Dec 8, 2011 - time periods: a forward transaction that allows consumers to select service options to hedge financial risks and a spot transaction that allows ...
Competitive Electricity Markets with Consumer Subscription Service in a Smart Grid Hung-po Chao1 December 8, 2011
Abstract This paper presents a theory of consumer subscription service, unifying priority service and dynamic pricing within a two-settlement system in ways that foster efficient risk management and competitive electricity markets. Consumer subscription service entails transactions in two time periods: a forward transaction that allows consumers to select service options to hedge financial risks and a spot transaction that allows consumers to secure electricity on demand. The difference between the forward subscription and the actual consumption is settled at spot prices determined in a competitive wholesale electricity market. Priority service offers consumers choices from a menu of reliability-differentiated service options with compensatory insurance for curtailments. Following the principles of revelation and competitive consistency, the priority service menu provides consumers opportunities to choose options that are consistent with their individual preferences and demand characteristics. Combining priority service and dynamic pricing, consumer subscription service is Pareto superior to an undifferentiated service design. Overall, consumer subscription service provides consumers incentives and tools for efficient demand management fostering price-responsive demand in competitive electricity markets. It facilitates an essential linkage between competitive wholesale and retail markets in ways that would enable consumers to be engaged in the process of the industry’s transition toward a smart grid future. Key Words: Consumer subscription service, electricity restructuring, smart grid, electricity pricing, priority service, spot pricing, dynamic pricing, uniform pricing, forward contract, risk hedging, price-responsive demand.
1
Director, Market Strategy and Analysis, ISO New England. This paper stems from my work for over two decades at Electric Power Research Institute (EPRI) in collaboration with Shmuel Oren, Stephen Peck and Robert Wilson. I am grateful to them as well as helpful comments from three anonymous referees and workshop participants at U.C. Berkeley, MIT, CRRI, EPRI, and Caltech in 2011. The opinions expressed in this paper are solely of my own and do not represent the positions of ISO New England or any other organizations.
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Competitive Electricity Markets with Consumer Subscription Service in a Smart Grid Hung-po Chao December 8, 2011
1. Introduction In 2000, the National Academy Engineering ranked “electrification” as the greatest engineering achievement of the 20th century. Combining electricity network with digital metering, communications and control technologies for better control of appliances and devices in factories, offices and homes, smart grid promises to transform the electricity network into the modern information age in ways that will rival any engineering achievement for the 21st century. As restructuring has expanded wholesale competition, smart grid opens the opportunities for retail competition with price-responsive demand. Emerging smart grid technologies enable a competitive market structure in which retail service companies would be motivated to offer differentiated electricity services for a heterogeneous consumers who may differ significantly in their willingness-to-pay for service attributes and reliability differentiated service offerings could be customized for different consumer segments.2 Competitive electricity markets are essential to promote efficient pricing and investment decisions in a smart grid. In 1978, the Public Utility Regulatory Policy Act (PURPA) introduced competition into electricity supply and initiated the process of electric industry restructuring. Electricity restructuring reflects the technological evolution underlying the industry cost structure, changed from one dominated by economies-of-scale with declining costs to one driven by diminishing returns with increasing costs. During the past three decades, however, the electricity restructuring has not been a smooth process. The California electricity crisis exposed the risks and flaws of electricity market designs that lack price-responsive demand. To date, the process has resulted in a hybrid market structure in the United States. On the one hand, competitive wholesale markets produce prices that generally reflect the dynamically time-varying marginal cost of generation. On the other hand, regulated retail rates largely retain the fixed uniform pricing structure inherited from the old vertically integrated utility. These rates tend to reflect the average cost-of-service over a period ranging from a few months to perhaps several years. 2
For example, some retail service providers may specialize in premium quality power to certain industrial and commercial customers for whom a reliable electric service is essential. Other retail service providers may offer interruptible service to those consumers who are willing to accept service interruptions in exchange for a lower electricity bill. Consumers should have open access to a menu of choices in subscribing differentiated services. Beyond that, electricity at real-time market prices could be available as the default service to all customers who want to have unrestricted access to electricity in excess of the amount covered by the basic service contracts. As a result, efficient rationing via spot pricing and priority service could replace indiscriminate load shedding and large-scale regional blackouts. See EPRI (1986).
2
Disconnect between wholesale and retail prices remains a major issue for electricity restructuring. A critical lesson learned from the 2000 California electricity crisis is that competitive electricity markets are not sustainable, when retail customers are completely disconnected from market prices. Consumer engagement is a key to the success of electricity restructuring. However, it is not enough to simply liberate the market and expect that consumers will automatically switch over in pursuit of economic efficiency. Consumers generally shy away from markets when products are complicated, supply is uncertain, prices are volatile, and information is lacking. Nevertheless, consumers do respond to economic incentives, when they make sense. To be engaged, consumers need to be an integral part of the franchise and have the power to shape the future. Despite well-documented benefits for dynamic pricing (Faruqui and Sergici, 2008), there are technological, economic and institutional barriers against price-responsive demand. First, the lack of advanced metering infrastructure presents a major technological barrier limiting a consumer’s ability to access spot prices. 3 Evidently, only the size of some large industrial and commercial customers can justify the expense of advanced metering infrastructure, communications, and enabling technologies at this time. Fortunately, over the past three decades, advances in digital technology have reduced the costs and increased the functionality of smart metering technologies and lowered the entry barrier. Recently, these technological barriers have been further reduced through federal and state policies. Second, the widespread practice of fixed uniform retail rate has been a strategic barrier that impedes price-responsive demand in wholesale and retail markets for a number of reasons. Fixed uniform retail rates provide consumers price insurances in such a way that consumers who are averse to price risks do not have incentives to switch to dynamic pricing. Moreover, by charging the same rate across a broad range of customers with diverse preferences, a fixed uniform retail rate inevitably creates cross-subsidies (Crew and Kleindorfer, 1978). Customers that consume most of their energy during low-cost, off-peak periods are charged the same price as those who consume most of their energy during high-cost, peak periods. As a result, a fixed uniform retail rate encourages lowvalue energy consumption when real-time wholesale energy prices are higher than retail rate, creating a tension between those customers who might prefer lower rates and those who prefer more reliable service. Finally, electric utilities have disincentives to promote price-responsive demand, because it generally reduces retail revenue. 4 Since 2000, the demand response issue has received much public attention, leading to new policy mandates. 3 In general public discussion, dynamic pricing often refers to a family of time-varying rates, including critical peak pricing, peak reduction credits, and real-time or spot pricing (Borenstein, Jaske and Rosenfeld, 2002). In this paper, dynamic pricing is considered equivalent to spot pricing. 4
Decoupling retail revenue from consumption levels has been introduced as a solution to these disincentives. The issue of retail rate reform has been debated within individual states. The California Public Utility Commission’s landmark decision in 2008 to adopt dynamic pricing as the default rate for all class customers is generally viewed as a positive step toward achieving priceresponsive demand.
