Assessing distribution grid costs of community solar projects

Miguel Heleno, Alan Valenzuela, Alexandre Moreira M. Heleno, A. Valenzuela and A. Moreira are with the Lawrence Berkeley National Laboratory, Berkeley, CA, USA (e-mail: {MiguelHeleno, AlanValenzuela, AMoreira}@lbl.gov).

Abstract

Abstract goes here…

Index Terms:

distribution expansion planning; large-scale distribution network; risk aversion; reliability.

Nomenclature

The mathematical symbols used throughout this paper are classified below as follows.

Sets

$H$: Set of indexes of storage units.
$H^{E}$: Set of indexes of existing storage units.
$H^{C}$: Set of indexes of candidate storage units.
${\cal L}$: Set of indexes of all line segments.
${\cal L}^{F}$: Set of indexes of fixed line segments.
${\cal L}^{FH}$: Set of indexes of feeder head transformers.
${\cal L}^{C}$: Set of indexes of candidate line segments for capacity upgrade.
${\cal L}^{OLTC}$: Set of indexes of line segments with existing or candidate OLTC transformers.
${\cal L}^{OLTC}{}^{E}$: Set of indexes of line segments with existing OLTC transformers.
${\cal L}^{OLTC}{}^{C}$: Set of indexes of line segments with candidate OLTC transformers.
$N$: Set of indexes of all buses.
$R^{line,options}_{l}$: Set of indexes of upgrade capacity options for line segment $l$ .
$R^{Solar,C}$: Set of indexes of candidate solar units.
$R^{Solar,E}$: Set of indexes of existing solar units.
$R^{Subs}_{n}$: Set of indexes of substations at bus $n$ . This set will have at most one index for each $n$ .
$T$: Set of all time periods.

Parameters

$\beta^{p}_{l}$: Auxiliary parameter related to the active power loss of line segment $l$ .
$\beta^{q}_{l}$: Auxiliary parameter related to the reactive power loss of line segment $l$ .
$\eta$: Round-trip efficiency of storage units.
$\varphi^{max}_{l}$: Maximum turns ratio of the OLTC transformer located at line segment $l$ .
$\varphi^{min}_{l}$: Minimum turns ratio of the OLTC transformer located at line segment $l$ .
$\psi$: Auxiliary parameter related to the reactive contribution capacity of storage units.
$\omega_{t}$: Weight of time $t$ .
$C^{CS,crt}_{t}$: Cost of curtailing solar generation output.
$C^{I}$: Imbalance cost.
$C^{FH,inv}_{l}$: Equivalent annual investment cost to upgrade the feeder head transformer located at line segment $l$ .
$C^{line,inv}_{l}$: Equivalent annual investment cost to upgrade the capacity of line segment $l$ .
$C^{OLTC,inv}_{l}$: Equivalent annual investment cost to install to upgrade line segment $l$ with an OLTC transformer.
$C^{st,inv}_{h}$: Equivalent annual investment cost to install storage $h$ .
${f}^{RTS}_{r,t}$: Capacity factor existing solar unit $r$ during time $t$ .
${f}^{CS}_{r,t}$: Capacity factor candidate solar unit $r$ during time $t$ .
${f}^{CS,inv}$: Total installed capacity of solar generation to be added to the system.
$\overline{F}_{l}$: Capacity of fixed line segment $l$ .
$\overline{F}^{FH}_{l}$: Initial capacity of feeder head transformer located at line segment $l$ .
$\overline{F}^{FH,upd}_{l}$: Potential upgrade capacity for feeder head transformer located at line segment $l$ .
$\overline{F}^{OLTC}_{l}$: Capacity of OLTC transformer located at line segment $l$ .
$\overline{F}^{option}_{l,r}$: Upgrade capacity option $r$ for line segment $l$ .
$\overline{g}^{RTS}_{r}$: Installed capacity of existing solar unit $r$ .
$\overline{P}^{in}_{h}$: Power charging capacity of storage unit $h$ .
$\overline{P}^{out}_{h}$: Power discharging capacity of storage unit $h$ .
$p^{load}_{n,t}$: Active power load of bus $n$ at time $t$ .
$q^{load}_{n,t}$: Reactive power load of bus $n$ at time $t$ .
$R_{l}$: Resistance of line segment $l$ .
$R^{option}_{l,r}$: Resistance associated with upgrade option $r$ for line segment $l$ .
$\overline{S}_{h}$: Duration of storage unit $h$ .
$v^{max}_{n}$: Maximum voltage of bus $n$ .
$v^{min}_{n}$: Minimum voltage of bus $n$ .
$v^{ref}_{n}$: Voltage reference at the feeder heads.
$X_{l}$: Reactance of line segment $l$ .
$X^{option}_{l,r}$: Reactance associated with upgrade option $r$ for line segment $l$ .
$\overline{x}^{CS,inv}_{r}$: Auxiliary parameter related to maximum investment in the capacity of candidate solar unit $r$ .
$\overline{x}^{st,inv}_{h}$: Auxiliary parameter related to maximum investment in the capacity of storage $h$ .

