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# Electricity Demand Response and Optimal Contract Theory

Part of the equation to achieve the 2015 United Nations Climate Change Conference (COP21) objective of limiting climate change effects to a 2-degree Celsius increase relies on the design of carbon-free electric systems. According to the International Energy Agency’s 2015 report on carbon emission from fuel combustion, more than a third of the world’s carbon emission for energy systems comes from power generation. The massive development of renewable energy sources worldwide, particularly solar and wind power, is helping us reach the COP21 objective. However, these sources are simultaneously reshaping the management of power systems.

### Decarbonation of Power Systems

Renewable energy sources are non-dispatchable and highly intermittent. The root mean square of the error forecast for the production of a wind farm in six hours can reach 20% of its installed capacity. These increases in uncertainty of power generation have put flexibility at the heart of system design for large-scale renewable energy sources.

One can increase the flexibility of power systems in two possible ways: acting on the generation side by adding batteries, or acting on the demand side by developing new demand response programs. We are interested here in the second tool. Many Organisation for Economic Co-operation and Development countries have made significant investments in the development of smart meters. Better communication with consumers is necessary to implement efficient demand-response programs. 45 million smart meters have already been deployed in Italy, Sweden, and Finland; there is an ongoing investment of 45 billion euros to reach the level of 200 million appliances in the EU-27, based on the European Commission’s Energy Efficiency Directive. Nevertheless, proposed demand-response schemes are generally used to shave peak-load demands. The need for flexibility in new power systems calls for a continuous assessment of large variations of net consumption over time addressed to the grid.

Recent progress in the theory of incentives and optimal contract allows researchers to design mechanisms that adapt demand to the flexibility capacity of power systems by incentivizing consumers to reduce the volatility of their consumption.

### Contract Theory and Electricity Demand Response

Contract theory is a field of microeconomics that analyses the interaction of economic agents linked by contract. This framework covers situations as different as relationships between stockholders and managers, managers and employees, and land owners and farmers. In each case, one side of the contract relation—the principal—is looking for an incentive mechanism that will lead the other side—the agent—to act within the principal’s best interests. The problem is complicated by the fact that the principal can only observe and contract on the results, and not on the agent’s efforts. Contract theory is thus about finding the optimal incentive mechanism to maximize the principal’s utility, while knowing that the agent will take advantage of the contract design only in her own interest. Of note is the 2016 Nobel Prize in economic sciences, which was awarded to Bengt Holmström and Oliver Hart for their crucial contribution to the theory of contract in continuous time.

Figure 1. Consumption in red and blue have quadratic variations. See article for details. Image credit: René Aïd, Dylan Possamaï, and Nizar Touzi.
We show here how contract theory concepts and methods can help successfully design efficient demand-response programs. We concentrate on a situation in which the principal is a power producer and the agent is a consumer. The power producer has to satisfy the consumer’s electricity demand for the following day. The key variable is the deviation from the predicted or baseline electricity demand; we denote it with  $$X_t$$ where t is an instant of the next day. This deviation procures a utility to the consumer $$f(X_t)$$ Moreover, the consumer can act on this deviation by reducing her mean consumption and volatility. In mathematical terms, this means that the process $$X_t$$ satisfies the following stochastic differential equation:

$dX_t^{a,b} = - \sum_i a_{i,s} ds + \sigma^i \sqrt{b_{i,s}} dW^i_s,\tag1$

where $$a_i$$ represents the agent’s effort on the mean consumption of electricity usage i, and $$b_i$$ denotes the effort on the volatility of usage i. A positive $$a_i$$ represents a reduction of demand on usage i, while $$b_i <1$$ indicates a reduction of the volatility of usage $$i$$. These efforts induce a cost represented by the function $$c(a,b).$$

For the producer, the deviation $$X_t$$ generates an extra cost $$(X>0)$$ or an economy $$(X<0),$$ denoted $$g(X).$$ Moreover, variation of the deviation $$X_t$$ over time also induces costs on the producer, whose flexibility capacity is limited. Figure 1 illustrates this point. Both consumptions are equal in energy, but the blue consumer presents a quadratic variation $$\langle X\rangle := \sum_t \big(X_{t+1}-X_t\big)^2$$ of $$650$$ while the red has a quadratic variation of only $$12$$. These variations incur a cost on the producer, whose generation plants are not flexible enough to follow such erratic behavior. We suppose that this cost is proportional to the quadratic variation with constant $$h$$.

