SIAM News Blog

Electricity Markets and Renewable Energy

By Michael Ferris and Andy Philpott

Despite ongoing reluctance by some governments to respond to anthropogenic climate change, many countries are pursuing policies to generate energy with very low carbon emissions. Technologies facilitating emission reduction via electricity systems include wind turbines, hydroelectric generation, rooftop solar power, concentrated solar power, geothermal power, and nuclear power. Other tools—like battery and pumped storage, heat pumps, carbon capture and storage, and enhanced transmission infrastructure—work in concert with emission-reducing devices to improve efficiency and reliability of supply across multiple time scales. Designing an efficient next-generation electricity system that is reliable, cost effective, and environmentally clean is a major challenge.

Incentive programs, such as feed-in tariffs that pay for excess electricity generated by rooftop solar panels, have encouraged growing investment in renewable energy generation. The near-zero short-run costs of wind and solar power generation—from freely-available wind and sunlight—are lowering electricity prices based on marginal supply costs. This poses difficulties for traditional electricity market generators reliant upon infra-marginal rents (the difference between revenue earned at system marginal prices and operating costs) to cover their fixed costs.

The aforementioned topics were central to discussions during “Electricity Systems of the Future: Incentives, Regulation, and Analysis for Efficient Investment” — an invited workshop at the University of Cambridge’s Isaac Newton Institute for Mathematical Sciences this spring. The workshop aimed to develop mathematical models to aid the design of next-generation electricity markets that would incentivize optimal-generation investments in environments with large quantities of renewable energy. Insight from attending mathematicians, economists, engineers, and representatives from the U.K. energy industry helped generate discussions relevant to a wide range of problems, motivated by the following three subjects:

  1. Optimization of generation capacity: what, where, and when one should build;
  2. Factors that might prevent investors in an electricity market from making socially-optimal decisions;
  3. Possible incentive structures to alleviate market failures.

Multiple countries are moving towards a deregulated, market-based electricity system in which competing agents make generation investments on a commercial basis. However, planners often consider system optimization problems to understand the best possible system design and operation. One can formulate the optimization of generation capacity as a stochastic program that chooses each generation technology’s capacity levels before operating in various conditions and states. More sophisticated multi-stage stochastic models design a sequence of investments over a multi-year planning horizon, where each investment decision is contingent upon an observed history of events — such as electricity demand growth from electric vehicles. Stochasticity at finer time scales accounts for variability in wind or solar generation. The objective is typically the expected social or overall benefit from investment creation and efficient operation. If demand for electricity is inelastic (i.e., insensitive to price changes), minimizing expected capital and operating costs will maximize expected social benefit.

We illustrate this concept with a two-stage model in which one selects investments of quantity \(x_k\) in technology \(k\) in the first stage at cost \(C_k\) per unit to optimize capital costs plus the expected cost \(\mathbb{E}_\omega (g_k y_k (\omega)) :=\) \(\Sigma_\omega p(\omega)g_k y_k (\omega)\) of electricity generation \(y_k(\omega)\). In this case, \(g_k\) is the operating cost, \(\omega\) denotes a random state of the world, and \(p(\omega)\) represents its probability. A social planner totals the costs and solves a model SP to meet demand \(d(\omega)\) in every future state of the world, possibly shedding load \(q(\omega)\) at some very high penalty cost \(V\):

\[ \textrm{SP}: \textrm{min} \:\:\Sigma_{a\in \mathcal{A}} \Sigma_{k \in a} [C_k x_k + \mathbb{E}_\omega (g_k y_k (\omega) + Vq(\omega))]\]

\[\textrm{s.t.} \: \: \: \: \: \: \: \Sigma_a \Sigma_{k \in a} y_k (\omega) + \: q(\omega) \ge d(\omega), \]

\[0 \le y_k(\omega) \le x_k + u_k, \]

\[0 \le q(\omega) \le d(\omega), \: \: \: x_k \ge 0.\]