Decision variables

$f^{p}_{l,t}$: Active power flow through line segment $l$ .
$f^{q}_{l,t}$: Reactive power flow through line segment $l$ .
$\overline{f}^{trf}_{l}$: Capacity of transformer in line segment $l$ .
$\overline{f}^{upd}_{l}$: Resulting capacity of candidate line segment $l$ for upgrade.
$g^{CS}_{r,t}$: Generation output of candidate solar unit $r$ during time $t$ .
$g^{CS,crt}_{r,t}$: Curtailed generation output of candidate solar unit $r$ during time $t$ .
$g^{RTS}_{r,t}$: Generation output of existing solar unit $r$ during time $t$ .
$p^{I,+}_{n,t}$: Active power surplus at bus $n$ during time $t$ .
$p^{I,-}_{n,t}$: Active power deficit at bus $n$ during time $t$ .
$p^{in}_{h,t}$: Active power charging of storage unit $h$ during time $t$ .
$p^{out}_{h,t}$: Active power discharging of storage unit $h$ during time $t$ .
$p^{subs}_{r,t}$: Active power output of substation $r$ during time $t$ .
$q^{I,+}_{n,t}$: Reactive power surplus at bus $n$ during time $t$ .
$q^{I,-}_{n,t}$: Reactive power deficit at bus $n$ during time $t$ .
$q^{subs}_{r,t}$: Reactive power output of substation $r$ during time $t$ .
$q^{st,+/-}_{h,t}$: Reactive power output of storage unit $h$ during time $t$ .
$soc_{h,t}$: State of charge of storage unit $h$ during time $t$ .
$soc^{t0}_{h}$: Initial state of charge of storage unit $h$ .
$v^{\dagger}_{n,t}$: Squared voltage of bus $n$ during time $t$ .
$v^{\dagger,mid}_{n,t}$: Auxiliary variable related to the squared voltage of bus $n$ during time $t$ considering an OLTC transformer.
$x^{CS,inv}_{r}$: Investment in solar unit $r$ .
$x^{FH,inv}_{l}$: Investment in the feeder head transformer located in line segment $l$ .
$x^{line,inv}_{l,r}$: Investment in option $r$ to upgrade line segment $l$ .
$x^{OLTC,inv}_{l}$: Investment to install an OLTC transformer on line segment $l$ .
$x^{st,bin,inv}_{h}$: Binary variable to select which storage unit will receive investment.
$x^{st,inv}_{h}$: Investment in storage unit $h$ .

I Introduction

Shared solar programs have emerged as a solution to provide accessibility to renewable energy benefits. It enables small residential and commercial consumers, including renters, owners of buildings with shaded roofs, or those with limited financial resources, to access clean and renewable energy without the need for personal rooftop panels.

Interconnection costs play a critical role in the successful implementation of community solar projects, significantly impacting their financial viability and operational feasibility. Regulatory issues related to interconnection costs often pose challenges and complexities for developers and participants.

