storage investment dec2020 - iu

On the Distributed Energy Storage Investment and Operations

Roman Kapuscinski*, Owen Q. Wu**, and Santhosh Suresh***

*Ross School of Business, University of Michigan, Ann Arbor, MI, [email protected]

**Kelley School of Business, Indiana University, Bloomington, IN, [email protected]

***McKinsey & Company, New York, NY, [email protected]

December 21, 2020

Problem Definition : We analyze a facility location problem for energy storage and compare the benefits

of centralized storage (next to the central energy generation site) versus localized storage (next to the

demand sites). We also analyze how much storage capacity to invest in, with the objective of minimizing the

total storage investment and energy generation costs. Practical Relevance: Energy storage has become

an indispensable part of power distribution systems. Understanding the trade-offs between centralized

storage versus localized storage provides guidelines for storage investment decisions. Methodology : We

use a stylized model to capture the critical features of an energy distribution system, including convex costs,

stochastic demand, storage efficiency, and line losses. We use dynamic programming to optimize storage

operations and derive the value of storage for the distribution system. Results: We find that there are

fundamental differences between centralization/localization decisions at the capacity investment stage and

the centralization/localization decisions at the storage operations level. At the operations level, centrally

stored energy offers more operational flexibility, which is consistent with the conventional understanding of

inventory pooling. However, at the investment level, we find that localized storage is preferred under various

circumstances. Managerial Implications: Storage investment should first be made at the demand

locations with positive minimum demand regardless of the level of demand variability, and then storage

investment should consider the trade-offs between centralized versus localized investment.

Key words: Distributed energy storage, capacity investment, inventory pooling, production smoothing

1. Introduction

Recent advancement in electricity storage technology has proven the feasibility of storing energy in

a distributed manner to reduce the total cost of energy services. The Electricity Storage Handbook

(Akhil et al. 2015) provides a detailed description of various storage technologies. Storage investment

cost, although declining, remains the main obstacle to the wide deployment of storage technologies

(Saboori et al. 2017). Bloomberg New Energy Finance (2018a) estimates that $620 billion will be

invested in energy storage globally from 2019 to 2040. Energy Information Administration (2020)

projects that energy storage capacity (excluding pumped hydro) in the U.S. will grow by at least six

times from 2019 to 2030, under all scenarios studied. These investment projects naturally raise the

questions about the locations, sizes and operations of storage facilities, and whether conventional

inventory management lessons and principles can be used.

In this paper, we focus on the most basic trade-offs in a distribution system to evaluate how

energy storage facilities should be located, sized, and operated. The system we consider consists of

one central generation node connected to multiple demand nodes via distribution lines. Our model,

although parsimonious, captures the elements that are important for storage investment decisions,

including stochastic demand, line losses, storage efficiency, and convex cost of energy generation.

Clearly, due to demand variability and the convex production cost, storage provides values in leveling

predictable demand fluctuations and buffering demand uncertainties. However, it is unclear, a priori,

how storage facilities should be distributed (i.e., located at the generation node or at the demand

nodes) to best provide these values. One might expect that centralizing storage is preferred because

of the conventional inventory or capacity pooling effect. However, although pooling benefits storage

operations, it does not necessarily apply to how we invest in energy storage facilities in a distribution

system. In this paper, we ask the following central question: What is the optimal locations and size

of storage investment, and what drives the relative benefits of centralized versus localized storage

investment?

While the dynamics of many inventory systems and supply chains are due to time delays, inven-

tory holding cost, and transportation cost, these are different in energy systems. There is effectively

no time delay in distributing energy. Corresponding to holding cost, there is energy loss during

storing and withdrawing energy. Corresponding to transportation cost, distributing energy incurs

line losses. Line losses are critical, because in absence of line losses, the energy distribution system

can be regarded as a single node and storage locations do not matter. Line losses in practice are not

insignificant. According to the International Energy Agency (2018), the transmission and distribu-

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tion losses account for 8 to 9% of total electricity generation in the world from 1990 to 2017, and

vary drastically across regions: e.g., Brazil 15.8%, Turkey 14.8%, Spain 9.6%, Canada 8.7%. In the

U.S., the line losses vary from 2% to 13% across different states (Wirfs-Brock 2015).

Cognizant of the energy storage characteristics that are different from physical goods, we are

particularly interested in understanding the relative benefits of centralized versus localized storage

capacity investment and whether the trade-offs are different when operating the storage. We find that

there are fundamental differences between storage capacity centralization/localization decisions and

inventory (energy) centralization/localization decisions. Under a given storage capacity investment,

centrally stored energy offers more operational flexibility, which is consistent with the conventional

understanding of inventory pooling. However, at the investment stage, we find that localized storage

is usually preferred. Specifically, in the absence of fixed cost, storage investment should first be made

at the demand nodes until storage capacity reaches the minimum demand level, regardless of the

level of demand variability. Further storage investment beyond the minimum demand level should

consider relative benefits of localized versus centralized storage. Localized storage continues to be

optimal when the demand correlation is high, and is asymptotically optimal as storage cost declines.

On the other hand, centralized storage allows energy to be stored centrally, improving storage usage

and reducing the need to transfer energy across demand nodes.