3
In 2005, the Congress passed the Energy Policy Act that gives FERC the Congressional mandate to address the demand response issue and promote price-responsive demand in organized wholesale electricity markets. After five years of extensive studies, the Federal Energy Regulatory Commission (FERC) issued a Notice of Policy Rule-making (NOPR) in March 2010 on a demand response compensation policy (FERC 2010). After a period of vigorous debates, in August 2011, the FERC issued Order 745 stipulating the final decision on the demand response compensation, while leaving some issues open for further studies. One such open issue is the choice of customer baseline. A customer baseline is a counterfactual estimate of consumer consumption level used in determining the amount of demand reduction. The choice of customer baseline affects consumer behavior and the effectiveness of demand response programs. As we will discuss later, consumer subscription service establishes a contractual customer baseline based on the revelation principle and consumer self-selection.5 This paper presents a theory of consumer subscription service, unifying priority service and dynamic pricing within a two-settlement system in ways that foster efficient risk management and competitive electricity markets. Consumer subscription service entails transactions in two time periods: a forward transaction that allows consumers to select forward contracts that hedge price risks and a spot transaction that allows consumers to adjust their procurement at prices determined in a competitive wholesale spot market. It offers each electricity consumer choices from a menu of reliability-differentiated service options with compensatory insurance for curtailments. By design, an incentive compatible priority service menu consists of a portfolio of service offerings that are priced in a manner consistent with competitive wholesale spot prices. The information made available in a liquid and competitive wholesale market would greatly facilitate the design of priority price menu. The priority service design introduces priceresponsive demand in ways that provide efficient incentives, hedge against price volatility, and mitigate market power for a competitive and efficient wholesale spot market. By combining priority service and dynamic pricing, a consumer subscription service design is Pareto superior to undifferentiated service with uniform pricing, ensuring that no consumers would be worse off. Consumer subscription service provides consumers incentives and tools for efficient demand management fostering competitive electricity markets in a smart grid world. As a result, consumer subscription service facilitates an essential linkage between competitive wholesale and retail markets in ways that would enable consumers to be engaged in the process of the industry’s transition toward a smart grid future. The concept of consumer subscription can be traced to Panzar and Sibley (1978) who introduced the concept of customer self-rationing.6 The theories of priority service and spot pricing have traditionally followed separate paths of development in the literature of peak-load pricing. 5
Economists generally recommend a contractual customer baseline. See Bushnell, Hobbs and Wolak (2010), Chao (2010, 2011) and Hogan (2010). 6 The self-rationing concept was incorporated in an experimental program, called Demand Subscription Service, at Southern California Edison in the early 1980s (EPRI, 1986 and Woo, 1990).
4
(Crew, Fernando and Kleindorfer, 1995) The concept of spot pricing was attributable to the seminal work by Vickrey (1971) and Schweppe et. al. (Schweppe, et. al. 1982, 1987), highlighting the critical role of price-responsive demand for dynamic pricing of electric services. The concept of priority service can be traced to Marchand (1974) and Tschirhart and Jen (1979) on interruptible service. Chao, Oren, Smith and Wilson (1986), Chao and Wilson (1987) and Wilson (1989) extended the theory of priority service to include reliability differentiation, consumer self-selection and efficient rationing. This paper presents a unified treatment of priority service and spot pricing highlighting their complementarities for efficient risk management in ways that promote price-responsive demand in competitive electricity markets. The analysis includes variable demand files and consumer risk aversion, which are largely missing in previous works but pertinent to the analysis of risk management. The overall organization of the remainder of the paper is outlined below. Section 2 provides the basic building blocks that include the consumer preference structure, the supply technology and the social welfare function. Supposing that service options can be flexibly assigned to individual devices and appliances in a smart grid world, we formulate consumer behavior and market demand based on consumer decisions at the micro-consumption-unit level. The supply technology mix and portfolio choice is determined by competitive investment decisions. Section 3 presents the priority service menu design with variable demand profiles. Section 4 addresses consumer subscription service, featuring consumer risk aversion and hedging behavior, and compares it with an undifferentiated service design. Section 5 discusses the role of consumer subscription service in facilitating price-responsive demand and linking wholesale and retail markets. Section 6 summarizes the main conclusions. The proofs are provided in the Appendix. 2. The Basic Model In this section, we describe components of the basic model, including the consumer preference structure, the supply technology and the social welfare function. The basic model incorporates a probabilistic formulation of the state-dependent nature of demand and supply. The states of nature are denoted by .7 Let E denote expectations with respect to the probability measure P . The time-related variation of demand and supply can be treated in the same manner as the stochastic variation. 2.1 Consumer preferences In this paper, we focus on heterogeneous consumers with diverse preferences and demand profiles. The consumers are indexed by their type, h H , where H 1,..., H represents the consumer population. Each consumer’s type is drawn randomly from the population, where 7
Formally, a probability space is represented by (, B , P ) , where is the sample set,
probability measure on
B. 5
B
is a σ-field on
, and P is a
consumer type is privately known, but the probability distribution is common knowledge. We use bold face letters to represent random variables. Let xh : xh () denote the variable demand profile and p : p( ) denote the state-dependent electricity prices or spot prices. We distinguish between electricity consumption and the consumption of all non-electric goods and services, where the latter are bundled into a single numeraire good, denoted by x0h . Each consumer’s preference is represented by a state-dependent utility function that depends on the consumption of electricity ( xh ) and the consumption of a numeraire good, x0h . We assume that the utility function is quasi-linear:8 u h (x h ) x 0h , where u h (x h ) is a continuously differentiable non-decreasing concave function with bounded derivatives that lie within the interval, V=[0,V]. Let the price of the numeraire good be unity, and let mh denote the total expenditure of consumer h. For real-time transactions, each consumer maximizes the utility function under budget constraint:9
Max u h ( x h ) x 0h px h x 0h mh , x 0h 0, x h 0 0 x h ,x h
Maxu h (x h ) px h x h 0 mh
.