In a community solar program, a utility or third-party owns a utility-scale PV array and sells portions of the array’s power (kilowatts or kW) or generation (kilowatt-hours or kWh) to multiple subscribers. These subscribers pay voluntarily for their portion of the array and then receive a credit on their electricity bill for their share of production. This bill credit for generation produced may also include payment for the associated renewable energy certificates (RECs), depending on program structure. Subscribers can pay for their share through either an up-front payment or an ongoing monthly payment, such as through a financing option [NREL_LMI].

Although community solar programs can be implemented within multiple financial mechanisms [KLEIN2021225], there are three base types of ownership types [SunShot]

In 2019, the Legislative Document 1711 in Maine, US, introduced a shared distributed generation procurement process that includes community solar projects, developed at the local level, limited to a maximum system size of 5 MW [DL1711]. The Sharing the Sun Community Solar Project Database reports 1187 projects, of which 51 are above the 5MW [SS_db].

I-A Literature Review

II Mathematical Formulation

Consider ${\cal L}={\cal L}^{C}\cup{\cal L}^{F}\cup{\cal L}^{FH}\cup{\cal L}^{OLTC}$ , ${\cal L}^{C}\cap{\cal L}^{F}\cap{\cal L}^{FH}\cap{\cal L}^{OLTC}=\emptyset$ , ${\cal L}^{C}\cap{\cal L}^{F}=\emptyset$ , ${\cal L}^{C}\cap{\cal L}^{FH}=\emptyset$ , ${\cal L}^{C}\cap{\cal L}^{OLTC}=\emptyset$ , ${\cal L}^{F}\cap{\cal L}^{FH}=\emptyset$ , ${\cal L}^{F}\cap{\cal L}^{OLTC}=\emptyset$ , ${\cal L}^{FH}\cap{\cal L}^{OLTC}=\emptyset$ . Also, ${\cal L}^{OLTC}={\cal L}^{OLTC,E}\cup{\cal L}^{OLTC,C}$ , $v^{\dagger}_{nt}=v^{2}_{nt}$ . For lines with OLTC, we have $v^{\dagger,mid}_{to(l),t}=v^{\dagger}_{to(l),t}\varphi^{2}_{l},\forall l\in{\cal L}^{OLTC}$ , where $\varphi^{2}_{l}$ and $to(l)$ are the turns ratio and $to(l)$ is the receiving bus of OLTC transformer $l$ , respectively.

II-A Objective

	$\displaystyle\underset{{\begin{subarray}{c}f^{p}_{l,t},f^{q}_{l,t},\overline{f}^{trf}_{l},\overline{f}^{upd}_{l},g^{CS}_{r,t},g^{CS,crt}_{r,t},\\ g^{RTS}_{r,t},p^{I,+}_{n,t},p^{I,-}_{n,t},p^{in}_{h,t},p^{out}_{h,t},p^{subs}_{r,t},\\ q^{I,+}_{n,t},q^{I,-}_{n,t},q^{subs}_{r,t},q^{st,+/-}_{h,t},soc_{h,t},\\ soc^{t0}_{h},v^{\dagger}_{n,t},v^{\dagger,mid}_{n,t},x^{CS,inv}_{r},x^{FH,inv}_{l,r},\\ x^{line,inv}_{l,r},x^{OLTC,inv}_{l},x^{st,bin,inv}_{h},x^{st,inv}_{h}\end{subarray}}}{\text{Minimize}}\sum_{h\in H^{C}}C^{st,inv}_{h}x^{st,inv}_{h}$
	$\displaystyle\hskip 30.0pt+\sum_{l\in{\cal L}^{OLTC,C}}C^{OLTC,inv}_{l}x^{OLTC,inv}_{l}$
	$\displaystyle\hskip 30.0pt+\sum_{l\in{\cal L}^{C}}\sum_{r\in R^{line,options}_{l}}C^{line\_option}_{l,r}x^{line,inv}_{l,r}\overline{F}^{option}_{l,r}$
	$\displaystyle\hskip 30.0pt+{\color[rgb]{0,0,0}\sum_{l\in{\cal L}^{FH}}C^{FH,inv}_{l}}x^{FH,inv}_{l}$
	$\displaystyle\hskip 30.0pt+\sum_{t\in T}\sum_{n\in N}C^{I}\omega_{t}(p^{I,+}_{n,t}+p^{I,-}_{n,t}+q^{I,+}_{n,t}+q^{I,-}_{n,t})$
	$\displaystyle\hskip 30.0pt+\sum_{t\in T}\sum_{r\in R^{Solar,C}}{\color[rgb]{0,0,0}C^{CS,crt}_{t}\omega_{t}}\cdot g^{CS,crt}_{r,t}$		(1)