2. Related Literature

Energy storage siting and sizing problem is relatively new to the research community. In an early

literature review, Hoffman et al. (2010) find the lack of models that optimize storage placement and

sizing, and call for filling this gap. To study the problem of storage siting and sizing, the minimum

construct needs to take into account storage losses, energy transfer between locations and line losses,

as well as the cost of energy generation. Existing literature on energy systems has evaluated some

of the trade-offs numerically, or analyzed special cases, such as single link between generation and

demand locations. The operations management literature has also analyzed some individual aspects

of the problem we consider. We describe these connections below.

Energy generation typically involves a convex production cost, as generators with low marginal

cost are typically dispatched first to meet the demand. Inventory management with convex purchas-

ing cost is first analyzed by Karlin (1960). A stream of research considers special cases of convex

production costs, mostly in piecewise linear format, e.g., Henig et al. (1997) consider designing con-

tracts that specify in advance the delivery frequency and volume, which effectively results in piecewise

linear convex cost. Recently, Lu and Song (2014) provide a detailed review of models with convex

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cost. These papers do not analyze transshipment and associated losses or storage efficiency (typical

for energy storage).

Energy storage size is typically limited due to its cost (refer to Section 1). Inventory management

with limited storage space, known as the warehouse problem, is introduced by Cahn (1948). Given

warehouse space but no storing/releasing speed limit, Charnes et al. (1966) show that the optimal

policy is a bang-bang type (either fill up the storage or sell all inventory). Traditional inventory

system models typically do not consider inventory losses that are crucial in energy systems, such as

energy losses during distribution, charging and discharging of storage. Secomandi (2010) extends

the traditional models by incorporating energy losses during storage operations. In this paper,

we consider not only inventory management under given storage size, but also storage investment

decisions that involve sizing and siting of storage facilities.

Inventory pooling, introduced by Eppen (1979), has been studied extensively in the literature.

We refer the readers to Federgruen and Zipkin (1984), Gerchak and Mossman (1992), Corbett and

Rajaram (2006), Berman et al. (2011) and the references therein. A stream of research analyzes

advantages of pooling across time or locations in the context of distribution systems and production-

inventory systems, such as Dada (1992), Ozer (2003), Benjaafar et al. (2005), Chen and Gavirneni

(2010), Gerchak (2016) and Yang et al. (2020). In this paper, there is a single decision-maker, who

considers pooling at two different levels—investment and operations. At the investment level, storage

sizing and siting decisions can be pooled or localized; at the operations level, if storage facilities exist

at both the central node and demand nodes, the decision-maker faces the trade-offs between pooling

inventory at the central storage (up to its size) or storing inventory at the local storage facilities (up

to their sizes).

Energy stored in one location can be sent to another location to serve the demand, which is

similar to the transshipment operations in other industries. Wee and Dada (2005) study a one-

warehouse-multiple-retailer system, where demand in a selling season is met by on-hand inventory

first and then by transshipment from other locations, with the objective of minimizing the total cost

of understocking, inventory holding, and transshipment. Multiperiod multilocation inventory man-

agement problem with transshipment is considered by Karmarkar (1987), Robinson (1990), Axsater

(1990), Zhao et al. (2008), among others. Hu, Duenyas, and Kapuscinski (2008) provide a detailed

review of this area. Despite the similarities, power distribution systems have unique features that

require different models. First, transporting energy incurs line losses, and charging/discharging en-

ergy storage involves storage losses. This feature is modeled in Zhou et al. (2015, 2019), and we

use the same cost form as in their work. Second, unlike other industries where production decisions

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are made before observing the demand, the lead time between power generation and consumption

is literally zero, and all demand has to be satisfied except for extraordinary situations. Thus, there

are no backorders or lost sales in our model.

In the energy research literature, one of the pioneering research papers on storage siting and sizing

is by Denholm and Sioshansi (2009). They study the case of one generation node (wind power) and

one demand node and consider the trade-off between co-locating storage with the generation or with

the demand, similarly to our paper. A key consideration of our research is multiple demand locations.

In recent years, while the problem of optimizing storage locations and sizes has attracted increasing

attention, most of the research effort has focused on developing computational methods to optimize

the storage locations and size for various applications. Carpinelli et al. (2013) consider the value of

storage in price arbitrage, loss reduction, voltage support, and network upgrade deferral and develop

a genetic algorithm to find optimal storage solutions. Ghofrani et al. (2013) focus on the value of

storage in mitigating intermittency of renewable generation and also develop a genetic algorithm to

compute the optimal placement of energy storage. Sardi et al. (2017) attempt to consider all possible

benefits and costs of storage and aim to numerically search for best storage investments based on

net present value. Other research in this domain includes Chen et al. (2011), Nick et al. (2014), and

Fortenbacher et al. (2018). Zidar et al. (2016) and Saboori et al. (2017) provide excellent reviews

of the literature. In a more recent review, Das et al. (2018) emphasize the importance of modeling

uncertainties in demand and renewable power generation. In this paper, we use a stochastic dynamic

programming approach, which complements the above body of numerical methods by analyzing the

storage operating policy and deriving insights about storage investment in distribution systems.