The producer needs to find an incentive scheme, denoted $$\xi,$$ that will prompt the consumer to reduce the average consumption and its variation. But the producer has no knowledge of what is happening in the house, being able to merely observe the total consumption. The contract $$\xi$$  can only depend on the observed values of $$X$$ and not on the efforts $$a, b.$$ When exposed to $$\xi,$$ the consumer aims to maximise the expected utility $$f(X_t)$$ minus the cost of effort $$c(a,b)$$ plus the payment $$\xi.$$  Namely, its purpose is to solve

$\sup_{a,b} \mathbb E\bigg[ \xi+\int_0^T \big( f(X_s^{a,b}) - c(a,b) \big) ds \bigg].\tag2$

The optimal controls of consumer $$a^\star(\xi)$$ and $$b^\star(\xi)$$ are functions of the contract scheme $$\xi.$$

On the other hand, the objective of the producer is to maximise his own utility. In mathematical terms, the producer is the following objective function:

$\sup_{\xi} \mathbb E\bigg[U\bigg(- \xi - \int_0^T g(X_t^\star) ds - h \langle X_t^\star \rangle_T \bigg) \bigg],\tag3$

where $$X_t^\star :=X_s^{a^\star(\xi),b^\star(\xi)}$$ represents the consumer optimal electricity deviation induced by contract $$\xi.$$ The producer first has to determine the optimal responses $$a^\star(\xi),b^\star(\xi)$$ of the consumer for any given $$\xi,$$ and then generate his own optimization knowing these responses. Contract $$\xi$$'s dependence on the observation of the whole trajectory $$X$$ complicates the problem. The producer will use all the available information contained in the variations of $$X$$ to determine whether—and to what extent—the agent is making an effort, or if the observations are just subject to random outcomes.

### Optimal Contract and Numerical Simulation

Possamaï and Touzi, along with Jakša Cvitanić, provide a general methodology to solve (3). They show that the optimal contract can be written as

$\xi = Y^{Z,\Gamma}_t := Y_0 +\int_0^t Z_s dX_s + \frac{1}{2}\Gamma_s d\langle X\rangle_s -\left(H(Z_s,\Gamma_s) + f(X_s) \right) ds,\tag4$

where $$Z$$ and $$\Gamma$$ are two stochastic processes to be chosen by the principal, and $$H(z,\gamma)$$ is the Hamiltonian of the agent:

$H(z,\gamma) := \sup_{a,b} \Big\{- \sum_i a_i z + \frac12 \sum_i \sigma_i^2 b_i^2 \gamma - c(a,b) \Big\}.$

This result brings the problem of the principal back to a standard stochastic control problem, where the objective function is given by (3), (4) replaces $$\xi,$$ the controls are the two processes $$Z$$ and $$\Gamma,$$ and the dynamics are given by (2) for $$X$$ and (4) for $$Y$$. The principal’s problem thus admits a nonlinear partial differential equation representation whose solution is approachable by numerical techniques.

Figure 2.Left. Electricity deviation of a rational consumer compared to a passive consumer Right. Associated payment for the rational consumer and for the passive consumer. Image credit: René Aïd, Dylan Possamaï, and Nizar Touzi.

We illustrate the resulting interaction between a producer and a consumer with numerical simulations on a one-day period. Figure 2 represents the consumption (left) and the payment (right) to a rational consumer (blue) who has signed and applied the contract, and a passive consumer (red) who has signed the contract but not applied it. Electricity deviation $$X^*$$ of a rational consumer is shown compared to a passive consumer $$\hat{X}$$. The right side shows associated payment $$Y^*$$ for the rational consumer and $$\hat{Y}$$ for the passive consumer. We notice that the rational consumer reduces both the deviation and its volatility when compared to the passive consumer, meaning that the contract produces the desired outcome. Moreover, the rational consumer receives a positive payment in all cases, while the passive consumer receives no payments and may face a penalty. As time goes by, the expected payment for the passive consumer becomes less and less volatile as the producer observes the consumption and infers that the consumer is making no effort.

### Perspectives

The model presented in this article is the first step towards a practical implementation of a demand-response contract at a large scale. Indeed, one should consider the case when a large number of agents must be controlled. Possamaï, Romuald Elie, and Thibaut Mastrolia have already investigated the instance of a single principal and a large number of agents, indicating that demand-response could possibly extend to more realistic situations. Despite its infancy, our model opens the door to the social engineering of power systems. Indeed, behavioral sciences are necessary to provide realistic yet tractable models of response function to price signals for a large population of consumers. Demand-response is on the list of required technologies to achieve an efficient zero-carbon electric system, and we hope to contribute to its development.

This article is based on René Aïd’s presentation of his joint work with Dylan Possamaï and  Nizar Touzi at the 2016 SIAM Conference on Financial Mathematics and Engineering, held last fall in Austin, Texas.