II-B Energy balance and reference voltage

	$\displaystyle\sum_{r\in R^{Subs}_{n}}p^{subs}_{r,t}+\sum_{l\in{\cal L}^{to}_{n}}f^{p}_{l,t}-\sum_{l\in{\cal L}^{fr}_{n}}f^{p}_{l,t}+\sum_{r\in R^{Solar,C}_{n}}g^{CS}_{r,t}$
	$\displaystyle\hskip 5.0pt+\sum_{r\in R^{Solar,E}_{n}}g^{RTS}_{r,t}+\sum_{h\in H_{n}}p^{out}_{h,t}-\sum_{h\in H_{n}}p^{in}_{h,t}-p^{load}_{n,t}$
	$\displaystyle\hskip 5.0pt-\sum_{l\in{\cal L}^{to}_{n}}\frac{\beta^{p}_{l}}{2}\Biggl{[}\sum_{n\in N}p^{load}_{n,t}-\sum_{r\in R^{Solar,E}}g^{RTS}_{r,t}-\sum_{r\in R^{Solar,C}_{n}}g^{CS}_{r,t}\Biggr{]}$
	$\displaystyle\hskip 5.0pt-\sum_{l\in{\cal L}^{fr}_{n}}\frac{\beta^{p}_{l}}{2}\Biggl{[}\sum_{n\in N}p^{load}_{n,t}-\sum_{r\in R^{Solar,E}}g^{RTS}_{r,t}-\sum_{r\in R^{Solar,C}_{n}}g^{CS}_{r,t}\Biggr{]}$
	$\displaystyle\hskip 85.0pt+p^{I,-}_{n,t}-p^{I,+}_{n,t}=0;\forall n\in N,t\in T$		(2)
	$\displaystyle\sum_{r\in R^{Subs}_{n}}q^{subs}_{r,t}+\sum_{l\in{\cal L}^{to}_{n}}f^{q}_{l,t}-\sum_{l\in{\cal L}^{fr}_{n}}f^{q}_{l,t}+\sum_{h\in H_{n}}q^{st,+/-}_{h,t}$
	$\displaystyle\hskip 5.0pt-q^{load}_{n,t}-\sum_{l\in{\cal L}^{to}_{n}}\frac{\beta^{q}_{l}}{2}\sum_{n\in N}q^{load}_{n,t}-\sum_{l\in{\cal L}^{fr}_{n}}\frac{\beta^{q}_{l}}{2}\sum_{n\in N}q^{load}_{n,t}$
	$\displaystyle\hskip 85.0pt+q^{I,-}_{n,t}-q^{I,+}_{n,t}=0;\forall n\in N,t\in T$		(3)
	$\displaystyle v^{\dagger}_{fr(l),t}=(v^{ref})^{2};\forall l\in{\cal L}^{FH},t\in T$		(4)
	$\displaystyle v^{{min}^{2}}_{n}\leq v^{\dagger}_{n,t}\leq v^{{max}^{2}}_{n};\forall n\in N,t\in T$		(5)