The literature has only recently started to explore the theory on optimal storage siting and sizing,

possibly because such problems are theoretically challenging: For every choice of storage sites and

sizes, there is an embedded problem of optimizing the charging and discharging of storage facilities,

which involve multiple dimensions of states and decisions. The first theoretical work in this area is by

Thrampoulidis et al. (2016). They focus on the value of storage in load shifting and choose storage

locations and size to minimize the total generation cost. They prove that the optimal solution should

place zero storage at generation-only nodes that connect to the rest of the network via a single link,

but they acknowledge that this result cannot be extended to nodes with multiple links, which is the

case we analyze. Tang and Low (2017) employ a “continuous tree” model for a distribution system

and aim to minimize the energy loss. They prove that, when all loads are perfectly correlated, it

is optimal to place storage near the leaves of the tree, as perfectly correlated loads is equivalent to

a single demand node. In this paper, we aim to understand the drivers of storage location choices

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and the relative benefit of centralized versus localized storage investment in a setting with multiple

demand nodes.

3. Model

We consider a storage capacity investment problem illustrated in Figure 1. A central production

facility at node 0 generates electricity and distributes to n demand nodes Ldef= {1, 2, . . . , n}. There

are three storage siting strategies. A localized (resp. centralized) investment installs storage only at

the demand nodes (resp. central node), and a mixed investment installs storage at both the central

and demand nodes. Each siting strategy also involves many sizing choices.

After the storage locations and sizes are determined, storage operations may be straightforward

if there is only one location such as Figure 1(b), but can also involve many possible operations

strategies, especially in the case of mixed investment in Figure 1(c). We will show in Section 5 that

under a mixed investment, storing energy will follow the principles of inventory pooling: store at the

central storage first and store at demand nodes only if the central storage is full. Interestingly, unlike

storage operational strategies, our main analysis in Section 4 will reveal that storage investment

usually favors a distributed (as opposed to pooled) manner.

Figure 1: Location choices for storage investment

(b) Centralized investment

0

1 �

0

1 �

0

1 �

(a) Localized investment (c) Mixed investment

2 2 2... ... ... ... ... ...

We consider a T -period planning horizon and let Tdef= {1, 2, . . . , T}. Each period t ∈ T represents

several hours in which the storage (e.g., lithium-ion battery) can be fully charged or discharged. Prior

to the first period, storage investment is made at one or multiple locations. Once storage facilities

are built, their sizes are fixed throughout T . The system planner’s objective is to minimize the sum

of storage investment cost and energy production cost (influenced by storage) over T .

We denote storage size by S = (S0, S1, . . . , Sn), where Si ≥ 0 is the storage size at node i. Denote

by st = (s0,t, s1,t, . . . , sn,t) the energy stored at the beginning of period t. The set of admissible

storage levels is Adef= {s : 0 ≤ s ≤ S}, t ∈ T . Figure 2 illustrates notations and their relations.

Unlike physical goods supply chains, there is no inventory holding cost in our system and the

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Figure 2: Network model and key variables

Energy flow

direction when the

associated term > 0

��

��

0

1

Storage level in period �:

begin: ��, end: ��,�

Demand ��, �

Generation



��(∆��,)

��, = ��, + ��(∆��,)

��(��,)

��(∆��,)



��(��,)

Demand ��,

��(∆��,)

��, = ��, + ��(∆��,)

... ... ... ...

dominant friction is the energy loss during storage charging and discharging, as modeled below. Let

α ∈ (0, 1] denote the one-way efficiency of the storage. That is, reducing storage level si,t by one

unit outputs α units of energy, and raising si,t by one unit requires α−1 units of energy.1 Thus, the

energy flow associated with an inventory change of ∆si,t ≡ si,t+1 − si,t is

ψα(∆si,t)def=

α−1∆si,t, if ∆si,t ≥ 0,

α∆si,t, if ∆si,t < 0,i ∈ {0} ∪ L, (1)

where ψα(∆si,t) > 0 is the energy inflow into storage and ψα(∆si,t) < 0 is the energy outflow.

We assume the demand is stochastic but steady over T , and do not consider the storage value in

deferring upgrades of lines and production capacity. Thus, we assume that the n distribution lines

are not capacity-constrained, i.e., all demands can be satisfied without storage.

Assumption 1 (i) The line capacity constraints are non-binding in all periods. (ii) Line loss is

linear in the amount of energy sent along the line.

In general, line loss is nonlinear in the energy transmitted. The linear approximation for line

loss is often used in the literature for economic analysis, see, e.g., Denholm and Sioshansi (2009),

Sadegheih (2009), Pereira and Saraiva (2011), Chamorro et al. (2012), and Zhou et al. (2019).

Let β ∈ (0, 1) denote the line efficiency in either direction. That is, when sending energy along

1The assumption of the same energy loss fraction in both ways brings notational and analytical convenience, butdoes not cause loss of generality. If charging efficiency α1 differs from discharging efficiency α2, we can set α =

√α1α2

and scale the storage size and storage levels accordingly.

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any of the n lines, 1− β fraction of energy is lost. Let ui,t denote the energy flow on line i reaching

demand node i in period t (illustrated in Figure 2), with ui,t > 0 for the energy received at node i

and ui,t < 0 for the energy leaving node i. The corresponding energy flow leaving or reaching the

central node 0 can be written as

ψβ(ui,t)def=

β−1ui,t, if ui,t ≥ 0,

βui,t, if ui,t < 0,i ∈ L. (2)

Note that ui,t < 0 occurs during “transshipment” (see Section 2 for related literature). The difference

between energy transfer and physical goods transshipment is that sending energy from one demand

node to another does not incur shipping cost, but incurs line losses.

