Optimal Electricity Distribution Pricing under Risk and High

Photovoltaics Penetration

Maxim Bichuch ∗ Benjamin Hobbs † Xinyue Song ‡ Yijiao Wang §

October 9, 2019

Abstract

Based on game and optimization theory, a Stackelberg game is studied between a centralized regulatorand multiple rational consumers in a competitive electricity market, under PV penetration and demand-side uncertainty. The proposed model integrates the demand dynamics, generation costs and networkoperation, allowing for the market benefit maximization and fully recovery of utility costs, to realize theoptimal sizing of distributed PV and pricing by the market regulator. Mathematical results on Nashequilibria and an algorithm for the game modeling are presented. This paper also gives numerical resultsto illustrate the market participants’ coupled decision, calibrated by PJM data.

AMS subject classification: 91G60.

JEL subject classification: C6,C72.

Keywords: Distributed generation, photovoltaic energy, net metering, optimization, game theory

1 Introduction

Depite the history and full practice of power grid in enegy supply, the Behind the Meter system (BTM)is gaining popularity especially in the new millennium. Typically located on the owner’s property, BTM isdescribed as a renewable energy generation unit uniquely serving the on-site use of its owners, concequentiallyreducing electricity purchase from the grid and protecting the owner from possible blackouts and unfavorablepurchase rates. Together with the penetration of BTM rises renewable Distributed Generation (DG). Oncontrast to the conventional centralized power generation, DG is defined to be an electric power sourceconnected directly to the distribution network or on the customer site of the meter (?), fueled by renewableor non-renewable resources. It is commonly net-metered, allowing for bi-directional flows that incorporatesDG as not only a supplement to household- or facility-wise power usage, but an important part of theregional energy generation sources. Consistent with the popularity of BTM, it is predicted by the US EnergyInformation Administration (EIA) that renewable DG capacity will increase more than ten-fold by 2040,dominating fossil fueled DG by a substantial margin, with solar Photovaitic (PV) being the leading player1. Among the many factors that drive the movement in BTM and DG, including technology, the DemandSide Management (DSM), and environmental awareness, the economics behind it serves as one of the bestquantified and modelled driver. According to the National Renewable Energy Laboratory (NREL), the PV

∗Department of Applied Mathematics & Statistics, Johns Hopkins University, Baltimore, MD 21218, USA mbichuch@jhu.edu.Work is partially supported by NSF grant DMS-1736414.†Department of Environmental Healath & Engineering Johns Hopkins University, Baltimore, MD 21218, USA

bhobbs@jhu.edu. Work supported by NSF grant DMS-1736414.‡Department of Applied Mathematics & Statistics, Johns Hopkins University, Baltimore, MD 21218, USA xsong11@jhu.edu.

Work supported by NSF grant DMS-1736414.§Department of Environmental Healath & Engineering Johns Hopkins University, Baltimore, MD 21218, USA

ywang303@jhu.edu. Work supported by NSF grant DMS-1736414.1https://www.eia.gov/outlooks/aeo/nems/2017/buildings/

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cost benchmark decreased more than three quarters in 2010-2017 period 2, making it a better and betterchoice even in today’s market where costs and emissions of general fuels are on average declining thanks tothe techonology. The trade-off between self-generated and purchased electricity in terms of financial utilitymotivates the PV decision of small-scale power consumers, as a participant in the asymmetric game betweenprice makers and price takers. Especially, the realistic interaction between DG and the retail rates poses aseries of questions to the distributed generators, utilities and grid operators, such as the so-called death spiralin DG, defined as the mutual promotion of the increased DG capacity and the rising retail price in order torecover utility-wise costs. One of the best-known example is the great failure of net-metered PV in Arizona,2016, when the government had to invalidate the previous pricing scheme where households get full creditsfor the sold-back solar generation, for the sake of electricity utilities. It is studied by a dozen of literatureshow such interaction results from and incentivizes generation/consumption strategies in liberalized markets,but it’s noteworthy that the role of the grid operator or regulator becomes critical in events alike. Thepresent paper focuses on the economics behind PV, proposing a game theory based model for consumers’PV strategies and the grid operator’s regulatory decisions.

Early papers attempt to formalize the concept of DG, especially by looking at its distinction fromthe conventional centralized power generation. Despite the comprehensive definition of DG and, to someextent, standardized terminologies for DG implementation in ?, ?, ? overview the techniques, benefits,issues and accompanying terms for small-scale DG, and in addition admit that there is no consensus on theconcept of DG, regarding of different points of view and emphases on problems to address. With renewablepower generation prevailing in the new millennium, the photovoltaic (PV) system is especially an up-to-dateresource among all renewable DG, as pointed out by ?, ?.

With the superiority of DG over purely centralized generation investigated in ?, ?, ?, new chanllengesare uprising due to the great diference in the two paradigms, especially because of the potential bidirectionalcommunication in the system. The phisical impact of DG on power grids is discussed in a series of papers (?,?, ?). But a more noteworthy perspective is based the planning models, on the capability for DG integratedsystems to supply power and mitigate prices, solving related problems including grid and market structuredesign. ? draw an analogue of the multidirectional and volatile flows in the decentralized smart grid tothe closed-loop supply chain theory, and point out the importance for policy makers of the hand-in-handdevelopment of the grid, including the tariff design, and the demand side. This insight has been shared by ?,?, who address the implementation of DG on an existing or expected power grid, especially with net demandanticipations.

Though dterministic models as simple as linear or quadratic programing (?, ?) are proposed for DGplanning, one of the major issues in the multidirectional interaction is the uncertainty in DG. Two types ofuncertainty are typically considered in the context of renewable energy supply: the variability of the energysource and the uncertainty in the market demand. A probabilistic point of view is taken in some modelsfor placement of DG units with renewables under risks in ?, ?, ?, ?. More interestingly, viewing from amarket-wide perspective, ?, ?, taking into consideration both the demand and supply risks, investigate theeconomic effects and incentives for generation with different sources.

The role of certain regulator becomes important in the overall operation of the power system and themarket with the participation of DG. The impact of regulation is summarized by ?, ?. But a more interestingquestioin is that in a much liberalized market with competitive DG integration, whether and how a regulator“standing outside the court” should participate for the system to fully function. ? focuses on the imbalancepenalty that the regulator imposes on the intrinsic instability of renewables with respect to both financial andphysical solutions, and other regulatory optimization models are presented (?). In particular, the uncertaintyin financial markets is handled by the market-wide DG of different participators considering an additionalcost and the balance energy volume.

The interaction of electricity rates and DG investment serves as a starting point for further analysison the systematic planning and design. ? compare three tariffs under residential PV penetration by socialwelfare maximization with technical constraints, taking into account network charges and ancillary services.It is concluded that the net-metering tariff is preferable if PV adoption comes with no substantial increaseof costs in the system. However, if there are additional system-wide costs incurred by distributed PV, thenet-metering amplifies market inequality by shifting the costs to non-adopting households.

2https://www.nrel.gov/docs/fy17osti/68925.pdf

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? devotes to approximate the magnitude of impact of distributed PV on retail rates, attributing it tothree basic drivers: the penetration level, the utilities’ avoided costs, and the customers’ bill savings. Withthe presence of other economic issues influencing the retail prices, a guidline is offered on when the DGinduced price impact, including the aforementioned cost shifting, is negligible.

Conversely, the impact of rate structures on the benefits from distributed PV is studied by ?. Thecustomer economics are quantified by future bill savings on electricity, modelled in market scenarios withdifferent retail rate assumptions. Designed to recover certain costs, the rate can contain a fixed componentand/or a flat/time-varying volumetric component, with other adjustment like price caps considered.

The awareness of future uncertainty in power and PV related costs, by ?, is proposed to be a keyparameter in the death spiral. ? investigate the two offsetting feedback loops in decentralized power systems:one between the adoption of DG and the increasing retail prices due to under-recovery of utility costs, andthe other one between the PV deployment and the reduced bill saving as a result of demand shifting, in hopethat the two feedback effects offest each other at the aggregate level and hence avoiding the death spiral.

? investigate a hypothetic fully distributed trading system, where market participants trade excessgeneration to individually attain optima. Though it differs from the current studied scenario where thereis presence of bulk suppliers, the methodology of the leader-follower game allows mathematical analysis forequilibria and incentives, offering a perspective to model the feedback cycle.

This paper complements the above literature on distributed generation by presenting a detailed hierar-chical model for the tariff design under uncertain demand and distributd PV generation. While maintainingthe full competitiveness of the wholesale market and the net-metering in microgrids, we allow a control ofthe economics of the power system by setting a regulator who has power to adjust the price in a certain way.Unlike the previous studies, which accounts for either DG planning or its price impact, we solve the two endsof the same problem at the same time, precluding the under-recovery of utility costs and under-adoption ofPV by constraints in the optimization. The two critical questions to be answered by the planning model,the sizing of DG and the pricing of power, are realized through a hierarchical Stackelberg game, in which thefeedback loop is presented, giving the market operator the option to view the death spiral and cost shiftingat a model level.

The present paper is organized as follow: Section ?? gives an overview of the model of the decisionprocess and related assumptions for each party. The main mathematical results are presented in Section ??,with proofs in Appendix??. In Section ??, we look at some numerical expetiments with real-world calibrateddata. At last, conclusions and proposals on future work are listed in Section ??.

2 Model Description

In this section, we describe the game between the buyers and seller of electricity in the market, andtheir individual decision processes. An equilibrium result will be given in next section.

We simplify the electricity market into three parties of participants: consumers, a passive player as thebulk seller of power, and a regulator of the electricity market as a centralized utilty who owns and operatesthe network. Each type of consumer possesses a dynamic demand and makes rational decisions on PV, whichadd up to the market-wide energy load. The bulk supplier is passive in that his action (the bulk energyprice) is fixed once the consumers’ decisions are revealed, described by a function of the load. The regulatorobserves the bulk supplier’s and consumers’ behaviors, and determines other components in the retail pricebesides the generation cost (which is recovered by the bulk price). Note the regulator in our context is notin term of legislation or automatic control, but is named in the perspective that it “regulates” the marketby adjusting prices.

To simplify the model while resembling the real-world process, three time scales are used: slots, daysand years. The assumed consumers’ demand evolution is discretized into each slot t = 0, 1, 2, . . . , whoselength is taken to be 15 min in accordance with the available date. To avoid enormous variance for long-runsimulation, we in addition assume that the demand is reset to a deterministic value at the beginning of eachday. The consumers’ and regulator’s PV and pricing decisions are made annually, at the beginning of eachyear y = 0, 1, 2, . . . . Let t ∈ Iy be the slots in year y. Notationally, we denote the length of time betweenslots by ∆T , the total number of slots in a day by P1, and the number of days in a year by P2.

In the Subsection ??, we briefly summarize the problems of each market participant and introduce nota-

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tion. Detailed assumptions and mathematical formulations are given in the following subsections: Subsection?? for the passive bulk supplier, Subsection ?? for consumers, and Subsection ?? for the regulator.