(1)
xh
In the following, we assume that the non-electric consumption in (1) is always positive, x0h 0 . That is, electricity consumption is unconstrained by the budget, and there are no income effects. Thus, we may drop mh from (1) and simply write the consumer utility function as u h (x h ) px h . The solution to the consumer decision problem in (1) dictates that the marginal utility equals the price,
uh (x h ) p .
(2)
The solution to (2) determines the individual consumer’s demand function, x h (p) which, in turn, defines the indirect utility function, w h (p ) u h ( x h (p )) px h (p ) .
(3)
The indirect utility function, w h (p) , is non-increasing and convex in p . From Shephard’s lemma, we derive the relationship between the consumer’s demand function and indirect utility function as follows,
8 See Varian (1984). A quasi-linear utility function is inherently a money metric utility function, which is broadly used in applied welfare economics. One of its advantages is that consumer’s surplus is an exact measure of social welfare rather than merely an approximation. However, this model ignores income effects when consumer’s choices are constrained by income levels. 9 We shall introduce von Neumann-Morganstern expected utility function for risk-aversion in the context of two-settlement transactions system with consumer subscription. Risk aversion has no effects on real-time decisions.
6
x h (p) wh (p)
(4)
We assume that in a smart grid world, each consumer has the ability to choose different service options at a micro-consumption-unit level of individual devices and appliances. Therefore, each micro-consumption-unit is treated as a decision entity with a consumer utility function, wh (v, p) (v p) xˆ h (v) . The micro-consumption-unit utility function can be integrated to obtain the consumer utility function as follows, V
0
V
w h (v, p)dv (v p)xˆ h (v)dv w h (p) . p
(5)
Therefore, the micro-consumption-unit model leads to no loss of generality. The market demand function is defined as the sum of individual consumer demand functions, H
D(p) x h (p)
(6)
h 1
The gross aggregate utility function is defined for a total consumption, Q, of the consumer population as follow, H H U (Q) Max u h (x h ) x h Q, x h 0, for h 1,..., H . xh h 1 h1
(7)
Given the market price, p, the aggregate utility function, defined as U(Q) pQ , and individual consumer utility functions are maximized simultaneously under the following set of conditions,
U(Q) uh (x h ) p, for h 1,..., H .
(8)
The consumers’ surplus, obtained from the integral of the market demand function, equals to the aggregate consumer utility function:
CS(p) D(p) dp p
H
H
x h (p) dp w h (p) h 1
p
(9)
h 1
H
u h (x h (p)) px h (p) U(D(p)) pD(p) h 1
Let CS E CS (p ) denote the expected consumers’ surplus. 2.2 Supply technology
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On the supply side, we assume that there is a continuum of investment opportunities with varying capacity cost, k, and operating cost c.10 The investment opportunities are represented by for the unit capacity cost, including the cost of capital, as a function of the operating cost, , . We assume that this function is decreasing in . If demand were certain and constant, the lowest cost technology, namely the base load unit, would be the only supply option that gets built to meet the load. But when load varies stochastically, and output is not storable, idled baseload capacity is expensive. Some peaking units may be off-line most of the time, running only for very low-probability contingencies each year. The efficient result is to invest in a balanced portfolio of technologies with varying capacity and operating costs that minimize the overall production cost. Let y (c) denote the installed capacity level of units with an operating cost c. Let y (c ) be the available capacity of operating cost c. The available capacity may be lower than the installed capacity level due to random outages, water flows or wind speeds. We suppose that in a competitive wholesale market, all available units will be run in economic merit order from the lowest operating cost to higher costs. The supply function can be obtained as the cumulative output from available units with operating costs below the price, p, p
S(p) y (c)dc .
(10)
0
The total capacity cost is
c
c
k (c ) y (c) dc .
(11)
The total operating cost is p
C(S(p)) cy (c)dc .
(12)
0
By differentiating (12), we can obtain the market equilibrium in which the marginal cost equals the market price, C(S(p)) p , if y (p ) 0 .
(13)
The producers’ expected surplus equals the expected short-run profit minus the capacity investment cost: PS E pS (p ) C (S (p )) k (c ) y ( c ) dc . c
c
(14)
10
The continuous formulation of supply technology simplifies the presentation. Chao (2011) presents an equivalent formulation with discrete technologies.
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2.3 Social welfare function As is standard in welfare economics, we define the social welfare function as the sum of consumers’ and producers’ surplus. We consider its formulation for two cases: dynamic pricing and uniform pricing. For the case with dynamic pricing, the spot price is determined by the market equilibrium condition,
D(p* ) S(p* ) .
(15)
For this case, the social welfare function can be written,
c
SW (p * ) CS PS E U ( D(p * )) C(S (p * )) k (c ) y (c ) dc . c
(16)
Dynamic pricing is known to achieve socially efficient allocation. As we show later, priority service achieves the same socially efficient allocation. Under uniform pricing, the price reflects the average cost, which equals the demand-weighted expected spot prices,
E p * D( p ) p . ED( p )
(17)
For this case, consumers respond to the uniform price rather than the spot price, and market demand may exceed the supply limit. When that occurs, non-price rationing or forced outage is inevitable. For the case with uniform pricing, the social welfare function should include the additional rationing cost, RC: 11 c SW ( p ) E U (Q ( p )) C(Q ( p )) k (c ) y (c ) dc RC .