Let dt = (d1,t, . . . , dn,t), where di,t is the demand at demand node i in period t. We assume the

demand process {dt : t ∈ T } is Markovian, and dt is realized at the beginning of period t and must

be satisfied in period t. Meeting all demands in the same period is feasible because there is sufficient

production capacity at node 0 and lead time for energy transfer is zero.

At the beginning of each period, observing the storage level st and demand dt, the system planner

decides the period-ending storage level st+1, which in turn determines the energy flows and required

generation as described below. At the demand nodes, the energy flow balance constraint is

ui,t = di,t + ψα(∆si,t), i ∈ L. (3)

Flow balance at node 0 implies that the required central generation quantity, denoted as qt, is a

function of demand dt and inventory change ∆st ≡ st+1 − st (also refer to Figure 2):

qt = q(∆st,dt)def= ψα(∆s0,t) +

∑i∈L

ψβ

(di,t + ψα(∆si,t)

). (4)

Because ψα(·) and ψβ(·) are convex and increasing functions, q(∆st,dt) is convex and increasing in

∆st. As generation quantity cannot be negative, the decision-maker must choose storage level from

{st+1 ∈ A : q(st+1 − st,dt) ≥ 0}, which is generally a non-convex set.

Let c(qt) denote the cost of producing qt in period t at the central node. The production satisfies

the following assumption.

Assumption 2 (i) c(qt) is convex and increasing in qt for qt ≥ 0; (ii) In every period t, generation

level qt can be adjusted to any non-negative level at negligible adjustment cost.

Managing physical goods inventory often involves an ordering cost, holding, and understocking

costs. In our system, the production cost is convex (Assumption 2) and all demands are met, while

energy losses occur during distribution and storage operations, which requires extra production to

compensate for these losses. The energy production cost over T is evaluated below.

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Under a given storage investment decision S, let Vt(st,dt;S) denote the minimum expected

discounted production cost from period t onward when the state is (st,dt), and let γ ∈ (0, 1] be the

discount factor. The optimal operating policy for given storage S is determined by the following

stochastic dynamic program:

Vt(st,dt;S) = minst+1

{c(q(st+1 − st,dt)

)+ γEt

[Vt+1(st+1,dt+1;S)

]}, t ∈ T , (5)

s.t. st+1 ∈ A, q(st+1 − st,dt) ≥ 0, t ∈ T , (6)

VT+1(· , · ;S) = 0,

where Et denotes the expectation with respect to future demand dt+1, conditioned on dt.

Recall that storage capacity is installed prior to period 1, and remains unchanged throughout

the operating stage T . Let |S|def=

n∑i=0

Si denote the total storage capacity invested, which requires

an upfront investment cost of p |S|, charged at the end of the installation period (or beginning of

period 1), where p is the investment cost per unit of storage capacity.

Without loss of generality, we assume that storage is fully charged after installation, i.e., s1 = S.

Thus, the system planner’s objective is to minimize the expected total cost:

minS≥0

{p |S| + V (S)

}, with V (S) ≡ EV1(S,d1;S), (7)

where the expectation is taken at the time when the storage investment decision is made. We can

extend the model to include a fixed cost at each storage site, but our analysis focuses on the linear

investment cost and aims to understand the basic trade-offs in storage investment decisions.

4. Optimal Investment Decisions

In this section, we consider the storage investment problem stated in (7). Our goal is to understand

the trade-off between localizing and centralizing storage investment (refer to Figure 1). One might

expect that investment analysis will require us to derive the optimal storage operating policies, but

it turns out that a significant portion of the properties of the optimal capacity investment depends

on only a few key facts. Therefore, we directly explore the properties of the value function below

(Lemmas 1 and 2) to serve our investment analysis.

4.1 Properties of the Value Function

We first derive the properties of the production cost function Vt(st,dt;S) and V (S). (Throughout

this paper, monotone and convex properties are not in strict sense, unless otherwise noted.)

Lemma 1 (i) Vt(st,dt;S) is decreasing and convex in (st,S) for any dt and t ∈ T .

(ii) The expected production cost V (S) is decreasing and convex in S.

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Intuitively, more stored energy and storage capacity reduce production cost. Lemma 1(i) confirms this

intuition and further shows that the production cost is jointly convex in stored energy and storage

capacity. Part (ii) of the lemma follows immediately from part (i) and confirms the diminishing

marginal benefit of storage capacity. The proof of Lemma 1 is nonstandard because constraint (6)

defines a non-convex feasible region for st+1. All proofs are included in the appendix.

The next lemma reveals that a storage facility should not release energy only to store it in another

location. Intuitively, there is no benefit from moving stored energy to incur line and storage losses.

Lemma 2 (i) In period t, suppose δ > 0 and st, st ∈ A satisfy sj,t = sj,t− δ and sk,t = sk,t+ β2δ at

two demand nodes j and k, and si,t = si,t at all other nodes. Then, Vt(st,dt) ≤ Vt(st,dt) for any dt.

(ii) In period t, suppose δ > 0 and st, st ∈ A satisfy sj,t = sj,t − δ and sk,t = sk,t + βδ, where either

j = 0 or k = 0, and si,t = si,t at all other nodes. Then, Vt(st,dt) ≤ Vt(st,dt) for any dt.

Lemma 2 is another building block for proving the investment-related results in the rest of this

section, including the next lemma.