Here, \(k \in a\) denotes technologies owned by agent \(a\). The solution to SP yields Lagrange multipliers for the first constraint of model SP, expressed as \(p (\omega) \pi (\omega)\) to demonstrate the way in which they account for the probability of \(\omega\). The social planner may also choose to add a constraint \(E\) on greenhouse gas emissions to this problem. For example,

\[\Sigma_a \Sigma_{k \in a} \mathbb{E}_\omega (\alpha_k y_k (\omega)) \le E,\: \: \:[\sigma]\]

where \(\alpha_k\) denotes a conversion factor to emissions and \(\sigma\) is the constraint’s multiplier. It is important to understand the implications of different forms of this constraint, which in the above equation confines average emissions. Variations that impose a bound on emissions in (almost) every state of the world—or with high probability—engender different policies [3]. A constraint imposed on non-renewable capacity \(x_k\) also produces different results (see Figure 1). A particular example shows that reducing capacity of thermal plants without extra renewable investment can lead to increased emissions from the hedging strategies of remaining plants.

Figure 1. The effects of increased renewable penetration on New Zealand’s electricity system in 2030. 1a. Impact on emissions when constraint is on capacity, energy, or carbon dioxide (CO2). 1b. Change in technology mix as renewable penetration increases. Figure courtesy of [3].

How do we incentivize agents to build the right plants? If all agents minimize expected costs using probabilities \(p(\omega)\), then a Lagrangian decomposition yields an emissions price \(\sigma\) and energy prices \(\pi (\omega)\) that decouple SP into agent problems 

\[P(a)\::\: \textrm{min}\:\:\: \Sigma_{k \in a}[C_k x_k + \mathbb{E}_\omega (g_k y_k (\omega)+ \sigma \alpha_k y_k(\omega) - \pi(\omega)y_k(\omega))]\]

\[\textrm{s.t.} \: \: \: \: \: \: \: 0 \le y_k (\omega) \le x_k + u_k, \: \: \: x_k \ge 0,\]

a consumer problem

\[P(d) \: : \: \textrm{min} \:\:\: \mathbb{E}_\omega ((V - \pi(\omega))q(\omega))\]

\[ \textrm{s.t.} \: \: \: \: \: \: \: 0 \le q(\omega) \le d(\omega),\]

and market-clearing complementarity conditions

\[0 \le \sum\limits_{a} \sum\limits_{k\in a} y_k (\omega) + \:q(\omega) - d(\omega) \perp \pi (\omega) \ge 0,\]

\[0 \le E - \sum\limits_{a} \sum\limits_{k\in a} \mathbb{E}_\omega (\alpha_k y_k (\omega)) \perp \sigma \ge 0.\]

This is a competitive equilibrium with the same solution as social planning.

Several caveats accompany this model. The equivalence between system optimization and equilibrium requires that the social planning optimization problem be convex. This precludes integer variables that might represent indivisible costs in investment or startup costs in operations. Alternative suggestions for out-of-market interventions and payments may create incentive problems and compromise many of the benefits that accompany efficient markets. The equivalence also demands that agents act nonstrategically, as though the price emerging from equilibrium is fixed and independent of their actions. Finally, all products in the social plan must be priced by the market under consideration, i.e., the market must be complete. This last criterion is critically absent when agents are reluctant to take risks and there are insufficient financial instruments to hedge their risk.

Much of the invited workshop was devoted to the study of incentives for efficient investment. Regulators set the value of lost load (\(V\) in formulation SP) as a price cap when demand is unable to respond to price spikes. Some of the load is shed when capacity cannot sufficiently meet demand; this renders price \(V\). In a competitive equilibrium with unrestricted entry, the number of periods each year when this occurs for a new peaking plant yields just enough profit on average to cover its annualized fixed cost. In energy-only markets, the theoretical choice of \(V\) is based on a desired level of reliability defined by an annual frequency of load shedding. Typical values of \(V\) are around $10,000/megawatt-hour.

