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Prosumers adopt distributed energy resources (DER) to cover part of their own consumption and to sell surplus energy. Although individual prosumers are too dispersed to exert operational market power, they may collectively hold a strategic advantage over conventional generation in selecting DER capacity via aggregators. We devise a bilevel model to examine DER capacity sizing by a collective prosumer as a Stackelberg leader in an electricity industry where conventional generation may exert market power in operations. At the upper level, the prosumer chooses DER capacity in anticipation of lower-level operations by conventional generation and DER output. We demonstrate that exertion of market power in operations by conventional generation and the marginal cost of conventional generation affect DER investment by the prosumer in a nonmonotonic manner. Intuitively, in an industry where conventional generation exerts market power in operations similar to a monopoly (MO), the prosumer invests in more DER capacity than under perfectly competitive operations (PC) to take advantage of a high market-clearing price. However, if the marginal cost of conventional generation is high enough, then this intuitive result is reversed as the prosumer adopts more DER capacity under PC than under MO. This is because the high marginal cost of conventional generation prevents the market-clearing price from decreasing, thereby allowing for higher prosumer revenues. Moreover, competition relieves the chokehold on consumption under MO, which further incentivises the prosumer to expand DER capacity to capture market share. We prove the existence of a critical threshold for the marginal cost of conventional generation that leads to this counterintuitive result. Finally, we propose a countervailing regulatory mechanism that yields welfare-enhancing DER investment even in deregulated electricity industries.
Over the past 40 years, the electricity industry in most OECD countries has experienced two structural reforms. First, it has gone from being a mostly state-regulated enterprise with vertically integrated investor-owned utilities to a decentralised one with separation of generation and retailing functions (Wilson 2002; Baek et al. 2014; Ajayi et al. 2017). Second, in the last decade, concerns about climate change have prompted decarbonisation of the power sector and electrification of wider energy use in other sectors. This orientation towards sustainability has been facilitated by policies for supporting renewable energy technologies and carbon pricing (von Hirschhausen 2014; de Leon Barido et al. 2020).
In particular, climate policy has catalysed the adoption of distributed energy resources (DER) (Burger and Luke 2017), such as rooftop solar photovoltaic (PV) panels (van Kooten and Mokhtarzadeh 2019) and plug-in electric vehicles (PEVs) (Fox et al. 2017). For example, small-scale PV generation by end users in the U.S. has increased fivefold in the past seven years.Footnote 1 While renewable-energy subsidies and targets have made DER technologies more economically attractive for residential and commercial entities, the rise of the so-called prosumer, i.e., an agent that both produces and consumes energy,Footnote 2 is further enabled by underpinning regulation. For example, FERC Order 2222Footnote 3 and EU Directive 2019/944 (Article 16)Footnote 4 ensure non-discriminatory access to electricity markets for DER providers. In this context, aggregators can pool DER capacity to participate more effectively in markets (Wang et al. 2019).Footnote 5 Thus, while a single prosumer may have limited influence in the electricity market, an aggregator with a diverse portfolio of DER can potentially exert market power (Iria et al. 2019; Yin et al. 2020).
We have two settings with a deregulated industry: one in which conventional generation acts perfectly competitively (PC) and another in which conventional generation alone exerts market power in a simultaneous-move game (MO). In either case, the prosumer invests in DER at the upper level, thereby leading to a bilevel problem. As a benchmark, we also have a central-planning setting (CP) in which all decisions, x, y, and z, are treated as if they were made by a single benevolent entity that maximises social welfare. Thus, CP is handled as a single-level optimisation problem.
Intuitively, Assumptions A1 and A2 bound consumer and prosumer willingness to pay for electricity in terms of the cost of DER investment, i.e., the assumptions rule out economically uninteresting results without DER adoption. Note that Assumption A4 is stronger than Assumptions A1 and A2 together, which lead to \(C\left( A+B\right) -2IC>0\) if added. Assumption A4 is necessary to ensure that DER adoption under CP is strictly greater than zero, cf. Proposition P1, thereby avoiding trivial results without DER adoption. Finally, Assumption A5 puts a lower bound on prosumer willingness to pay in terms of consumer willingness to pay to avoid irrelevant results without both strictly positive consumption and production by prosumers in both PC and MO settings, cf. Propositions P3 and P6.
PS is the revenue from prosumer sales, py, plus the gross benefit to prosumers from consumption, \(B\left( z-y\right) -\frac12\left( z-y\right) ^2\), minus the cost of DER investment, Iz.
Since the cost of electricity purchased by the consumer, \(p\left( x+y\right)\), cancels the revenue terms accruing to the conventional generation, px, and the prosumer, py, social welfare may be expressed as in (1).Footnote 7
Intuitively, the socially optimal solution is to ensure that the marginal benefit of consumption equals marginal cost, whether from conventional generation (2) or DER output (3). Note that the latter condition implies that the opportunity cost of forgone consumption by the prosumer, \(B-\left( z-y\right)\), equals the market-clearing price, \(A-\left( x+y\right)\). This way, the marginal cost of DER generation equals its marginal cost of investment (4). Thus, the electricity price is set by the marginal cost of DER investment (9). It is also possible to prove that the optimal DER investment is monotonically increasing in the marginal cost of conventional generation, C, cf. Proposition P2:
Under PC, both conventional generation and the prosumer act as price takers at the lower level when making their electricity sales. At the upper level, the prosumer acts as a Stackelberg leader when deciding upon its DER investment. We solve for the subgame-perfect Nash equilibrium at the lower level first before obtaining the optimal DER investment at the upper level.
Note that (19) equates the marginal revenue from DER capacity expansion to its marginal cost. The SOSC may be verified because the partial derivative of (19) with respect to z is less than zero. We can readily solve (19) to obtain the optimal DER investment by a prosumer under PC, which is an interior solution as shown in Proposition P3:
Under PC, optimal DER investment by the prosumer increases monotonically in the marginal cost of conventional generation as long as \(4I-A \ge 0\). However, if \(4I-A 0\), below (above) which optimal DER investment by the prosumer increases (decreases) in the marginal cost of conventional generation.
Under MO, conventional generation exerts market power in operations at the lower level, while the prosumer is still a price taker in electricity-market operations. Note that the lower level is a simultaneous-move game (von der Fehr 2010), which leads to a Nash equilibrium. At the upper level, the prosumer again acts as a Stackelberg leader when deciding upon its DER investment.
Note that (27) equates the marginal revenue from DER capacity expansion to its marginal cost. The SOSC may also be verified because the partial derivative of (27) with respect to z is less than zero. We can again solve (27) to obtain the optimal DER investment by a prosumer under MO and verify that the solutions are interior in Proposition P6:
Under MO, optimal DER investment by the prosumer increases monotonically in the marginal cost of conventional generation as long as \(4I-A \ge 0\). However, if \(4I 0\), then optimal DER investment by the prosumer decreases monotonically in the marginal cost of conventional generation.
The results in Sects. 4.1 and 4.2 exhibit prosumer investment that is not aligned with the welfare-maximising results under central planning, cf. Sect. 3. To increase welfare in decentralised industries, regulators may propose incentive-alignment mechanisms to entice prosumers to modify their capacity adoption. One such measure could be a simple subsidy, S [in $/MW], on the investment cost of DER, which makes the effective cost of DER investment \(\left( I-S \right) z\) for a prosumer. Note that it may be possible for S to be high enough that it makes the effective cost of DER investment negative. 2ff7e9595c
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