(18)
c
Where Q( p ) MinD( p ), S(c )
The term, RC, represents the cost of non-price rationing, which includes the value of lost consumption, spoilage during outages and other administrative costs.
In general, dynamic pricing is more efficient than uniform pricing, i.e., SW (p* ) SW ( p) . Uniform pricing tends to produce over-consumption when the spot price is high and underconsumption when the spot price is low. Moreover, extra social costs are incurred for non-price 11
*
weighted expected spot price. See Chao (1983).
9
*
*
*
In comparison, the optimal ex ante uniform price equals p E p D' ( p ) E D' ( p ) , which is the marginal-demand-
rationing during a supply shortage. With competition, investors are expected to earn zero profits in the long run, and consumers are expected benefit from dynamic pricing. Then, why do many consumers still prefer uniform pricing? A main concern is that under dynamic pricing, consumers are exposed to price risks. Some consumers may prefer uniform pricing due to risk aversion. Moreover, some consumers may adhere to uniform pricing because they benefit from the cross subsidies embedded in uniform pricing with significant electricity consumption during the peak periods when the spot prices are high. We will return to these topics after the discussion of priority service mechanism in the next section. 3. Priority Service Mechanism Priority service refers to a menu of contingent forward delivery contracts. Each customer’s selection of a contract from the menu determines the customer’s service priority. In each contingency, the service is delivered in order of the selected priorities. Fundamentally, priority service embodies the dual features of product differentiation and efficient rationing (Chao and Wilson, 1987; Wilson 1989). In the following, we analyze the priority service menu design based on an approach that recognizes its close relationship with the spot market. A priority service menu consists of service options, each characterized by a strike price and a demand profile θ Θ We assume that Θ is a Hilbert space that includes all possible demand profiles. Each service option is a contract for the delivery of a specified demand profile according to a priority ranking that is increasing with the strike price, V .12 We denote the menu of priority service options by M ( , θ, Ps ( , θ)) | V , θ Θ , where Ps ( , θ) is the price schedule for service option with the strike price and demand profile . In the following, assuming that a competitive wholesale market is available, we focus on a priority service design that is compatible with the spot prices established in the wholesale market. We focus on consumer decisions at the micro-consumption-unit level. To simplify notation, we index each consumption unit by the marginal value of consumption (v) that reflects consumer’s willingness-to-pay and the demand profile (q), and will drop the consumer index h when it creates no confusion. Suppose that for each consumption unit, (v, q) , the consumer chooses a service option (v, q) and a demand profile θ(v, q) . Given the spot prices that are available in a competitive wholesale market, each priority service option is structured as a contingent contract that ensures delivery of service, if and only if the spot price is below the strike price. If the spot price is above the strike price, the service will be curtailed; in that case, however, the consumer will be compensated by an amount equal to the strike price, . Therefore, from a buyer’s viewpoint, each priority service option represents a 12
The standard theory of priority service has two restrictive assumptions. First, the demand profile is constant. Second, consumers are risk neutral. In Sections 3 and 4, these assumptions are relaxed.
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curtailable service contract13 with interruption insurance. From a seller’s viewpoint, each priority service contract entails a call-off option with a strike price so that the supplier can curtail the service when the spot price exceeds the strike price. The price menu is constructed by leveraging on the spot prices in a competitive wholesale market. In comparison, the standard approach relies on the revelation principle based on the information provided through customer self-selection. We define the rationing function as a random variable that assumes a value of 1 when the service is delivered and a value of 0 when the delivery is curtailed. Consistent with the merit-order optimal dispatch procedure adopted in the system operation, the rationing function is an indicator function,
1, if p ; ρ( ) 0, if p .
(19)
That is, a service will be delivered, if and only if the spot price, p , is lower than the strike price, . The product of the rationing function and the demand profile, ρ( )θ , is called the delivery
profile. We shall call the expected delivery profile, ( ) E ρ ( )θ , the service delivery function. The probability distribution of the spot price defines the service reliability function: r ( ) E ρ( ) Prp .
(20)
If the demand profile is unity, i.e., θ 1 , then the service delivery function is identical to the service reliability, i.e., ( ) r ( ) . A priority service menu is consistent with competitive spot prices, if there is zero profit from arbitrage between priority service options and the spot prices established in competitive wholesale markets. Such a menu design can be constructed by invoking the revenue equivalence theorem in competitive markets: the expected payment under priority service should be the same as the expected procurement cost in a spot market. Such a menu design can be described as following the principle of competitive consistency. Proposition 1. A priority service menu M ( , θ, Ps ( , θ)) | V , θ that is consistent with competitive spot prices is given by
Ps ( , θ) E θ p θ (u )du .
0
(21)
13
Curtailable service is a familiar idea. During the 1980’s, for example, PG&E in California offered a curtailable service for large customers that were compensated if they reduced their load when the frequency of the supply was deficient. Effectively, curtailable service contracts give the supplier a call option to interrupt the service at high prices. A key feature of priority service mechanism is that consumers are allowed to choose from a range of options that could be offered by multiple suppliers competitively rather than only a single option by a single supplier.
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Suppose that a consumer of type (v, q ) selects service option, ( , θ) . If the service is delivered, the actual consumption would be Min{q, θ} ; otherwise, the curtailment compensation for nondelivery is θ . Therefore, the consumer utility function can be written as φ( , θ | v, q ) E ρ ( )vMin{q, θ} 1 ρ ( ) θ Ps ( , θ ) .
(22)
The consumer utility maximization problem can be stated as,
(v, q ) Max φ( , θ | v, q ) V , θ Θ.