Lemma 3 For any given S = (S0, S1, . . . , Sn), we have

(i) V (Sc) ≤ V (S), where Sc =(S0+β

−1∑

i∈L Si, 0, . . . , 0);

(ii) V (Sl) ≤ V (S), where Sl = (0, S1+βS0, . . . , Sn+βS0).

Lemma 3(i) states that, for any given investment S that is not centralized, there exists a cen-

tralized investment Sc that yields a lower expected production cost but requires a higher investment

cost: |Sc| ≥ |S| since β ∈ (0, 1). Therefore, if S is a mixed or localized investment, simply pooling

the total capacity |S| at the central node may not reduce the production cost; a higher capacity |Sc|

at the central node is need to ensure production cost reduction.

Localized storage investment does not dominate in general, either. Lemma 3(ii) finds a localized

investment Sl that yields a lower expected production cost than S, but also requires a higher invest-

ment cost: |Sl| ≥ |S| if β ≥ 1/n, which holds if n ≥ 2 and line efficiency β ≥ 0.5 (which is usually

the case). However, for a special case with only one demand node (n = 1), Lemma 3(ii) implies that

|Sl| < |S|. Therefore, localized investment is optimal, which is stated below.

Corollary 1 If there is only one demand node (n = 1), then localized storage investment is optimal.

In what follows, we define the optimal centralized and localized investment decisions as

Sc∗ ∈ argmin{V (S) + p |S| : S0 ≥ 0, Si = 0, i ∈ L

}, (8)

Sl∗ ∈ argmin{V (S) + p |S| : S0 = 0, Si ≥ 0, i ∈ L

}. (9)

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The optimal investment S∗ ∈ argmin{V (S) + p |S| : S ≥ 0

}may coincide with Sc∗ or Sl∗, or may

be a mixed investment.

Before furthering analysis, let us intuitively consider how line losses affect the economic value of

storage: One unit of energy output from a local storage can serve one unit of local demand, or only

β2 units of demand at other demand nodes; one unit of energy output from the central storage can

serve β units of demand at any location. Thus, the economic value of storage is affected by the fact

that some stored energy is wasted in line losses.

4.2 Impact of Minimum Demand on Storage Investment

To demonstrate the complexity in storage investment decisions and highlight some counter-intuitive

insights, consider an example in Figure 3.

Example 1 Nodes 1 and 2 have independent demands fluctuating between zero and a high level,

while node 3 has constant demand. Suppose we invest in a small unit of storage capacity. Where

should we place this storage?

Figure 3: A storage location choice problem

�

0

1 2

Demand 1 Demand 2 Demand 3

3

�

Convex generation cost

Lin

e l

oss

�

Intuitively, it is beneficial to place the storage at node 0 because it can buffer the combined

variabilities from nodes 1 and 2. The downside is that part of the energy released from the storage

becomes lost along the lines, implying that part of the storage capacity is wasted in storing line

losses. Alternatively, we can place the storage at the source of the variability, e.g., at demand

node 1. However, if the demand at node 2 is high while the demand at node 1 is zero, using energy

stored at node 1 to smooth production will require sending energy from node 1 back to node 0,

incurring an extra line loss. Considering the above trade-offs between storage capacity loss and line

loss, one may conclude that the storage capacity should be split among nodes 0, 1, and 2, depending

on the relative benefits.

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Let us consider the seemingly inferior choice of placing the storage at node 3. When the demands

at nodes 1 and 2 are zero, the storage at node 3 stores energy. When the demand is high at node 1

or 2 or both, the storage at node 3 releases energy to reduce the amount of energy sent from node 0

to node 3, effectively reducing the generator’s output and smoothing production without incurring

extra line losses, because no energy is sent back to node 0. Furthermore, the storage at node 3 stores

energy to be consumed at node 3, avoiding wasting storage space for storing line losses. Placing the

small storage at node 3 with constant demand is actually optimal.

Although this example allows us to reason the optimal placement of a small storage, we need

theoretical support for the optimal storage investment in general, which we begin to formalize in

Lemma 4 and Proposition 1 below.

Observe that in the above example, the minimum demand at node 3 is positive, which turns out

to be one of the important factors for storage investment decision. To proceed our analysis, we define

dmini as the minimum demand at node i ∈ L.

Lemma 4 Suppose dminj > 0 for a given demand node j ∈ L. Then,

(i) If S and S satisfy Sj = Sj + βδ < α−1dminj and S0 = S0 − δ for some δ > 0, and Si = Si for

i ∈ L, i 6= j, then V (S) = V (S).

(ii) If S and S satisfy Sj = Sj + δ < α−1dminj and Sk = Sk − δ for some k ∈ L and δ > 0, and

S0 = S0, Si = Si for i ∈ L, i 6= j, k, then V (S) ≤ V (S).

Lemma 4(i) states that we can maintain the same production cost, while reducing the total storage

capacity by replacing δ units of storage at node 0 with βδ units of localized storage, as long as the

increased local storage capacity is below the minimum demand. Lemma 4(ii) suggests that we can

maintain the same storage capacity while reducing the production cost by shifting storage capacity

across the demand nodes, as long as the increased local storage capacity is below the minimum

demand.

Proposition 1 (i) If the optimal localized investment Sl∗ satisfies αSl∗j < dmin

j for some j ∈ L, then

S∗ = Sl∗, and any other investment with S0 > 0 is suboptimal.

