1

Sharing Economy and Optimal InvestmentDecisions for Distributed Solar Generation

Rodrigo Henriquez-Aubaa,b, Patricia Paulic, Dileep Kalathild, Duncan S. Callawayb and Kameshwar Poollab

Abstract—Distributed generation is a versatile way to promoterenewable integration in electricity grids. In particular, rooftopsolar photovoltaic (PV) has seen an impressive growth in thelast years. This paper presents three different models to addressthe PV investment problem faced by households: (a) the status-quo of net-metering or feed-in tariffs, (b) a sharing economymodel, and (c) a cooperative PV investment decision faced byan aggregator participating in wholesale electricity markets. Inthe status-quo case, firms can sell back their excess generationto the utility at a feed-in or retail tariff subject to the constraintthat they cannot be net producers of electricity on an annualbasis. In our sharing economy model, firms can pool their excessPV generation and trade it in a spot market among themselves,but the collective do not get paid for selling electricity back tothe utility. In the cooperative case, a centralized entity decideswhere to optimally invest PV among the participants. Our maingoal in studying these three cases is that net-metering and feed-in programs are under threat and being phased out, whichplaces future distributed PV investment at risk. In this event,we argue that new business models, such as the sharing economymodel or a cooperative participation, offer a plausible pathway tomaintain and even accelerate distributed PV investment. Severalcase studies with numerical results are presented to discuss theproperties of the three proposed models.

Index Terms—Sharing Economy, Distribution Systems, PVGeneration, Aggregator.

NOMENCLATURE

A. Sets

N Set of firms or households, indexed by k =1, . . . , n.

T Set of time slots, indexed by t = 1, . . . , T .M Set of periods that represents a collective of con-

secutive time slots, indexed by m = 1, . . . ,M , thatusually represents months or years.

Tm Subset of time slots that belong on a period m.

B. Parameters

`k(t) Random load of firm k in time slot t in kW.wk(t) Effective random irradiance at firm k in time slot t

in kW/m2.πs Capital cost of solar PV per m2 amortized over T

time slots.

This research is supported by the National Science Foundation undergrants EAGER-1549945 and CPS-1646612, and by the National ResearchFoundation of Singapore under a grant to the Berkeley Alliance for Researchin Singapore.

a Corresponding author. rhenriquez@berkeley.edub University of California, Berkeley, CA 94720.c Technische Universitat Darmstadt, Germany 64289.d Texas A&M University, College Station, TX 77843.

πg(t) Wholesale price of electricity in $/kWh.πd(t) Retail tariff for electricity in distribution in $/kWh.πnm(t) Net-metering price for selling electricity to the grid

in $/kWh.amaxk Maximum area available to install PV panels for the

firm k.acapk Area necessary to install PV panels for the firm

k to be a zero-energy firm (in expectation) in theanalyzed time horizon.

C. Variables

ak Panel area in m2 installed for user k.a Vector that contains the PV installed ak for all users.

D. Notation

We define the average expected value as ET [x|y] =1T

∑t∈T E[x(t)|y(t)] and the positive part function as

f(x)+ = max(f(x), 0).

I. INTRODUCTION

INVESTMENT in renewable sources has seen a substantialgrowth in recent years. For example, in the last 5 years,

the renewable generation capacity has seen an average annualgrowth of 8.5% [1]. In particular, investment in photovoltaic(PV) has seen a significant growth worldwide, from 40 GWin installed capacity in 2010 to 303 GW in 2016 [2]. In theU.S., PV has an average annual growth of 68% over the lastdecade, and in the five-year period between 2012 and 2017,U.S. solar industry related employment grew by 16% annually,adding 131,000 jobs [3]. Much of this important growth isfrom small-scale distributed systems.

Distributed generation (DG) can play a major role in fu-ture electricity grids to meet the energy demand, promotingrenewable sources and reducing the greenhouse gas emissionsassociated in the power grid. In this sense, rooftop solar PVhas been the main driver in recent decades.

Three key aspects push the growth in behind-the-meter PVsystems. First, integrated costs of PV have reduced by 50%over the last 6 years [4], [5], while levelized costs are below$0.16 per kWh for distributed PV in many regions [6]. Second,subsidies improve the private economics of PV (for example,in the US, the federal investment tax credit [7]). Third, feed-in tariffs and net-metering policies – in which utilities creditcustomers for production in excess production – mandateutilities to purchase excess solar generation at premium orretail rates, which are higher than utilities’ avoided cost ofpurchasing energy in wholesale markets.