2.1 Notation

We use the following mathematical notation: Vectors and matrices are denoted by bold letters. Forexample, X ∈ RI is an I-dimensional vector, and X(−i) is the (I − 1)-dimensional vector resulting fromdeleting the i-th item from X. The notation [·]i=1,...,I,j=1,...,J and [·]i=1,...,I respectively specifies a (I × J)-

dimensional matrix and I-dimensional (column) vector, and diag(

[·]i=1,...,I

)a n× n matrix, with diagonal

element (i, i) being the i-th coordinate of the vector. Additionally, 1I×J is a I × J matrix with all elementsone. The mean and standard deviation of a random variable X are denoted by µ(X) and σ(X), and themean and covariance matrix of a random vector X by µ(X) and Σ(X). Y + := max(Y, 0) means taking thenon-negative part of Y ∈ R.

To fully characteriz the consumers’ behavior, we abstract them into types m = 1, . . . ,M , each consisting

of Nm identical agents. An agent of type m has a dynamic demand d(m)t in kW at time t, which, for simplicity,

is assumed to be exogenous of the prices in the market. At the beginning of year y, he determines howmuch PV capacity to add to his system and hence the accumulated total PV capacity, denoted respectively

by ∆K(m)y and K

(m)y and measured in kW. The PV generation is primarily used to fulfil his individual

electricity demand, and any excess amount can be sold at a credit to the grid if permitted. In other

words, at time t ∈ Iy, his net demand to fulfill by purchase is(d

(m)t −K(m)

y

)+

and his net generation

to sell, when permitted, is(K

(m)y − d(m)

t

)+

. We also denote the corresponding vectors to be the demand

and generation of all types’ agents: dt =[d

(m)t

]m=1,...,M

, Ky =[K

(m)y

]m=1,...,M

. We denote the simple

summation of all consumers’ demand at time t by Dt (dt) :=

M∑m−1

Nmd(m)t , and the net aggregate demand

by Dt(dt,Ky) :=

M∑m−1

Nm

(d

(m)t −K(m)

y

)+

. For convenience, we may omit the arguments and simply write

Dt, Dt, unless necessary to show.A net metering model is used for the charges on consumers’ electricity usage. The price that a consumer

agent m faces is netted at every time index t, consisting of two (three) parts: a fixed cost C0 in ($/15 min)regardless of his energy consumption, a proportional cost for each kilo-Watt he purchases and a possibleproportional cost for the excess generation he sells to the grid. We assume the two proportional costs arebased on the stack price and adjusted by some adds-on decided by the regulator. The stack price (or thebulk market power price, as engineers may put it) is determined by the bulk seller, primarily to recover the(variable) costs at which electricity is generated at power plants. Given a net aggregate market demanda.k.a. the load, the bulk supplier solves the dispatch problem of minimizing the overall generation costs,and thereby decides the stack price as the dual variable of the generation plan. As a result, at each timet, the stack price is a mapping from the net aggregate demand Dt to an amount in $/kW, and we denotethis mapping by s(Dt). This stack price is not controlled by the regulator, rather, it’s a decision variableof the bulk supplier, a passiver player in the market game. Though, the regulator possesses certain marketpower, in that it can adjust the proportional prices by imposing two “adds-on” on consumers’ bills: C1 forelectricity purchased from the grid, and −C2 for PV generation sold to the grid. Namely, the consumeragent pays s(Dt) + C1 dollars to buy 1kW from the grid and claims s(Dt) − C2 dollars to sell 1kW to thegrid. For instance, C1 is the distribution fee, and C2 a deductible in the PV pay-back credit. We denotethe triple of pricing variables C := (C0, C1, C2), which is determined by the regulator and gives a pricingscheme in the power market. To guarantee the revenue of the power plants, it is assumed that C0, C1 ≥ 0.

Given a pricing scheme C, the agent m forms a prediction of his year-y personal utility. This predictionis based on not only his own future comsumption and generation, but also the possible power rates, which

is decided by the aggregate demand. We denoted it by pu(m)(d(m)t t∈Iy ,K

(m)y ,∆K

(m)y ; Dtt∈Iy | C), with

the prescript p ∈ n, s respectively denoting the non-selling case, where excess PV generation is simply

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discarded, and the selling case, where the excess is sold to the grid. The agent, who already owns a PV

capacity of K(m)y−1 kW at the beginning of year y, then maximizes this personal utility function to determine

∆K(m)y , the amount of capacity to add to his system. Note the personal utility function is dependent

on both the agent m’s behavior across the year, as shown by d(m)t t∈Iy ,K(m)

y ,∆K(m)y , and the aggregate

demand resulting from all agents’ decisions, as shown by Dtt∈Iy . We introduce the Nash game amongconsumers where each consumer decides the optimal PV capacity to maximize the individual utility, withother consumers’ behaviors taken into account. Indeed, the optimal PV capacity in year y of a single type-m

agent is dependent on the PV decisions of all the other agents. We denote this reliance by pk(m)∗y (K

(−m)y ),

and its value by pK(m)∗y , with the vector form pK∗y =

[pK

(m)∗y

]m=1,...,M

for market-wide decisions. The

resulting optimal PV increment at the beginning of year y is denoted by p∆K(m)∗y .

The above Nash game is based on a given pricing scheme. The regulator’s role is to design the optimalpricing variables: it determines C such that certain regulatory goals are achieved, once all consumers haveindividually optimized their PV decisions under this optimal C.

The notation are tabulated in Table ??. (We only show scalar notation, and the corresponding vectornotation are omitted.)

Consumers’ Demand

d(m)t Demand of a type-m agent at time t

Dt Market’s aggregate demand at time t

Dt Net aggregate market demand at time t

PV Generation

∆K(m)y PV increment of a type-m agent at the beginning of year y

K(m)y PV capacity of a type-m agent in year y

K(m)∗y Optimal PV capacity for a type-m agent in year y

Pricing Variables

C0 Fixed fee, nonnegative

C1 Proportional fee, nonnegative

C2 Proportional sales deductible

(a) Variables

s(Dt) Stack price, given the aggregate demand Dt

pu(m)(d(m)t t∈Iy ,K

(m)y ,∆K

(m)y ; Dtt∈Iy | C)

Predicted annual utility of a type-m agent in year y under

pricing scheme C, with individual demand d(m)t t∈Iy and

PV decision K(m)y ,∆K

(m)y , and net aggregate market demand

Dtt∈Iy ; p ∈ n, s denoting whether excess PV can be sold

pk(m)∗y (K

(−m)y )

Optimal PV capacity of a type-m agent in year y, as a

function of others’ decisions K(−m)y , p ∈ n, s

(b) Mappings

Table 2.1: Notation

The decision making and information flows at time t ∈ Iy are illustrated in Figure ??, with one boxfor each market participant (marked bold) and its decision to make. The oval inside each square shows thedecision variable or result of the corresponding participant, and the arrows represent the information flowsbetween participants.

We first look at the lower right part with a dashed boundary, which describes the Nash game amongconsumers, when the pricing scheme C is given. To begin with, the mapping s(·) for the stack price is passively

determined. If the PV capacity K(m)y is decided for all consumers m = 1, . . . ,M at the beginning of year

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Regulator:Maximize overall expected utility across market such thatrevenue and PV penetration are adequate

C

Consumer Agent 1:Maximize personal expected utility

K(1)y

Consumer Agent M:Maximize personal expected utility

K(M)y

...

Bulk Supplier:Minimize total generation costs such thatmarket demand is supplied

s(·)

Nash game

C

u(M)∗y (C),

K(M∗)y (C)

u(1)∗y (C), K

(1)∗y (C)

d(1)t , K

(1)y

s(Dt)

d(M)t , K

(M)y

Figure 2.1: Diagram of Interaction

y, together with the demand time series d(m)t t∈Iy , it results in the net aggregate demand Dt and whence

the stack price s(Dt). The predicted annual utility of each agent m = 1, . . . ,M can thereby be calculated.The optimal PV decision of the type-m agent is then calculated by individual utility maximization. It willbe shown in a later section that a Nash equilibrium can be eventually attained in this game. We denote in

the diagram the equilibrium PV decision by K(1)∗y (C), . . . ,K

(M)∗y (C), and the corresponding utility values

by u(1)∗y (C), . . . , u

(M)∗y (C), to emphasize its reliace on the pricing scheme.

Then we move to the rest part of the diagram, which is a Stackelberg game between the consumers

and the regulator. We assume that the regulator perfectly predicts K(m)∗y (C), u

(m)∗y (C)m−1,...,M , once a

feasible C is given. It then adjusts C such that the markte’s utility is maximized, while the power sellermakes a certain revenue and a desired PV penetration is guaranteed. The optimal pricing scheme C∗ isdefined as the eqilibrium of this game.

In the next three subsections, we look one by one at the market participants and their decisions, andgive formal mathematical description.

2.2 Bulk Supplier’s Model

We start with the simplest participant, the bulk supplier, whose only role is to determine the stack pricefunction s(·).

We set the bulk market price equal to the cost at which power is generated at power plants. Theminimization of the total generation costs at plants results in the simple merit order dispatch for the bulkprice: cheap resources are first used to produce power, and expensive resources are used only if the marketdemand cannot be fulfilled with cheaper ones. Denote the marginal cost of plant l (from low to high) byαl, and the corresponding cumulative production capacity by βl, and the bulk price is a step function of the

form s(Dt) =

L∑l=1

al1Dt∈[βl−1,βl), with al non-decreasing in l. For example, if the net aggregate demand

falls into the first step (β0 ≤ Dt < β1), then the cheapest resource (typically renewable energy) will be usedand result in a cost of α1. For convenience, let β0 = −∞, βL = ∞, and we tabulate the parameters at the

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end of this section in Table ??.

2.3 Consumer’s Model and Decision

In this subsection we profile the market demand, introduce PV decisions and compute personal utility.A stochastic process is used to model an agent’s demand process. Due to its inherent self-reverting

nature and to keep the discretization, we assume a discretized multivariate Ornstein-Uhlenbeck process

dt+1 − dt = κt(θt − dt)∆T + ΣtρZt√

∆T , t = 0, 1, . . . , (2.1)

with θt, κt, Σt respectively being the vector of long-term mean, the vector of reverting speed to mean andthe (diagonal) matrix of volatility at time t, and ρ being the ”half correlation” matrix among consumertypes, the lower triangular Cholesky decompositionof the correlation matrix. The stochastic noise at time t,Zt, is assumed to be independent and standard normally distributed.

We also assume periodicity of the demand dynamics: the demand is reset to an initial value at thebeginning of each day, the mean, reversion and volatility only vary across different days, and the corre-

lation is fixed all the time. More precisely, θt =[θ

(m)t

]m=1,...,M

, κt = diag

([κ

(m)t

]m=1,...,M

), Σt =

diag

([σ

(m)t

]m=1,...,M

)are fixed for all t that belong to the same day, and ρ is fixed across all the days,

with ρρT =

1 ρ12 . . . ρ1M

ρ12 1 . . . ρ2M

......

