Abstract: Deregulation in the power industry has encouraged the utilities to change their practice, from regional monopoly to open market competition, to enhance the operational efficiency. The competitions in the production, transport, and sale of energy will lead to more uncertainties in the industry. Particularly, since the setup cost to sell energy in the retail and the wholesale markets is low, the marketing sector of energy services will face more competition. In this paper, various uncertainties that may be faced by a utility in an open market are discussed. In addition, several risk management tools are explored to provide ways to manage the uncertainties of serving customer energy services in a deregulated power industry. Finally, examples utilizing sensitivity analysis and parametric analysis are presented.
Keywords: deregulation, risk management, parametric analysis, sensitivity analysis.
I. INTRODUCTION
Calls for competition in the power industry, from the wholesale to the retail level, has made deregulation an attractive option around the world. A study of this new evolving market structure reveals the need for a more acceptable framework that would ultimately satisfy regulatory bodies, customers, and suppliers, alike. One approach is the application of a brokerage system to the power industry to promote competition. To accomplish this, existing vertically integrated utilities need to be broken up. The framework of an energy market can be found in [1].
The deregulation in the power industry provides many research opportunities. One of these opportunities is studying how the regulated power industry should re-strategize itself to operate in a deregulated, competitive environment. In particular, this paper emphasizes on energy services, illustrating the various uncertainties that may be encountered by a utility and the various risk management and assessment tools that may be adopted by the utility. In particular, an example is provided to evaluate sensitivity analysis and parametric analysis.
In
section II, a linear programming model for a utility serving customers’ energy
demand using contracts purchased through the open market is proposed. In section III, various uncertainties
encountered by the utility are described.
In
section IV, several risk management and assessment tools are proposed to incorporate the uncertainties into the decision making. Section V provides an example evaluating the potential of sensitivity analysis and parametric analysis. Finally, section VI concludes this paper, emphasizing the importance of researching risk management tools in the deregulated power industry.
II. MEETING CUSTOMERS DEMAND WITH ENERGY CONTRACTS
Auction market provides a competitive environment where energy contracts may be traded freely. Dekrajangpetch [2] characterizes an energy auction market based on how the market is implemented. Ng [3] further addresses various contract specifications to be considered by a utility buying energy contract to serve customer energy services. In this paper, the contract specifications outlined in [3] are adopted.
In a competitive market environment, utilities strive to maximize profits from serving customer energy services. In this paper, the utility is assumed to have no generation capability. However, the utility has adopted load management programs to lower the volatility of the customer demand and to lower the cost of maintaining the customers’ desired reliability level. A profit-based scheduling model is shown in (1).
where
(1)
is the amount of energy to be purchased from the open market
at period j. 
is the total customer
demand at period j, 
is the deferred
direct load control (DLC) demand at period j,
is the paid back DLC
demand at period j, 
is the energy stored
at period j, and 
is the energy
released from the energy storage system (ESS) at period j. The models for 
, 
, 
, and 
can be found in
[4]. 
is the per-unit price
of energy on the open market at period j. 
is the rate customers
are charged at period j. 
is the rebate given
to the customers participating in the DLC program at period j and 
is the operating cost
of ESS at period j.
When buying energy contracts, the utility has to determine the reliability level of the purchased energy. 95% reliable energy for example, means that all purchased energy will be delivered to the buyer 95% of the time while no energy will be delivered 5% of the time. Figure 1 shows a contract of 5 MWhr having a 95% reliability level.
Figure 1. A contract
delivering 5MWhr @ 95% reliability level.
In
addition, the utility needs to be aware of how much the purchased energy may
vary, the volatility, within the delivery duration. Volatility refers to the percentage change allowed in the demand
during a specific duration. For
instance, if a utility purchases an energy contract allowing 5% volatility, the
customer demand can fluctuate within the 5% range when energy is delivered. Figure 2 shows a contract of P MWhr allowing b volatility level during
delivery duration. Any customer demand
exceeding 
level will not be
served under the contract specification.
In addition, when customer demand falls below 
, contract seller does not have to compensate the contract
buyer for the amount of energy not delivered.