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Nevertheless, subsidies and tax credits are being phasedout, and utilities strongly oppose to feed-in tariffs or net-metering policies because it enables customers to avoid costsof infrastructure, reserves and reliability. In addition, thesepolicies are criticized due to the increased cost shift fromowners of distributed generation to other utility ratepayers.Depending on the net avoided cost of utilities and net-meteringpolicies, increasing penetration of PV can potentially increasethe retail price of electricity [8].

In this work we explore the impacts and effects of PVinvestment decisions of residential consumers under differentmarket structures. In particular, how users may change theiroptimal investment decision depending on tariff structures,such as net-metering policies or a new scheme based on asharing-based structure. We submit that a sharing economybusiness model offers a plausible alternative to sustain behind-the-meter PV growth, even if net-metering policies are phasedout. Sharing models have already transformed industries likeridesharing (Uber, Lyft) and lodging (Airbnb) using peer-to-peer platforms. This paper explores how community of homescan share excess generation among them. In our earlier work,we have studied a sharing economy business model of firmssharing installed electricity storage to arbitrage against time-of-use tariffs [9]. In addition, this work extends previousresults of our sharing model for PV investment decisions pre-sented in [10]. There has been several pilot programs that arecurrently studying sharing economy models for transactions ofexcess solar generation. For example, the Brooklyn Microgridin the US or the NRGcoin in Europe, are projects that areexploring peer-to-peer energy transactions using blockchain-based framework [11], [12].

There is a substantial amount of work on optimal sizing andsiting of DG. Papers such as [13]–[18] focus on engineeringimpacts, cost minimization or utility profit maximization.Game theoretical approaches [19]–[21], examine peer-to-peerenergy trading schemes among consumers and producers.However, research in this area only tackles the problem fromthe perspective of individual customers or of distributionsystem operators. Besides our previous work [10], we areunaware of efforts to address how PV investment decisionschange when users are able to share, or aggregate, their solarproduction. While the main idea of aggregating variable energyproduction has already been applied in other contexts, [22]–[26], sharing and aggregation still has to be studied as analternative to feed-in tariffs or net energy metering policiesfor individual consumers.

This paper proposes different optimization models to deter-mine the optimal investing of PV under various pricing andsharing schemes, explicitly considering the uncertainty frombehind-the-meter energy consumption and solar generation.The paper considers a first interaction among how new pricingschemes could impact utilities and affect those same users,depending on the penetration of this new schemes. The maincontributions of this work are:• Three optimization models that decides the optimal in-

vestment decisions of rooftop photovoltaic solar, underthree different market structures and customers’ partici-pation.

• We derive analytic results on the solution of the proposedmodels under specific conditions, that provides intuitionon the main drivers that promotes the rooftop PV invest-ment.

• Thorough analysis for different numerical simulations,using real data on load data and solar production. Resultsshow that sharing schemes can promote PV investmentin comparison to standalone scenarios, while wholesalemarket models can drastically modify their PV investmentin presence of distribution fees.

The rest of the paper is organized as follows. Section IIpresents the problem set-up and mathematical formulationof the models proposed. In Section III, several numericalexperiments are presented and analyzed. Finally, conclusionsare drawn in Section IV.

II. PROBLEM SET-UP FOR FIRMS

Consider a connected community N of n firms (users) orhouseholds indexed by k. These firms are considering insta-llation of rooftop solar photovoltaic systems. The investmentdecision for firm k is the panel area ak for its PV system. Weconsider a multi-year investment horizon broken into smalltime intervals (e.g. 1 hr), on which time slots are indexed byt. We use a simple linear model of investment costs: the priceof PV panels amortized over their lifetime is πs $/m2 per timeslot. This can be derived by combining commonly used $/wattPV levelized cost figures with production models (see SectionIII).