. . ....

ρ1M ρ2M . . . 1

, ρij ∈ [−1, 1], i, j = 1, . . . ,M .

For PV updates, we have the following rules:

1. A type-m agent decides to increase his PV capacity by ∆K(m)y at the beginning of year y, and the

generation capacity in year y is defined as K(m)y = K

(m)y−1 + ∆K

(m)y , for all y ≥ 1 (Assume that there is zero

capacity at the beginning);

2. Once installed, the panels cannot be uninstalled, and will not retire. That is, ∆K(m)y ≥ 0;

3. The total price for installing ∆K(m)y , denoted by Q(m), is averaged into each time interval across the

year;4. As in practice, the PV panels only generate during specific time periods t ∈ P, when the solar energy

is ample.

We then move on to the utility of a type-m agent, based on which he makes decision on ∆K(m)y . For

an agent of type m, we assume an exponential utility function with risk aversion γm > 0, with x being hisnet expenditure

U (m)(x) := − 1

γ(m)eγ

(m)x.

Explicitly, the expenditure, with the prescript n for the case where customers do not sell their excess PVgeneration and s for the case where they sell it, is a function of the stack price s, his net demand d, his netgeneration K, and the incremental PV capacity ∆K, given a pricing scheme C

nx(m)(d,K,∆K, s | C) := C0 + (s+ C1)(d−K)+ +Q(m)

P1P2∆K,

sx(m)(d,K,∆K, s | C) := C0 + (s+ C1)(d−K)+ − (s− C2)(K − d)+ +Q(m)

P1P2∆K.

Note that the expenditure can be negative if the representative consumer effectively gets paid by the grid.Assume that the consumer has a preference for high utility in the nearer future, characterized by the

time discounting factor r(m) ≥ 0. To decide the PV decision, which lasts for the following whole year, theannual expected utility function given the pricing scheme C is defined, for p ∈ n, s, as

pu(m)(d(m)t t∈Iy ,K(m)

y ,∆K(m)y ; Dtt∈Iy | C)

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:=∑t∈Iy

e−r(m)(t−P2nb t

P2nc)∆TE

[U (m)

(px(m)

(d

(m)t ,K(m)

y ,∆K(m)y , s

(Dt

)| C))]

∆T. (2.2)

Note above annual expected utility function is dependent on both the agent m’s behaviors, as shown

by d(m)t t∈Iy ,K(m)

y ,∆K(m)y , and the aggregate demand resulting from all agents’ decisions, as shown by

Dtt∈Iy . We introduce the game among consumers where each consumer decides the optimal PV capacityto maximize the individual utility, with other consumers’ behaviors taken into account. The optimal PVdecision in year y, with p ∈ n, s respectively denoting the decision when excess PV generation is simplydiscarded and when the excess is sold to the grid, is denoted by

p∆K(m)∗

y = arg max∆K

(m)y ≥0

pu(m)(d(m)t t∈Iy ,K(m)

y ,∆K(m)y ; Dtt∈Iy | C).

The parameters involved in the consumers’ game are summarized at the end of this section in Table ??.

2.4 Price Regularization

One of our ultimate goals is to design an optimal pricing scheme for the regulator, which is characterizedby the pricing triple C = (C0, C1, C2). We assume that the regulator correctly anticipates the consumers’PV decision, stated in Subsection ??. That makes it a Stackelberg leader in the multi-party game amongthe regulator and the consumers of different types.

Put formally, the overall decision making in year y is described by the following hierarchical Stackelberggame, where the regulator serves as a leader and all consumers as followers:

At the beginning of year y, the regulator first releases a pricing scheme C, and the history PV levelKy−1 is known to public. Each consumer, depending on wheter sale of excess PV is allowed, subsequently

make the decisionp∆K

(m)∗y = arg max

∆K(m)y ≥0

pu(m)(d(m)t t∈Iy ,K

(m)y ,∆K

(m)y ; Dtt∈Iy | C), m = 1, . . . ,M . With

Ky−1 known, the resulting optimal PV capacity K(m)∗y , is dependent on others’ decision K

(−m)y , denoted by

K(m)∗y = k

(m)∗y (K

(−m)y ),m = 1, . . . ,M . In the vector form, the market’s optimal PV decision is k∗y(Ky) :=[

k(m)∗y (K

(−m)y )

]m=1,...,M

. Though all consumers have no information on others’ generation decisions at the

beginning of the game, a Nash equilibrium can be eventually attained, such that every consumer achieveshis optima given others’ decisions fixed. Such an equilibrium is expected to be found for each given C. (SeeTheorem ?? for the proof.)

In each game, the regulator is able to corretcly predict how this equilibrium is produced and what itloos like. It then initiates the next Nash game by altering the triple C, with the objective of maximizing themarket’s total utility, while (on average) fully recovering the operation cost Λ, and guranteeing the minimumpenetration Ky of PV generation. That is the “outer layer”, a single-leader Stackelberg game: while everyconsumer knows the mapping from their PV generation to the electricity retail price, and accordingly chooseshis optimal generation until the Nash equilibrium is reached, the regulator does its own optimization, todetermine the pricing scheme, and iterate so until a Stackelberg equilibrium is reached. It is shown that anequilibrium exists for this leader-follower game.

We can set the target revenue to be higher than Λy by making C0 larger. So the interesting case iswhen it is on the boundery. Put mathematically, for p ∈ n, s, given the demand dynamics dtt∈Iy andall other parameters, we solve the following two-level optimization problem, with the feasible region for Cbeing nΩ = C = (C0, C1) : C0 ≥ 0, C1 ≥ 0, sΩ = C = (C0, C1, C2) : C0 ≥ 0, C1 ≥ 0:

pC∗ = arg minC∈nΩ

M∑m=1

N (m) · pu(m)(d(m)t t∈Iy ,K(m)

y ,∆K(m)y ; Dtt∈Iy | C) (2.3)

s.t.E

∑t∈Iy

e−r(t−P2nb tP2nc)∆T

M∑m=1

(pxm(d

(m)t ,

pK(m)y ,

p∆K(m)

y , s(Dt) | C) +Q(m)

P1P2

) = Λy,

M∑m=1

N (m) ·pK(m)∗y ≥ Ky,

8

wherepK(m)∗y =

pk(m)∗y (K(−m)

y ) = arg max∆K

(m)y ≥0

pu(m)(d(m)t t∈Iy ,K(m)

y ,∆K(m)y ; Dtt∈Iy | C), m = 1, . . . ,M.

The parameters are summarized in Table ??.

αl marginal costs of power plants in nondecreasing order, l = 1, . . . , Lβl accumulative production capacity of power plants, l = 0, 1, . . . , L; β0 = −∞, βL =∞

(a) Parameters for Bulk Supplier’s Model

Demand Dynamics

θt vector of demand mean at time t

κt matrix of reverting speed to mean at time t

Σt matrix of volatility at time t

ρ”half correlation” matrix, the lower triangular Cholesky decomposition of the correlation matrix,fixed over time

Zt stochastic noise at time t, independent and standard normally distributed

PV Generation

Qm price for PV installlation in $/kW

K(m) maximum installation each year

P time periods when PV generation is allowed

Utility Calculation

γ(m) risk aversion of a type-m agent

r(m) discount factor of utility of a type-m agent

(b) Parameters for Consumer’s Model

Λy target revenue of the power supplier in year y

Ky target minumum PV panetration in year y

(c) Parameters for Regulator’s Model

Table 2.2: Model Parameters

3 Mathematical Results

We introduce some mathematical results in this section. The firse subsection gives the closed-formobjective of the consumers’ optimization, based on the distribution of the demand. In the second subsection,some equilibrium results are given. Numerical simulations are provided in the Section 4.

3.1 Closed-form Objective

For proof of the results in this subsection, please refer to Appendix ??.First we introduce the following notations:We use the superscript (m) to denote the quantities for consumers of type m, and the superscript (−m)

for all consumers but type m. For example, D(m)t is the total demand of type-m consumers at time t, and

D(−m)t the total demand of all the others. Also, we denote the market’s total PV generation capacity in year

y by Gy := NTKy, and similarly decompose it into Gy = G(m)y +G

(−m)y .

The closed-form solution to (??) is given by induction:

dt =(1M×M − κt∆T )t\P2dnb tP2c + κt,j

t\P2∑i=0

(1M×M − κt∆T )t\P2−i−1θb tP2c+i

9

+

t\P2∑i=0

(1M×M − κt∆T )t\P2−i−1ΣtρZt√

∆T ,

with ‘\’ being taking the mod, and ‘b·c’ being rounding down.The distribution of consumers’ demand follows:

dt ∼ N(µ (dt) ,Σ (dt)), with

µ (dt) = (1M×M − κt∆T )t\P2dnb tP2c + ∆Tκt

t\P2∑i=0

(1M×M − κt∆T )t\P2−i−1θnb tP2c+i, (3.1)

Σ (dt) =

t\P2∑i=0

(1M×M − κt∆T )t\P2−i−1ΣtρρTΣt(1M×M − κt∆T )t\P2−i−1.

With above the distributions of D(m)t , D

(−m)t are readily available. We denote the mean and standard

deviation of Dt as µ (Dt) and σ (Dt), the mean and standard deviation of D(m)t as µ

(D

(m)t

)and σ(D

(m)t ),

and the correlation coefficient D(m)t and D

(−m)t between as ρ

(D

(m)t , D

(−m)t

).

Also, we have the functions f(x;µ, σ) to be the probability density of the univariate normal distributionwith mean µ and standard deviation σ, and f2(x, y;µX , µY , σX , σY , ρ) for the bivariate normal distributionwith marginal mean µX , µY , marginal standard deviation σX , σY and correlation ρ. Especially, we have the(marginal) standard normal density functions φ(x) = f(x; 0, 1), φ2(x, y, ρ) = f2(x, y; 0, 0, 1, 1, ρ), and the

corresponding probability distribution functions Φ(x) =

x∫0

φ(z)dz, Φ2(x, y, ρ) =

x∫0

y∫0

φ2(z1, z2, ρ)dz1dz2.

To calculate the expected utility, a main difficulty in integration is that Dt =

M∑m=1

Nm(d(m)t − K(m)

y ·

1t∈P)+ is not linear. To detour that, we us Dt :=

M∑m=1

Nm(d(m)t −K(m)

y · 1t∈P), instead of directly using Dt.

Intuitively, this simplification is natural when the individual demand is sufficiently high, which is formallyjustified in the last part of this subsection. With such simplification, we obtain the following closed-formexpressions.