References [3,5] further discuss the auction mechanisms, contract
specifications and show how the two criteria may be formulated into mathematical
equations and added as constraints in (1).
Figure 2. The customer demand during a particular duration.
Equation (1) maximizes the profit that the utility may enjoy from serving the customer demand using energy contracts and load management programs. It does not consider the impact of excess capital or the needs to raise capital. When the cost of borrowing becomes crucial in the utility operation, (1) is insufficient in capturing the cost of borrowing. To incorporate the cost of borrowing, a cash management-based model may be used to replace the objective function in (1). Reference [5] further explores the cash management-based model. However, in this paper, only the profit-based model as shown in (1) will be adopted for discussion.
III. UNCERTAINTIES OF SERVING CUSTOMER DEMAND IN A DEREGULATED POWER INDUSTRY
In general, the factors influencing the profitability of the utility, suggested by (1), can be categorized into three:
· Market – energy spot price, forward/future/option prices for energy, and reliability of the purchased energy. These market factors affect the cost of serving customer demand.
· Customer – customer demand, number of customers, rate structure (tariff at which customers are paying), and energy reliability level desired by the customers. These customer factors affect the revenue of serving customer demand.
· Load Management Program – controllable energy, number of participants, rebate structure (incentives to encourage customers participation), and reliability of the load management devices. The load management programs increase the capital cost, but at the same time improve the profitability of serving customer demand and relax the customer demand.
Some of these factors are uncertain in nature. For instance, the customer demand is uncertain because the utility may only estimate the customer demand at all time. However, some uncertainties in these factors are time dependent. For instance, in the near future, the utility is certain on the tariff at which it is charging the customers; however, in the long run, depending on the competition level, the utility may have to lower or increase the tariff it is charging the customers, rendering the tariff structure uncertain.
In
a deregulated environment, the number of uncertainties will increase, especially
when the utilities have to compete against each other to attract customers and
to purchase energy through the open market.
For instance, the number of customers that the utility serves will vary
since these customers are free to choose their service providers. Further discussions on the uncertainties of
serving customer energy services in the deregulated environment can be found in
[5].
Equation
(1) can be easily solved using the linear programming should all parameters are
deterministic. However, as suggested,
with all types of uncertainties embedded in (1), adopting linear programming
becomes questionable. Other approaches
to replace linear programming may be desirable, even if these approaches are
only to provide alternative solution.
IV. RISK MANAGEMENT AND ASSESSMENT TOOLS
In this section, various operation research tools intended for linear constraints and linear objective model are investigated. They include sensitivity analysis and parametric analysis, mean-variance analysis, stochastic linear programming, fuzzy linear programming, and value at risk. To facilitate the discussions, Equation (1) is further simplified as shown in (2).
subject to:
(2)
where cT
is the cost coefficients associated to the decision variables, x
while 
and 
refer to the set of
constraints on the load management programs and the contracts specifications.
1. Sensitivity analysis and parametric analysis
Sensitivity analysis and parametric analysis are the two most readily available approaches to assess and manage the risk if the solution methodology used to optimize the model is the linear programming. They are post-optimality analysis available from the optimal solution reached by the linear programming model.
Sensitivity analysis examines the effect of relaxing some of the constraints on the value of the optimal objective without having to resolve the problem. The analysis [6, 7] includes change in the objective function, cT, change in the right-hand-side vector, b, change in the constraint matrix, A, addition of a new constraint, and addition of a new variable. Sensitivity analysis is used only when there are not many changes to be analyzed. Otherwise, sensitivity analysis is no different than solving a new linear programming problem. Furthermore, to conduct sensitivity analysis, the changes must be added one by one. For example, to analyze the impact on the changes in two cost coefficients, the technique requires first analyzing the impact of the first coefficient. Then, the impact of the second coefficient can be evaluated based on the original optimal solution or the optimal solution reached after considering the impact of the first coefficient.