Firm k receives irradiance wk(t) in slot t. If it invests in akof panel area, this home generates akwk(t) kWh of electricityin slot t. Thus, we say that wk(t) captures the effectiveirradiance including factors such as panel efficiency, inverterefficiency and relative declination of incident sunlight. Itvaries with time of day and technology choice. The electricitydemand for home k in time slot t is `k(t). We treat `k and wkas non-stationary random sequences. The firm demand `k(t)is first served by its own local PV production. The deficit ornet-load (`k(t) − akwk(t))+ is bought from the utility at aspecific tariff. Depending on the situation, any surplus may besold back through different schemes. This situation is depictedin Figure 1.

panelarea ak

firm kirradiancewk

load `kakwk

PV gen

`k − akwk

net-load

Fig. 1. Set-up: irradiance, PV generation, and net-load for a firm.

We consider three different problems for studying the in-vestment of PV panels from firms or households. These threeproblems are depicted in Figure 2.

A. Standalone problem

We consider a single firm that have to decide how muchm2 of PV panels install depending on its own load, irradiance,

3

firm 1 firm k..... .....

Distribution System

πnm πd

Utility

πnm πd πnm πd

firm n firm 1 firm k..... ..... firm n

Distribution Systemand Sharing Platform

πeqπeq

Utility

Collective of firms

firm 1 firm k..... ..... firm n

Dist. System

Collective of firms

Aggregator

Utility

πg

dist.fees

dist.fees

dist.fees

Fig. 2. Three different problems for PV installation in distribution. (a) left: standalone problem, (b) middle: sharing scheme model, (c) right: collective onwholesale markets.

costs and pricing as observed on the left side of Figure 2. Theoptimization problem for each user k, P sa

k , can be written as:

minak

Jk(ak) = πsa+ E[πd(t)(`k(t)− akwk(t))+]

− E[πnm(t)(akwk(t)− `k(t))+] (1a)

subject to:

0 ≤ ak ≤ amaxk (1b)

ETm [`k(t)− akwk(t)] ≥ 0, ∀m ∈M (1c)E[akwk(t)− `k(t)] ≤ Pmax

k , ∀t ∈ T (1d)

The objective function (1a) minimizes the installation costs onPV plus the costs of purchasing energy to the grid minus theincome of selling energy at net-metering price. Equation (1b)limits the investment decision to be less than the maximumarea available to install PV. Equation (1c) doesn’t allow firmk to be a net producer through a period (usually a month ora year). Equation (1d) imposes that the investment decisionshould not violate in expectation distribution limits, a valuethat tipically depends on the reliability of the distributionsystem. Observe that both (1c) and (1d) can be cast as upperbounds of the variable ak as:

ak ≤ETm [`k(t)]

ETm [wk(t)]= acap

m , ∀m ∈M (1e)

and:

ak ≤Pmaxk + E[`k(t)]E[wk(t)]

= apcapt , ∀t ∈ T (1f)

Defining mk = min{amaxk , acap

1 , . . . , acapM , apcap

1 , . . . , apcapT

},

the optimization problem can be simply cast as:

minak

Jk(ak) s.t. 0 ≤ ak ≤ mk

Theorem 1: Assume πnm(t) = πd(t) ∀t ∈ T , and that pricesπd are independent of the irradiance wk. Then, the optimalPV investment decisions of household k under the standalonemodel P sa

k are given by the threshold policy:

a∗k =

{mk if E[wk(t)] > E

[πs/πd(t)

]0 else

Proof: See [10].

In the case of discounted net-metering prices πnm < πd,the objective function Jk(ak) is convex (while not linear),and thus, the optimal investment decision a∗k can be easilycomputed using historical data to form empirical expectations.

B. Collective of firms under sharing

Now, in this scenario, we study the PV investment decisionof a firm under a sharing economy business model. Thesituation is depicted on the middle of Figure 2.

1) Individual firms: Each firm k may have a deficit of net-load (`k−akwk)+ in some time slots. This energy deficit canbe purchased from homes who have a surplus in the collective,or from the utility at a price πd $/kWh. In the same way,firms can have excess net generation (akwk − `k)+ in sometime slots. This can be sold to other homes, or returned tothe utility under net-metering. To simplify our analysis, weconsider the sharing economy investment model with no net-metering, i.e. πnm(t) = 0, ∀t. In effect, our goal is to arguethat this model can supplant net-metering in some (likely)future scenario when utilities withdraw these programs.