(23)
,θ
We denote by * , θ* * (v, q), θ* (v, q) the optimal solution to the above utility maximization problem. In the standard theory of priority service, the priority service menu design follows the revelation principle. According to the revelation principle, the priority service menu M is incentive
compatible, if the optimal decision reveals the consumer type, i.e., * , θ* v, q , for all v V , q . We show below that the principle of competitive consistency implies incentive
compatibility and is thus aligned with the revelation principle. Proposition 2. A priority service menu design is incentive compatible, if it is consistent with competitive spot prices. In general, the revelation principle does not produce a unique menu design, because the price schedule can be shifted by a constant without violating incentive compatibility. However, under the requirement of revenue equivalence, incentive compatibility and competitive consistency produce an identical menu design. To ensure revenue equivalence, we augment incentive compatibility with the requirement that the priority charge is zero for consumption with zero valuation, i.e., Ps(0,)=0. The following result summarizes the relationship between the competitive consistency and the revelation principles. Proposition 3: For a given priority service menu M ( , θ, Ps ( , θ)) | V , θ , the following statements are equivalent: 1) The priority service menu is incentive compatible with Ps(0,)=0 2) The priority service menu is consistent with competitive spot prices 3) The priority price schedule is
Ps ( , θ) E θ p θ (u )du .
0
(24)
Traditionally, priority service and dynamic pricing are considered alternative approaches that can achieve the same socially optimal allocation and consumer benefits. In the unified framework, they are complements instead of competing substitutes. A liquid and competitive spot market 12
greatly provides useful market information that facilitates priority service menu design. As we will discuss in the following section, priority service provides heterogeneous consumers differentiated risk management products that facilitate price-responsive demand for a competitive spot market. 4. Consumer Subscription Service In this section, we discuss the conceptual framework of consumer subscription service combining priority service and dynamic pricing into a unified market design. Consumer subscription service entails a two-settlement transaction system consisting of a subscription service and a supplemental default service. As a forward transaction, the subscription service allows each consumer to choose specific levels of reliability-differentiated service options from a priority service menu. As a spot transaction, the supplemental default service allows consumers to adjust their consumption levels. The differences between the subscribed forward contracts and the actual consumption are settled at spot prices established in the wholesale market. Essentially, priority service allows consumers to choose customized service options under forward contracts at pre-determined prices, while dynamic pricing is applied to the real-time deviations relative to the baselines established in the forward contracts based on consumer’s subscription decision. In this section, we continue to focus on consumer decisions at the microconsumption-unit level. 4.1 A two-stage consumer decision model Suppose that for each micro-consumption-unit, (v, q) , the consumer subscribes a priori a service option (v, q) and a demand profile z(v, q) , prior to the actual consumption decision in terms of a demand schedule, x(v, q) . For simplicity, these decisions may be referred to as , z, and x. To incorporate risk aversion, we use a von Neumann- Morgenstern expected utility function formed by taking the expectation of a Bernoulli utility function, U(.), over the consumer utility function, i.e., EU (φ( , z | v, q)) (Arrow, 1971). We assume that the Bernoulli utility function U(.) is a continuously differentiable, increasing and concave function. A consumer’s decision process is conducted in of two stages. First, the consumer subscribes a priority service option (, z) and then chooses the demand schedule x, after the spot price, p, is revealed. If the spot price turns out to be lower than the strike price of the subscribed service option, then the subscribed service will be delivered, i.e., ρ( ) 1 . In this case, the difference between the actual consumption level, x, and the subscribed demand profile, z, would be settled at the spot price for a net payment of p(x z) , .14 On the other hand, if the spot price is higher than the strike price, the service will be curtailed, i.e., ρ( ) 0 , but the consumer will be compensated for curtailment of the scheduled delivery by an amount, z. In this case, the 14
The resale may require an administrative fee. Here, we ignore the administrative consideration and focus on the hedging value.
13
consumer will have to procure the actual consumption, x, at the spot price, p , for a net payment of px z . The decision problem for the second stage can be formulated as, φ( , z | v, q )
Maxρ( )vx p ( x z ) 1 ρ( ) vx px z Ps ( , z ) 0 x q x
(25)
Max(v p ) x ρ( ) p z z Ps ( , z ) 0 x q x
(v p ) q p z z Ps ( , z )
The last step of (25) follows from the observation that the optimal consumption decision in the second stage is to consume the maximum amount of q, if and only if the spot price is below the marginal value of consumption, i.e., x ρ( )q .15 In effect, consumer subscription mitigates the financial risks in spot transaction and shifts the transaction into a forward hedging contract under priority service. Without loss of generality, we consider a two-part pricing structure that can achieve a full hedge:
Ps ( , θ) (θ ) (u )du .
(26)
0
The first term on the right side of (26) is a contingent adjustment payment that modifies the insurance contract by replacing the expected demand with the actual demand profile as the basis for compensation. Since the adjustment yields zero expected payment, the price schedule remains arbitrage free. As will be shown below, the price schedule is incentive compatible. The objective for the subscription decision problem in the first stage is to maximize a von Neumann- Morgenstern expected utility function as follows:
(v, q) Max EU φ( , z | v, q) V , z Θ Max E U (v p) q p z z (u)du V , z Θ 0
.
(27)
The optimal subscription decision for (27) is denoted by * (v, q), z * (v, q) * , z * .
Proposition 4. The price schedule (26) is incentive compatible, i.e., * , z * (v, q) and the optimal subscription decision leads to a perfect hedge. The subscribed service option becomes a perfect hedge when the random variables in the argument of the Bernoulli utility function vanish. For consumers who are risk neutral, they are indifferent about shifting the financial risks from spot transactions to forward hedging contracts. However, for consumers who are risk averse, hedging could make a significant difference. 15
In contrast, for a pure priority service, the actual consumption is x ρ( ) Min{z, q} .