Historically, price caps in many markets have been kept lower than necessary for reliability targets. This practice arises partly from regulators’ desire to curb possible exercise of market power (where agents or firms strategically choose generation levels \(y_k (\omega)\), or government motivation to keep prices low. If \(V\) is low, peaking plants need more frequent load shedding to make enough money to cover their fixed costs. Low \(V\) thus means less reliability, unless generators are compensated for providing capacity. This is the so-called missing money problem.

A capacity market is one way to provide the missing money. These markets typically take the form of procurement auctions, in which system operators seek to acquire required electricity generation capacity at minimum cost. One caveat is the need for regulators to determine the system’s required level of capacity. While some jurisdictions treat all capacity equally, certain types (e.g., one megawatt of flexible plant) are clearly more valuable than others (e.g., one megawatt of wind capacity), thus necessitating adjustments for effectiveness.

The argument for capacity markets becomes stronger as renewable energy increases. As mentioned before, most renewable plants have almost no operating costs; this means that increasing renewable penetration lowers wholesale electricity prices. One must recover the capital cost for solar panels, wind turbines, and backup peaking plants in shortage periods. Several years without shortage periods might pass before a year with substantial shortage (when no income is earned) transpires. Over a long period of time, a capacity market can provide more certainty to investors who seek to build renewable generation.

The workshop also explored new mathematical techniques for modeling capacity expansion in risky markets, with particular emphasis on risky competitive equilibrium [2, 4]. In this scenario, agents are endowed with risk measures that convert random operating surpluses into certainty-equivalent values. Expectation is a risk-neutral measure; more interesting models use the rich class of coherent risk measures [1] to understand the relationship between competitive equilibrium and system optimization.

Future work in this area must link energy models to other sectors of the economy and consider pricing information related to abatement. Such integrated assessment models could investigate carbon taxes to encourage investments in combined heat and power technologies, anaerobic digestion, and forms of compensated demand reduction. For example, the use of carbon capture with storage combined with renewable biomass generation actually removes carbon dioxide from the atmosphere.

Michael Ferris presented similar research during an invited talk at the 2019 SIAM Conference on Computational Science and Engineering, which took place in Spokane, Wash., earlier this year.

Michael Ferris discusses his research around renewable electricity, based on his invited talk at the 2019 SIAM Conference on Computational Science and Engineering.

Acknowledgments: This research was supported in part by a grant from the Department of Energy and the New Zealand Marsden Fund.

[1] Artzner, P.,  Delbaen,  F., Eber,  J.-M., & Heath, D. (1999). Coherent measures of risk. Math. Fin., 9(3), 203-228.
[2] Ferris, M.C., & Philpott, A.B. (2018). Dynamic risked equilibrium. Submitted to Oper. Res. 
[3] Ferris, M.C., & Philpott, A.B. (2019). 100% renewable electricity with storage.  Submitted to Oper. Res. 
[4] Ralph, D., & Smeers, Y. (2015). Risk trading and endogenous probabilities in investment equilibria. SIAM J. Opt., 25(2), 2589-2611.

Michael Ferris is the Stephen C. Kleene Professor of Computer Science at the University of Wisconsin, Madison and director of the Data Sciences Hub at the Wisconsin Institutes for Discovery. His research is concerned with algorithmic and interface development for large-scale problems of mathematical programming. Ferris has worked on many applications of both optimization and complementarity, including cancer treatment planning, energy modeling, economic policy, and traffic and environmental engineering. Andy Philpott is a professor of operations research at the University of Auckland and director of the Electric Power Optimization Centre. His research interests span linear, nonlinear, and stochastic programming and their application to operations research problems, including optimal planning under uncertainty, capacity expansion in telecommunications and power networks, and optimal operation of hydroelectric power systems.

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