1 Introduction↩︎

The increasing adoption of renewable energy, particularly solar photovoltaic systems, plays an important role in modern power systems. While utility-scale solar projects contribute significantly to overall capacity, distributed rooftop installations are playing an increasingly prominent role in reshaping the electricity landscape. Notably, as of 2022, approximately 140 GW of solar capacity had been installed in the U.S., with more than 35% of it coming from installations by households and commercial entities [1].

Unlike centralized projects planned and operated by utilities, distributed solar investments are driven by decisions of individual households and businesses. Several factors influence these decisions. A key factor is the financial return from the investment, driven by the expected revenue earned in short-term distribution-level markets over the panels’ lifespan, where that revenue is determined by the markets’ equilibria. Meanwhile, these short-term equilibria depend on the aggregate invested panel capacity available in the markets, which is shaped by the long-term investment equilibrium that emerges from interactions among individual investors. Therefore, there is a feedback loop between short-term markets equilibria and long-term investment equilibrium, as each influences the other. A precise characterization of this relationship is crucial for understanding the emergence of investment equilibria. This insight is not only important for utilities and independent system operators, who should account for distributed capacity growth in their planning studies, but also for policymakers and market designers aiming to understand the long-term impact of diverse market designs on the solar panel adoption. This motivates us to study the following critical open question: How do short-term market equilibria, arising from diverse solar market mechanisms, shape the corresponding long-term distributed investment equilibrium?

1.1 Organization and Contributions↩︎

To address this problem, we consider a large population of individual investors and aim to characterize the long-term investment equilibrium that emerges from the previously described feedback loop. We first develop an investment model, formulated as a non-atomic game, that links long-term investment decisions to expected revenues under a given market design (Section 2). To characterize these revenues, we introduce a unified modeling framework for short-term markets (Section 3). This framework allows us to formally define and compare different market designs by capturing key features such as product differentiation and ex-ante/ex-post trading. We then apply it to three representative mechanisms that differ in these key features (Section 4): (a) a single-product real-time market, (b) a product-differentiated real-time market, and (c) a contract-based panel market. For each, we first derive analytical characterizations of the short-term market equilibria and then use the investment model to derive the associated long-term equilibrium aggregate capacities (Section 5). Our analysis, initially assuming homogeneous operation periods, is extended to handle heterogeneous operation periods (Section 6), followed by numerical experiments (Section 7).

We contribute to the literature in the following key ways:

(a) The non-atomic game-based investment model encapsulates how individual investors impact each other through their aggregate investment decisions, offering an analytically tractable way to link short-term market equilibria to long-term aggregate investment equilibrium.

(b) The proposed unified modeling structure provides closed-form characterizations of short-term market equilibria across diverse mechanism designs, accommodating both ex-ante and ex-post trading as well as heterogeneous consumer preferences.

(c) These analytical characterizations enable us to theoretically establish that the single-product real-time market, where solar energy is pooled with conventional grid energy and traded ex-post, leads to under-investment. In contrast, the product-differentiated market, where solar energy is traded ex-post and may be sold at a premium due to heterogeneous user preferences, yields the socially optimal capacity. Finally, the contract-based market, where ex-ante panel capacity is traded instead of realized generation, can result in over-investment when users’ extra valuations for solar are low.

1.2 Related Literature↩︎

This work is related to the intersection of two broad lines of literature: distribution-level electricity market mechanisms and long-term investment in solar panels. Most existing studies on the investment side of the literature adopt a centralized perspective, focusing on the optimal strategy of a single decision-maker such as a utility planner or an individual investor (see, e.g., [2]–[6]). However, this line of work neglects the interdependencies of investment decisions in distributed settings, where the actions of one investor can influence the returns of others through shared market mechanisms.

On the market side, a variety of distribution-level electricity market designs have been proposed. Understanding how these structures differ is essential for our analysis, as their design features shape the expected revenue of investors. In this context, prior research has proposed a range of market designs, with particular emphasis on real-time pricing and service contract-based approaches. In a broad sense, real-time pricing mechanisms involve the determination of time-varying electricity prices based on the ex-post realized values of supply and demand. In contrast, service contract-based mechanisms typically offer less volatile prices in exchange for a fixed level of electricity service, relying on ex-ante statistics such as those related to random solar generation. Extensive research has explored the relative merits of these retail market designs, examining factors such as economic efficiency [7]–[10], price and bill volatility [11]–[13], and environmental impacts [14]. In addition, some studies have examined another facet of market design: whether electricity from solar and conventional sources should be treated as differentiated products in electricity markets, given their distinct operational characteristics and emissions [15]–[19]. Despite the contributions of all these reviewed studies, they all focus primarily on short-term performance criteria and overlook the long-term impacts of their designed markets on investment decisions.

The most closely related paper to ours is [20], which analyzes the equilibrium panel investment capacities under feed-in-tariff and sharing economy markets. Nonetheless, they adopt a finite-player game to analyze the setup involving large investors. In contrast, we focus on the setting with a large number of small investors. Through a non-atomic game formulation, we can analytically compare equilibrium capacities driven by various market mechanisms, for which they have to resort to numerical simulation.

This paper generalizes our previous conference work [21] by (a) extending our theoretical results to setups with heterogeneous operation periods, (b) presenting rigorous mathematical proofs, and (c) demonstrating our theoretical results by performing numerical experiments.

2 Investment Model↩︎

A large group of potential investors considers installing solar panels and becoming panel owners. Their decisions weigh upfront capital and installation costs against expected revenues from participating in short-term electricity markets over the panels’ lifespan. An overview of the setup is shown in Fig. 1.

2.1 Connecting Planning and Operation Timescales↩︎

Investors typically consider a 25-year lifespan for solar panels [22]. However, estimating expected revenue at the time of investment is challenging, as reliable long-term forecasts of solar output and market conditions are rarely available. A common approach is to use a shorter planning period (e.g., one year), estimate revenues from historical data, and scale³ the result by the number of such intervals over the panel’s lifespan. The appropriate length of the planning period should be selected based on the availability and granularity of historical data.

When the planning period is long, there may be multiple operation periods within it. We consider a finite-horizon discrete-time planning window consisting of operation periods \(\mathcal{T} = \{1, \dots, T\}\), indexed by \(t\), where \(T\) is the total number of such periods. For each operation period \(t \in \mathcal{T}\), we model the short-term operation conditions (e.g., supply and demand) as constants and incorporate variability (e.g., of solar generation) across different operation periods by sampling from a probability distribution. Thus, the expected revenue remains consistent across operation periods. We estimate it for a single representative period (e.g., as shown in Fig. 1), then scale it up to the planning horizon and ultimately to the full lifespan. The total expected revenue is therefore obtained by multiplying the representative period’s revenue by a scaling factor \(\widetilde{T}\), which captures the number of such periods over the panel’s lifespan and incorporates any interest-rate adjustments.

2.2 Panel Investment Game↩︎

Considering a large number of small investors each of whom does not have market power, we model the group of investors as the continuous interval \(\mathcal{I}_\mathrm{inv} := [0,1]\). We consider the case where each investor \(i\in \mathcal{I}_\mathrm{inv}\) decides whether to install a solar panel with a fixed capacity \(\overline{c}\). Let the decision of investor \(i\) be \(x_i\in \{0,1\}\). The total panel capacity that the investors will install is then \[\label{eq:agg:c} c = \overline{c} \int_{\mathcal{I}_\mathrm{inv}} x_i \,\, \mathrm{d}i.\tag{1}\]

Let the capital and installation cost for a unit panel capacity be \(\pi_0\). We assume that the panel upfront cost is attractive, at least when its solar production can be sold at the utility rate: \[\label{eq:pi0cond} {\color{black} c}\pi_0 \le \widetilde{T}\pi_\mathrm{u} \mathbb{E} [{\color{black} c}G],\tag{2}\] where \(\pi_{\mathrm{u}} > 0\) is the fixed price at which the utility provides backstop electricity to buyers, and \(G\) denotes solar generation per unit panel capacity, typically measured in \(\mathrm{W}/\mathrm{m}^2\) or \(\mathrm{GW}/\mathrm{km}^2\). This solar generation is modeled as a random variable with a cumulative distribution function \(F_G\) and a probability density function \(f_G\). For simplicity, we assume that \(f_G\) is differentiable and that \(G\) is supported on \(\mathbb{R}_+\) with finite first and second moments, and \(f_G(g) > 0\) for all \(g \in \mathbb{R}_+\).