II-C Fixed lines segments

	$\displaystyle-\overline{F}_{l}\leq f^{p}_{l,t}\leq\overline{F}_{l};\forall l\in{\cal L}^{F},t\in T$		(6)
	$\displaystyle-\overline{F}_{l}\leq f^{q}_{l,t}\leq\overline{F}_{l};\forall l\in{\cal L}^{F},t\in T$		(7)
	$\displaystyle f^{q}_{l,t}\leq cotan\biggl{(}\biggl{(}\frac{1}{2}-e\biggr{)}\frac{\pi}{4}\biggr{)}\biggl{(}f^{p}_{l,t}-cos\biggl{(}e\frac{\pi}{4}\biggr{)}\overline{F}_{l}\biggr{)}$
	$\displaystyle\hskip 35.0pt+sin\biggl{(}e\frac{\pi}{4}\biggr{)}\overline{F}_{l};\forall l\in{\cal L}^{F},t\in T,e\in\{1,\dots,4\}$		(8)
	$\displaystyle-f^{q}_{l,t}\leq cotan\biggl{(}\biggl{(}\frac{1}{2}-e\biggr{)}\frac{\pi}{4}\biggr{)}\biggl{(}f^{p}_{l,t}-cos\biggl{(}e\frac{\pi}{4}\biggr{)}\overline{F}_{l}\biggr{)}$
	$\displaystyle\hskip 35.0pt+sin\biggl{(}e\frac{\pi}{4}\biggr{)}\overline{F}_{l};\forall l\in{\cal L}^{F},t\in T,e\in\{1,\dots,4\}$		(9)
	$\displaystyle v^{\dagger}_{to(l),t}-\biggl{[}v^{\dagger}_{fr(l),t}-{2}(R_{l}f^{p}_{lt}+X_{l}f^{q}_{lt})\biggr{]}=0;\forall l\in{\cal L}^{F},$
	$\displaystyle\hskip 200.0ptt\in T$		(10)

II-D Candidates line segments for reinforcement

	$\displaystyle\overline{f}^{upd}_{l}=\sum_{r\in R^{line,options}_{l}}x^{line,inv}_{l,r}\overline{F}^{option}_{lr};\forall l\in{\cal L}^{C}$		(11)
	$\displaystyle\sum_{r\in R^{line,options}_{l}}x^{line,inv}_{l,r}=1;\forall l\in{\cal L}^{C}$		(12)
	$\displaystyle x^{line,inv}_{l,r}\in\{0,1\};\forall l\in{\cal L}^{C},r\in R^{line,options}_{l}$		(13)
	$\displaystyle-\overline{f}^{upd}_{l}\leq f^{p}_{l,t}\leq\overline{f}^{upd}_{l};\forall l\in{\cal L}^{C},t\in T$		(14)
	$\displaystyle-\overline{f}^{upd}_{l}\leq f^{q}_{l,t}\leq\overline{f}^{upd}_{l};\forall l\in{\cal L}^{C},t\in T$		(15)
	$\displaystyle f^{q}_{l,t}\leq cotan\biggl{(}\biggl{(}\frac{1}{2}-e\biggr{)}\frac{\pi}{4}\biggr{)}\biggl{(}f^{p}_{l,t}-cos\biggl{(}e\frac{\pi}{4}\biggr{)}\overline{f}^{upd}_{l}\biggr{)}$
	$\displaystyle\hskip 22.0pt+sin\biggl{(}e\frac{\pi}{4}\biggr{)}\overline{f}^{upd}_{l};\forall l\in{\cal L}^{C},t\in T,e\in\{1,\dots,4\}$		(16)
	$\displaystyle-f^{q}_{l,t}\leq cotan\biggl{(}\biggl{(}\frac{1}{2}-e\biggr{)}\frac{\pi}{4}\biggr{)}\biggl{(}f^{p}_{l,t}-cos\biggl{(}e\frac{\pi}{4}\biggr{)}\overline{f}^{upd}_{l}\biggr{)}$
	$\displaystyle\hskip 22.0pt+sin\biggl{(}e\frac{\pi}{4}\biggr{)}\overline{f}^{upd}_{l};\forall l\in{\cal L}^{C},t\in T,e\in\{1,\dots,4\}$		(17)
	$\displaystyle-(1-x^{line,inv}_{lr})M\leq v^{\dagger}_{to(l),t}-\biggl{[}v^{\dagger}_{fr(l),t}-{2}(R^{option}_{lr}f^{p}_{lt}$
	$\displaystyle\hskip 2.0pt+X^{option}_{lr}f^{q}_{lt})\biggr{]}\leq(1-x^{line,inv}_{lr})M;\forall l\in{\cal L}^{C},t\in T,$
	$\displaystyle\hskip 155.0ptr\in R^{line,options}_{l}$		(18)