2) Spot market for sharing excess PV generation: We studya simple spot market for sharing excess PV generation fromfirms. The collective supply S and demand D of sharedelectricity are

S =

n∑k=1

(akwk − `k)+, D =

n∑k=1

(`k − akwk)+.

Consider the time slot t. If the supply S(t) exceeds thecollective demand D(t), firms with excess solar generationwill compete against each other, and as a result, the sharedelectricity will trade at the floor price offered by the utilityπnm(t) = 0. On the other side, if the collective demand D(t)exceeds the collective supply S(t), homes with a net deficit ofelectricity will compete against each other, and so, the sharedelectricity will trade at the ceiling price πd(t). These effectsare illustrated in Figure 3. As a result, the spot market pricefor shared electricity in this competitive scenario is:

πeq(t) =

{πnm = 0 if S(t) > D(t)πd(t) else (2a)

4

pric

e

energyD S

supplyschedule

demandschedule

clearingpriceπd

πnm

pric

e

energyDS

demandschedule

clearingprice

πd

πnm

Fig. 3. Clearing price for shared electricity: (a) left: S > D, (b) right:S < D.

We stress that this clearing price is random - it depends onmarket conditions in each time slot.

At each time slot t ∈ T , we define the collectiveload L(t) =

∑k `k(t), the collective generation G(t) =∑

k akwk(t) and the collective net-load X(t) = L(t)−G(t).Observe the following (at any time slot t):

S −D =∑k∈N

(akwk − `k)+ −∑k∈N

(`k − akwk)+

=∑k∈N

akwk − `k = G− L = −X

Thus, the spot market price for shared electricity (2a) can bere-written as:

πeq(t) =

{πnm = 0 if X(t) < 0πd(t) if X(t) ≥ 0

(2b)

That is, if the collective net-load is positive, the spot marketprice is the ceiling price πd(t). Otherwise, the price is tradedat the net-metering price (that is set to zero).

3) Optimal investment decisions under sharing: The ex-pected cost faced by firm k (per time slot) has three compo-nents:

Jk(ak | a−k) = πsak︸︷︷︸capital cost

+E[πeq(`k − akwk)+

]︸ ︷︷ ︸cost of buying deficit

− E[πeq(wk − ak`k)+

]︸ ︷︷ ︸revenue from sharing surplus

(2c)

Note that the objective function for firm k depends on the spotmarket price πeq, and hence on the investment decisions a−kof other households. This induces a PV investment decisiongame. The social cost is then given by J(a) =

∑k ak.

As a first step we consider the case of common irradiancewk = w, with no bound on maximum panel area supported ateach firm. The result is the following:

Theorem 2: Assume that all firms receive common irradi-ance wk = w,∀k and a fixed retail price πd,∀t. Then:(a) this game admits a unique Nash equilibrium,(b) the optimal total investment A =

∑k a∗k is the unique

solution of:

0 = πs − πd

T

∑t∈T

E [p(t)w(t) | L(t) > Aw(t)] (2d)

where p(t) = Prob{L(t) > Aw(t)},

(c) at this Nash equilibrium, the optimal investment of firmk is:

a∗k = A · E[`k | L = Aw]

E[L | L = Aw](2e)

(d) this Nash equilibrium supports the social welfare.Proof: With the assumption of no net-metering, commonirradiance and fixed retail prices, equation (2c) reduces to:

Jk(ak|a−k) = πsak

+πd

T

∑t∈T

p(t)E[`k(t)− akw(t) | X(t) ≥ 0]

We first show that a Nash equilibrium exists. Consider firmk, which investment decision is ak and fix the decisions ofall the firms a∗−k from equation (2e), with A the solution ofequation of (2d). To show that a∗ is a Nash equilibrium, wehave to show that a∗k obtained from equation (2e) is indeeda global minimizer of Jk(ak | a∗−k). Let α =

∑j 6=k a

∗j the

investment of the other firms, and so A = ak + α. Observethat the condition X ≥ 0 can be rewritten as:

L−Aw ≥ 0 → L ≥ (α+ ak)w

and so, considering any time slot t, in terms of the net-loadxk = `k − akw, the cost function can be written as:∑

t∈T

{πs

Tak +

πd

Tp(t)E[xk(t) | X(t) ≥ 0]