Theorem 3.1. With the simplification with Dt, we have the expected annual utility in the case where PVgeneration is only used to fulfill personal demand,

nu(m)(d(m)t t∈Iy ,K(m)

y ,∆K(m)y ; Dtt∈Iy | C)

=− rtγ(m)

∑t∈Iy\P

L∑l=1

q(νl) (Φ (d∗ (νl, βl))− Φ (d∗ (νl, βl−1)))− rtγ(m)

∑t∈P∩Ty

(Φ (E∗) +

L∑l=1

q(νl)e−K(m)

t νl

(Φ2

(c∗(νl, βl), d

∗(νl, βl), ρ∗(D

(m)t , D

(−m)t

))− Φ2

(c∗(νl, βl−1), d∗1(νl, βl−1), ρ∗

(D

(m)t , D

(−m)t

))− Φ2

(c∗(νl, βl), D

∗(νl), ρ∗(D

(m)t , D

(−m)t

))+ Φ2

(c∗(νl, βl−1), D∗(νl), ρ

∗(D

(m)t , D

(−m)t

)))); (3.2)

and in the case where selling of excess PV generation is permitted,

su(m)(d(m)t t∈Iy ,K(m)

y ,∆K(m)y ; Dtt∈Iy | C)

=− rtγ(m)

∑t∈Iy\P

L∑l=1

q(νl) (Φ (d∗ (νl, βl))− Φ (d∗ (νl, βl−1)))− rtγ(m)

∑t∈P∩Iy

L∑l=1

(q(νl)e

−K(m)t νl

(Φ2

(c∗(νl, βl), d

∗(νl, βl), ρ∗(D

(m)t , D

(−m)t

))− Φ2

(c∗(νl, βl−1), d∗(νl, βl−1), ρ∗

(D

(m)t , D

(−m)t

))10

− Φ2

(c∗(αl, βl), D

∗(νl), ρ∗(D

(m)t , D

(−m)t

))+ Φ2

(c∗(νl, βl−1), D∗(νl), ρ

∗(D

(m)t , D

(−m)t

) ))+ q(νl)e

−K(m)t νl

(Φ2

(c∗(νl, βl), D

∗(νl), ρ(D

(m)t , D

(−m)t

))− Φ2

(c∗(νl, βl−1), D∗(νl), ρ

(D

(m)t , D

(−m)t

))− Φ2

(c∗(νl, βl), d

∗(νl, βl), ρ∗(D

(m)t , D

(−m)t

))+ Φ2

(c∗(νl, βl−1), d∗(νl, βl−1), ρ∗

(D

(m)t , D

(−m)t

))+ Φ (d∗(νl, βl))− Φ (d∗(νl, βl−1))

)), (3.3)

where

ρ∗(D

(m)t , D

(−m)t

)=ρ(D

(m)t , D

(−m)t

)σ(D

(m)t ) + σ(D

(−m)t )

σ (Dt),

νl = γ(m)(αl + C1), νl = γ(m)(αl − C2), rt = − ∆T

γ(m)e−r(m)(t−P2nb t

P2nc)∆T+γ(m)

(C0+PV (m)

P1P2∆K(m)

y

),

E∗ =N (m)K

(m)y − µ

(D

(m)t

)σ(D

(m)t )

, D∗(ν) = E∗ − σ(D(m)t )ν

N (m), q(ν) = e

(σ(D(m)t ))

2ν2

2(N(m))2+µ(D(m)

t )νN(m) ,

c∗(ν, b) = −ρ∗(D

(m)t , D

(−m)t

)σ(D

(m)t )ν

N (m)+b+Gy −N (m)K

(m)t − µ

(D

(−m)t

)σ(D

(−m)t )

,

d∗(ν, b) = −

(σ(D

(m)t ) + ρ

(D

(m)t , D

(−m)t

)σ(D

(−m)t )

)σ(D

(m)t )ν

N (m)σ (Dt)+b+Gy − µ (Dt)

σ (Dt).

For the justification of approximating Dt with Dt in caculating the annual expected utility, note that theonly resulting differece is the bulk price s(Dt) and s(Dt). With (??) being a finite summation, the total error

is bounded by bounds of the error resulting from approximating U (m)(px(m)

(d

(m)t ,K

(m)y ,∆K

(m)y , s(Dt) | C

))with U (m)

(px(m)

((d

(m)t ,K

(m)y ,∆K

(m)y , s(Dt) | C

)), p ∈ n, s. Define the term-wise error at time t by

perrt :=∣∣∣E [U (m)

(px(m)

((d

(m)t ,K

(m)y ,∆K

(m)y , s(Dt) | C

))− U (m)

(px(m)

((d

(m)t ,K

(m)y ,∆K

(m)y , s(Dt) | C

))]∣∣∣.We give a sufficient condition on the OU process parameters such that perrt is small. Recall νL =

γ(m) (αL + C1), ν0 = γ(m) (α0 − C2).

Corollary 3.2. Suppose d(i)

P2b tP2c ≥ 0, κ

(i)t ≤ 1

∆T , i = 1, . . . ,M . Also suppose C2 ≤ α0. For arbitrary ε > 0,

perrt ≤ ε for both p ∈ n, s, if

(1− κ(i)max∆T )P2

(d

(i)min + κ

(i)minθ

(i)min

)≥ K(i)

y + σ(i)max

√2Γ+, i = 1, . . . ,M,

with

Γ = (4ν2L (σmax)

2+ 3νL (dmax + P2θmax + νLσmax)− γ(m)

(C0 +

Q(m)

P1P2

)− 2 log

(ν2Lε

2α2LM

),

where d(i)min = min

td

(i)

P2b tP2c, θ

(i)min = min

tθ

(i)t , κ

(i)min = min

tκ

(i)t , κ(i)

max = maxtκ

(i)t , and dmax = max

i,td

(i)

P2b tP2c,

θmax = maxi,t

θ(i)t , σmax = max

i,tσ

(i)t .

The assumptions in Corollary ?? are natural: The daily initial demand is, by definition, supposed tobe nonnegative. Intuitively, if the reversion is too fast, the demand process would be primarily led by thenormal noise, making the OU process assumption less meaningful. The corollary says that if the process isreverting at a moderate speed, the error of the approximation can be arbitrarily small if the long term meanis appropriately large for sufficiently small volatility.

In fact, the corollary states that, as long as the OU process is modelled such that the mean of eachconsumer’s demand is sufficiently higher than his PV generation compared with the standard deviation, our

11

closed form expressions in Theorem ?? give a good estimate of the objective, the expected annual utility.We will use this estimate to optimize for the game equilibrium in Section ?? and give illustrative numericalresults.

3.2 Equilibrium Analysis

First we show by induction that effectively we can only look at one year to determine the ultimate PVgeneration capacity:

In year 1, K(m)1 = ∆K

(m)1 , and the optimization for this year is simply

maxK

(m)1

pu(m)(d(m)t t∈Iy ,K(m)

y ,∆K(m)y ; Dtt∈Iy | C).

In year y, supposeK(m)y−1 is given by arg max

K(m)y−1

pu(m)(d(m)t t∈Iy ,K(m)

y ,∆K(m)y ; Dtt∈Iy | C). The optimization

for year y is then

maxK

(m)y

pu(m)(d(m)t t∈Iy ,K(m)

y ,∆K(m)y ; Dtt∈Iy | C)

= e−γ(m) PV (m)

P1P2K

(m)y−1 pu(m)(d(m)

t t∈Iy ,K(m)y ,∆K(m)

y ; Dtt∈Iy | C)

which is equivalent to optimizing only within the current year.If we assume that the demand evolvement is annually cyclical, and the pricing scheme unchanged over

years, the problem boils down to one single optimization arg maxK

(m)y ≥0

pu(m)(d(m)t t∈Iy ,K(m)

y ,K(m)y ; Dtt∈Iy | C).

We study two equilibria in our hierarchical game model: the Nash equilibrium on PV generation strate-gies for the lower-level game among consumers, and the equilibrium on prices of the regulator. First wedefine and show that a Nash equilibrium can be attained with a given regulator’s strategy.

Definition 3.1. (Nash Equilibrium of Consumers’ Game) A solution K∗y ∈ E is called a Nash

equilibrium solution if the following holds for all m = 1, ...M , for all K(m)y ≥ 0,

pu(m)(d(m)t t∈Iy ,K(m)∗

y ,K(m)∗y ; Dt(dtt∈Iy ,K(m)∗

y )t∈Iy | C)

≥ pu(m)(d(m)t t∈Iy ,K(m)

y ,K(m)y ; Dt(dtt∈Iy ,K(m)

y )t∈Iy | C),

where Ky equals K∗y with the m-th entry replaced by K(m)y .

That is, no single player can improve his utility by a unilateral change in his decision.

Proposition 3.3. Fix the pricing scheme C. If the PV installation fee is high enough such that Q(m) >2α0 + C1 − C2, then the optimal PV capacity or a type-m agent,

pK(m)∗y = arg max

K(m)y ≥0

E[pu(m)

(d(m)t t∈Iy ,K(m)

y ,K(m)y ; Dtt∈Iy | C

)],

is bounded from above for both p ∈ n, s, regardless of other consumers’ PV strategy K(−m)y .

Theorem 3.4. (Existence of Nash Equilibrium of Consumers’ Game) With the pricing scheme anddemand evolvement given, the consumers’ optimization for the PV generation strategy has a Nash equilibriumsolution.

Proof. The above equilibrium point is equivalent to the fixed point K∗y = k∗y(K∗y ).We have shown that the the searching region E is compact, on which the consumers’ optimizatin problem

k∗y is defined and the optima is searched.For every single consumer’s PV strategy, the influence of other consumers’ PV generation is effec-

tively quantified by Gy, which is upper bounded by some G. Rewrite pu(m)(dtt∈Iy ,Ky,K(m)y ;C) to be

12

vm(K(m)y , Gy), m = 1, ...M . By the closed-form expressions in Section 2, vm is continuous in both argu-

ments. The problem is now k∗y(K∗y ) = [ arg max0≤K(m)

t ≤Gvm(K(m)

y , Gy)]m=1,...,M . By the Maximum Theorem,

K(m)∗y (Gt) := arg max

K(m)y ∈[0,Gy ]

vm(K(m)y , Gy) is upper-hemicontinuous.

The continuity of k∗y(Ky) follows. Note E is convex and compact. By Brouwer Fixed Point Theorem,there exists a fixed point K∗y such that K∗y = k∗y(K∗y ).

We also show the overall equilibrium of this bi-level game exists, if we impose a uniqueness or concavityassumption on the consumers’ game.

Theorem 3.5. (Existence of Leader’s Equilibrium in No-Excess-Sale Case) Assume the demandof any consumer type is not perfectly correlated with the demand of all other types. If the consumer’s Nashequilibrium is unique given any feasible leader’s strategy, or the objective of each consumer agent is concavein his PV decision, then the leader’s problem admits a Nash equilibrium.

Proof. This is a single-leader hierarchical game: the leader conjectures the followers’ (consumers) strategygiven its decision, and uses the followers’ equilibrium as a constraint in its own optimiation.