II-E Tap changing

	$\displaystyle\frac{v^{\dagger,mid}_{to(l),t}}{(\varphi^{max}_{l})^{2}}\leq v^{\dagger}_{to(l),t}\leq\frac{v^{\dagger,mid}_{to(l),t}}{(\varphi^{min}_{l})^{2}};\forall l\in{\cal L}^{OLTC}\cup{\cal L}^{FH},$
	$\displaystyle\hskip 200.0ptt\in T$		(19)
	$\displaystyle-x^{OLTC,inv}_{l}M\leq v^{\dagger,mid}_{to(l)}-v^{\dagger}_{to(l),t}\leq x^{OLTC,inv}_{l}M;$
	$\displaystyle\hskip 130.0pt\forall l\in{\cal L}^{OLTC,C},t\in T$		(20)
	$\displaystyle x^{OLTC,inv}_{l}\in\{0,1\};\forall l\in{\cal L}^{OLTC,C}$		(21)

II-F Feeder head and OLTC transformers

	$\displaystyle-\overline{f}^{trf}_{l}\leq f^{p}_{l,t}\leq\overline{f}^{trf}_{l};\forall l\in{\cal L}^{OLTC}\cup{\cal L}^{FH},t\in T$		(22)
	$\displaystyle-\overline{f}^{trf}_{l}\leq f^{q}_{l,t}\leq\overline{f}^{trf}_{l};\forall l\in{\cal L}^{OLTC}\cup{\cal L}^{FH},t\in T$		(23)
	$\displaystyle f^{q}_{l,t}\leq cotan\biggl{(}\biggl{(}\frac{1}{2}-e\biggr{)}\frac{\pi}{4}\biggr{)}\biggl{(}f^{p}_{l,t}-cos\biggl{(}e\frac{\pi}{4}\biggr{)}\overline{f}^{trf}_{l}\biggr{)}$
	$\displaystyle\hskip 5.0pt+sin\biggl{(}e\frac{\pi}{4}\biggr{)}\overline{f}^{trf}_{l};\forall l\in{\cal L}^{OLTC}\cup{\cal L}^{FH},t\in T,$
	$\displaystyle\hskip 165.0pte\in\{1,\dots,4\}$		(24)
	$\displaystyle-f^{q}_{l,t}\leq cotan\biggl{(}\biggl{(}\frac{1}{2}-e\biggr{)}\frac{\pi}{4}\biggr{)}\biggl{(}f^{p}_{l,t}-cos\biggl{(}e\frac{\pi}{4}\biggr{)}\overline{f}^{trf}_{l}\biggr{)}$
	$\displaystyle\hskip 5.0pt+sin\biggl{(}e\frac{\pi}{4}\biggr{)}\overline{f}^{trf}_{l};\forall l\in{\cal L}^{OLTC}\cup{\cal L}^{FH},t\in T,$
	$\displaystyle\hskip 165.0pte\in\{1,\dots,4\}$		(25)
	$\displaystyle 0\leq\overline{f}^{trf}_{l}\leq\overline{F}^{FH}_{l}+x^{FH,inv}_{l}\overline{F}^{FH,upd}_{l};\forall l\in{\cal L}^{FH}$		(26)
	$\displaystyle\overline{f}^{trf}_{l}=\overline{F}^{OLTC}_{l};\forall l\in{\cal L}^{OLTC}$		(27)
	$\displaystyle v^{\dagger,mid}_{to(l),t}-\biggl{[}v^{\dagger}_{fr(l),t}-{2}(R_{l}f^{p}_{lt}+X_{l}f^{q}_{lt})\biggr{]}=0;$
	$\displaystyle\hskip 110.0pt\forall l\in{\cal L}^{OLTC}\cup{\cal L}^{FH},t\in T$		(28)
	$\displaystyle x^{FH,inv}_{l}\in\{0,1\};\forall l\in{\cal L}^{FH}$		(29)