14
In the above analysis, perfect hedge hinges on the assumption that the priority service menu includes a complete collection of variable demand profiles so that consumers can choose customized options that perfectly match their individual characteristics. In reality, complete service menu is impossible, and the demand may not be totally heterogeneous (for example, all air conditioning users share similar demand profiles.). Therefore, in implementation, the priority service menu may include a much smaller set of standard demand profiles. For illustrative purposes, we examine partial hedging under a priority service menu design that includes a single standard demand profile. Let σ denote the standard demand profile with E{}=1 and z denote the subscription level. To focus on the optimal hedging behavior, we 2 assume that the Bernoulli utility function is of a quadratic form, Uφ φ φ ,16 and that
consumers have chosen the strike prices that reflect their willingness-to-pay. Then, we may simplify the problem in (27) as follows,
(v, q) Max EU φ( z | v, q) z 0
. Max E U (v p) (q zσ) z (u)du 0 z 0
(28)
Proposition 5. The optimal subscription level for priority service menu with a standard demand profile , if positive, is given by
z*
Cov (v p) q, (v p) σ . Var (v p) σ
(29)
If the individual demand profile is proportional to the standard demand profile, then the optimal subscription decision results in a perfect hedge. That is, if q q σ , then z * q . An example We illustrate the previous results with an example. We assume that the demand and supply functions are given by D( p) αD( p ) αp S ( p ) β S ( p ) βp
.
(30)
where α and β are random variables independently and uniformly distributed on the unit interval [0,1]; and are, respectively, the elasticities of demand and supply. The market equilibrium is given by, 16
The quadratic utility function is widely used as a convenient representation of individual choice under uncertainty. But it has well known limitations. For example, it exhibits increasing absolute risk aversion. This means that an individual’s holding of risk assets will decrease with wealth, or risky assets are inferior goods, an implication that violates the empirical observations (Arrow, 1965).
15
p* α1/( )β 1/( ) and Q* α /( )β /( ) .
Suppose that 0.5 and 1.5 . The expected spot price is p E p* 1.333 . The service reliability function, or the probability that the service will be delivered, is. v2 , if 0 v 1 r (v) Prp v 2 1 1 2 , if v 1 2v
.
(31)
The priority price schedule that will induce a consumer with constant demand profile to selfselect the given service option, is
v3 v 6 , if 0 v 1 v Ps (v,1) v r (u )du . 0 4 1 , if v 1 3 2v
(32)
Notice that Ps (v,1) / v 1 r (v) 0 . Therefore, the priority price schedule Ps (v,1) is an increasing function of v. Similarly, for consumers with a demand profile θ 2α , the service delivery function and the priority price schedule are, v4 if 0 v 1 , (v ) 3 1 22 , if v 1 3v
.
(33)
v5 v , if 0 v 1 v . Ps (v, θ) v (u )du 15 0 8 2 , if v 1 5 3v
(34)
Notice that Ps (v, θ) / v 1 (v) 0 . We consider a case in which all consumers have the same demand profile as the standard service offering: q== In this case, everyone will choose the priority service option that forms a perfect hedge. The price for the uniform service is p E θp* 8 / 5 . By switching from uniform service to priority service, the net benefit for a consumer with valuation v is
16
v5 if 0 v 1 15 , 8 2 8 B(v) v , if 1 v 5 5 3v 2 8 , if v 3v 5
.
(35)
The net benefit is positive for every v, and it reaches a maximum value of 5/12 at v p 8 / 5 . 4.2 Comparison with undifferentiated service In this section, we compare the consumer subscription service design based on priority service and dynamic pricing with a subscription service design with undifferentiated service and uniform pricing. Under undifferentiated service, the utility offers a uniform service at a fixed price and a supplemental default service for actual consumption that exceeds the subscription level. The supplemental service provides a volumetric insurance, which meets the regulatory requirement of obligation-to-serve. For simplicity, we assume that the standard service offering has a constant demand profile, and the fixed price equals the expected spot price, p Ep . The supplemental price p equals the average procurement cost from the spot market, p E p * D(p * ) E D(p * ). The supplemental price is generally higher than the standard price, and the difference, ( p p) , is the volumetric insurance premium. Under uniform service, the decision problem for each consumer is to choose z that maximizes the expected consumer utility as follows,
Uniform (v, q) Max E U v p z (v p ) q z z 0
(36)
The subscription level, z, is positive only if v p . If we substitute priority service for the uniform service, then the decision problem becomes
Pr iority (v, q) Max E U v Ps (v) z (v p) q z z 0
(37)
We show that no consumer can be worse off by switching from uniform service to priority service. Proposition 6: Priority service is Pareto superior to uniform service, i.e., Pr iority (v, q) Uniform (v, q) , for all v [0, V ] and q Θ . Proposition 6 extends previous results in priority service to the two-settlement system with riskaverse consumers and variable demand profiles. For these consumers, priority service is more 17
attractive than uniform service because it allows differentiated risk hedges. Some consumers with high willingness-to-pay may select high service reliability and pay a premium price, and those with a lower willingness-to-pay may select lower service reliability and pay less. Proposition 7: In the absence of risk aversion, all consumers would switch from uniform pricing to dynamic pricing in equilibrium. Proposition 7 indicates that uniform pricing is unsustainable when consumers are risk neutral. Intuitively, with uniform pricing, there are always some consumers who are cross-subsidizing others. These consumers should be better off switching to dynamic pricing, unless they are averse to price risks. The migration should continue until all consumers have switched to dynamic pricing. Although with risk aversion, some consumers might be willing to pay a premium for a price hedge, however, such consumers would be better off hedging price risks through priority service. Therefore, combining priority service and dynamic pricing, consumer subscription service could provide an attractive alternative for all consumers and promote priceresponsive demand. For public utilities and regulators, the implementation of new tariff is often constrained by its distributional effects for consumers. In general, Pareto-improving is considered a favorable design feature. A good example is the introduction of two-part tariff. Pareto improving change is possible by introducing an appropriately designed two-part tariff is introduced as an option available to customers who are on a linear and uniform tariff (Wilson 1993).17 This suggests that if the two-settlement system of consumer subscription service is introduced as an option available to consumers who are on an undifferentiated service and uniform tariff, consumers may make the transition to consumer subscription service voluntarily with minimum regulatory intervention.