Given the leader’s decision, with the uniqueness of the follower’s equilibrium or the concavity of thefollowers’ objectives, the variational inequality[

∂

∂K(m)y

pu(m)(d(m)t t∈Iy ,K(m)∗

y ,K(m)∗y ; Dtt∈Iy | C

)]Tm=1,...,M

(Ky −K∗y ) ≥ 0, ∀Ky feasible

is a necessary and sufficient condition of the equilibrium. ? present an existence result for the leader’sequilibrium of problems alike. With a single leader, the consistency requirement is automatically satisfied,and the leader’s equilibrium is guaranteed if the consistently-conjectured optimization problem, as stated

in the paper, admits a global minimizer, which is true if ρ(D

(m)t , D

(−m)t

), ρ∗

(D

(m)t , D

(−m)t

)∈ (−1, 1) the

negative objective of the leader is coercive in (C,Ky) ∈ R2+M .

The assumption on the demand correlation in above theorem is natural in real-world data, and can beeasily checked once the parameters are callibrated. The equilibrium can be computed with Gauss-Seideltype heuristic.

However, in the excess-sale-permitted case, a general equilibrium may not exist or be very unstable, aswe will show by a numerical example in Section ??. Still, an equilibrium exists if consumers are allowedmore control on their PV decisions: if they are able to curtail their selling of PV generation, then similarresults to above hold.

We assume the curtailment decision is also annually made, and the optimization model is slightlymodified by adding one more decision variable η ∈ [0, 1] to the consumers problem:

C∗ = arg minC

M∑m=1

N (m) · pu(m)(d(m)t t∈Iy ,K(m)

y ),K(m)y ); Dtt∈Iy | C

)(3.4)

s.t.E

∑t∈Iy

e−r(t−P2nb tP2nc)∆T

M∑m=1

(pxm(d

(m)t ,K(m)

y ,∆K(m)y , s(Dt) | C) +

Q(m)

P1P2

) = Λy,

M∑m=1

N (m)K(m)∗y ≥ Ky,

where

K(m)∗y is obtained by solving max

∆K(m)y ≥0,η∈[0,1]

pu(m)(d(m)t t∈Iy ,K(m)

y ),K(m)y ); Dtt∈Iy | C

), m = 1, . . . ,M.

The Nash equilibrium for consumers’ game exists in above setting:

13

Theorem 3.6. (Existence of Nash Equilibrium for Selling Percentage) If we add another decisionvariable υ(m) ∈ [0, 1] for the type-m agent, and allow him to sell in year y his PV generation of the amount

υ(m)K(m)y , a Nash equilibrium exists over the decision couple (K

(m)y , υ(m))m=1,...,M .

Proof. It follows the same line as the above proof, with(υ(m)

)m=1,...,M

∈ [0, 1]M added to the decision

space.

And the Stackelberg equilibrium is similarly deduced.

4 Numerical Results

In this section, we will give some numerical results in our simulated market with a centralized utility andthree types of consumers: residential, commercial and industrial. The first subsection gives the parameterswe use in simulation, calibrated from data. The algorithm is breifly introduced in the second subsection,and the simulated strategies are given numerically in the last subsection.

4.1 Parameterization and Algorithm Formulation

For the demand simulation, the parameters κtt, θtt, Σt, ρ are estimated by MLE, under the in-dependent normal noise assumption. Due to availability of data, we use history records in Washington DC3, which include electricity consumption in kW for various consumer types at 15-min intervals. For fullcharacterization, please see Appendix ??.

For computation, we employ the following algorithm:First we build a black box solver for optimal PV generation under given pricing scheme -

1. Initialize Ky(0)

2. Mean-field game in optimization for K: In the kth iteration,

(a) Calculate Dt(k − 1) := NT (dt −Ky(k − 1) · 1t∈P)+ for all j, n

(b) Optimize Ky(K(m)y ) =

[arg max∆K

(m)y ≥0

E[pu(m)(d(m)

t t∈Iy ,K(m)y ,∆K(m)

y ; Dtt∈Iy | C)]]m=1,...,M

Loop until Ky(K(m)y ) stabilizes. Set it as K∗y (C0, C1, C2)

Then with the above solver, solve the optimization problem (??).

4.2 Environmental Benefit

Recent PJM area has witnessed natural gas takes the place of coal as one of the most important resourcesfor power generation. Data from EIA(2018)4 suggest the following supply curve, where the lower costs fornatural gas coal relative to coal result in a tendency for cleaner fuels at a lower costs in merit order.

Simulations show that as a result of the optimal PV decision, the general power production relies lesson dirtier fuels, for instance, coal, and moves toward a cleaner direction. Figure ?? shows the mean market-wide demand on a typical day in March, where demand in the green area is fulfilled with clean resources likenuclear and natural gas.

3http://www.buildsmartdc.com/buildings4https://www.eia.gov/electricity/annual

14

1091 2798 7146 11858

Cumulative Supply(MW)

2.14

30.53

37.45

126.43

Renew-

ableNuclear

Natural Gas

Coal

Petroleum

(a) (b)

Figure 4.1: (a):Bulk Market Supply Curve, 2018 (Vertical axis shows the price in $/MW.)(b):Impact of PV on Aggregate Demand (Vertical axis shows the aggregate demand in MW.)

4.3 Equilibrium Results

For equilibrium analysis, we use the bulk price in Appendix ??. This is an older data than the price inlast subsection, and we use this set for the exploratory purpose in this subsection, because it better fits themathematical equilibrium results, and is more robust when we modify demand dynamics to explore possibleconsequences. The results shown in this subsection is by no means an exact predictor for current or futurepower markets, but rather an illustrative investigation on what possible situations PV adopters and theirpower utility would face, when dynamics take different characteristics.

The discounting factors and risk aversions are respectively taken to be 1% and 10% equivalently to allconsumer types. We aim for a total cost recovery of Λy=$500,000/h.

We investigate three scenarios, where different pricing schemes are employed as a result of consumers’ PVgeneration at equilibrium. The major difference between scenarios are in the demand dynamics, controlledby the long-term mean and the volatility. We will see different PV strategies are adopted under differentcharacterization of demand.

4.3.1 Moderate Case

In this scenario, the exact parameter set callinrated from the data is used for consumers’ demanddynamics. The consumers’ demand is summarized in Table ??.

It turns out to be a typical situation where consumers determine their optimal PV strategy based onthe demand and price they face, and will serve as our base case. Results, as shown in Table ??, suggest thatconsumers would generally settle on a PV generation level which partially offsets their demand on average,and has the potential to provide with an exra earning when the market price is favorable. Around 60% ofthe total power is provided by self-generated PV, which is a rather optimistic anticipation. Around 75% ofthe total costs are recovered by the proportional fees.

4.3.2 Favorable-PV Case

We are particularly interested in modeling the death spiral, where consumers are greatly incentilizedto adopt DG because of high retail rates, and the power utility in turn has to further increase the price tocompensate for the lost sales.

More volatile demand dynamics are simulated, in the sense that risk-averse consumers are likely to turnto PV against the huge uncertainty in the market demand and hence in retail rates. In case where sellingexcess PV generation is allowed, to make consumers better adapted to the potential huge drop in retail

15

Residential Commercial Industrial

Average Demand 1.33 9.05 131.24Max Demand 3.06 13.44 236.40

(a) Individual Demand Summary

K∗Residential K∗Commercial K∗Industrial C0 C1 C2

Excess Generation Sale Forbidden 0.85 4.54 78.34 28.81 0.075 N.A.Excess Generation Sale Allowed 0.93 5.79 86.92 30.08 0.082 0.022

(b) Consumers’ and Regulator’s Decisions at Equilibrium

Table 4.1: Equilibrium Summary for Moderate Case

prices, we allow them to sell a proportion of their self-generation. (Imagine a household system where theowner installs PV to offset his demand, and is ready to sell when applicable. But by foreseeing the overallmarket demand, he realizes that it may not be profitable to sell full excess generation whenever available.It is not realistic that our smart PV owner adjusts frequently how much to sell to dynamically maintainoptimality. Instead, we assume once in a year, he is able to set a proportion of all his future PV generation,such that he could sell this much of his excess generation.)

Results, as shown in Table ??, suggest the PV penetration is higher than the base case, accounting for70% of the total power consumption.

Residential Commercial Industrial

Average Individual Demand 3.76 25.17 243.92Max Individual Demand 7.11 48.35 322.51

(a) Individual Demand Summary

K∗Residential K∗Commercial K∗Industrial C0 C1 C2

Partial Excess Sale Forbidden 2.89 19.41 200.01 7.92 0.11 0.152Partial Excess Sale Allowed 5.13*(55%) 40.20*(50%) 350.51*(68.5%) 7.91 0.115 0.153

(*: The PV generation amount is no substantial, as it can be curtailed by seling percentages (shown in brackets).)

(b) Consumers’ and Regulator’s Decisions at Equilibrium

Table 4.2: Equilibrium Summary for Favorable-PV Case

A fragile equilibrium is achieved: more than 90% of the operation costs are recovered through theproportional fees, which makes it ecnomical for consumers to generate PV electricity to lower their purchase.Meanwhile, the regulator, who is clever enough to learn from the 2016 termination of net metering in Arizona,decides to reduce consumers’ possible profit from PV, by imposing a high dedictible in the price at whichPV electricity is sold to the grid, such that consumers wouldn’t be lured to install so much PV that theretail power price is decreased to an unacceptably low level.

Indeed, this equilibrium is far from stable, in that even a minor change in prices or demand can result indramatic fluctuations. Further, if we force a low C2 or a high PV penetration goal Gy, the system exhibitsa death-spiral-type behavior, shown as an alternative increase in PV adoption and retail price, and presentsno explicit equilibrium.

16

4.3.3 Unfavorable-PV Case

In this case, the centralized utility faces such a situation, where for any slight increase in the proportionalfees, consumers are greatly incentivized to install PV, and in turn reduce the revenue of the utility. As aresult, the utility is forced to recover most of its operation costs by the fixed charge imposed on consumers,regardless of their PV generation. And due to the relatively low proportional fees, consumers tend to avoidPV compared with the base case.

This type of pricing scheme is likely to occur when the demand is relatively stable, such that the utilityforesees the consequence of a price change generally comes with little uncertainty and decides whether it’soptimal to recover by raising the fixed charge.

The results are summarized in Table ??, with the volatility artificially decreased from the originalparameter sets. Only about 18% of the total energy demand is fulfilled by PV. The costs are mainlyrecovered through the fixed costs, which accounts for around 80% of the total recoverd costs.