}that can be written in terms of the distributions as (omittingthe sum over t):

πs

Tak+

πd

T

∫ ∞`k=0

∫ ∞w=0

∫ ∞L=ak+α

xkf`k,w,L(`k, w, L)d`kdwdL

By taking the derivative and using the Leibniz rule we got:

dJkdak

=∑t∈T

{πs

T− πd

TE[p(t)w(t)|L ≥ (ak + α)w]︸ ︷︷ ︸

φt(ak)

− πd

TfL(ak + α)E[`k − akw | L = (ak + α)w]︸ ︷︷ ︸

ψt(ak)

}

Observe that for a∗k obtained from (2e), we obtain a∗k+α = A.Then from (2d),

∑t φt(a

∗k) = 0. Thus:

dJkdak

(a∗k) = −πdfL(A)E[`k − a∗kw | L = Aw]

= −πdfL(A)E[`k|L = Aw]− E[`kAw|L = Aw]

E[L|L = Aw]

= 0

that implies the derivative vanishes at a∗k. Now observe thatthe function φt(ak):

πs

T− πd

T

∫ ∞`k=0

∫ ∞w=0

∫ ∞L=ak+α

wf`k,w,L(`l, w, L)d`kdwdL

is monotone increasing on ak, since increasing ak decreasethe value of the integral, and hence, due to the negative sign,increase the value of φt(ak). In addition, ψt(ak) is monotone

5

non-increasing, since increasing ak can only increase the solarproduction, reducing the expected net-load (if L = (ak+α)w).Since ψt(ak) also vanishes at a∗k we have for our functions:

φt(ak)

< 0 if ak < a∗k= 0 if ak = a∗k> 0 if ak > 0

ψt(ak)

≥ 0 if ak < a∗k= 0 if ak = a∗k≤ 0 if ak > 0

Combining our results we obtain:

dJkdak

< 0 if ak < a∗k= 0 if ak = a∗k> 0 if ak > 0

that proves that a∗k is the global minimizer of Jk(a∗k | a∗−k),and hence establishing that a∗ = (a∗1, . . . , a

∗n) is a Nash

equilibrium with solution obtained from equations (2d) and(2e). To show that supports social welfare, consider the socialcost and using that all firms receive same irradiance:

J(a) = πs∑k∈N

ak +πd

T

∑t,k

E [`k(t)− akw(t) | X(t) ≥ 0]

= πsA+πd

T

∑t∈T

E [L(t)−Aw(t) | X(t) ≥ 0]

Taking the derivative and setting it to zero, equation (2d)is obtained, that shows that the Nash equilibrium supportssocial welfare. �

Now we explore a more realistic condition of diverseirradiance and assuming that each firm k can invest at mostmk m2 of panel area due to physical limitations at the site(such as rooftop size or power limitations). Assuming that alarge number of n firms participate in PV sharing we will haveasymptotically (in n) perfect competition, in which no singlefirm can influence the statistics of the clearing price πeq. Wehave our main result:

Theorem 3: In the sharing economy model, under theperfect competition model and no net-metering, the optimalPV investment decisions of firm k are given by the thresholdpolicy:

a∗k =

{mk if

∑t E[πd(t)p(t)wk(t) | X(t) > 0

]> πsT

0 else

where X(t) is the net-load random sequence, p(t) =Prob{X(t) > 0}, and T is the number of time slots.

Proof: See [10].Remark: On the case on which the load `k and irradiance

wk are stationary random sequences, p(t) = p is independentof t. In this case, for fixed retail prices πd = πd, the thresholdpolicy of Theorem 3 becomes:

a∗k =

{mk if µk = E[wk | L > G] > θ0 else

(2f)

where the threshold is the unique solution of θ = πs/(p · πd).The quantity µk = E[wk | L > G] captures the value of PVfor firm k. This firm will invest the maximum if µ > θ, andnot invest otherwise.

4) Computing of the threshold θ: Computation of θ in (2f)is not trivial. Observe that θ determines the PV investment offirms, which influences the statistics of the collective net-loadX , that, in turn, affects θ. To solve this issue, in our previouswork [10] we proposed a convergent algorithm that jointlycomputes the threshold θ and the investment decisions a∗.The algorithm iteratively updates the set of firms that investand do not invest in order to determine the threshold θ andthe decision a∗ based on the threshold policy (2f).

