Incentive effects of paying demand response in wholesale electricity markets

Hung-po Chao; Mario DePillis

doi:10.1007/s11149-012-9208-1

Journal of Regulatory Economics

June 2013, Volume 43, Issue 3, pp 265–283 | Cite as

Incentive effects of paying demand response in wholesale electricity markets

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Hung-po ChaoEmail author
Mario DePillis

Original Article

First Online: 15 January 2013

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Abstract

Recently issued U.S. Federal Energy Regulatory Commission regulations require comparable treatment of demand reduction and generation in the wholesale electric market so that they are compensated at the same market clearing price. The new regulations measure demand reduction as a reduction from a “customer baseline,” a historically based estimate of the expected consumption. In this paper, we study the incentive effects on the efficiency of the demand response regulation using a static equilibrium model and a dynamic extension of the model. Our analysis provides three main results. Firstly, our analysis shows that the demand reduction payment will induce consumers to (1) inflate the customer baseline by increasing consumption above the already excessive level during normal peak periods and (2) exaggerate demand reduction by decreasing consumption beyond the efficient level during a demand response event. This result persists when applied to alternative baseline designs in a dynamic model. Secondly, we study alternative policy remedies to restore the efficiency of demand response regulation and introduce a new approach to define the customer baseline as a fixed proportion of an aggregate baseline. In particular, the aggregate baseline approach can significantly weaken or eliminate the incentive to inflate the baseline. Finally, we illustrate that if the baseline inflation problem is solved and demand and supply functions are linear, the current policy can produce a net social welfare gain. However, the welfare improvement requires that demand reduction be paid only when the wholesale price is at least twice the fixed retail rate. This argues that the policy should include a sufficiently high threshold price below which demand response is not dispatched.

Keywords

Demand response Customer baseline Electricity restructuring Incentive effects Wholesale electricity market

JEL Classification

L43 L51

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Appendix A: the reference model

In this appendix, we describe the reference model for comparing alternative baseline designs under the current demand response policy in the wholesale electricity market. The reference model does not include the demand response policy. The symbols that will be used in the model are summarized in Table below:

$t\in \left\{ 1,\ldots ,T\right\} $: the time index

$\theta _t \in \left\{ {n,e} \right\} $: the state of nature at time $t$, where $n$ represents a normal period and $e$ represents an event period

$\lambda _{t}$: the probability of the arrival of an event in each time period t

$q_t^\theta $: the electricity consumption at time t in state $\theta $ (in MWh)

$\bar{{q}}_t$: the customer baseline at time t

$\alpha $: the parameter in the autoregression formulation

$x_t$: the generation output at time t (in MWh)

$p^{R}$: the fixed retail rate

$p_{t\theta }^W$: the wholesale price at time t in state $\theta $

$U_t(q_t^\theta )$: the gross utility function at time t in state $\theta $

$D_t(p)$: the demand function at time t

$C_t(x_t)$: the generation cost function at time t

$S_{t\theta }(p)$: the supply function at time t in state $\theta $

The reference model is used as a benchmark in comparing the performance of alternative baseline designs. The basic model structure consists of a demand function, supply function and market equilibrium, which are described below.

First, we assume that each consumer pays a fixed retail rate, $p^{R}$, and the electricity consumption at time t in state s is described by the demand function, $q_t=D_t(p^R )$. Suppose that the utility function is quasi-linear, the demand function can be obtained as a result of utility maximization as follows,

$$\begin{aligned} D_t(p^R )=Arg\mathop {Max}\limits _{q_t } \left\{ {U_t (q_t )-p^{R}q_t } \right\} \end{aligned}$$

(31)

The optimality condition yields the equality between the retail price and the marginal utility of consumption, which equals the inverse demand function,

$$\begin{aligned} {U}^{\prime }_t (q_t )=p^R \end{aligned}$$

(32)

Next, we assume that each generator is paid a time-varying wholesale price, $p_{t\theta }^W $. The electricity generation is described by the supply function, $x_t=S_t(p_t^W )$. The above relationship is consistent with the result of maximizing the expected producer’s surplus, or expected net revenue, as shown below,

$$\begin{aligned} S_t(p_t^W )=Arg\mathop {Max}\limits _{x_t } \left\{ {p_t^W x_t^W -C_t (x_t )} \right\} \end{aligned}$$

(33)

The optimality condition yields the equality between the wholesale price and the marginal cost of generation, as follows,

$$\begin{aligned} {C}^{\prime }_t (x_t )=p_t^W \end{aligned}$$

(34)

Lastly, under the market balance condition, we have $x_t=q_t$, and the social welfare function can obtained from combining (30) and (33).