Residential Commercial Industrial

Average Demand 1.33 9.05 131.24Max Demand 1.86 9.98 142.46

(a) Individual Demand Summary

K∗Residential K∗Commercial K∗Industrial C0 C1 C2

Excess Generation Sale Forbidden 0.23 1.45 28.15 62.71 0.050 N.A.Excess Generation Sale Allowed 0.24 1.45 28.42 63.11 0.045 0.049∗∗

(*: In this case the actual value of C2 is not substantial, as there is typically no PV excess generation. )

(b) Consumers’ and Regulator’s Decisions at Equilibrium

Table 4.3: Equilibrium Summary for Unfavorable-PV Case

5 Conclusions

Renewable distributed generation is playing an increasingly important role in the present energy market,with PV being a particular example. Together with the economic and environmental benefits from PV,challenges to the present network are uprising, especially on the allocation of DG and the appropriate pricingscheme, with new considerations like the death spiral and cost shifting. A game theory model is presented onthe hierarchical decision making in a competitive market, considering both utility maximization of consumersand cost recovery of the regulator. Mathematically the equilibrium of such procedure is shown to exist andavailable by iteratieve algorithm. Numericl results show that the pricing and generating strategies dependon characteritics of the demand dynamics, and account for the positive feedback cycle between the retailprice and distributed generation. Moreover, a shift from highly polluting fossil fuels to cleaner energy canbe expected with appropriate incentives.

Some interesting future research lines are expected to be continued in further work. As DSM becomesa critical part in energy planning and environmental concerns, it is meaningful to consider the demandresponse to prices. A possible method is to include a simple reaction function of consumers to the stackprice, besides the presumed demand dynamics applied in this paper. The aggregation of prosumers enablesend users to make PV decision and inter-trade in the unit of local blocks, which enhances the efficiency andeconomics of DG. It is also expected that an optimal or near optimal aggregation can be given in futurework.

17

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19

Appendix A Proof of therorems and corollaries

For proof of Theorem ??, we first introduce the following lemma

Lemma A.1. For arbitrary a, b,

∞∫−∞

φ(x)Φ(ax+ b)dx = Φ(b√

a2 + 1), (A.1)

c∫−∞

φ(x)Φ(ax+ b)dx = Φ2(c,b√

a2 + 1,− a√

a2 + 1). (A.2)

Proof. Denote I(a, b) =

∞∫−∞

φ(x)Φ(ax+ b)dx, J(a, b, c) =

c∫−∞

φ(x)Φ(ax+ b)dx.

By Dominating Convergence Theorem,

Ib(a, b) =

∞∫−∞

φ(x)φ(ax+ b)dx =1√

2π√a2 + 1

e− b2

2(a2+1) ,

Jb(a, b, c) =

c∫−∞

1

2πe− 1

2 (a2+1)(x2+ 2aba2+1

x+ b2

a2+1)dx.

Note limb→−∞

I(a, b) = limb→−∞

I(a, b, c) = 0.

I(a, b) =

b∫−∞

Ib(a, s)ds = Φ(b√

a2 + 1),

J(a, b, c) =

b∫−∞

c∫−∞

1

2πe− 1

2 (a2+1)(x2+ 2aa2+1

xy+ 1a2+1

y2)dxdy = Φ2(c,

b√a2 + 1

,− a√a2 + 1

).

Proof. (Theorem ??) For both p ∈ n, s, pu(m)(d(m)t t∈Iy ,K(m)

y ,∆K(m)y ; | Dtt∈Iy | C) will be readily

available once, for every t ∈ Iy, we have E[U (m)

(px(m)

(d

(m)t ,K(m)

y ,∆K(m)y , s

(Dt

)| C))]

. That is, we are

interested in the expected utility of three types: the utility when PV generation is unavailable, available forprivate use only, and available for sale. Below we respectively derive the closed-form expression of the threetypes of utility.

Expression I: E[U (m)

(px(m)

(d

(m)t ,K(m)

y ,∆K(m)y , s

(Dt

)| C))]

with t ∈ Iy \ PThis is the expected utility at a single time slot during non-generation periods. The effective part inside

the expectation is

E[eγ

(m)(s(Dt)+C1)d(m)t

]=

L∑l=1

E[eνld

(m)t 1βl−1+Gy≤Dt<βl+Gy

].

With d(m)t =

D(m)t

N(m) , Dt = D(m)t + D

(−m)t , and (D

(m)t , D

(−m)t ) bivariant normally distributed, by simple

rearrangement and change of variables,

E[eνld

(m)t 1βl−1+Gy≤Dt<βl+Gy

]

20

= E

[eνlD

(m)t

N(m) 1βl−1≤D(m)t +D

(−m)t <βl+Gy

]

=

∫∫βl−1+Gy≤x+y≤βl

eνl

N(m)xf2

(x, y;µ(D

(m)t , µ(D

(−m)t ), σ(D

(m)t ),

(σ(D

(−m)t )

)2

, ρ(D

(m)t , D

(−m)t

))dxdy

= q(νl)

∞∫−∞

φ(y) (Φ (d0(y; νl, βl))− Φ (d0(y; νl, βl−1))) dy,

where d0(y; ν, b) =

b+Gy−µ(Dt)

σ(D(m)t )

−(σ(D

(m)t )+ρ

(D

(m)t ,D

(−m)t

)σ(D

(−m)t )

)νl

N(m) −(σ(D

(−m)t )

σ(D(m)t )

+ ρ(D

(m)t , D

(−m)t

))y√

1−(ρ(D

(m)t , D

(−m)t

))2.

By (??) in Lemma ??,

E[eνld

(m)t 1Dt∈[βl−1+Gy,βl+Gy)

]= q(νl) (Φ (d∗ (νl, βl))− Φ (d∗ (νl, βl−1))) .

As a result, for t ∈ Iy \ P, p ∈ n, s

E[U (m)

(px(m)

(d

(m)t ,K(m)

y ,∆K(m)y , s

(Dt

)| C))]

= − 1

γ(m)eγ(m)

(C0+Q(m)

P1P2∆K(m)

y

)E[eγ

(m)(s(Dt)+C1)d(m)t

]= − 1

γ(m)eγ(m)

(C0+Q(m)

P1P2∆K(m)

y

)L∑l=1

q(νl) (Φ (d∗ (νl, βl))− Φ (d∗ (νl, βl−1))) .

Epression II: E[U (m)

(nx(m)

(d

(m)t ,K(m)

y ,∆K(m)y , s

(Dt

)| C))]

with t ∈ Iy ∩ PThis is the expected utility at a generating time slot with excess PV generation discarded. We are

interested in the quantity

E[eγ

(m)(s(Dt)+C1)(d(m)t −K(m)

y )+]

= E[1d(m)

t <K(m)y

]+

L∑l=1

e−νlK(m)y E

[eνld

(m)t 1βl−1+Gy≤Dt<βl+Gy, d(m)

t ≥K(m)y

].

By the normality of d(m)t ,

E[1d(m)

t <K(m)y

]= Φ(E∗).

By rearrangement and change of variables, each term in the summation is

E[eνld

(m)t 1βl−1+Gy≤Dt<βl+Gy, d(m)

t ≥K(m)y

]= E

[eνlD

(m)t

N(m) 1βl−1+Gy≤D(m)

t +D(−m)t <βl+Gy,

D(m)t

N(m)≥K(m)

y

]

=

∫∫βl−1+Gy≤x+y≤βl+Gy,

x≥N(m)K(m)y

eνl

N(m)xf2

(x, y;µ

(D

(m)t

), µ(D

(−m)t

)), σ(D

(m)t ), σ(D

(−m)t ), ρ

(D

(m)t , D

(−m)t

))dxdy

= q (νl)

c∗(αl,βl)∫−∞

φ(y)Φ (d1(y, αl, βl)) dy −c∗(νl,βl−1)∫−∞

φ(y)Φ (d1(y, νl, βl−1)) dy −c∗(νl,βl)∫

c∗(αl,βl−1)

φ(y)Φ (d2(y, νl)) dy

,

21

where d1(y; ν, b) =

b+Gy−µ(Dt)

σ(D(m)t )

−(σ(D

(m)t )+ρ

(D

(m)t ,D

(−m)t

)σ(D

(−m)t )

)νl

N(m) −(σ(D

(−m)t )

σ(D(m)t )

+ ρ(D

(m)t , D

(−m)t

))y√

1−(ρ(D

(m)t , D

(−m)t

))2,

d2(y; ν) =

N(m)K(m)y −µ

(D

(m)t

)σ(D

(m)t )

− σ(D(m)t )ν

N(m) − ρ(D

(m)t , D

(−m)t

)y√

1−(ρ(D

(m)t , D

(−m)t

))2.

By (??) in Lemma ??,

E[eνld

(m)t 1βl−1+Gy≤Dt<βl+Gy, d(m)

t ≥K(m)y

]= q(νl)

(Φ2

(c∗(νl, βl), d

∗(νl, βl), ρ∗(D

(m)t , D

(−m)t

))− Φ2

(c∗1(νl, βl−1), d∗1(νl, βl−1), ρ∗

(D

(m)t , D

(−m)t

))−Φ2

(c∗(νl, βl), D

∗(νl), ρ(D

(m)t , D

(−m)t

))+ Φ2

(c∗(νl, βl−1), D∗(νl), ρ

(D

(m)t , D

(−m)t

))).

Hence for t ∈ Iy ∩ P,

E[U (m)

(nx(m)

(d

(m)t ,K(m)

y ,∆K(m)y , s

(Dt

)| C))]

= − 1

γ(m)eγ(m)

(C0+Q(m)

P1P2∆K(m)

y

)E[eγ

(m)(s(Dt)+C1)(d(m)t −K(m)

y )+]

= − 1

γ(m)eγ(m)

(C0+Q(m)

P1P2

)(Φ(E∗) +

L∑l=1

e−νlK(m)y q(νl)

(Φ2

(c∗(αl, βl), d

∗(νl, βl), ρ∗(D

(m)t , D

(−m)t

))− Φ2

(c∗1(νl, βl−1), d∗1(νl, βl−1), ρ∗

(D

(m)t , D

(−m)t

))− Φ2

(c∗(νl, βl), D

∗(νl), ρ(D

(m)t , D

(−m)t

))+ Φ2

(c∗(νl, βl−1), D∗(νl), ρ

(D

(m)t , D

(−m)t

)))).

Epression III: E[U (m)

(sx(m)

(d

(m)t ,K(m)

y ,∆K(m)y , s

(Dt

)| C))]

with t ∈ Iy ∩ PThis is the expected utility at a generating time with excess PV generation sold to the grid. We are

interested in the quantity

E

[eγ(m)

((s(Dt)+C1)

(d(m)t −K(m)

y

)+−(s(Dt)−C2)

(K(m)y −d(m)

t

)+)]

= E[eγ(m)(s(Dt)+C1)

(d(m)t −K(m)

y

)1d(m)

t ≥K(m)y

]+ E

[e−γ(m)(s(Dt)−C2)

(K(m)y −d(m)

t

)1d(m)

t <K(m)y

]= E

[eγ

(m)(s(Dt)+C1)(d(m)t −K(m)

y )+]− E

[1d(m)

t <K(m)y

]+

L∑l=1

e−νlK(m)y E

[eνld

(m)t 1βl−1+Gy≤Dt<βl+Gy, d(m)

t <K(m)y

].