C. Collective of users in wholesale markets

In this scenario we explore the PV investment decisionsof the collective, when it is coordinated and participate inwholesale market for purchasing and selling energy to the grid.We study the problem in which a centralized planner, like anaggregator, can decide the optimal PV investment for each firmak by promoting through incentives the optimal investmentdecisions of the participants. In this case, the collective ofusers are subjected to the wholesale market price πg(t).

1) Decision problem for the planner: The problem for thecentralized planner can be written as:

minaJc =

∑k∈K

{πsak + E[πg(t)(`k(t)− akwk)]

}+ d fee(a)

s.t. 0 ≤ ak ≤ mk, ∀k = 1, . . . , n.

In this problem we consider that the collective is still under thedomain of a utility, and hence a distribution fee (d fee(a)) isconsidered. The distribution fees are modeled as a combinationof three different schemes:

d fee(a) = α1E

{∣∣∣∣∣∑k∈K

`k(t)− akwk(t)

∣∣∣∣∣}

+ α2EM

{maxt∈Tm

∣∣∣∣∣∑k∈K

`k(t)− akwk(t)

∣∣∣∣∣}

+ α3n

The first term represents a volumetric cost for using thegrid (independent if its injecting or withdrawing). The secondterm depicts a demand charge, in which the collective paysa fee over a period (usually a month) that depends on themaximum consumption in that period. Finally the third termrepresents a connection cost that depends on the total users inthe collective.

2) Solution under a price taker aggregator: It is assumedthat πg(t) does not depend on the investment decisions of thecollective, i.e. the collective is considered a price-taker entity.Thus, the previous problem is a convex problem, that can berecast as a LP by adding sufficient slack variables to replacethe absolute value and max functions of the distribution fees.

III. SIMULATIONS

In this section we use simulations to assess how solaradoption differs in the proposed three models: (a) a standalonenet-metering scenario, (b) a shared solar model and (c) acollective participating in wholesale markets. We consider a

6

0 0.2 0.4 0.6 0.8 10

50

100

Ratio of net-metering price and retail tariffγ = πnm/πd

Diff

eren

ces

inPV

inve

stm

ent

D(γ)

[%]

Fig. 4. Differences of PV investment between both models in Case 1.

representative collective of 1000 firms that can invest in PV.All models are solved using Python and cvxpy [27] and thecode used is publicly available in Github (citeGithub).

A. Data

Load and effective irradiance. In this simulation weuse real profiles of electricity load from households fromPecan Street’s Dataport website [28] from 2017 as empiricalexpectations for our firms. From the same database we obtainreal profiles of real solar production, that are normalizedusing the nominal capacity. From 194 real profiles of loadand solar generation, we generate 1000 synthetic profiles byrandomly selecting combinations of solar and load time seriesand adding independent white noise with variance σ2 = 0.2(in the case of solar profiles, the noise is only added whenthere is actual production).

Prices. From [4] we use a levelized cost of 2.80 $/WDCfor installed PV system (including installation and inverter).A typical panel of 165× 99 cm (1.64 m2) with a rated powerof 300 WAC [29]. With this, each firm faces an investmentcost of 512.2 $/m2. Using a discount rate of r = 5% and atime horizon of 20 years, we obtain an annuity of 41.1 $/m2,that yields a price of πs = 0.0047 $/m2 per time slot. Weuse a retail price πg = 0.18 $/kWh and a net-metering priceπnm = γπg for the standalone model, with γ ∈ [0, 1]. Thewholesale electricity prices πg(t) are obtained using historicaldata from CAISO in 2017 [30]. We decided to use CAISO datathat has low prices at sunny hours to stress the feasibility ofa investment decision model in a wholesale market model. Inall study cases of sharing economy models, the net-meteringprice is set to zero (no net-metering). In both sharing economyand wholesale market models no annual cap on production isconsidered for firms.