Parametric analysis analyzes how the optimal solution changes as several parameters change simultaneously over some ranges. The analysis includes simultaneous changes in the objective function, or the right-hand-side vector. Parametric analysis is not suitable when the risk manager intends to change the constraint matrix, either by changing the parametric values, or by adding new constraints and activities.
An example adopting sensitivity analysis and parametric analysis will be presented in section V.
2. Mean-variance analysis
The mean-variance analysis assumes the decision analysis is based on the expected income on the investment, which is the optimal objective value shown in (2), and the associated variance that represents the risk of investment. There are various models to represent the mean-variance analysis [7, 8]. The two commonly used models are shown in (3) and (4).
subject to:
(3)
subject to:
(4)
E
and V are the expected value and
covariance matrix of the objective function, 
, in (2).
3. Fuzzy linear programming
Since fuzzy linear programming
approach is developed based on the fuzzy logic extension theory, the approach
is introduced last of all. Key concepts
of fuzzy linear programming can be found in [9, 10]. In describing the fuzzy linear programming, the concept of satisfaction of criteria is
important. The criteria can either be
the constraints or the objectives. In
general, the objective is to find 
that would satisfy
the set of criteria as shown in (5).
(5)
The value of 
represents the
aspiration level of the objective function.
The goal of fuzzy linear programming is to determine a solution that
would satisfy the criteria mostly when uncertainty or fuzziness exists in (5),
i.e., by minimizing the risk of violating the set of criteria. From economic point of view, the goal of
fuzzy linear programming is to minimize the risk of violating the set of
criteria in (5). The formulation is
similar to mean-variance analysis with the exception that fuzzy set theory is
utilized to formulate the problem
There are two distinct differences between the two. First, in mean-variance analysis, the uncertainty is stochastic, and in fuzzy linear programming, the uncertainty is fuzzy. Stochastic uncertainty is mainly used to describe vagueness due to the lack of information, where the future state of the system might not be known completely. Fuzzy uncertainty is mainly used to describe vagueness concerning the description of the semantic meaning of the events, phenomena, or statements themselves [10]. Second, mean-variance analysis requires solving quadratic programming model while fuzzy linear programming requires solving only linear programming model. To solve fuzzy linear programming using linear model, the fuzzy uncertainties among the criteria are assumed not correlated.
4. Stochastic linear programming
The stochastic linear programming approach has been researched since 1950s [11]. The approach borrows its technique from statistics. The classical approach to the stochastic linear programming problem is a two-stage linear program with recourse [11].
The requirement for a two-stage setting is due to the fact that one set of variables has only uncertain parameters associated with it, while the other set of variables has both deterministic and uncertain parameters associated with it. Thus, one part of the problem solves for the variables that have only uncertain parameters associated with it, while the other part of the problem solves for the variables that have both uncertain and deterministic parameters associated with it.
5. Value at Risk (VaR) analysis
Value at risk (VaR) is a risk assessment tool that has been used by the financial institutions to evaluate the risk of holding a portfolio of assets. The VaR approach differs from all previously discussed approaches. The previously presented four approaches decide the set of actions to be taken for given uncertainties. VaR, however, assumes that certain actions have been decided. The issue is to determine the monetary risk of taking such actions.
There are in general three techniques that can be used to evaluate VaR. The first technique is the historical simulation. Historical simulation applies the historical data to evaluate the VaR. The second technique is the covariance technique. To apply the covariance technique, the correlation matrix of the uncertain factors is assumed available. The third technique is the Monte Carlo simulation. Monte Carlo simulation involves artificially generating a very large set of events, correlated changes, from which VaR is derived [12].
6. Remarks
Each of these presented approaches has strengths and weaknesses. These approaches may be compared and contrasted in light of the following aspects: (1) applicability, (2) technical requirement, and (3) time requirement.