5. Linking Retail and Wholesale Electricity Markets 5.1 Price-responsive demand The retail market could be connected with the wholesale market in much the same way as the forward market is connected with the spot market. In the retail market, the forward contracts in the priority service menu are constructed on the basis of the spot prices expected in the wholesale market. Consumer subscription produces a portfolio of priority service contracts with a schedule of strike prices, which reflect the incremental values of consumption. These contracts provide individual consumers hedges against price risks. The aggregate schedule of these contracts forms a price-elastic demand function, H
V
Ds ( p ) z h* (v)dv . h 1
(38)
p
17
It is emphasized that electing the two-part tariff must be optional, not required.
18
In real time, the demand delivered under subscription service decreases in as the spot price, p, because low value consumption units would be curtailed before those with high values. In the wholesale market, this aggregate demand function and the residual real-time demand function will together form the total demand function, D( p) . By providing price elasticity in the realtime market demand, consumer subscription provides an effective measure to mitigate market power fostering a competitive wholesale market.
5.2 Forward procurement mechanism Given the technical and institutional barriers to price-responsive demand (FERC 2006), the imperfections in price-responsive demand necessitate the use of price cap that may lead to situations in which the market revenue may be inadequate to compensate investors for achieving optimal investments (EPRI 2005). This creates, so called, the “missing money” problem. Mechanisms, such as capacity procurement auctions, are used to restore investment incentives for the purpose of attracting an efficient portfolio of capacity investments. This issue is generally known as the resource adequacy problem that calls for forward capacity procurement (Oren 2005b). Additionally, forward procurement mechanism may be needed to acquire adequate long-term reserves to fulfill the system reliability management requirements. A well-designed forward procurement mechanism has a number of desirable features. First, it should ensure ample generation capacity to support consumers’ subscription decisions. Second, it should encourage long-term contracting that provides mutual hedge for the utility and the generators for efficient capacity investments. Long term contracts can assure sustainable prices to stimulate new investment. Third, it should mitigate market power in spot markets. Lastly, it should facilitate competitive wholesale and retail markets by transferring transactions from spot market to forward market where supply is more elastic and contestable with opportunities for new investment and entry. In forward markets, supply elasticity is substantially greater, and correspondingly the market power of incumbent suppliers is less. More importantly, capacity expansion and new entry are more feasible on a longer time frame before the spot markets occur. In particular, the procurement auction designs for capacity obligations proposed by Chao and Wilson (2004) and Oren (2005a) complement the consumer subscription service approach in meeting those requirements. The capacity requirements can be structured in terms of option contracts with a range of different strike prices that provide the utility financial hedges in meeting consumers’ demand for subscription service. The portfolio of option contracts obtained from all energy offers effectively creates an elastic aggregate supply function. In the wholesale spot market, capacity options are dispatched in merit order based on the energy offer price. The capacity options will be called to provide energy when prices exceed the strike prices of these options. On the other hand, when the price exceeds the consumer’s willingness-to-pay, the utility can limit the price rise by curtaining the customers and paying them the curtailment compensations in the subscription service contracts, thus reinforcing market competition and limiting the total energy payment required to fulfill the obligation to consumers’ subscription. 19
The priority service options and capacity options provide price-elastic demand and supply functions, which will be cleared along with the residual price-responsive demand and supply to set the spot energy market price. These price-elastic demand and supply would provide a natural antidote to the incentive to exercise market power that may remain among the residual suppliers to inflate their energy offer prices above marginal costs. As a result, each supplier will be motivated to make competitive energy offers that reflect the actual marginal cost in its offer prices.
5.3 Long-term investments With price-responsive demand, the competitive wholesale and retail markets achieve the maximum social welfare. The socially optimal capacity expansion can be obtained through decentralized investment decisions in competitive wholesale and retail markets. With free entry, the optimal capacity investments would yield zero long-term profits in the wholesale spot market,
k ( c ) E α p c .
(39)
This implies that the investment costs should be recovered by the value of call option with a strike price equal to the operating cost of the technology. As we show below, with consumer subscription service, the value of the call option equals the price differential between the top service priority and the marginal service priority valued at the operating cost of the technology. Proposition 8. An efficient investment portfolio satisfies the following set of conditions: k (c ) Ps (V , α ) Ps (c, α ) , for all c c, c
(40)
The priority price menu Ps (,) in (40) is given by (21). Proposition 8 establishes the basic investment criteria for achieving the optimal capacity portfolio with consumer subscription service in competitive wholesale and retail markets. With price-responsive demand, investors are enabled to recover investment costs in competitive markets, thus resolving the “missing money” problem.
5.4 Dynamic interactions between markets and system control The two-settlement transaction system complements a two-tiered system architecture created by electricity restructuring, with an economic layer of market transactions that interacts with the engineering layer of system control and reliability management (Chao and Peck, 1998; Chao et. al., 2005). Consumer subscription service will affect in important ways the dynamics of such interactions, including the procurement of ancillary services and capacity resources to meet reliability requirements and the provision of price-responsive demand that enables efficient price
20
signals. However, an in-depth analysis of the dynamic economic and engineering interactions is beyond the scope of this paper, and it remains an open topic for future research.
6.
Conclusion
In this paper, we develop a theory of consumer subscription service by unifying priority service and dynamic pricing within a two-settlement system in ways that foster efficient risk management and competitive electricity markets. Within such a system, consumers can hedge price risks through priority service and reap economic benefits through dynamic pricing. The priority service menu design follows the principle of competitive consistency and the revelation principle. It provides heterogeneous consumers an array of differentiated service options that can be customized through subscription choices to match their individual demand characteristics. Combining priority service and dynamic pricing, consumer subscription service is Pareto superior to an undifferentiated service design. It promotes price-responsive demand in competitive wholesale and retail market in ways that preserve economic efficiency, hedge against price volatility, and mitigate market power. Overall, consumer subscription service facilitates an essential linkage between wholesale and retail markets in ways that would enable consumers to be engaged in the process of the industry’s transition toward a smart grid future.
Appendix PROOF of Proposition 1: For the arbitrage profit to be zero, the price of each priority service option should be equal to the expected total of procurement cost and curtailment compensation,
Ps ( , θ) Eρ( )p 1 ρ( ) θ E θ p θ .