B. Results and Discussion

The following section focuses on studying the differencesof PV investment and solar production for standalone, sharingand wholesale economy models. Case 1 analyzes how stan-dalone and sharing models differ when maximum panel areaavailable to invest is limited for each user. Case 2 explores

1 5 10 15 20Time [h]

Net

load

[kW

]

X(t)x330(t)

24-1.5

-1-0.5

00.51

1.52

2.53

40050060070080090010001100120013001400

Fig. 5. Comparison of net-load of the firm with dataid 330 (blue) and thecollective (red) on February 1st for Case 1.

the standalone and sharing models when users have enoughavailable area to install PV that can be potentially be netproducers in a year. That is, in Case 2 we study how the annualcap acap

k can play an important role on optimal investmentdecisions in both market models. Finally, Case 3 explores,given current wholesale market prices, the effects of energyand demand charges when solar costs are lower enough tomake the wholesale market model attractive for firms to investin PV. We stress that under current levelized cost of distributedsolar and current wholesale market prices in California, thataverage around 0.05 $/kWh (and as low of 0.025 $/kWh insunny hours) [30], no investment in PV occurs under thewholesale markets model.

1) Case 1. Limited maximum panel area mk: Our firstanalysis considers a scenario in which all firms face limitedspace to install PV panels, that is amax

k ≈ β1acapk , ∀k, with

β1 = 0.2. In this case the limitation comes from the areaavailable to install PV mk = amax

k , and not the annual capacapk of being a net producer through the year. We study the

case of varying γ, that is, for the standalone case, the net-metering price as a proportion of the retail tariff. Recall thatin the sharing economy model, no net-metering is considered.

Comparison of PV investment among models is depictedin Fig. 4. The figure show the difference between the totalinvestment in PV between the models, calculated as:

D(γ) =

∑k a∗,sharingk −

∑k a∗,standalonek (γ)∑

k a∗,standalonek (γ)

[%]

For γ < 1, the sharing economy model yields higher invest-ments in PV than the standalone model, as observed. Thereason is as follows. For a low maximum panel area, optimalinvestment decisions of the collective in the sharing modelwill never yield a negative collective net load, at any hourof the day. This implies that each firm inside the collectivethat invests in PV always faces a full retail price whenselling excess generation. On the contrary, for the standalonecase, even at small area installations, firms have hours on

7

0 100 200 300 400 500 600 700 800 900 1000

Houses sorted by average irradiance

0

1000

2000

3000

4000

5000

6000

7000

8000C

umul

ativ

e ar

ea in

vest

ed [

m2 ]

SharingStandalone

E[wk(t)]

Fig. 6. Comparison of total investment decision with γ = 0.9 and β2 = 1.5for Case 2. A low rank on the horizontal axis corresponds to a firm with highirradiance.

which their own net-load is negative as depicted in Fig. 5.In those hours, firms face a discounted selling price, thataffects their private economics, reducing their optimal PVinvestment. In the case γ = 1, both models yield sameinvestment results. This is expected, since in this scenario ofequal tariff, πd = πnm, and always positive collective net-load,p = Prob{L > G} = 1, firms always face full retail tariff (inboth models) for buying and selling energy. Thus, Theorem1 and Theorem 3 are equivalent, providing the same optimalPV investment solution.

2) Case 2. Effects of the annual cap acapk : In this section

we study a case when the annual cap plays an important rolein the standalone model in comparison to the sharing model.That is, we set amax

k = β2acapk ,∀k, with β2 > 1. The previous

implies that in the sharing model, firms face a limitation givenby the maximum area available to invest amax

k = mk = β2acapk .

However, in the standalone model firms face a limitation onPV investment due to the annual cap mk = acap

k . Fig. 6 depictsthe comparison of cumulative investment on PV between thetwo models using γ = 0.9 and β2 = 1.5 for the standalonemodel.

Here, we observe two different phenomena with the invest-ment of firms between both models. Firms with relatively highirradiance (low rank in Fig. 6) would like to invest more in thestandalone case, but they are not allowed due to the annual caplimitation. We say that these firms under-invest in comparisonto the sharing case, as observed in the higher slope of the blueline of Fig. 6, that means that those firms are investing more inthe sharing model. On the other hand, in the standalone modelsome firms still invest when they don’t invest in the sharingcase. We say that these firms over-invest in comparison to thesharing case, as observed in the slope of the red line of Fig.6, for k ∈ [120, 260].