(ii) Let S∗ be an optimal solution to (7). If S∗0 > 0, then S∗

i ≥ α−1dmini for all i ∈ L.

Proposition 1(i) provides a criterion for verifying the global optimality of a localized optimal

investment. Importantly, the criterion is simple in the sense that it requires αSl∗j < dmin

j to be true for

only one demand node. Proposition 1(ii) implies that positive minimum demand precludes centralized

investment (Figure 1(b)) from being optimal. This result on the suboptimality of centralized storage

investment is in stark contrast to the conventional wisdom that it is beneficial to pool capacity at

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the central node.2

4.3 Demand Correlation and Storage Placement

In this section, we explore a second driver for storage investment decisions—correlation of demands

across the demand nodes. We use a simple example below to illustrate the impact of demand

correlation and other effects in later sections.

Example 2 Suppose there are n = 2 demand locations. In even-numbered periods, both nodes

have zero demand d2t = (0, 0), while in odd-numbered periods, demand d2t−1 can be (10, 10),

(0, 20), (20, 0), with probabilities z, 1−z2 , 1−z

2 , respectively, where z ∈ [0, 1]. This is the simplest

demand model that removes the minimum demand effect identified in Section 4.2 and reflects the

key characteristic of energy demand—its cyclic pattern. The odd and even periods represent peak

and off-peak periods, respectively. Off-peak demand is zero and peak demand may be zero (to remove

minimum demand effect); the peak demand has considerable variability across nodes with known

aggregate demand. Clearly, a higher z corresponds to a higher demand correlation, with z = 1 for

perfect correlation (demand alternates between 0 and 10 for both nodes) and z = 0.5 corresponding

to zero correlation (it can be verified that the covariance of the demand series is 50z − 25).

Because V (S) is convex in S (Lemma 1) and the system is symmetric with respect to the two

demand nodes, we can focus our attention on symmetric investment, i.e., S1 = S2, without losing

optimality. Furthermore, we can write the long-run discounted production cost function explicitly:

V (S) = mins∈[0,S1]

1

1− γ2

[z

(c(20− 2αS1

β− αS0

)+ γc

(2S1αβ

+S0α

))

+ (1− z)

(c(20− αS1

β− βαs − αS0

)+ γc

(S1 + s

αβ+S0α

))],

where 20−2αS1

β− αS0 is the production quantity when demand is (10, 10) and all stored energy is

released to meet the demand, 2S1

αβ+ S0

αis the production quantity in the next period to fully charge

all storage, 20−αS1

β− βαs − αS0 is the production quantity when demand is either (0, 20) or (20, 0)

and we fully discharge storage at the node with demand 20 while we reduce inventory by s ∈ [0, S1]

(to be optimized) at the node with zero demand, and S1+sαβ

+ S0

αis the production quantity in the

next period to fully charge all storage.

Figure 4 shows the optimal storage investment as a function of the demand correlation. As

demand correlation increases, the optimal storage investment shifts from centralized investment to

mixed, and then to localized investment. It can also be seen that the total amount of storage

investment is higher if the centralized investment is optimal, because extra capacity is needed to

2We will show in Section 5 that inventory pooling is still optimal at the operations level.

12

Figure 4: Optimal storage investment under various demand correlations

α = 0.9, β = 0.9, c(q) = q2, γ = 0.99

S1 S2

S0

0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

S1 S2

S0

0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

(b) Storage cost � = 400(a) Storage cost � = 100

Demand correlationDemand correlation

Storage capacityStorage capacity

Centralized capacity

Localized capacity Centralized

capacity

Localized capacity

account for the line losses.

In this example, in the absence of the minimum demand effect, when the demand correlation is

low, the conventional wisdom about the benefit of capacity pooling remains salient. However, the

key difference is that the pooling benefit does not come from lower capacity investment (in fact, the

optimal centralized capacity is even higher than the optimal localized capacity), but come from the

operating flexibility, which we will detail in Section 4.4.

Furthermore, because centralized capacity investment involves a higher capacity, it becomes less

desirable when the storage capacity is more expensive, as seen from comparing Figure 4(a) and (b).

In fact, an even higher capacity cost p can render centralized storage investment suboptimal for the

entire range of demand correlation. On the other hand, a lower storage cost will never change the

optimality of the localized storage investment under perfectly correlated demand, as is proven in the

following proposition, for the general setting with any number of demand nodes.

Proposition 2 If for any i ∈ L and t ∈ T , di,t = ki d1,t for some constant ki > 0, then S∗ = Sl∗.

Note that multiple demand nodes with perfectly correlated demands can be treated as a single

demand node, in which case Corollary 1 implies that localized investment is optimal. Localized

investment tends to be optimal for highly positively correlated demand. Intuitively, as discussed at

the end of Section 4.1, energy transfer across demand nodes reduces economic value of locally stored

energy, but such transfer is rarely needed if demands are highly positively correlated across nodes.

13

4.4 Benefits of Centralized Storage Investment

In this section, we detail the benefits of centralized investment in terms of its operating flexibility.

The benefit of centralized storage can be best understood by examining the operational inefficiencies

in the optimal localized investment Sl∗, as illustrated in the example below.