(A1)
By definition, the following two terms are equivalent,
( p) θ ρ(u )θdu .
(A2)
0
Substituting (A2) in (A1) and interchanging the expectation and the integral, we obtain, Ps ( , θ) E θ ρ(u )θdu (u )du . 0 0
PROOF of Proposition 2: Substituting (21) in (22), we obtain the consumer utility function,
21
(A3)
φ( , θ | v, q )
E ρ( )vMin{q, θ} θ p θ E ρ ( )vMin{q, θ} θ ρ( ) p θ. E ρ ( )vMin{q, θ} pθ
(A4)
E ρ ( )v p θ | 0 θ q
Solving the linear programming problem in (A4), we obtain the optimal solution for the demand profile as θ* q , when p v , and θ* 0 , when p v , or equivalently, θ* ρ(v)q . Thus, we can write as
φ( , θ * | v, q ) E ρ( )v p q .
(A5)
Then, the optimal solution for the strike price is * v , because if v , then the consumer utility can only be smaller when the spot price lies between and v, and remains the same if
v or v . With * v , the two demand profiles, θ* ρ(v)q and θ* q yield the same
utility. Therefore, we have proven that * , θ* v, q is optimal. PROOF of Proposition 3: Proposition 1 establishes that statement 2 and statements 3 are equivalent. Proposition 2 establishes that statement 1 follows from statement 2. What remains to be shown is that statement 3 follows from statement 1. Suppose that the priority service menu M is incentive compatible with Ps(0,)=0. Then, for u , we have
( , θ) ( , θ | , θ) (u , θ | , θ) Eρ(u ) p θ . Eρ(u )u p θ ρ( ) u θ (u , θ) q (u )( u )
(A6)
Rearranging terms in (A6), we have
( , θ) (u, θ) q(u )( u )
(A7)
Switching and u in (A7), we obtain
( , θ) (u, θ) q( )( u ) .
(A8)
Combining (A7) and (A8) gives the following inequalities
22
q(u )( u ) ( , θ) (u, θ) q( )( u ) .
(A9)
Dividing (A9) by ( u ) and then passing to the limit as u approaches produces a differential equation,
( , θ) q( )
(A10)
The condition of Ps(0,)=0 implies the initial condition that (0, θ) 0 . Integrating (A10) with
(0, θ) 0 yields,
(v, θ) v Ps (v, θ) (u )du v
(A11)
0
PROOF of Proposition 4:
First, we observe that if * , z * (v, q) , then uncertainty disappears in argument of the Bernoulli utility function:
φ(v, q; v, q) E (v p) q .
(A12)
In the following, we show that the solution (v, q) yields the highest von-Neumann-Morgenstern expected utility is achieved when the uncertainty in the Bernoulli utility function vanishes, and this occurs when ( * , z * ) (v, q) . It follows from the concavity of the Bernoulli utility function that for any ( , z ) , we have
EU φ( , z; v, q)
U E(v p) q p z z P ( , z)
E U (v p) q p z z Ps ( , z) .
(A13)
s
From Proposition 2, we can write
U E(v p) q v p z vz P (v, z) . U E (v p) q p z z Ps ( , z)
U E (v p) q U φ(v, q; v, q)
s
(A14)
Combining (A13) and A(14) yields,
EU φ( , z; v, q) EU φ(v, q; v, q) .
(A15) 23
PROOF of Proposition 5: With a quadratic Bernoulli utility function, the decision problem in (28) can be written as, (v, q) Max E U (v p) (q zσ) z (u )du 0 z 0
Max E (v p) q E (v p) (q zσ) zE (v p) σ z 0
.
(A16)
2
If z* 0 , the optimality condition can be written as follows,
σ 2zVar (v p) σ 0
2E (v p) q (v p) σ E (v p) σ 2zE (v p) σ E (v p) σ
2Cov (v p) q, (v p)
. (A17) 2
Solving (A17) yields the optimal subscription level,
z*
Cov (v p) q, (v p) σ . Var (v p) σ
(A18)
PROOF of Proposition 6: Comparing (36) and (37), we only need to proof the following,
v Ps (v)z (v p) z , for all v [0, V ] and
z 0.
(A19)
If v p , the consumer’s surplus is zero under undifferentiated service, but under priority service, it is always non-negative:
v Ps (v) R(v) 0 (v p) .
(A20)
If v p , the consumer’s surplus can be written,
v Ps (v) R(v) E(v p) E(v p) (v p) .
(A21)
From (A20) and (A21), we obtain A(19). PROOF of Proposition 7: Suppose that for some m < H, consumers h 1,..., m prefer uniform pricing in equilibrium. Without loss of generality, we assume that their subscription levels are zero. We denote, 24
Epxh (v) Epq h (v) ph ( v ) , for h 1,..., m , Exh (v) Eq h (v)
(A22)
and p
Epx ( p) p (v) Eq (v)dv . Ex ( p) Eq (v)dv m
m
h 1 m
V
h 1 p
h
m
h 1
h
h
h
(A23)
V
h
h 1 p
Given a heterogeneous consumer population, ph (v) cannot be constant. There exists a consumer i with vi such that pi (vi ) p .
If v i p , we have
E (vi p) q i (vi ) 0 E (vi p) q i (vi ) .
(A24)
If v i p , we note that pi (vi ) p and thus have,
E (vi p) q i (vi ) E(vi p)q i (vi ) E(vi pi (vi ))q i (vi ) E (vi p) q i (vi ) . (A25) From (A24) and (A25), we have shown,
E (vi p) q i (vi ) E (vi p) q i (vi ) , for all vi [0, V ]. This means that the consumer prefers dynamic pricing, contradicting the assumption that the consumer prefers uniform pricing. Therefore, no consumer could be better off choosing uniform pricing. PROOF of Proposition 8: The result follows from the free entry condition (39) and a straightforward mathematical manipulation,
E α V V p αc c p k (c ) E α p * c
*
*
(A26)
Ps (V , α ) Ps (c, α )
25
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