Applicability refers to how easy the uncertainty may be addressed by the various approaches. The mean-variance, sensitivity analysis, and parametric analysis allow studies on the correlated changes in either the cost coefficient, c, or the right-hand-side vector, b, only. Even though the sensitivity analysis evaluates the perturbation in c, A, and b, it evaluates the perturbation one at a time. It would be easier to resolve the linear programming problem if too many perturbations are needed in the sensitivity analysis. The stochastic linear programming approach studies the uncertainties in c, A, and b simultaneously. The fuzzy linear programming approach utilizes a broad range of techniques. The approach, in general, studies the changes in c, A, and b simultaneously. However, to use the fuzzy linear programming, either all parameters within a constraining equation are considered fuzzy (i.e., being perturbed) or none at all are fuzzy. The VaR approach is capable of including risk factors that are hard to model. To evaluate such financial risk using approaches other than VaR, linearizing the risk factor is needed. However, the VaR approach can determine such nonlinear financial risk without any prior simplification. Unfortunately, to evaluate the VaR, a decision choice must be made. This means that some other approaches must be used in addition to the VaR approach to provide a basis of valuation.
Technical requirement refers to the degree of knowledge needed in analyzing the uncertainty using the different approaches. Mean-variance, sensitivity analysis and parametric analysis, and fuzzy linear programming approaches are based on the linear programming technique. Understanding the concept of linear programming is sufficient to model an uncertain operational problem using these approaches. Fuzzy linear programming approach requires additional technical expertise as the approach borrows its concept from the fuzzy set extension theory. The approach is, nevertheless, similar to the sensitivity analysis and parametric analysis. To use the stochastic linear programming approach requires knowledge of Bender decomposition and Monte Carlo simulation techniques. As a result, the approach is more math-oriented than the rest. Understanding the Monte Carlo simulation technique is sufficient in the VaR approach
Time requirement refers to the time needed to analyze using the different approaches. The mean-variance, sensitivity analysis and parametric analysis, and fuzzy linear programming approaches are based on the linear programming technique. Sensitivity analysis and parametric analysis is the easiest approach since solving (2) has already provided information needed to analyze using the approach. Mean-variance analysis and fuzzy linear programming require remodeling (2). However, since linear programming technique can still be used to solve the model derived from the two approaches, the time requirement is considered moderate. The stochastic linear programming and VaR approaches require the Monte Carlo simulation to obtain the random changes. The time requirements for the two approaches are considerably higher. Since stochastic linear programming approach requires solving the model iteratively, time requirement for the approach is the highest.
V. AN EXAMPLE USING SENSITIVITY ANALYSIS AND PARAMETRIC ANALYSIS
Sensitivity analysis
analyzes only the desired price change in contract type 2 at a particular
period. The effect of a simultaneous
change in the market price is not considered.
To consider the simultaneous change, parametric analysis is used. To analyze the changes in the market prices,
the direction of perturbation needs first to be determined. In this paper, the standard deviations of
the market prices are used as the direction of perturbation. Equation (6) shows the market prices after
including the direction of perturbation and Table 3 shows the results of solving
(6). As 
increases, the
purchasing scheme changes. When 
, DLC program becomes desirable and all controllable customer
demand at the 5th period is deferred.
|
Variable |
Solving (A.1) |
|
|
42.44 |
|
|
0 |
|
|
40.42 |
|
|
0 |
|
|
42.44 |
|
|
0 |
|
|
0 |
Table 2. Desired price drop in contract type 2.
|
Period |
Desired price drop in contract type 2 before any purchase |
Desired price of contract type 2 should be lower than |
|
|
5 |
$5.63/MW-period |
$60.65/MW-period |
|
|
6 |
$5.51/MW-period |
$59.38/MW-period |
|
|
7 |
$5.68/MW-period |
$61.21/MW-period |
|
(6)
Table 3. Using market price s.t.d as direction of perturbation.