$$\begin{aligned} SW=\sum _{t=1}^T {\left[ {U_t(q_t)-C_t (q_t )} \right]-} \sum _{t=1}^T {\left[ {p^{R}q_t^ -p_t^W q_t } \right]} \end{aligned}$$

(35)

Under the revenue neutrality requirement, the last term in (35) equals zero. Therefore, the retail price can be expressed as the demand-weighted wholesale prices for the price-insensitive demand:

$$\begin{aligned} p^{R}=\frac{\sum _{t=1}^T {p_t^W q_t} }{\sum _{t=1}^T {q_t}} \end{aligned}$$

(36)

Given this revenue neutrality condition, the social welfare function can be simplified as follows:

$$\begin{aligned} SW=\sum _{t=1}^T {U_t(q_t)-C_t (q_t)} \end{aligned}$$

(37)

The reference model is summarized by Eqs. (35)–(37).

Finally, to ensure revenue neutrality in the basic model with demand response, the retail rate should be adjusted to cover the demand reduction payment. Therefore, we modify Eq. (36) to let the retail rate equal the average cost that covers the wholesale costs and demand reduction payments:

$$\begin{aligned} p^{R}&= \frac{\sum _{t=1}^T {\left[ {\lambda _t \left[ {p_{te}^W q_t^e +p_{te}^W \left( {q_t^n -q_t^e } \right)} \right]+(1-\lambda _t )p_{tn}^W q_t^n } \right]} }{\sum _{t=1}^T {\left[ {\lambda _t q_t^e +(1-\lambda _t )q_t^n } \right]} }\nonumber \\&= \frac{\sum _{t=1}^T {\left[ {\lambda _t p_{te}^W +(1-\lambda _t )p_{tn}^W } \right]q_t^n } }{\sum _{t=1}^T {\left[ {\lambda _t q_t^e +(1-\lambda _t )q_t^n } \right]} } \end{aligned}$$

(38)

The market equilibrium for the case with demand response policy is obtained by solving Eqs. (2), (3), (34) and (38).

Appendix B: dynamic baseline designs

There are several methods of estimating the customer baseline based on past observations. The two challenges of the estimation techniques are first that using observations from too far back in history risks introduction of behavior and weather that are no longer relevant. Second, hours with in which demand response takes place must be excluded from the sample because they do not represent normal behavior.

To address this, designers have come up with two methods of estimation, both of which use approximately 10 observations. The first, used by ISO-NE, is an autoregressive approach. It uses a weighted average of the most recent baseline and the most recent consumption observation. The second, approach is a uniformly weighted moving average of the most recent observations.

In the following, we examine these two approaches to baseline design to see if the inter-temporal interactions that link consumer decisions across different time periods alter the insights of the static model.

Autoregressive process

Let $\bar{q}_t $denote the customer baseline at time t. In each period, we assume that the baseline evolves according to the following autoregressive process,

$$\begin{aligned} \bar{{q}}_{t+1} =\left\{ {{\begin{array}{ll} (1-\alpha )\bar{{q}}_t +\alpha q_t^n ,&\text{ in} \text{ a} \text{ normal} \text{ period} \\ \bar{{q}}_t ,&\text{ in} \text{ an} \text{ event} \text{ period} \end{array} }} \right.. \end{aligned}$$

(39)

In the above process, we denote by $\alpha $ the weighting factor for the current consumption level and 1 – $\alpha $ for the old baseline level.

Given that the demand reduction payment is incurred only in a demand response event, the consumer decision problem in (1) turns into a dynamic programming problem,

$$\begin{aligned} V_t (\bar{{q}}_t )&= \mathop {Max}\limits _{q_t^n ,q_t^e }\left( {\text{1-}\lambda _t } \right)\left[ {U_t (q_t^n )-p^{R}q_t^n +V_{t+1} \left( {(1-\alpha )\bar{{q}}_t +\alpha q_t^n } \right)} \right] \nonumber \\&+\lambda _t \left[ {U_t (q_t^e )-p^{R}q_t^e +p_{te}^W \left( {\bar{{q}}_t-q_t^e } \right)+V_{t+1} (\bar{{q}}_t )} \right] \end{aligned}$$

(40)

Therefore, each consumer solves the dynamic programming problem in (40) subject to the baseline adjustment process in (39).