Example 3 Consider Example 2 with z = 0 and p = 100. That is, the demands are most negatively

correlated: In odd-numbered periods, d2t+1 = (0, 20) or (20, 0) with equal probability. In this setting,

it can be verified that the optimal localized investment Sl∗ is to invest storage capacity dedicated to

each demand node. That is, stored energy at node 1 will never be used to serve demand at node 2

and vice versa. The optimal localized investment is to invest a total of |Sl∗| = 16.1 units of storage

capacity, much higher than the optimal capacity |S∗| = 9.4 units shown in Figure 4(a). Under Sl∗,

although energy does not transfer across demand nodes to incur extra line losses, the storage capacity

is not shared by the demand nodes and thus under-utilized.

Now change z = 0.5 while keeping everything else unchanged. We find that the optimal localized

investment is to invest a total of |Sl∗| = 8.6 units of capacity, partially shared by the demand nodes

as follows. When demand is (0, 20), the storage at the node with demand 20 is fully discharged while

the storage at the node with zero demand is partially discharged and the released energy is sent back

to the central node to smooth production. Because this strategy incurs extra line losses, it turns out

that the centralized investment is still the optimal investment in this case.

The above example reveals two operating benefits of centralized investment. The first benefit

is that centrally stored energy can serve demand at any demand node without incurring extra line

loss associated with transferring energy across demand nodes. The second benefit is that investing

in centralized storage can avoid under-utilizing storage capacity dedicated to each demand node.

Dedicated localized storage investment results in over-investment and under-utilization of storage

capacity, because the storage is used to smooth production less frequently.

In general, when neither localized nor centralized investment is optimal, mixed investment be-

comes optimal by striking a balance between the flexibility of centralized storage and the proximity

of localized investment to the demand.

Finally, we point out that if the minimum demand is positive at either demand node, then

the minimum demand effect discovered and described in Section 4.2 dominates the above trade-offs

between localized and centralized investment, as Proposition 1 implies that investment should first go

to the demand nodes with positive minimum demand. When storage capacity reaches the minimum

demand, an incremental investment will then face the trade-offs discussed in Sections 4.3 and 4.4.

14

4.5 Impact of Storage Cost in Storage Investment

While the cost of storage remains high, storage technologies keep evolving with expectation of cost

declines, which will impact the optimal storage investment decisions. Therefore, in this section, we

write S∗(p) to emphasize the dependence of the optimal investment on the unit investment cost p.

Proposition 3 (i) The minimum total cost V (S∗(p)) + p |S∗(p)| increases in p;

(ii) The optimal total storage capacity investment |S∗(p)| decreases in p, but S∗i (p) may not decrease

in p.

As expected, Proposition 3 confirms that, as storage cost declines, more investment in storage

capacity will take place (part (ii)), reducing the total cost (part (i)). However, storage cost decline

also affects the relative values of centralized versus localized storage. Centralized storage tends

to store more energy that will become line losses. As storage cost declines, this disadvantage is

less prominent and the optimal storage investment may shift toward centralized storage capacity,

resulting in less localized investment.

We illustrate the above effects using the previous example in Figure 4. Comparing Figure 4(a)

and (b) by focusing on the demand correlation range between 0.4 and 0.7. Within this range, a

lower storage capacity cost certainly induces a higher total capacity investment, but within the mix,

we actually see less localized storage capacity. In other words, although the total investment |S∗(p)|

decreases in p, the components of S∗(p) may not decrease in p, as the optimal storage locations shift.

The above observation begs the question of whether centralized investment will be the preferred

choice when storage cost keeps declining. To answer that question, we consider a futuristic case where

storage cost becomes very low and some demand nodes may have significant local power generation

so that dit may be negative. As shown in Proposition 4 below, when p→ 0+, both Sl∗(p) and Sc∗(p)

are asymptotically optimal if the demand at each node is nonnegative. However, if demand can be

negative, this asymptotic optimality holds only for the localized investment Sl∗(p), and generally

does not hold for the centralized investment Sc∗(p). Therefore, localized storage investment is still

preferred if storage cost becomes very low.

Proposition 4 (i) limp→0+

[V (S∗(p)) + p |S∗(p)| − V (Sl∗(p))− p |Sl∗(p)|

]= 0,

(ii) If dt ≥ 0, t ∈ T , then limp→0+

[V (S∗(p)) + p |S∗(p)| − V (Sc∗(p))− p |Sc∗(p)|

]= 0.

For part (ii), a very low p suggests that installing storage at node 0 costs little extra, even though

some of the stored energy will become line losses. However, when the demand is negative (i.e.,

locally generated energy exceeds local demand), localized storage can store energy locally, whereas

15

centralized investment is no longer asymptotically optimal, since transferring excess local energy to

the central storage and transferring back to serve future demand incur line losses.

5. Optimal Operating Policy for Given Storage Investment

The previous section suggests that unpooling storage capacity investment tends to be a preferred

choice under various circumstances. In this section, we show that, after any given mixed storage

investment is made, pooling inventory at the central storage facility (up to its size) is still preferred,

which is consistent with the conventional understanding of inventory pooling. This can be seen

from the structural properties of the optimal storage operating policy, which are very intuitive. The

technical proof of these optimal structures, however, is quite long and we refer the reader to Suresh

(2014) for the proofs.