|
Variable |
|
|||
|
|
|
|
|
|
|
|
42.44 |
41.36 |
21.00 |
0 |
|
|
0 |
0 |
19.94 |
40.94 |
|
|
40.42 |
40.85 |
40.85 |
40.85 |
|
|
0 |
0 |
0 |
0 |
|
|
42.44 |
43.04 |
43.04 |
43.04 |
|
|
0 |
0 |
0 |
0 |
|
|
0 |
24.00 |
24.00 |
24.00 |
VI. SUMMARY
This paper describes the uncertainties that a utility serving customer demand in a deregulated power industry may face. Given uncertainties in a deregulated environment, linear programming may not be sufficient in providing the utility the desired solution on how to compete in the new environment or to maintain a sustainable profitability. Thus, various risk management and assessment tools are presented and reviewed, in hope to provide alternative solutions for the utility to consider. The pros and cons of these tools are also discussed based on their applicability, technical requirements, and time requirements. This paper ends with an example showing how sensitivity analysis and parametric analysis may be applied. The difficulties of using the sensitivity analysis and parametric analysis in the designated problem are also pointed out.
VII. REFERENCES
[1] G. B. Sheblé, Computational Auction Mechanisms for Restructured Power Industry Operation. Boston: Kluwer Academic Publishers, 1999.
[2] S. Dekrajangpetch, “Auction Development for the Price-based Electric Power Industry,” Doctoral dissertation, Iowa State University, Ames, December, 1999.
[3] K. –H. Ng, G. B. Sheblé, “Serving Customer Energy Services using Contracts and Load Management Programs,” submitted for IEEE publication.
[4] K. –H. Ng, Reformulating Load Management under Deregulation, Master’s thesis, Iowa State University, Ames, May 1997.
[5] K. H. Ng, “Operational Planning of an Energy Service Company,” Doctoral dissertation, Iowa State University, Ames, December, 2000.
[6] M. S. Bazaraa, J. J. Jarvis, and H. D. Sherali, Linear Programming and Network Flows. New York: John Wiley & Sons, Inc., 1990.
[7] F. S. Hillier and G. J. Lieberman, Introduction to Operations Research. New York: McGraw-Hill, Inc.,
1995.
[8] P. B. R. Hazell and R. D. Norton, Mathematical Programming for Economic Analysis in Agriculture. New York: Macmillan Publishing Company, 1986.
[9] H. J. Zimmermann, Fuzzy Set
Theory – And Its Applications. Boston: Kluwer-Nijhoff Publishing, 1985.
[10] H. Tanaka and K. Asai, “Fuzzy linear programming problems with fuzzy numbers,” Fuzzy Sets and Systems (11), 1984.
[11] G. Infanger, Planning Under Uncertainty: Solving Large-Scale Stochastic Linear Programs. Danvers, Massachusetts: Boyd & Fraser Publishing Company, 1994.
[12] P. Best, Implementing Value at
Risk. Chichester: New York, 1998.
VIII. BIOGRAPHIES
subject to:
(A.1)
Table 1. Parameters used in (A.1).
|
Factor |
Description |
Variable name |
Value |
Remark |
|
Customer |
variable tariff |
|
$90/ MW-period |
deterministic |
|
fixed tariff |
– |
– |
– |
|
|
demand |
|
42 MW |
uncertain |
|
|
|
40 MW |
uncertain |
||
|
|
42 MW |
uncertain |
||
|
reliability |
|
0.04 |
deterministic |
|
|
decision variables |
– |
– |
– |
|
|
Market |
price of energy |
|
$59.40/MW-period |
uncertain |
|
|
$66.28/MW-period |
uncertain |
||
|
|
$58.16/MW-period |
uncertain |
||
|
|
$64.89/MW-period |
uncertain |
||
|
|
$59.95/MW-period |
uncertain |
||
|
|
$66.89/MW-period |
uncertain |
||
|
reliability |
|
0.05 |
uncertain |
|
|
|
0.03 |
uncertain |
||
|
variability |
|
0.05 |
deterministic |
|
|
|
0.10 |
deterministic |
||
|
contract duration |
– |
1 period |
– |
|
|
decision variables |
k = 1, 2 |
– |
– |
|
|
DLC |
variable rebate |
|
$2.50/MW deferred |
deterministic |
|
fixed rebate |
– |
0 |
– |
|
|
deferrable demand |
|
0.044 MW/unit |
uncertain |
|
|
reliability |
|
0.03 |
deterministic |
|
|
Decision variable |
|
– |
– |
, 
, 
, 
, 