We assume that the utility and cost functions are stationary over time and that the time horizon is sufficiently long so that we can focus on stationary equilibrium.⁵ We obtain the optimality conditions after dropping the time index as follows.

$$\begin{aligned} {U}^{\prime }(q^n )&= p^{R}-\alpha {V}^{\prime }(\bar{{q}}). \end{aligned}$$

(41)

$$\begin{aligned} {U}^{\prime }(q^e )&= p^{R}+p_e^W . \end{aligned}$$

(42)

From the Envelope Theorem, we have,

$$\begin{aligned} {V}^{\prime }(\bar{{q}})=(1-\lambda )(1-\alpha ){V}^{\prime }(\bar{{q}})+\lambda \left[ {p_e^W +{V}^{\prime }(\bar{{q}})} \right]. \end{aligned}$$

(43)

Simplifying (43), we obtain,

$$\begin{aligned} \alpha (1-\lambda ){V}^{\prime }(\bar{{q}})=\lambda p_e^W . \end{aligned}$$

(44)

Substituting (43) in (41) yields,

$$\begin{aligned} {U}^{\prime }(q_n )=p^{R}-\frac{\lambda }{1-\lambda }p_e^W \end{aligned}$$

(45)

Moving-average process

Let us define $\mathbf{q}_t=\left( {q_{-1}^n ,q_{-2}^n ,\ldots ,q_{-L}^n } \right)$ where $q_{-j}^n $ denotes the consumption level in the $\text{ j}$th normal period prior to time period t, and $\gamma =1/\text{ L}$ the constant weight applied to each previous period’s consumption to calculate the moving average customer baseline. In a moving average process, the baseline is updated according to the following equation,

$$\begin{aligned} \bar{{q}}_t =\gamma \left( {\sum _{s=1}^L {q_{-s}^n } } \right). \end{aligned}$$

(46)

As in the auto-regressive formulation, each consumer solves the dynamic programming problem in (40), but now subject to the baseline adjustment process in (46) rather than (39). Then, the consumer’s dynamic decision problem can be written as follows,

$$\begin{aligned} V_t (\mathbf{q}_t )&= \mathop {Max}\limits _{q_t^n ,q_t^e }\;\left( {\text{1-}\lambda _t } \right)\left[ {U_t (q_t^n )-p^{R}q_t^n +V_{t+1} \left( {\mathbf{q}_{t+1} } \right)} \right] \nonumber \\&+\lambda _t \left[ {U_t (q_t^e )-p^{R}q_t^e +p_{te}^W \left( {\bar{{q}}_t -q_t^e } \right)+V_{t+1} (\mathbf{q}_t )} \right] \end{aligned}$$

(47)

We make the same assumption as previously that the utility and cost functions are stationary over time and that the time horizon is sufficiently long so that we can focus on stationary equilibrium. The optimality conditions are similar to (41) through (43) in (48) through (51) as follows.

$$\begin{aligned} {U}^{\prime }(q^{n})&= p^{R}-\frac{\partial V(\mathbf{q})}{\partial q_{-1}^n } \end{aligned}$$

(48)

$$\begin{aligned} {U}^{\prime }(q^{e})&= p^{R}+p_e^W . \end{aligned}$$

(49)

From the Envelop Theorem, we have,

$$\begin{aligned} \frac{\partial V(\mathbf{q})}{\partial q_{-j}^n }&= (1-\lambda )\frac{\partial V(\mathbf{q}^{n})}{\partial q_{-(j+1)}^n }+\lambda \left( {\gamma p_e^W +\frac{\partial V(\mathbf{q}^{n})}{\partial q_{-j}^n }} \right) \end{aligned}$$

(50)

$$\begin{aligned} \frac{\partial V(\mathbf{q})}{\partial q_{-L}^n }&= \lambda \left( {\gamma p_e^W +\frac{\partial V(\mathbf{q})}{\partial q_{-L}^n }} \right) \end{aligned}$$

(51)

Solving (50) and (51), we obtain,

$$\begin{aligned} \left( {1-\lambda } \right)\frac{\partial V(\mathbf{q})}{\partial q_{-j}^n }=\gamma (L-j+1)\lambda p_e^W \end{aligned}$$

(52)

Substituting (52) in (48) for the case with $j$=1 yields,

$$\begin{aligned} {U}^{\prime }(q^{n})=p^{R}-\frac{\lambda }{1-\lambda }p_e^W \end{aligned}$$

(53)

Note that the effective price to the consumer in Eqs. (49) and (53) are identical to those in Eqs. (42) and (45).

Therefore, we conclude that the stationary equilibrium results for the two dynamic models are identical to the market equilibrium in the static model.

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Hung-po Chao
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Email author
Mario DePillis
- 1

1.ISO New EnglandHolyokeUSA

Incentive effects of paying demand response in wholesale electricity markets

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