The difference of the first two terms is already available. In the same manner, we calculate each termin the summation

E[eνld

(m)t 1βl−1+Gy≤Dt<βl+Gy, d(m)

t <K(m)y

]=

∫∫βl−1+Gy≤x+y<βl+Gy,

x<N(m)K(m)y

eνl

N(m)xf2

(x, y;µ(D

(m)t , µ(D

(−m)t ), σ(D

(m)t ), σ(D

(−m)t ), ρ

(D

(m)t , D

(−m)t

))dxdy

= q (νl)

c∗(νl,βl)∫c∗(νl,βl−1)

φ(y)Φ (d2(y; νl)) dy +

∞∫c∗(νl,βl)

φ(y)Φ (d1(y; νl, βl)) dy −∞∫

c∗(νl,βl−1)

φ(y)Φ (d1(y, νl, βl−1)) dy

22

= q(νl)(− Φ2

(c∗(νl, βl), d

∗(νl, βl), ρ∗(D

(m)t , D

(−m)t

))+ Φ2

(c∗(νl, βl−1), d∗(νl, βl−1), ρ∗

(D

(m)t , D

(−m)t

))+ Φ2

(c∗(νl, βl), D

∗(νl), ρ(D

(m)t , D

(−m)t

))− Φ2

(c∗(νl, βl−1), D∗(νl), ρ

(D

(m)t , D

(−m)t

))+ Φ (d∗(νl, βl))

− Φ (d∗(νl, βl−1))).

Hence for t ∈ Iy ∩ P,

E[U (m)

(sx(m)

(d

(m)t ,K(m)

y ,∆K(m)y , s

(Dt

)| C))]

= − 1

γ(m)eγ(m)

(C0+Q(m)

P1P2∆K(m)

y

)E

[eγ(m)

((s(Dt)+C1)

(d(m)t −K(m)

y

)+−(s(Dt)−C2)

(K(m)y −d(m)

t

)+)]

= −L∑l=1

((Φ2

(c∗(αl, βl), d

∗(νl, βl), ρ∗(D

(m)t , D

(−m)t

))− Φ2

(c∗1(νl, βl−1), d∗1(νl, βl−1), ρ∗

(D

(m)t , D

(−m)t

))−Φ2

(c∗(νl, βl), D

∗(νl), ρ(D

(m)t , D

(−m)t

))+ Φ2

(c∗(νl, βl−1), D∗(νl), ρ

(D

(m)t , D

(−m)t

)))· eνlK

(m)y q(νl)

+(−Φ2

(c∗(νl, βl), d

∗(νl, βl), ρ∗(D

(m)t , D

(−m)t

))+ Φ2

(c∗(νl, βl−1), d∗(νl, βl−1), ρ∗

(D

(m)t , D

(−m)t

))+ Φ2

(c∗(νl, βl), D

∗(νl), ρ(D

(m)t , D

(−m)t

))− Φ2

(c∗(νl, βl−1), D∗(νl), ρ

(D

(m)t , D

(−m)t

))+ Φ (d∗(νl, βl))

− Φ (d∗(νl, βl−1)))· e−νlK

(m)y q(νl)

)· 1

γ(m)eγ(m)

(C0+Q(m)

P1P2

).

Collecting all the pieces, we have the desired formulae.

Before going into the error approximation in Corollaty ??, we introduce a lemma and a proposition,looking at the limiting behavior of perrt, p ∈ n, s with respect to the normally distributed damand. :

Lemma A.2. For arbitrary a,

∞∫a

x2f(x;µ, σ)dx =(µ2 + σ2

)(1− Φ

(a− µσ

))+ (a+ µ)

σ√2πe−

(a−µ)2

2σ2 .

Proof. Note −e−(a−µ)2

2σ2 =

∞∫a

d(e−(x−µ)2

2σ2 ) = −√

2π

σ

∞∫a

(x − µ)f(x;µ, σ)dx. By rearranging and change of

variables, we have∞∫a

xf(x;µ, σ)dx = µ

(1− Φ

(a− µσ

))+

σ√2πe−

(a−µ)2

2σ2 .

By integration by parts,

∞∫a

x2f(x;µ, σ)dx = µ

∞∫a

xf(x;µ, σ)dx− σ√2π

∞∫a

xde−(x−µ)2

2σ2

= µ

(µ+

σ√2πe−

(a−µ)2

2σ2 − µΦ

(a− µσ

))+

aσ√2πe−

(a−µ)2

2σ2 + σ2

(1− Φ

(a− µσ

))= (µ2 + σ2)

(1− Φ

(a− µσ

))+ (a+ µ)

σ√2πe−

(a−µ)2

2σ2 .

Proposition A.3. Assume µ(d(i)t ) ≥ K

(i)y , i = 1, . . . ,M , and ν0 ≥ 0. For a set of fixed standard deviation

σ(d(i)t )i=1,...,m, nerrt → 0, as µ(d

(i)t )→∞ for all i.

23

Proof. First of all, note Dt ≥ Dt and s(Dt) ≥ s(Dt).We now look at the non-selling case. For a fixed demand and generation, U (m) (nx (d,K,∆K, s | C)) is

differentiable in s and has derivative ∂∂sU

(m) (nx (d,K,∆K, s | C)) = γ(m)(d−K)+U (m) (nx (d,K,∆K, s | C)).

By the mean value theorem, there exists a z ∈ [s(Dt), s(Dt)] such that, with B = e2γ(m)

(C0+Q(m)

P1P2∆K(m)

y

),

nerr2t

≤(E[∣∣∣U (m)

(nx(m)

(d

(m)t ,K(m)

y ,∆K(m)y , s(Dt) | C

))− U (m)

(nx(m)

(d

(m)t ,K(m)

y ,∆K(m)y , s(Dt) | C

))∣∣∣])2

=(E[∣∣∣γ(m)(d

(m)t −K(m)

y )+U (m)(nx(m)

(d

(m)t ,K(m)

y ,∆K(m)y , s | C

))(s(Dt)− s(Dt)

)∣∣∣])2

≤ E[∣∣∣γ(m)(d

(m)t −K(m)

y )+U (m)(nx(m)

(d

(m)t ,K(m)

y ,∆K(m)y , s | C

))∣∣∣2]E [∣∣∣s(Dt)− s(Dt)∣∣∣2]

= BE[(d

(m)t

)2

e2γ(m)(s+C1)(d(m)t −K(m)

y )+]E[∣∣∣s(Dt)− s(Dt)

∣∣∣2]

Substitute d(m)t ∼ N

(µ(d

(m)t ), σ(d

(m)t )

). Note γ(m) (s(D) + C1) ≤ νL, ∀D, and by Lemma ??,

∞∫0

x2f(x;µ, σ)dx ≤

µ2 + σ2 +µσ√2π≤ (µ+ σ)2 for µ, σ ≥ 0. For the first term we have,

E[(

(d(m)t −K(m)

y )+)2

e2γ(m)(s+C1)d(m)t

]≤ E

[((d

(m)t −K(m)

y )+)2

e2νLd(m)t

]

=

∞∫K

(m)y

(x−K(m)

y

)2

e2νL(x−K(m)y )f

(x;µ

(d

(m)t

), σ(d

(m)t

))dx

= e2(νLσ

(d(m)t

))2+2νL

(µ(d(m)t

)−K(m)

y

) ∞∫0

x2f

(x;µ

(d

(m)t

)−K(m)

y + 2νL

(σ(d

(m)t

))2

, σ(d

(m)t

))dx

≤ 1

ν2L

e4ν2L

(σ(d(m)t

))2+3νL

(µ(d(m)t

)−K(m)

y

)+νLσ

(d(m)t

),

where the last inequality is because µ(d

(m)t

)≥ K(m)

y and ex ≥ x2, ∀x ≥ 0.

For the second term, note P(s(Dt) > s(Dt)

)≤ P

[M⋃i=1

d(i)t < K(i)

y

]≤

M∑i=1

P[d

(i)t < K(i)

y

]. The

famous tail probability bound for each marginal normal distribution is given in literatures like ?: For

x ≥ 0, P[d

(i)t ≤ µ(d

(i)t )− xσ(d

(i)t )]≤ e−

x2

2 . So under the assumption d(i)t ≥ K

(i)y , i = 1, . . . ,M , we have

P[d

(i)t < K

(i)y

]≤ e− 1

2

(µ(d

(i)t )−K(i)

y

σ(d(i)t )

)2

. Now the following holds

E[∣∣∣s(Dt)− s(Dt)

∣∣∣2] = E[(s(Dt)− s(Dt)

)2∣∣∣∣ s(Dt) > s(Dt)

]P[s(Dt) > s(Dt)

]

≤ α2L

M∑i=1

P[d

(i)t < K(i)

y

]≤ α2

L

M∑i=1

e− 1

2

(µ(d

(i)t )−K(i)

y

σ(d(i)t )

)2

24

Collecting both pieces, we have

nerr2t ≤ F

(µ(d

(i)t )i=1,...,M , σ(d

(i)t )i=1,...,M

)(A.3)

:=α2LB

ν2L

e4ν2L

(σ(d(m)t

))2+3νL

(µ(d(m)t

)−K(m)

y

)+νLσ

(d(m)t

) M∑i=1

e− 1

2

(µ(d

(i)t )−K(i)

y

σ(d(i)t )

)2

.

For fixed σ(i)t , i = 1, . . . ,M , each term decays at the rate O

(e−cµ(d

(i)t )2

)for some constant c. We have

the desired convergence of nerrt → 0 as µ(d(i))t →∞ as a result.The proof for serrt, assuming that ν0 ≥ 0, is much similar. With ∂

∂sU(m) (sx (d,K,∆K, s | C)) =

γ(m)(d(m)t −K(m)

y )U (m) (nx (d,K,∆K, s | C)),

nerr2t ≤ BE

[(d

(m)t −K(m)

y )2e2γ(m)

((s+C1)d

(m)t −(s−C2)K

(m)t

)]E[(s(Dt)− s(Dt)

)2]

, with z ∈ [s(Dt), s(Dt)].

For the first multiplicand, substitute d(m)t ∼ N

(µ(d

(m)t ), σ(d

(m)t )

),

E[(d

(m)t −K(m)

y )2e2γ(m)

((s+C1)d

(m)t −(s−C2)K

(m)t

)]≤ E

[(d

(m)t −K(m)

y

)2

e2νL

(d(m)t −K(m)

y

)1d(m)

t ≥K(m)y

]+ E

[(d

(m)t −K(m)

y

)2

e−2ν0

(K(m)y −d(m)

t

)1d(m)

t <K(m)y

].

