

 

     (5)

 

The function u (v) calculates certain monetary equivalent (CMV) of any given pay off.  In other words, the pay off is modulated by risk function.  All the tree calculations are then performed on CMEs and the equivalent EMV at any node can be calculated by using the inverse of the risk preference function.

 

Overall procedure

 

The overall procedure for solving contract selection problem using decision analysis is described in Figure 6.

 

1.  Construct the influence diagram and generate the corresponding decision tree.

2.  Determine probability of each path emanating from chance node as described in section 3.  That is,

(a) Use reliability models to compute f (Pg) and f (Pl). 

(b) Run probabilistic load flow for the system to compute f (Tc ≥ Tcmax) using equation (2).

3. Calculate production cost F (p) using LP formulation of equation (3) for all the path of decision tree.  The net pay off for each path is given as:

                   Pay off  = F (p) + (EVP)*Tp + I* Ip

4. Calculate the CME on every pay off using the risk preference function given in equation (5).

5. Calculate the expected CME for every contract using the CME as terminal payoff.

6. Choose the contract with the least value of expected CME.

Figure 6.  DA approach for contract selection

 

 

5.  AN ILLUSTRATIVE EXAMPLE

 

An illustrative example is presented in this section.  For simplicity, the problem is limited to a constant load (single time period) case.  The data is given in Table 1 and 2.  The data corresponds to the tree shown in Figure 5. 

 

Table 1.  Reliability data

 

Load Data

Generation Data

Transmission Data

MW

Prob.

Unit

State

Prob.

Trans.

Outage

700

0.1

1

0

0.05

T1

0.1

800

0.8

 

300

0.30

T2

0.2

 

 

 

600

0.65

 

 

 

 

2

0

0.05

 

 

 

 

 

400

0.95

 

 

 

Table 2.  Costing data

 

Generation Data

Unit

Minimum Capacity

Maximum

Capacity

Fixed

Cost

Linear

Cost

Quadratic Cost

1

150

600

561.0

7.92

0.001562

2

100

400

310.0

7.85

0.001940

 

Contract Cost

T1

8 $/MW

T2

7 $/MW

Contingent Contract Cost

CC1

20 $/MW

CC2

7 $/MW

 

 

The decision tree was computed using the procedure described in Figure 6.  The summary of expected production costs for different alternatives are shown in Table 3.  It is clear that T2 is the best alternative.

 

Table 3.  Summary of Analysis

Alternative                  Expected cost

 

Do nothing                        $ 8412.30

         T1                                   $ 7960.91

         T2                                   $7931.06

 

 

 

6.  CONCLUSION

 

An extended decision analysis method is developed to extend contract selection.  The decision situation is well described by an influence diagram.  The necessary data needed for the proposed model are described.  Techniques to generate the model data are discussed with the proper reference.  One of the important attempts in the work is to integrate system security consideration in order to evaluate the cost of system adequacy.  However, the choice of risk preference function and several parameters like the probability of normal system operation are issues of future work. 

 

7.  REFERENCES

 

1.      “MIDAS, Multi Objective Integrated Decision Analysis System, EPRI Journal, vol. 12 May’87: pp 57-58.

2.      Decision Systems International, “ Bulk Power Marketing Decisions Under Uncertainty” sponsored by EPRI Palo Alto, CA Aug’92.

3.      R A Howard, “ Decision Analysis: Practice And Promise” Management Science, 1988, pp 679- 675

4.      F F Wu and Y K Tsai, “Probabilistic Dynamic Security Assessment Of Power Systems: Part 1 - Basic Model”  IEEE trans. on CAS, May’ 83: 148-149.

5.      F Alvarado, Y Hu, D. Ray,  Stevenson and E. Cashmam, “Engineering Foundation For The Determination Of Security Costs”  IEEE trans. on PWS, Aug’91: 1175-1182.

6.      J McCalley and B Krause, “Rapid Transmission Capacity Margin Determination For Dynamic Security Assessment Using Artificial Neural Networks” IEEE Transactions on Power Systems.

7.      R S Ru, P Toy and K F Schenk, “ Expected Energy Production Cost  By Methods of Moments,”  IEEE Transactions on PAS, 1980, pp 1917-1980.

8.      G Gross, N V Garapic and B. Mcnutt, “The Mixture Of Normal Approximation Techniques For Equivalent Load Duration Curve,”  IEEE Transactions on PAS, 1988, pp 368-374.

9.      X. Wang and J. R. McDonald,  “Modern Power System Planning”  McGraw-Hill, London 1994

10.  J T Day,  “Forecasting Minimum Production Costs With Linear Programming”  IEEE Transactions on PAS, April 1971, pp 814-823.

11.  J. A. White, M. H. Agee and K. E. Case,  “Principles Of Engineering Economic Analysis”  John Wiley & Sons, New York, 1989.