The expected return for each investor \(i\) is influenced not only by their own decision, \(x_i\), but also by the collective actions of other participants and the specific short-term market mechanism assumed for the representative operation period. Thus, the payoff for investor \(i\) from installing panel capacity \(\overline{c}\) and selling the generated electricity in a short-term market \(m\) (where \(m\) serves as an index for different short-term market mechanisms) is given by the expected total revenue minus the investment cost, expressed as: \[\label{eq:inv:po} \Pi^\mathrm{inv}_{m,i} (x_i, c) = \left[ (\Pi_{m}^\mathrm{s}(c)/c)- \pi_0 \right] \overline{c}\,x_i,\tag{3}\] where \(\Pi_m^\mathrm{s}(c)\) represents the expected total revenue received by solar panel owners under the short-term market \(m\).

Given the players \(\mathcal{I}_\mathrm{inv}\), actions \(x = \{x_i\}_{i \in \mathcal{I}_\mathrm{inv}}\), and payoff 3 , we have defined the solar panel investment game, which is an aggregate game where the dependence of investor \(i\)’s payoff on the other investors’ decisions is only through the total panel capacity \(c\). A Nash equilibrium (NE) for this game is defined as:

Under this condition, no investor will have an incentive to unilaterally deviate from the NE decision given the aggregate capacity that is consistent with everyone’s investment decision.

Building on this investment model, we next examine several representative solar market mechanisms to assess how their structural differences influence investors’ payoffs and, in turn, the resulting long-term NE aggregate capacities.

3 Unified Modeling Framework for Short-term Market Mechanism Analysis↩︎

We present generic modeling elements applicable across diverse market settings and then provide a detailed description of the specific market mechanisms analyzed in this paper.

3.1 Common Model Elements↩︎

Consider an operation period (e.g., the representative operation period in Fig. 1) where the expected revenue of investors is shaped by short-term conditions in the distribution-level solar energy market. The market consists of buyers (i.e., electricity consumers), sellers⁴ (i.e., panel owners), and a utility company that provides backstop supply when solar generation alone cannot fully satisfy demand. Similar to the investment game, we model the groups of a large number of buyers and sellers as continuous intervals.

Define the set of buyers by the interval \(\mathcal{I}_\mathrm{b}:= [0,1]\). In our analysis, we consider a situation where environmentally conscious electricity consumers have an extra heterogeneous valuation for solar energy over the conventional energy supplied by the utility. For a buyer \(i \in \mathcal{I}_\mathrm{b}\), we denote their extra valuation for each unit of load served by solar as \(v_i \in \mathbb{R}_+\), referred to as the solar premium. The distribution of \(v_i\) can be characterized by a function \(F_V\) defined as \(F_V(v) = \int_{i \in \mathcal{I}_\mathrm{b}}\, \mathbb{1}\{v_i \le v\} \, \, \mathrm{d}i\), where \(\mathbb{1}\) is the indicator function. For simplicity, we assume that \(F_V\) is differentiable and strictly increasing on \([0, \overline{v}]\), where \(\overline{v} \ge 0\). We denote its derivative by \(f_V\), supported on \([0, \overline{v}]\), and assume \(f_V(v)\) is bounded away from zero for all \(v \in [0, \overline{v}]\). Let the total electric load of all consumers be \(L\). We assume \(L\) is inelastic, deterministic, and equally distributed across all consumers.

Let the set of sellers be \(\mathcal{I}_\mathrm{s} := [0,1]\). Note that although only a subset of \(\mathcal{I}_\mathrm{inv}\) decides to invest and thereby constitutes \(\mathcal{I}_\mathrm{s}\), we relabel this subset to form a normalized interval \([0,1]\). We specifically examine the scenario where the sellers are homogeneous⁵. Given the aggregate capacity \(c\) from 1 , the homogeneity of sellers and unit length of \(\mathcal{I}_\mathrm{s}\) imply that the panel capacity of each seller is also \(c\), as \(c=\bar{c} \int_{\mathcal{I}_{\mathrm{inv}}} x_i \, \mathrm{d}i=c \int_{\mathcal{I}_s} \, \mathrm{d}i\). Thus, the total solar generation from all sellers is \(cG\). We now proceed with detailed modeling of the selected markets.

3.2 Real-time Pricing Mechanisms↩︎

We elaborate on two alternative types of real-time pricing mechanisms proposed in the existing literature. In both mechanisms, the market price is determined by the realized values of supply and demand. The primary distinction between these two mechanisms lies in whether solar generation is treated as the same product as utility-supplied electricity. This aspect becomes crucial when consumers place additional value on renewable generation. We denote the mechanism where solar is traded as a distinct product as the product-differentiated real-time market (prt), while the other is referred to as the single-product real-time market (srt).

Within the product-differentiated real-time market, the traded product is solar electricity. Each seller \(i\in \mathcal{I}_\mathrm{s}\) can sell any quantity of solar generation within the set \(\mathcal{Q}_i^\mathrm{s} = [0, cG]\), which depends on the realized value of \(G\). The seller payoff function is \[\Pi_i^\mathrm{s}(q_i^\mathrm{s},\pi) = \pi q^\mathrm{s}_i,\nonumber\] where market equilibrium price \(\pi\) and cleared quantity \(q^\mathrm{s}_i\) will be defined momentarily. Given the inelastic load \(L\), a buyer \(i \in \mathcal{I}_\mathrm{b}\) can purchase any quantity of solar within the range \(\mathcal{Q}^\mathrm{b}_i= [0,L]\). Any remaining demand not met by solar will be fulfilled by the utility backstop at the price \(\pi_{\mathrm{u}}\). The buyer payoff function is \[\Pi_i^\mathrm{b}(q_i^\mathrm{b},\pi) = (v_i- \pi) q^\mathrm{b}_i - \pi_\mathrm{u} (L-q^\mathrm{b}_i),\nonumber\] where \(q_i^\mathrm{b}\) is the cleared quantity for buyer \(i\). Denote \(q^\mathrm{s}:= \{q^\mathrm{s}_i\}_{i \in \mathcal{I}_\mathrm{s}}\) and \(q^\mathrm{b}:= \{q^\mathrm{b}_i\}_{i \in \mathcal{I}_\mathrm{b}}\).

We use the following equilibrium notion to characterize the market outcome:

The CE conditions are conceptually straightforward: the resulting market outcome, comprising the equilibrium price and traded quantities, must be stable. In other words, no buyer or seller should have an incentive to unilaterally adjust their quantity, given that individual actions do not influence the market price. This price-taking assumption is consistent with our non-atomic model, where each agent is infinitesimal and cannot affect the price unilaterally.

With the equilibrium price \(\pi\) and the cleared quantities \(q^\mathrm{s}_i\) for sellers in the product-differentiated real-time market, the total expected revenue for sellers with an aggregate panel capacity \(c\) over the panels’ lifespan can be calculated as \[\label{prt:total:rev} \Pi^\mathrm{s}_\mathrm{prt}(c) = \widetilde{T} \,\mathbb{E}\left[\int_{\mathcal{I}_\mathrm{s}} \Pi^\mathrm{s}_i (q_i^\mathrm{s},\pi)\,\, \mathrm{d}i\right].\tag{4}\]

We next describe the single-product real-time market, which also trades electricity. Unlike the product-differentiated real-time market, in this market both solar energy and utility-supplied energy are pooled together as a single product, regardless of the source of generation. Thus, consumers cannot express a preference or pay a premium for solar, even if they assign it a higher value. In fact, it can be shown that the outcome of this market coincides with that of the product-differentiated real-time market in the special case where \(v_i \equiv 0\). We denote the total expected revenue for sellers in the single-product real-time market as \(\Pi^\mathrm{s}_\mathrm{srt}(c)\).