II-G Solar

	$\displaystyle g^{RTS}_{r,t}=\overline{g}^{RTS}_{r}f^{RTS}_{r,t};\forall r\in R^{Solar,E},t\in T$		(30)
	$\displaystyle g^{CS}_{r,t}+g^{CS,crt}_{r,t}=x^{CS,inv}_{r}f^{CS}_{r,t};\forall r\in R^{Solar,C},t\in T$		(31)
	$\displaystyle\sum_{r\in R^{Solar,C}}x^{CS,inv}_{r}=f^{CS,inv}$		(32)

II-H Storages

	$\displaystyle soc^{t0}_{h}=soc_{h,t\_last};\forall h\in H$		(33)
	$\displaystyle soc_{h,t}=soc^{t0}_{h}+\eta p^{in}_{h,t}-p^{out}_{h,t};\forall h\in H,t=t\_ini$		(34)
	$\displaystyle soc_{h,t}=soc_{h,t-1}+\eta p^{in}_{h,t}-p^{out}_{h,t};\forall h\in H,$
	$\displaystyle\hskip 160.0ptt\in T\|t\neq t\_ini$		(35)
	$\displaystyle 0\leq soc_{h,t}\leq\overline{S}_{h}\cdot\overline{P}^{in}_{h};\forall h\in H^{E},t\in T$		(36)
	$\displaystyle 0\leq soc_{h,t}\leq\overline{S}_{h}x^{st,inv}_{h}\overline{P}^{in}_{h};\forall h\in H^{C},t\in T$		(37)
	$\displaystyle 0\leq p^{in}_{h,t}\leq\overline{P}^{in}_{h};\forall h\in H^{E},t\in T$		(38)
	$\displaystyle 0\leq p^{in}_{h,t}\leq x^{st,inv}_{h}\overline{P}^{in}_{h};\forall h\in H^{C},t\in T$		(39)
	$\displaystyle 0\leq p^{out}_{h,t}\leq\overline{P}^{out}_{h};\forall h\in H^{E},t\in T$		(40)
	$\displaystyle 0\leq p^{out}_{h,t}\leq x^{st,inv}_{h}\overline{P}^{out}_{h};\forall h\in H^{C},t\in T$		(41)
	$\displaystyle 0\leq x^{st,inv}_{h}\leq x^{st,bin,inv}_{h}\overline{x}^{st,inv}_{h};\forall h\in H^{C}$		(42)
	$\displaystyle-\psi\overline{P}^{in}_{h}\leq q^{st,+/-}_{h,t}\leq\psi\overline{P}^{in}_{h};\forall h\in H^{E},t\in T$		(43)
	$\displaystyle-\psi x^{st,inv}_{h}\overline{P}^{in}_{h}\leq q^{st,+/-}_{h,t}\leq\psi x^{st,inv}_{h}\overline{P}^{in}_{h};\forall h\in H^{C},$
	$\displaystyle\hskip 200.0ptt\in T$		(44)

II-I Solar and storage colocation enforcement

	$\displaystyle 0\leq x^{CS,inv}_{r}\leq\overline{x}^{CS,inv}_{r}x^{st,bin,inv}_{h(r)};\forall r\in R^{Solar,C}$		(45)
	$\displaystyle\sum_{h\in H^{C}}x^{st,bin,inv}_{h}=1$		(46)
	$\displaystyle x^{st,bin,inv}_{h}\in\{0,1\};\forall h\in H^{C}$		(47)

where $h(r)$ is the index of the candidate storage to be co-located with the solar investment of index $r$ .

III Case studies

IV Conclusions

These are my conclusions…

Acknowledgment

This work has been funded by the U.S. Department of Energy, Office of Electricity, under the contract DE-AC02-05CH11231.