The first term has been dealt with. We now look at he second term. With γ(m) (s(D)− C2) ≥ ν0, ∀D,

E[(d

(m)t −K(m)

y

)2

e−2ν0

(K(m)y −d(m)

t

)1d(m)

t <K(m)y

]

=

K(m)y∫

−∞

(K(m)y − x

)2

e−2ν0(K(m)y −x)f

(x;µ

(d

(m)t

)−K(m)

y , σ(d

(m)t

))dx

= e2(ν0σ

(d(m)t

))2−2ν0

(K(m)y −µ

(d(m)t

)) ∞∫0

x2f

(x;K(m)

y − µ(d

(m)t

)− 2ν0

(σ(d

(m)t

))2

, σ(d

(m)t

))dx

= e2(ν0σ

(d(m)t

))2−2ν0

(K(m)y −µ

(d(m)t

))(K(m)y − µ

(d

(m)t

)− 2ν0

(σ(d

(m)t

))2

+ σ(d

(m)t

))2

≤ 1

ν20

e4ν2

0

(σ(d(m)t

))2+3ν0

(µ(d(m)t

)−K(m)

y

)−ν0σ

(d(m)t

).

The last inequality is because of the assumption ν0 ≥ 0.Combined with the bound of tail probability,

serr2t ≤

α2LB

ν20

exp

(4ν2

0

(σ(

d(m)t

))2

+ 3ν0

(µ(

d(m)t

)−K(m)

y

)− ν0σ

(d

(m)t

)) M∑i=1

exp

−1

2

(µ(d

(i)t )−K

(i)y

σ(d(i)t )

)2

+ F(µ(d

(i)t )i=1,...,M , σ(d

(i)t )i=1,...,M

)≤ 2F

([µ(d

(i)t )]i=1,...,M

,[σ(d

(i)t )]i=1,...,M

)(A.4)

The last inequality is beacuse of the monotonicity of the function f(x) =ex

x2, x ≥ 0.

25

Proof. (Corollary ??) From (??), we can bound the mean and variance of a type-i customer demand by

µ(d(i)t ) = (1− κ(i)

t ∆T )t\P2d(i)

nb tP2c + ∆Tκ

(i)t

t\P2∑j=0

(1− κ(i)t ∆T )t\P2−j−1θ

(i)

nb tP2c+j

≤ d(i)

P2b tP2c +

t−P2b tP2c−1∑

j=0

θ(i)

P2b tP2c+j ≤ dmax + P2θmax,

µ(d(i)t ) ≥ (1− κ(i)

max∆T )P2d(i)min + ∆Tκ

(i)min · P2(1− κ(i)

max∆T )P2θ(i)min

= (1− κ(i)max∆T )P2

(d

(i)min + κ

(i)minθ

(i)min

),

σ(d(i)t ) =

∆T

t\P2∑j=0

(1− κt∆T )2(t\P2−j−1)(σ(i)t )2

12

≤(

∆T · P2(σ(i)t )2

) 12

= σ(i)t ≤ σ(i)

max.

The error in (??) (and hence (??)) is then further bounded

perr2t ≤ 2F

([µ(d

(i)t )]i=1,...,M

,[σ(d

(i)t )]i=1,...,M

)

≤ 2α2LB

ν2L

exp(

4ν2L (σmax)

2+ 3νL

(µmax −K(m)

y

)+ νLσmax

) M∑i=1

exp

−1

2

(µ(d

(i)t )−K

(i)y

σmax

)2 (A.5)

A sufficient condition for (??) to be bounded by ε2 is µ(d(i)t ) ≥ K(i)

y + σ(i)max

√2Γ+, i = 1, . . . ,M . Note that

the assumption in Proposition ?? is automatically satisfied under this condition.

Proof. (Proposition ??) We want to show that pky(m)∗(K

(−m)y ) is upper bounded for both p ∈ n, s,

∀K(−m)y ≥ 0.First we claim that it suffices to show sky

(m)∗(0) is bounded from above. Note for any PV decision

K(m)y , for each realization of the demand dt, the benefit that agent m obtains by increasing 1kW more PV

generation, is the largest when K(−m)y = 0, because that is when the stack price is the highest possible. That

benefit becomes smaller when any other consumers increase their PV from 0, making PV less attractive.

As a result, the optimal PV for agent m is lthe argest if K(−m)y = 0, and that holds for every path.

Hence the maxima of the expected utility is also the largest if K(−m)y = 0. That is, pky

(m)∗(K(−m)y ) ≤

pky(m)∗(0), ∀K(−m)

y ≥ 0, for both p ∈ n, s. Similarly, allowing sale of excess PV generation makes PV

more attractive, i.e. nky(m)∗(K

(−m)y ) ≤ sky

(m)∗(K(−m)y ), ∀K(−m)

y ≥ 0.

Then we show sky(m)∗(0) = arg max

K(m)y ≥0

su(m)(dtt∈Iy ,K(m)y em,K

(m)y ;C) is bounded from above, with em

being the vector with a 1 in the m-th coordinate and 0’s elsewhere. Fix a generation level K(m)y = K for the

type-m agent. We devide the scenarios into two parts: where the stack price is the lowest one, and where itis not. With our aproximation of Dt ≈ Dt, denote the set A(K) = Dt(dt, Kem) > b1.

For arbitrary ∆K > 0, note A(K + ∆K) ⊂ A(K) and whence 1A(K+∆K)C ≥ 1A(K)C . Denote c :=

γ(m)

(Q(m)

P1P2− (2α0 + C1 − C2)

).

E[U (m)

(sx(m)

(d

(m)t , K + ∆K, K + ∆K, s

(Dt

(dt, (K + ∆K)em

)); C))]

≤ E[U (m)

(sx(m)

(d

(m)t , K + ∆K, K + ∆K, s

(Dt

(dt, (K + ∆K)em

)); C))

1A( ¯K+∆K)C

]= E

[U (m)

(sx(m)

(d

(m)t , K + ∆K, K + ∆K,α0; C

))1A(K+∆K)C

]26

≤ E[U(sx(m)

(d

(m)t , K + ∆K, K + ∆K,α0; C

))1A(K)C

]= E

[− 1

γ(m)eγ(m)

(C0+(α0+C1)(d

(m)t −K−∆K)+−(α0−C2)(K−d(m)

t +∆K)++Q(m)

P1P2(K+∆K)

)1A(K)C

]

≤ E

[− 1

γ(m)eγ(m)

(C0+(α0+C1)

((d

(m)t −K(m)

y )+−∆K)−(α0−C2)

((K(m)

y −d(m)t )++∆K

)+Q(m)

P1P2(K+∆K)

)1A(K)C

]= ec∆KE

[U (m)

(px(m)

(d

(m)t , K, K, s

(Dt(dt, K)

); C))

1A(K)C

]= ec∆K

(E[U (m)

(px(m)

(d

(m)t , K, K, s

(Dt(dt, K)

); C))]

−E[U (m)

(px(m)

(d

(m)t , K, K, s

(Dt(dt, K)

); C))

1A(K)

]). (A.6)

Specifically, E[U (m)

(px(m)

(d

(m)t , K, K, s

(Dt(dt, K)

); C))

1A(K)

]is negligible for sufficiently large

K: (E[∣∣∣U (m)

(sx(m)

(d

(m)t , K, K, s

(Dt

(dt, (K)em

)); C))

1A(K)

∣∣∣])2

≤ E[∣∣∣U (m)

(sx(m)

(d

(m)t , K, K, s

(Dt

(dt, (K)em

)); C))∣∣∣2]E [1A(K)

]≤ E

[∣∣∣U (m)(sx(m)

(d

(m)t , K, K, s

(Dt

(dt, (K)em

)); C))∣∣∣2]P [A(K)

]. (A.7)

From Theorem ??, we see E[∣∣∣U (m)

(px(m)

(d

(m)t ,K(m)

y ,K(m)y , s

(Dt

); C))∣∣∣2] grows at the rate ofO

(ec1K

(m)y

),

while P(A(K(m)y )) decays at O

(ec2(K

(m)y )

2)by the normal tail probability, where c1, c2 are constants irrele-

vant of K(m)y . So for arbitrarily ε > 0, there exists a Kε such that (??) is bounded by ε.

We combine (??) (??) to see, for arbitrary ε > 0, there exists a Kε such that, if we choose ∆Kε =

log

E[U (m)

(sx(m)

(d

(m)t , Kε, Kε, s

(Dt

(dt, (Kε)em

)); C))]

E[U (m)

(sx(m)

(d

(m)t , Kε, Kε, s

(Dt

(dt, (Kε)em

)); C))]

+ ε

, then for any δ > 0,

E[U (m)

(sx(m)

(d

(m)t , Kε + ∆Kε + δ, Kε + ∆Kε + δ, s

(Dt

(dt, (Kε + ∆Kε + δ)em

)); C))]

≤ ecδE[U (m)

(sx(m)

(d

(m)t , Kε, Kε, s

(Dt

(dt, (Kε)em

)); C))]

.

Above holds for all t ∈ Iy, and under the assumption in the proposition, c > 0. su(dtt∈Iy ,K(m)y em,K

(m)y ; C)

is a finite weighted summation of E[U (m)

(sx(m)

(d

(m)t ,K

(m)y ,K

(m)y , s

(Dt

(dt,K

(m)y em

)); C))]

, so its

maxima sk(m)∗y (0) must be finite.

Appendix B Parameters

The parameters for the demand dynamics are calibrated to 2016-2017 data in Washington DC, originallycollected to study smart meter utilization. The data are in 15-min intervals, and consist of 14 residences(apartments), 5 industrial sites, and 5 commercial sites. We assume 365 days for each year, and the twoyears are used in parallel to estimate a parameter set across one year. (That is P1 = 365, P2 = 96 in ournotation.) Missing data are substituted with the moving average. In order to simulate the overall householddemand in PJM area, the residential data are modified by a factor matching the average consumption oftypical local households.

27

The daily initial demand d(m)t t=P2n,n=1,...,P1

are estimated by the average daily initial value overconsumers of the type. We assume each month possesses a set of long-term mean θtt=0,1,...,P1P2−1, es-timated by the average demand at each time. Further, we assume that the reversion and volatility areidentical through a single day, respectively being κtt=0,1,...,P1−1 and σtt=0,1,...,P1−1, and calibrate themthrough Maximum Likelihood Estimation (MLE): With the initial value and long-term mean calculated

above, optimize for each couple (κ(m)t , σ

(m)t ) such that the likelihood of the normal noise is maximized.

Part of the calibration results are exhibited in Figure ??.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hour

0

0.2

0.4

0.6

0.8

1

1.2

Jan

Feb

March

April

May

June

July

Aug

Sep

Oct

Nov

Dec

(a)

1/1 3/1 6/1 9/1 12/1

Date

0

1

2

3

4

5

6

7

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

(b)

Figure B.1: Parameter Calibration, Residential Type as an Example(a): Long-term Mean θt (Vertical Axis shows value in kW)(b): Reversion κt (left axis, in kW/15 min) and Volatility σt (right axis, in kW)

The bulk price as shown in Figure ?? is used to better accomodate our exploratory simulation accountedfor in Subsection ??. It is taken from Maryland energy data, and modified to match up-to-date marketinformation.

938 2745 7904 10304 13615

Cumulative Supply(MW)

14

35

49

91

119

Renew-

able

Nuclear

Natural Gas

Coal

Petroleum

Other Fossils

Figure B.2: Bulk Market Price (Vertical axis shows the price in $/MW.)

28