3.3 Contract-based Market↩︎

Unlike the real-time solar markets, the contract-based market (cb) operates well in advance of the actual delivery period. Consequently, service contracts are traded based on the expected, rather than realized, solar generation. For concreteness, we follow the terminology in [13] and focus on a setting where the contract is tied to panel capacity, meaning a panel owner rents out units of capacity and the buyer receives the realized solar output associated with that rented capacity over the duration of the contract. Since both the traded quantity and price are determined prior to the realization of generation, this market differs fundamentally from real-time pricing mechanisms. Nevertheless, for notational consistency, we will reuse the notation from the previous subsection to describe outcomes in the contract-based setting.

Since seller \(i \in \mathcal{I}_\mathrm{s}\) can rent out any portion of their panel capacity \(c\), the set of feasible amount s to sell is \(\mathcal{Q}_i^\mathrm{s} =[0, c]\). Meanwhile, as buyer \(i\) can in principle lease any panel capacity, we have \(\mathcal{Q}_i^\mathrm{b} = \mathbb{R}_+\). Given the market equilibrium price \(\pi\) for panel capacities, the payoff for seller \(i \in \mathcal{I}_\mathrm{s}\) is \(\Pi_i^\mathrm{s}(q_i^\mathrm{s},\pi) = \pi q^\mathrm{s}_i\), and the payoff for buyer \(i \in \mathcal{I}_\mathrm{b}\) is \[\begin{align} \label{eq:payoff:cb} \Pi_i^\mathrm{b}(q_i^\mathrm{b},\pi) = v_i \mathbb{E}\min\{q^\mathrm{b}_i G, L \}-\!\pi q^\mathrm{b}_i -\!\pi_\mathrm{u} \mathbb{E} (L\!-\!q^\mathrm{b}_i G)_+, \end{align}\tag{5}\] where \((z)_+:= \max(z,0)\), and \(q_i^\mathrm{s}\) and \(q_i^\mathrm{b}\) are the cleared quantities (i.e., panel capacities) for the seller and buyer, respectively.

Adapting the same CE concept as in Definition 2, with the traded product and payoff functions updated, and given the equilibrium price \(\pi\) and seller cleared quantity \(q^\mathrm{s}\), we can express the total sellers revenue for the contract-based market as \[\label{cb:total:rev} \Pi^\mathrm{s}_\mathrm{cb}(c) =\widetilde{T} \int_{\mathcal{I}_\mathrm{s}} \Pi^\mathrm{s}_i (q_i^\mathrm{s},\pi)\,\, \mathrm{d}i.\tag{6}\]

Now that we have completed the modeling of these market mechanisms, we proceed to analyze how the corresponding CE can be derived for each.

4 Competitive Equilibria of Solar Markets↩︎

In this section, we derive the expected revenue for panel owners as a function of panel capacity \(c\).

When supply is abundant, i.e., the total realized solar generation exceeds demand, the resulting prices and allocations are identical under both types of real-time markets. This result follows from the fact that solar supply fully covers demand, rendering utility support unnecessary. As a consequence, the zero marginal cost of solar generation results in a zero equilibrium price. When the solar supply is insufficient to cover the load, the market clearing outcomes of the two market types become different. Since the utility backstop is required in this case, the market price for the single-product real-time market will be \(\pi_\mathrm{u}\). When solar is traded as a separate product, however, we see that the scarce solar production \(cG\) is allocated among the buyers with higher solar premiums first. The solar market price turns out to have a premium over the utility-supplied energy, driven by the competition among the buyers.

We now proceed to characterize the CE for the contract-based market. For each seller \(i \in \mathcal{I}_\mathrm{s}\), the utility-maximizing choice within the feasible set for \(q_i^\mathrm{s}\) is \(q_i^\mathrm{s} = c\). On the buyer side, for a given market price \(\pi\), let \(d_i^\star(\pi)\) denote the capacity that maximizes \(\Pi_i^\mathrm{b}(q_i^\mathrm{b}, \pi)\), as defined in 5 .

To analytically characterize \(d_i^\star(\pi)\), we define the expectation of a truncated version of \(G\) as the following differentiable, non-increasing function: \[\label{eq:funcg} \mu_{G_\mathrm{tr}}(d_i):= \mathbb{E} \left[G \mathbb{1} \bigl\{d_iG \le L\bigl\}\right].\tag{7}\] Let an extension of the inverse of this function be \[\tilde{\mu}_{G_\mathrm{tr}}^{-1}(z) := \begin{cases} {\mu}_{G_\mathrm{tr}}^{-1}(z), &if0 \le z \le \mathbb{E} G,\\ 0, &ifz > \mathbb{E} G, \end{cases} \nonumber\] where \({\mu}_{G_\mathrm{tr}}^{-1}(z):=\mathrm{sup}\{d_{i}|\mu_{G_\mathrm{tr}}(d_i)=z\}\). We can now characterize \(d_i^\star (\pi)\) as the following lemma.

This lemma allows us to obtain the CE for the contract-based market participation game.

5 Equilibrium Panel Capacities↩︎

Building on the short-term market outcomes derived in the previous section, we next study the investment model to characterize the resulting equilibrium panel investment capacities.

The underlying idea behind ?? is straightforward: regardless of the specific market mechanism, the aggregate capacity reaches equilibrium when the last infinitesimal investor who decides to invest earns zero net profit, meaning \(\Pi^\mathrm{inv}_{m,i}(x_i, c)\) defined in 3 is zero for that investor. In other words, the equilibrium capacity can be found at the point where the capital cost curve intersects the expected revenue curve, both expressed as functions of capacity \(c\). This relationship is illustrated in Fig. 2.

The existence of a solution to ?? , and consequently to ?? , is ensured under assumption 2 . If the capital cost \(\pi_0\) is too high, the expected return may fall short of covering the investment costs, even when \(c\) is small. When assumption 2 is violated, the stated equilibrium condition no longer applies, and \(c_{\mathrm{srt}}^{\mathrm{ne}}=0\).

While Lemma 4 enables us to use numerical simulations to compare the equilibrium capacities for each market mechanism, deeper insight comes from a theoretical comparison of the resulting aggregate capacities. To enable a more meaningful comparison, we first define a benchmark capacity as follows.

5.1 Benchmark: Social Welfare Optimal Panel Investment↩︎

In addition to the market-driven investment outcomes, we also consider the social welfare optimal panel investment capacity as a benchmark, defined as the aggregate capacity at which the total payoff of consumers and investors is maximized.

To this end, we first need to define how to optimally allocate solar energy, when it is scarce, among the collection of consumers in an operation period. Given the realization of solar generation \(G\), we can characterize an optimal allocation as a solution to the following optimization problem: \[\tag{8} \begin{align} \!\!\max_{\sigma:\mathcal{\mathbb{R}}_{+}\mapsto \{0,1\}} \quad& \int_{0}^{\bar{v}} v_i \sigma(v_i)\,\, \mathrm{d}F_V(v_i) \tag{9}\\ s.t.\quad\quad & \int_{0}^{\bar{v}} \sigma(v_i)\,\, \mathrm{d} F_V(v_i) \le\min\{cG/L, 1\},\tag{10} \end{align}\] where, for each \(v_i\), the binary variable \(\sigma(v_i)\) indicates whether the consumer’s load is fully served by solar (\(\sigma(v_i)=1\)) or not (\(\sigma(v_i)=0\)). The optimal value \(v^\star(c,G)\) then characterizes the maximum average solar premium that can be achieved given the solar capacity \(c\) and the realized \(G\). The social welfare optimal panel capacity is then a solution to the following optimization problem \[\begin{align} \label{social:cap} \max_{c\in \mathbb{R}_+} \enspace & \widetilde{T}\,\mathbb{E} \left[v^\star(c,G) L-\pi_\mathrm{u} (L-cG)_+\right]- \pi_0 c. \end{align}\tag{11}\]

The following result provides an analytical characterization of this capacity.