The aforementioned effects imply that the distribution offirms that invest in the sharing case is much smaller, but thetotal area invested is higher. In this case, only the firms with fa-vorable PV irradiance invest in PV, and become the “supliers”for the remaining firms of the collective. For the remainingfirms become more challenging to satisfy the threshold policywhen the total PV area invested increase, due to more hourswith negative net load L < G, that impacts both the value of

50 60 70 80 90 100

Overnight cost of PV [$/m2]

0

2

4

6

8

10

12

14

Tot

al P

V in

vest

men

t [m

2 ]

104

40

No demand chargeDemand charge: [$/kW]Maximum PV availability

α2 = 10

Fig. 7. Comparison of total PV investment in Case 3, for β3 = 2, withpurely energy charges (blue line) and energy and demand charge (red line).

PV µk, and the threshold θ.3) Case 3. Effects on demand charge on wholesale market

pricing: In this section we explore a scenario on whichthe amortized PV installation costs are comparable to thewholesale market prices. As aforementioned before, giventhe current distributed PV installation costs (overnight costof 512.2 $/m2), no PV investment occurs due to the lowwholesale electricity prices in sunny hours.

For that purpose, we consider a scenario of high availabilityto invest on which mk ≈ β3a

capk ,∀k, with β3 > 1. Fig. 7

analyze the PV investment decisions in the wholesale marketsmodel, with β3 = 2, against low overnight costs of PV.Without any distribution fees, the optimal investment decisionsare quite sensitive to the PV costs, going from the maximumavailable to zero investment in PV, as depicted in blue.

Now, if a monthly demand charge of 10 $/kWp is considered(to cover distribution fees), the solution changes drastically.Without PV investment, demand charges represent 45% oftotal costs of the collective. Thus, in this scenario there exists atradeoff between energy and demand charge that the collectivemust face. As showed in the red line in Fig. 7, at a higher PVcost, there is more investment in PV than the pure energy case.Solar generation can be used to reduce the demand charge, thatin most months occur around 4-6 pm. On the contrary, at lowercosts of PV, there is a smaller investment in PV in comparisonto the pure energy case. In this scenario, more investment doesnot further reduce the demand charge, since now the peakdemand is outside sunny hours, hence modifying the optimalinvestment decision for the collective in comparison to theonly energy charge case.

IV. CONCLUSIONS

In this paper we study optimal investment decision fordistributed PV under three models: (a) a standalone net-metering model with annual production cap, (b) a sharingeconomy model where firms can trade excess generation ina local spot market and (c) a wholesale market participationmodel, on which an aggregator can promote the investmentin PV. In the standalone model each firm make independent

8

investment decisions based on their own load and irradianceprofiles. Under the sharing scheme, optimal investment de-cisions are coupled as they affect the clearing price on thelocal spot market. For a large number of firms we haveasymptotically perfect competition, and we showed that theoptimal investment are determined by a threshold policy. Inthe wholesale market model, an entity decides the optimalinvestment decision for each firm, that are coupled since itfaces distribution fees for the entire collective.

Several numerical experiments using load and irradiancedata were presented to investigate how optimal PV investmentdecisions may differ depending on area availability, annualproduction caps and distribution fees. Results show that forlow total PV investment, both sharing and standalone modelyield same results under full net-metering scenario. For dis-counted net-metering pricing, the sharing model promotes ahigher investment in PV than the standalone framework. Inthe case of high total PV investment, there exists hours wherethe collective net-load is negative, changing the distributionof which firms invests in the sharing model in comparison tothe standalone case. In addition, we showed that given currentcosts of PV and low wholesale energy prices, no PV invest-ment occurs, that highlights the importance of high retail tariffsto make distributed PV investment attractive for firms. Finally,we also showed how the inclusion of demand charge, in ascenario of low PV installation costs, can drastically changethe optimal PV investment in a wholesale market participationmodel, and must be considered if a future scenario includesthese types of distribution fees for collectives.

Analyzing how the impacts of different solar investmenthave in the power flow in the distribution system is aninteresting topic for an extension. Future work might alsoexplore a sharing model framework to buy and sell energyfor isolated communities or when possible contingencies occurand the collective is disconnected from the grid, and the retailtariff does not play any role. In addition, exploring how energysales are impacted for a utility when firms invest in PV underdifferent market structures is a future step. Finally, consideringfairness and equity issues inside the collective is one of thenext step in this research.

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