5.2 Analytical Comparison of Equilibrium Capacities↩︎

As ?? , ?? , and ?? have similar structures, comparison among \(c^\mathrm{ne}_\mathrm{srt}\), \(c^\mathrm{ne}_\mathrm{prt}\), and \(c_{\mathrm{opt}}\) is not difficult. However, comparing \(c^\mathrm{ne}_\mathrm{cb}\) with other equilibrium capacities turns out to be challenging in general. For this purpose, we re-parametrize the solar premiums of consumers and introduce a sequence of scaled versions of the problem as follows. For each \(i\in \mathcal{I}_\mathrm{b}\), fix \(\tilde{v}_i\) and let \(v_i = \epsilon \tilde{v}_i\) for \(\epsilon\ge 0\). Given a fixed distribution \(F_{\widetilde{V}}\) for \(\tilde{v}_i\), changing the value of \(\epsilon\) corresponds to shifting the distribution of the solar premiums that the population of buyers has for solar energy. While a large \(\epsilon\) reflects that buyers are highly environmentally conscious and willing to pay substantial premiums for solar energy over utility electricity, a small \(\epsilon\) indicates a limited willingness to do so. These scaled versions of the problem lead to the main results of this section:

The first interesting result from ?? in Theorem 1 is that the single-product real-time market, in general, leads to an under-investment in solar panels relative to social welfare optimal panel capacity. In fact, with \(\epsilon>0\), it is easy to identify a distribution \(F_{\widetilde{V}}\) such that \(c^\mathrm{ne}_\mathrm{srt}< c_\mathrm{opt}\), and therefore the single-product real-time energy market leads to under-investment. While one could argue that the single-product real-time market is easier to implement, particularly given the limitations of current metering infrastructure, it results in a loss of social welfare when viewed through the lens of long-term investment equilibrium.

The second observation form ?? is that the product-differentiated real-time market is guaranteed to achieve the socially optimal panel investment. This stems from the fact that the product-differentiated market indeed allocates solar energy in the same way as the social welfare optimal allocation in 8 . This leads to the same expected revenue for solar panel owners and therefore the same condition for the resulting aggregate panel capacities.

When the electricity users are unwilling to pay much more for solar energy (i.e., when \(\epsilon\) is small), we establish a qualitative relation between the contract-based market equilibrium capacity, \(c^\mathrm{ne}_\mathrm{cb}\), and the social welfare optimal capacity, \(c_\mathrm{opt}\). With a typical value of \(\pi_0\) so that \(c^\mathrm{ne}_\mathrm{srt}\) is relatively large, ?? suggests that the contract-based market will consistently lead to over-investment. Even though this analytical result is established only for small \(\epsilon\), our numerical results suggest that the over-investment effect extends to large values of \(\epsilon\) (Section 7.2). Taking a static point of view, this may be viewed as sub-optimal for the problem setting considered in this paper. However, if we incorporate potential economies of scale in panel manufacturing, over-investment may result in a smaller \(\pi_0\) in the future and thus bring benefits to the society. Therefore, under a dynamic investment model, a policymaker may favor the contract-based market.

Finally, and perhaps most surprisingly, ?? indicates that all market mechanisms yield the same equilibrium panel capacity when users place no additional value on solar energy relative to utility-supplied electricity. This scenario may arise if utility-supplied energy is also fully renewable, or if users exhibit no specific preference for solar generation. However, given current levels of renewable penetration, non-zero values of \(v_i\) are expected for a considerable portion of users [24]. As a result, the under/over-investment effects that market mechanisms have on the long-term solar equilibrium capacities are likely to remain substantial in the near future.

6 Heterogeneous Operation Periods↩︎

So far, we have assumed that the values/distributions of key parameters, such as loads and solar irradiation, are identical across operation periods. In practice, however, these values/distributions may vary over time due to factors such as seasonality, time of day, and day of the week. In this section, we extend our analysis to incorporate this temporal heterogeneity.

In our extended model, each period \(t\) is characterized by a distinct set of parameters: the total consumer load \(L_t\), the solar generation per unit panel \(G_t\) with associated probability density function \(f_{G_t}\), and the utility backstop price \(\pi_{\mathrm{u}_t}\). Under this setup, real-time markets are cleared independently for each \(t \in \mathcal{T}\), and the expected revenues of solar panel investors are determined accordingly. In contrast, contract-based markets involve long-term agreements, where consumers rent panels from suppliers once for the entire \(\mathcal{T}\).

Unlike the homogeneous case, where the expected revenue is identical across periods, here the revenues vary across operation periods. Hence, we compute them individually and then take their average. As in the homogeneous case, we then scale this average by \(\widetilde{T}\) to obtain the expected total revenue \(\Pi_m^{\mathrm{s}}(c)\).

We start by characterizing the CE of short-term markets and associated revenue for investors under this new setup.

6.0.1 CE of solar markets↩︎

The results from Lemma 1 and Table 1, with parameters \(G\), \(\pi_{\mathrm{u}}\), and \(L\) indexed by \(t\), apply to the real-time markets in each operation period \(t \in \mathcal{T}\). As a result, the expected total revenue for investors is \[\begin{align} &\Pi^\mathrm{s}_\mathrm{prt} = \sum_{t\in\mathcal{T}} \frac{\widetilde{T}}{T} \mathbb{E} \left [ \left(\pi_{\mathrm{u}_{t}}+\overline{F}_V^{-1}\left(\frac{cG_{t}}{L_{t}}\right)\right)\,G_{t}\,\mathbb{1} \bigl\{cG_{t} \leq L_{t} \bigl\}\right],\tag{12}\\ &\Pi^\mathrm{s}_\mathrm{srt} = \sum_{t\in\mathcal{T}}\frac{\widetilde{T}}{T} \mathbb{E} \left [\pi_{\mathrm{u}_{t}}\,G_{t}\,\mathbb{1} \bigl\{cG_{t} \leq L_{t} \bigl\}\right ]. \tag{13} \end{align}\]

In the contract-based market, \(d_i^\star(\pi)\) is a solution to \[\max_{d_i \ge 0} \enspace \sum_{t\in\mathcal{T}}\left[v_i\mathbb{E}\min\{d_i G_{t}, L_{t} \}- \pi_{\mathrm{u}_{t}} \mathbb{E} (L_{t}-d_i G_{t})_+\right]- \pi d_i.\nonumber\]

Similar to 7 , for each individual \(t \in \mathcal{T}\), we can define \(\mu_{G_{t,\mathrm{tr}}}(d_i):= \mathbb{E} \left[G_{t} \mathbb{1} \bigl\{d_iG_{t} \le L_{t}\bigl\}\right]\). Using this we can analytically characterize \(d_i^\star(\pi)\) as a solution to \(\sum_{t\in\mathcal{T}} \left(\pi_{\mathrm{u}_{t}}+v_{i}\right)\mu_{G_{t,\mathrm{tr}}}\left(d_{i}\right) = \pi.\) The left-hand side of this equation represents the consumer’s expected total valuation for one unit of rented panel capacity over the entire planning period. We treat this quantity as a monotonically decreasing function, denoted by \(w_i(d_i)\), and define an extension of its inverse as \[\widetilde{w}_{i}^{-1}(z) = \begin{cases} w_{i}^{-1}(z), &if0 \le z \le \sum_{t\in\mathcal{T}} \left(\pi_{\mathrm{u}_{t}}+v_{i}\right) \mathbb{E} G_{t},\\ 0, &ifz > \sum_{t\in\mathcal{T}} \left(\pi_{\mathrm{u}_{t}}+v_{i}\right) \mathbb{E} G_{t}, \end{cases} \nonumber\] where \(w_{i}^{-1}(z):=\mathrm{sup}\{d_{i}|w_{i}(d_{i})=z\}\). Thus, we get \(d_i^\star (\pi) =\widetilde{w}_{i}^{-1}\left(\pi\right)\), which results in \[\label{d:star:tv} \widehat d^\star(\pi) = \int d_{i}^\star(\pi)\,\, \mathrm{d}i=\int \widetilde{w}_{i}^{-1}\left(\pi\right)\,\, \mathrm{d}i.\tag{14}\]

Given this result, the conditions for the CE of the contract-based market given in Lemma 3 apply to this new setup. Thus, the total investors revenue is \(\Pi^\mathrm{s}_\mathrm{cb} = \frac{\widetilde{T}}{T} c\pi\).

6.0.2 NE panel capacities↩︎

Lemma 4 holds under this setup, with ?? , ?? , and the right-hand side of ?? replaced by 12 , 13 , and \(\widehat{d}^\star\left(\frac{\pi_0 T}{\widetilde{T}}\right)\) obtained from 14 , respectively.

6.0.3 Social welfare optimal investment↩︎

Different operation periods may have non-identical levels of loads and expected solar generation per unit panel capacity. Thus, for each \(t \in \mathcal{T}\), the optimal allocation of solar energy among consumers, denoted by \(\sigma_{t}: \mathbb{R}_{+}\mapsto \{0,1\}\), and the corresponding maximum average solar premium, denoted by \(v_{t}^\star(c,G_{t})\), should be obtained individually by solving 8 with an added subscript \(t\) to parameters \(L\), \(G\), and function \(\sigma\). Once \(v_{t}^\star(c,G_{t})\) is obtained for all operation periods, the social welfare optimal capacity can be characterized by:

6.0.4 Analytical comparison of equilibrium capacities↩︎

Building on these results, we now extend the main findings of Theorem 1 to this new setting:

7 Numerical Experiments↩︎

7.1 Case Study and Problem Description↩︎

We conduct numerical experiments in this section to assess our theoretical results. Given California’s leading role in solar adoption, we focus our analysis on this state and consider a large group of potential investors in late 2023 deciding whether to install solar panels across various locations within it. To simplify the analysis, we assume a single pooled distribution-level market for the entire state and consider the three representative market frameworks discussed earlier.

We consider a 25-year planning horizon, with hourly operation periods. While our analytical results are applicable to any number of heterogeneous operation periods, we simplify the case study by grouping operation periods into two representative categories. The first group corresponds to daytime hours, between sunrise and sunset, when solar irradiation is available. The second group includes the remaining hours, during which solar irradiation is minimal or absent. This setup is equivalent to considering \(\mathcal{T} = \{1, 2\}\), where \(t = 1\) corresponds to the first group and \(t = 2\) to the second, with appropriate weights applied to each group in the expected revenue calculations.

Solar irradiation data from 2000 to 2022 for San Francisco, California, are used to model the distributions \(f_{G_t}(g)\) for all \(t \in \mathcal{T}\) [25]. To account for panels’ efficiency, all irradiation values are scaled by a factor of 0.2 to obtain the effective solar irradiation [26]. A kernel density estimator is then used to fit a probability density function to the gathered data, modeling the daytime irradiation density \(f_{G_{1}}(g)\). The resulting density function is evaluated against the empirical distribution obtained from the histogram of the data in Fig. 3. Hours with zero or negligible effective irradiation (below \(0.1~\mathrm{W}/\mathrm{m}^{2}\)) are grouped into the second operation period, represented by a random variable \(G_{2}\) that equals zero with probability 1.

We use survey data from [27] on consumers’ willingness to pay a premium for renewable electricity and convert these values into the per-kWh solar premium \(v_i\) used in our model. As the data is relative to the average electricity bill for the users, we assume an average monthly household energy consumption of \(600~\mathrm{kWh}\) and convert the data into solar premium for \(1 \, \mathrm{kWh}\) of solar energy. We also adjust the data by applying the compound inflation rate in the U.S. from 1999 to 2023. We then fit a truncated exponential density function to the data to model \(f_V(v_i)\), which resulted in a mean of 2.86 $/kWh and an upper bound of 16.57 $/kWh. Both \(\pi_{\mathrm{u}_{1}}\) and \(\pi_{\mathrm{u}_{2}}\) are set at 29 $/kWh, the average retail electricity price in California in 2023 [28]. Moreover, we set \(L_{1}=27 \, \mathrm{GWh}\) and \(L_{2}= 29 \, \mathrm {GWh}\) [29]. We consider an average capital and installation cost of $2.7 per watt [30] of solar panels, which includes labor and equipment costs.

7.2 Numerical Results↩︎

7.2.1 Validity of Theorem 2↩︎

Given this setup, we first compute the expected revenues of panel investors under the real-time markets using 12 and 13 . Meanwhile, for the contract-based market, we obtain the aggregate demand function, i.e., \(\widehat{d}^\star(\pi)\), using 14 . Leveraging the characterizing equations for the heterogeneous case, we then obtain the NE aggregate investment capacity under the three considered short-term markets as summarized in Table 2.

First, observe that this result mirrors the comparison outcome established in ?? for case 1 (general case) in Theorem 1, and likewise in Theorem 2. Moreover, although in this setup we do not scale the distribution of consumers’ premiums, i.e., \(\epsilon = 1\) is not in the neighborhood of zero, and assumption 15 does not hold, we still obtain the same comparison result as stated in ?? , corresponding to case 2 (small solar premium) of Theorem 1, and likewise in Theorem 2. Specifically, the relation \(c_\mathrm{opt} \lesssim c^\mathrm{ne}_\mathrm{cb}\) in that case holds here with a strict inequality, i.e., \(c_\mathrm{opt} < c^\mathrm{ne}_\mathrm{cb}\). We then consider the scenario where consumers assign no premium to solar energy, i.e., \(\epsilon=0\), which yields identical aggregate capacities across all market mechanisms, as shown in Table 2 and implied by ?? under case 3 (no solar premium) of Theorems 1 and 2.

In the following two subsections, we analyze how variations in consumers’ solar premiums and in panel capital and installation costs affect the comparison of equilibrium aggregate capacities across market mechanisms.

7.2.2 Sensitivity analysis with respect to consumers’ solar premiums↩︎

and analyze the resulting investment outcomes. The results are shown in Fig. 4.

As consumers’ solar premiums increase, this figure reveals two key insights. First, the single-product real-time energy market results in a larger gap compared to the social welfare optimal capacity. This is primarily due to this market’s inability to capture the additional revenue potential from environmentally conscious consumers who are willing to pay a premium for clean energy. Second, the contract-based market consistently results in over-investment, reflected in the widening gap between \(c^\mathrm{ne}_\mathrm{prt}\) and \(c^\mathrm{ne}_\mathrm{cb}\).

7.2.3 Sensitivity analysis with respect to capital and installation costs↩︎

We vary the capital and installation costs for a unit panel capacity and analyze how these changes affect the equilibrium aggregate capacities, as shown in Fig. 5.

An immediate interpretation of this figure is that increasing upfront costs reduces the aggregate installed capacity of solar panels, while maintaining the inequality \(c_{\mathrm{srt}}^{\mathrm{ne}} < c_{\mathrm{opt}} = c_{\mathrm{prt}}^{\mathrm{ne}} < c_{\mathrm{cb}}^{\mathrm{ne}}\) up to a certain threshold. Beyond this point, a sufficiently high \(\pi_0\) renders solar investments unprofitable in the single-product market, causing inequality 2 to break down and resulting in \(c^\mathrm{ne}_\mathrm{srt} = 0\), as shown in the right-most inset subfigure of Fig 5. In contrast, due to the added revenue from environmentally conscious consumers, both \(c^\mathrm{ne}_\mathrm{prt}\) and \(c^\mathrm{ne}_\mathrm{cb}\) remain positive for some larger values of \(\pi_{0}\). However, once \(c^\mathrm{ne}_\mathrm{srt} = 0\), the underlying assumptions for the theoretical results no longer hold. This breakdown is reflected in the numerical results, which indicate a shift from the expected ordering: \(c^\mathrm{ne}_\mathrm{prt} = c_{\mathrm{opt}} > c^\mathrm{ne}_\mathrm{cb} > c^\mathrm{ne}_\mathrm{srt} = 0\). However, given the historical decline in solar installation and capital costs and ongoing technological advances, such extreme price levels are unlikely. Thus, in most realistic cases, we expect \(c_{\mathrm{cb}}^\mathrm{ne} > c^\mathrm{ne}_\mathrm{prt} = c_{\mathrm{opt}} > c^\mathrm{ne}_\mathrm{srt}\) to hold, consistent with our earlier theoretical insights.

8 Concluding Remarks↩︎

This paper examines how the design of short-term electricity markets influences long-term investment in distributed solar panels, where investors’ decisions are interdependent through the expected revenues earned from participating in these markets over the panels’ lifespans. To analyze this relationship, we develop an investment model that links short-term market equilibria to the resulting long-term investment outcomes. As a key analytical tool, we introduce a unified modeling framework that derives closed-form expressions for short-term equilibrium outcomes across a broad class of market mechanisms. These expressions then serve as inputs to our investment model. Then, we focus on three representative short-term market mechanisms from the literature: a single-product real-time market, a product-differentiated real-time market, and a contract-based panel market, each reflecting a distinct approach to pricing and trading in distribution networks. For each, we apply our unified modeling framework followed by the investment model to derive analytical expressions for the corresponding long-term equilibrium panel capacity. We then theoretically establish that the product-differentiated real-time market leads to socially optimal investment, while the single-product market results in under-investment. Interestingly, when solar premiums of consumers are small, our theoretical analysis reveals that the contract-based market leads to over-investment, a pattern that our numerical results confirm even at higher levels of consumer premiums.

Future directions include integrating centralized utility planning with the distributed investment problem and incorporating policy tools, such as tax incentives or feed-in tariffs, into our analysis of long-term investment equilibria.

9 Proofs for Section 4↩︎

10 Proofs for section 5↩︎

10.1 Proof of Lemma 4 and Proposition 1↩︎

10.2 Proof of Theorem 1↩︎

We establish each case individually:

11 Proof of Lemma 5 and Lemma 6↩︎

11.1 Proof of Lemma 5↩︎

We can use 7 to reformulate 17 as \[\label{prt:eps:g} \underbrace{ \frac{\pi_{0}}{\widetilde{T} } - \pi_{\mathrm{u}}\,\mu_{G_\mathrm{tr}}(c^\mathrm{ne}_\mathrm{prt}) - \epsilon\, k(c^\mathrm{ne}_\mathrm{prt}) }_{:=\phi_{\mathrm{prt}}(\epsilon, c^\mathrm{ne}_\mathrm{prt})} = 0,\tag{28}\] where \[k(c^\mathrm{ne}_\mathrm{prt}):=\mathbb{E} \left[ \overline{F}_{\widetilde{V}}^{-1}\left(\frac{c_\mathrm{prt}^{\mathrm{ne}}G}{L}\right) \,G\,\mathbb{1} \bigl\{c^\mathrm{ne}_\mathrm{prt}G \leq L \bigl\}\right].\nonumber\]

For given values of \(\pi_0\), \(\pi_{\mathrm{u}}\), and \(\epsilon\), the solution to \(\phi_{\mathrm{prt}}(\epsilon,c^\mathrm{ne}_\mathrm{prt})=0\) determines \(c^\mathrm{ne}_\mathrm{prt}\). We first show that if \(\epsilon\) is sufficiently close to \(0\), then the corresponding \(c^\mathrm{ne}_\mathrm{prt}\) lies in a neighborhood of \(c_0\).

To emphasize the dependence of the solution \(c^\mathrm{ne}_\mathrm{prt}\) on \(\epsilon\), we write \(c^\mathrm{ne}_\mathrm{prt} = \Psi(\epsilon)\), where \(\Psi: [0, 1] \to \mathbb{R}\). From ?? , it follows that \(\Psi(0) = c_0\). Furthermore,

the following lemma shows that \(\Psi(\epsilon)\) is Lipschitz continuous with constant \(L_{\Psi}\).

Equipped with Lemma 7, we have

\(|\Psi(\epsilon) - \Psi(0)| \leq L_{\Psi} |\epsilon - 0|,\) which implies \[\label{prt:error} |c^\mathrm{ne}_\mathrm{prt} - c_0| \leq L_{\Psi} \epsilon.\tag{29}\]

In other words, if \(\epsilon\) is sufficiently close to \(0\), then \(c^\mathrm{ne}_\mathrm{prt}\) remains close to \(c_0\). This proximity allows us to expand \(\phi_{\mathrm{prt}}(\epsilon, c^\mathrm{ne}_\mathrm{prt})\) in a Taylor series at \((0, c_0)\): \[\begin{align} \label{prt:Taylor} \phi_{\mathrm{prt}}(\epsilon,c^\mathrm{ne}_\mathrm{prt})=&-\epsilon\,k\left(c_{0}\right)-\pi_{\mathrm{u}}\, (c^\mathrm{ne}_\mathrm{prt}-c_0)\,\mu_{G_\mathrm{tr}}'(c_0)\nonumber \\ &+O(\lVert (\epsilon ,c^\mathrm{ne}_\mathrm{prt})- (0, c_0)\rVert ^2)=0. \end{align}\tag{30}\] Using 29 , we can bound the higher-order terms as \[\lVert (\epsilon ,c^\mathrm{ne}_\mathrm{prt})- (0, c_0)\rVert ^2 \leq \epsilon ^{2} \left(1+L_{\Psi}^2\right),\nonumber\] which can be applied in 30 to reformulate it as \[\begin{align} \label{prt:Taylor:exp} \phi_{\mathrm{prt}}(\epsilon,c^\mathrm{ne}_\mathrm{prt})=&-\epsilon\,k\left(c_{0}\right)-\pi_{\mathrm{u}}\, (c^\mathrm{ne}_\mathrm{prt}-c_0)\,\mu_{G_\mathrm{tr}}'(c_0)\nonumber \\ &+O(\epsilon^{2})=0. \end{align}\tag{31}\] Isolating \(c^\mathrm{ne}_\mathrm{prt}\) then yields ?? . It remains to prove Lemma 7.

11.2 Proof of Lemma 6↩︎

Using 7 , we can reconstruct 20 as: \[\underbrace{ \frac{\pi_{0}}{\widetilde{T}} - \pi_{\mathrm{u}} \, \mu_{G_{\mathrm{tr}}}\left(d_i^\star\left(\frac{\pi_{0}}{\widetilde{T}}\right)\right) - \epsilon \tilde{v}_i \, \mu_{G_{\mathrm{tr}}}\left(d_i^\star\left(\frac{\pi_{0}}{\widetilde{T}}\right)\right) }_{:=\phi_{\mathrm{cb},{i}}\left(\epsilon,d_i^\star\left(\frac{\pi_0}{\widetilde{T}}\right)\right)} = 0. \nonumber\] The idea mirrors the previous case: if \(\epsilon\) is close to \(0\), then the solution \(d_i^\star\left(\frac{\pi_0}{\widetilde{T}}\right)\) to \(\phi_{\mathrm{cb},{i}}\left(\epsilon, d_i^\star\left(\frac{\pi_0}{\widetilde{T}}\right)\right) = 0\) lies near \(c_0\). To prove this, note from ?? that \(d_i^\star \left(\frac{\pi_0}{\widetilde{T}}\right) = \tilde{\mu}_{G_\mathrm{tr}}^{-1}\left(\frac{\pi_0}{\widetilde{T}\left(\pi_{\mathrm{u}}+\epsilon \tilde{v}_i\right)}\right)\) and \(c_0=\tilde{\mu}_{G_\mathrm{tr}}^{-1}\left(\frac{\pi_0}{\widetilde{T}\, \pi_{\mathrm{u}}} \right)\). Thus, \[\left\lvert d_i^\star\left(\frac{\pi_0}{\widetilde{T}}\right) - c_{0} \right\rvert= \left\lvert \Delta_{\mathrm{cb},i}\left(\epsilon\right)\right\rvert,\] where \[\label{Delta:cb:i} \Delta_{\mathrm{cb},i}\left(\epsilon\right):=\tilde{\mu}_{G_\mathrm{tr}}^{-1}\left( \frac{\pi_0}{\widetilde{T}\left(\pi_{\mathrm{u}} + \epsilon \tilde{v}_i\right)} \right) - \tilde{\mu}_{G_\mathrm{tr}}^{-1}\left( \frac{\pi_0}{\widetilde{T}\, \pi_{\mathrm{u}}} \right),\tag{40}\] and it remains to bound \(\lvert \Delta_{\mathrm{cb},i}\left(\epsilon\right) \rvert\), which diminishes as \(\epsilon \rightarrow 0\).

To proceed, define the function \(r_{i}(\epsilon) := \frac{\pi_0}{\widetilde{T}\left(\pi_{\mathrm{u}} + \epsilon \tilde{v}_i\right)}\). Then, \(\Delta_{\mathrm{cb},i}\left(\epsilon\right) = \tilde{\mu}_{G_\mathrm{tr}}^{-1}(r_{i}(\epsilon)) - \tilde{\mu}_{G_\mathrm{tr}}^{-1}(r_{i}(0)).\) We now show that \(\tilde{\mu}_{G_\mathrm{tr}}^{-1}(r_{i}(\cdot))\) is Lipschitz continuous by proving that both \(r_{i}(\cdot)\) and \(\tilde{\mu}_{G_\mathrm{tr}}^{-1}(\cdot)\) are individually Lipschitz continuous. Note that both of these functions are decreasing, continuous, and differentiable on their respective domain. Thus, the Lipschitz constant for \(r_{i}(\epsilon)\) is given by \(L_{r_{i}} = \max_{\epsilon} \left| \frac{\mathrm{d}r_{i}\left(\epsilon\right)}{\mathrm{d}\epsilon}\right| = \frac{\pi_0 \tilde{v}_i}{\widetilde{T} \, \pi_{\mathrm{u}}^2}.\) Similarly, the Lipschitz constant for \(\tilde{\mu}_{G_\mathrm{tr}}^{-1}(\cdot)\) is obtained as \[L_{\tilde{\mu}_{G_\mathrm{tr}}^{-1}}=\max_{y} \quad \left | \frac{\mathrm{d}\tilde{\mu}_{G_\mathrm{tr}}^{-1}\left(y\right)}{\mathrm{d} y}\right |=\max_{y} \quad \left |\frac{\big(\tilde{\mu}_{G_\mathrm{tr}}^{-1}(y)\big)^3}{L^2 f_{G}\left(\frac{L}{\tilde{\mu}_{G_\mathrm{tr}}^{-1}(y)}\right)}\right |,\nonumber\] where \(\frac{\pi_{0}}{\widetilde{T} \left(\pi_{\mathrm{u}}+\bar{v}\right)} \leq y \leq \frac{\pi_{0}}{\widetilde{T} \, \pi_{\mathrm{u}}}\), and therefore \(\tilde{\mu}_{G_\mathrm{tr}}^{-1}(y)\) is bounded as \(\tilde{\mu}_{G_\mathrm{tr}}^{-1}\left(\frac{\pi_{0}}{\widetilde{T} \, \pi_{\mathrm{u}}}\right) \leq \tilde{\mu}_{G_\mathrm{tr}}^{-1}(y) \leq \tilde{\mu}_{G_\mathrm{tr}}^{-1}\left(\frac{\pi_{0}}{\widetilde{T} \left(\pi_{\mathrm{u}}+\bar{v}\right)}\right)\). Finally, since the composition of two Lipschitz continuous functions is also Lipschitz continuous, the function \(\tilde{\mu}_{G_\mathrm{tr}}^{-1}(r_{i}(\epsilon))\) is Lipschitz continuous with constant \(L_{\mathrm{cb},i}= L_{\tilde{\mu}_{G_\mathrm{tr}}^{-1}} \,L_{r_{i}}\). Therefore, \[\frac{\left|\tilde{\mu}_{G_\mathrm{tr}}^{-1}(r_{i}(\epsilon)) - \tilde{\mu}_{G_\mathrm{tr}}^{-1}(r_{i}(0))\right|}{ \epsilon} \leq L_{\mathrm{cb},i},\] and combining this with 40 , we conclude that \[\label{epsilon95cb:lim} 0\leq \Delta_{\mathrm{cb},i}=\left |\Delta_{\mathrm{cb},i}\right | \leq \epsilon \, L_{\mathrm{cb},i}\tag{41}\] where the first inequality holds because \(\Delta_{\mathrm{cb},i}(r_{i}(\epsilon))\) is an increasing function with respect to \(\epsilon\).

Equipped with this result, we derive the Taylor expansion of \(\phi_{\mathrm{cb},i}\left(\epsilon, d_i^\star\left(\frac{\pi_0}{\widetilde{T}}\right)\right)\) at \(\left(\epsilon, d_i^\star\left(\frac{\pi_0}{\widetilde{T}}\right)\right) = (0, c_0)\) as: \[\begin{align} \label{h:Taylor} \phi_{\mathrm{cb},i}\left(\epsilon, d_i^\star\left(\frac{\pi_0}{\widetilde{T}}\right)\right)= & -\left(d_i^\star\left(\frac{\pi_{0}}{\widetilde{T}}\right)-c_0\right) \pi_{\mathrm{u}}\mu_{G_\mathrm{tr}}'(c_0) \nonumber \\ & -\epsilon \tilde{v}_{i} \mu_{G_\mathrm{tr}}(c_0) \nonumber \\ &+O\left(\left\lVert \left(\epsilon ,d_i^\star\left(\frac{\pi_{0}}{\widetilde{T}}\right)\right)- (0, c_0)\right\rVert ^2\right)\nonumber \\ &=0. \end{align}\tag{42}\] We have \[\left\lVert \left(\epsilon ,d_i^\star\left(\frac{\pi_{0}}{\widetilde{T}}\right)\right)- (0, c_0)\right\rVert ^2 = \epsilon^2 + \underbrace{ \left(d_i^\star\left(\frac{\pi_{0}}{\widetilde{T}}\right)-c_0\right)^2 }_{\left(\Delta_{\mathrm{cb},i}\right)^{2}}. \nonumber\] Then, based on 41 , we obtain \[\left\lVert \left(\epsilon ,d_i^\star\left(\frac{\pi_{0}}{\widetilde{T}}\right)\right)- (0, c_0)\right\rVert ^2 \leq \epsilon ^2 \left(1+\left(L_{\mathrm{cb},i}\right)^{2}\right).\nonumber\]

Using this result in 42 , we can reformulate 42 as \[\begin{align} \label{vphsgjqt} \phi_{\mathrm{cb},i}\left(\epsilon, d_i^\star\left(\tfrac{\pi_0}{\widetilde{T}}\right)\right) = & -\left(d_i^\star\left(\tfrac{\pi_{0}}{\widetilde{T}}\right)-c_0\right)\pi_{\mathrm{u}}\mu_{G_\mathrm{tr}}'(c_0) \nonumber \\ & -\epsilon \tilde{v}_{i} \mu_{G_\mathrm{tr}}(c_0) +O\!\left(\epsilon^{2}\right)=0. \end{align}\tag{43}\]

Isolating \(d_i^\star\!\left(\tfrac{\pi_{0}}{\widetilde{T}}\right)\) yields ?? , and applying ?? directly then gives ?? , thereby completing the proof.

12 Proofs for Section 6↩︎

12.1 Proof of Proposition 2↩︎

The expression for \(v_t^\star(c, G_t)\) is derived individually for each \(t \in \mathcal{T}\) using the same method as in the proof of Proposition 1. Substituting these expressions into ?? yields: \[\begin{align} \max_{c\in \mathbb{R}_+} \enspace \sum_{t\in\mathcal{T}} \frac{\widetilde{T}}{T} \mathbb{E} \Biggr[&L_{t}\mathbb{1}\!\left\{cG_{t} \leq L_{t} \right\}\!\int_{0}^{F_{V}^{-1}\left(1-\frac{c G_{t}}{L_{t}}\right)} v_i \, \,\mathrm{d}F_{V}(v_{i})\nonumber\\ & +L_{t}\mathbb{E}[V]-\pi_\mathrm{u_{t}}\mathbb{(}L_{t}-cG_{t})_+\Biggr] - \pi_0 c. \nonumber \end{align}\]

The objective is concave in \(c\) since its second derivative is given by \[-\sum_{t\in\mathcal{T}}\int_0^{L_{t}/c} \frac{\left(g_{t}\right)^2}{L_{t}} \, \frac{f_G\left(g_{t}\right)}{f_V\left(F_V^{-1}\left(1 - \tfrac{cg_{t}}{L_{t}}\right)\right)} \, \,\mathrm{d}g_{t} \leq 0,\] which ensures that the problem is convex. Therefore, the first-order optimality condition is sufficient to analytically characterize \(c_{\mathrm{opt}}\) as ?? , completing the proof.

12.2 Proof of Theorem 2↩︎

We prove each case individually:

13 Proof of Lemma 8 and Lemma 9↩︎

13.1 Proof of Lemma 8↩︎

We can rewrite 44 as: \[\label{prt:eps:k:g:tv} \underbrace{ \pi_0 \frac{T}{\widetilde{T}} - \sum_{t\in\mathcal{T}}\left[\pi_{\mathrm{u}_{t}}\,\mu_{G_{t,\mathrm{tr}}}(c^\mathrm{ne}_\mathrm{prt}) - \epsilon\, k_{t}\left(c^\mathrm{ne}_\mathrm{prt}\right)\right] }_{:=\phi_{\mathrm{prt}}\left(\epsilon, c^\mathrm{ne}_\mathrm{prt} \right)} = 0.\tag{51}\] Here, we overload our notation by letting \(\phi_{\mathrm{prt}}\left(\epsilon, c^\mathrm{ne}_\mathrm{prt} \right)\) denote the left-hand side of the equation and defining \[k_{t}(c^\mathrm{ne}_\mathrm{prt}) := \mathbb{E} \left[ \overline{F}_{\widetilde{V}}^{-1}\left(\frac{c_\mathrm{prt}^{\mathrm{ne}} G_{t}}{L_{t}}\right) \, G_{t} \, \mathbb{1} \left\{c^\mathrm{ne}_\mathrm{prt} G_{t} \leq L_{t} \right\} \right], \nonumber\] and \(\Psi: [0,1] \to \mathbb{R}\) as the function mapping \(\epsilon\) to the solution to 51 . Similar to the proof of Lemma 7, we can show that \(\Psi\) is Lipschitz continuous⁶. Using this result, we then follow the same steps as in the proof of Lemma 5, with appropriately adjusted terms in the Taylor expansion of the \(\phi_{\mathrm{prt}}(\epsilon, c^\mathrm{ne}_\mathrm{prt})\), whose derivation is straightforward and leads to ?? , completing the proof. \(\qquad \qquad \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \, \, \, \, \blacksquare\)

13.2 Proof of Lemma 9↩︎

We begin by rewriting 50 in the following form: \[\label{cb:h:tv} \underbrace{ \pi_0 \, \frac{T}{\widetilde{T}} - \sum_{t\in\mathcal{T}} \left[\left(\pi_{\mathrm{u}_{t}} + \epsilon\,\tilde{v}_{i}\right) \mu_{G_{t,\mathrm{tr}}}\left(d_i\left(\frac{\pi_0 T}{\widetilde{T}}\right)\right) \right] }_{:= \phi_{\mathrm{cb},i}\left(\epsilon, d_i\left(\frac{\pi_0 T}{\widetilde{T}}\right)\right)} = 0,\tag{52}\] where we overload our notation and denote the left-hand side by the function \(\phi_{\mathrm{cb},i}\left(\epsilon, d_i \left(\frac{\pi_0 T}{\widetilde{T}}\right)\right)\). For a given \(\pi_0\), the solution to \(\phi_{\mathrm{cb},i}\left(\epsilon, d_i \left(\frac{\pi_0 T}{\widetilde{T}}\right)\right) = 0\) gives \(d_i^\star\left(\frac{\pi_0 T}{\widetilde{T}}\right)\).

To emphasize the dependence of \(d_i^\star\left(\frac{\pi_0 T}{\widetilde{T}}\right)\) on \(\epsilon\), we further overload our notation and define \(\Psi_i: [0, 1] \to \mathbb{R}\) such that \(d_i^\star\left(\frac{\pi_0 T}{\widetilde{T}}\right) = \Psi_i(\epsilon)\) and \(c_{\mathrm{srt}}^{\mathrm{ne}} = \Psi_i(0)\). Similarly, we can show that \(\Psi_i(\epsilon)\) is Lipschitz continuous for all \(i \in \mathcal{I}_\mathrm{b}\). Hence, we obtain \(\left| d_i^\star\left(\frac{\pi_0 T}{\widetilde{T}} \right) - c_0 \right| = \left| \Psi_i(\epsilon) - \Psi_i(0) \right| \leq L_{\Psi_i} \epsilon\), which allows us to bound the higher-order terms in the Taylor expansion of \(\phi_{\mathrm{cb},i}\left( \epsilon, d_i\left(\frac{\pi_0 T}{\widetilde{T}} \right) \right)\) around \((0, c_0)\), yielding ?? . Combining this result with 14 gives ?? , thereby completing the proof of Lemma 9. \(\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \blacksquare\)

References↩︎

---

1. M. Davoudi and J. Qin are with the Elmore Family School of Electrical and Computer Engineering, Purdue University. X. Lin is with the Department of Information Engineering, The Chinese University of Hong Kong. Emails: {mdavoudi , jq, linx}@purdue.edu↩︎

2. M. Davoudi and J. Qin are with the Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, USA. X. Lin is with the Department of Information Engineering, the Chinese University of Hong Kong. Emails: mdavoudi@purdue.edu, jq@purdue.edu, xjlin@ie.cuhk.edu.hk↩︎

3. This step may include interest rate adjustments and net present value computations.↩︎

4. We acknowledge the possibility that electricity prosumers can be both buyers and sellers at the same time. For the market mechanisms that we will consider in this paper, a prosumer can be modeled as the superposition of a buyer and a seller. This observation may not apply to other markets where buying and selling prices are non-uniform.↩︎

5. Suppliers with heterogeneous panel characteristics have been studied in, e.g., [13].↩︎

6. Establishing 32 with adjusted bounds is straightforward, as 33 holds for each operation period with appropriate modifications. Once continuity of \(\Psi(\epsilon)\) is shown, Lipschitz continuity follows by adapting the steps of the proof of Lemma 7, using adjusted terms and bounds. Since all expressions appear in summation form, the extension is direct.↩︎