To describe the structure of the optimal policy, we decompose the problem in (5)-(6) into a master

problem and a subproblem. The master problem decides the production quantity qt:

Vt(st,dt;S) = minqt

{c(qt) + γWt(qt, st,dt;S) : qt ∈ Q(st,dt)

}, (10)

and the subproblem finds the optimal use of qt by deciding the period-ending storage levels:

Wt(qt, st,dt;S) = minst+1

{Et

[Vt+1(st+1,dt+1;S)

]: st+1 ∈ A(qt)

}, qt ∈ Q(st,dt), (11)

where st+1 is chosen from an iso-production surface A(qt), defined as

A(qt)def=

{st+1 ∈ A : q(st+1 − st,dt) = qt

}, (12)

and qt is chosen from Q(st,dt) ≡[qt, qt

], where qt = q(S − st,dt) is the maximum production in

period t, which satisfies the demand and fully charge all storage, and qt =(q(−st,dt)

)+is the

minimum production, which satisfies the remaining demand after the storage is discharged to meet

as much demand as possible.

Let s∗t+1(qt, st,dt;S) denote an optimal solution to the subproblem (11). Solving (11) gives the

minimum expected cost-to-go function Wt(qt, st,dt;S), which is decreasing and convex in qt for any

given (st,dt,S) and t ∈ T (proof is straightforward based on Lemma 1). Because both the minimum

expected cost-to-go Wt(qt, st,dt) and production cost c(qt) are convex in qt, the master problem

(10) is a one-dimensional convex optimization problem that is straightforward to solve. Therefore,

the rest of this section focuses on describing the structure of s∗t+1(qt, st,dt;S), the solution to the

subproblem (11).

We intuitively describe the structures of the optimal solutions as follows. Demand in any given

period can be met by two sources of energy: stored energy (generated in the previous periods) and

16

current energy (generated in the current period). Typically, storing energy incurs more energy loss

than sending energy from the central node to the demand nodes (i.e., α ≤ β). Thus, it is more cost

efficient to use the current generation to meet as much demand as possible. If the current generation

is insufficient to cover the entire demand, stored energy is released to satisfy the remaining demand;

if the current generation exceeds the demand, the excess energy is stored. The optimal way to store

or release energy is described below.

Let qot =∑i∈L

di,t/β denote the current generation level that exactly meets the demand in period t.

If the current generation q > qot , we use q to satisfy demand entirely and store the excess generation,

q − qot , in the following order. First, charge the central storage, shown as step 1© in Figure 5.

Second, if the central storage is full, charge the local storage to levels as “balanced” as possible,

illustrated by steps 2©- 4©. Intuitively, it is optimal to charge the central storage before charging

the local storage because centrally stored energy provides more flexibility in meeting future demand

than locally stored energy. When charging local storage, in the case of symmetric demand nodes

(i.e., same storage size and symmetric demand distributions), equalizing the storage levels across the

demand nodes minimizes the expected future cost. (In general, balanced local storage levels are not

necessarily equal.)

Figure 5: Optimal storage level s∗t+1(q, st,dt) for q ≥ qot and symmetric demand nodes

��,��,�

��,�

��,�

��

��

Storage

level

0

① ②

③

④

Central

storage 0

Local

storage 1

Local

storage 2

Local

storage 3

③

④ ④

If the current generation is insufficient to satisfy the demand, i.e., q < qot , we meet the demand

by using the current generation q and then discharging the storage located as close to the demand as

possible. That is, storage should be discharged in the sequence of local storage first, central storage

next, and remote storage last (remote storage for any given demand node i refers to the storage at

any other demand node j 6= i). Such sequence of using inventory is also found to be optimal for

17

physical goods supply chains; see Wee and Dada (2005).

Figure 6(a) shows that local storage is discharged to more balanced levels first (steps 1©- 3©),

and then we continue discharging the local storage at a node only if the demand at that node is not

satisfied (steps 3©- 5©). If the current generation q and the local storage are insufficient to meet all

demands, i.e., q <∑i∈L

(di,t − si,tα)+/β, then the central storage is discharged (step 6©) and finally, if

needed, remote storage (at nodes 1 and 3) is used (steps 7©- 8©) to meet the remaining demand (at

node 2). Using remote storage to meet demand involves extra line losses due to the distance, but

such a strategy can be optimal when the overall demand in a period is so high that the generation

cost reduction outweighs the extra line losses.

Figure 6: Optimal storage level s∗t+1(q, st,dt) for q ≤ qot and symmetric demand nodes

Note: In this example, demand at nodes 1 and 3 can be satisfied by their respective local storage, whiledemand at node 2 cannot be satisfied by local storage alone.

(a) Discharge local storage first (b) Then discharge central and remote storage

� �,� = ��,� − ��,�/�

� ��

��,�

� ,�

��,�

Central

storage 0

Local

storage 1

Local

storage 2

Local

storage 3

��,�

0

Storage

level

①

②

③

④

⑤

②

③

④

③ ��,�

⑦

⑧

⑥

⑧

��,� � �,� = ��,� − ��,�/�

Storage

level

��,�

� ��

Central

storage 0

Local

storage 1

Local

storage 2

Local

storage 3

0

��

In short, energy is stored centrally before stored locally, while locally stored energy is used before

centrally stored energy. Therefore, consistent with our prior expectation, for given storage capacities

at all nodes, it is optimal to pool inventory at the central location whenever possible.

The optimal structure also provides a huge computational benefit for evaluating the expected

production cost, noting that the dynamic program in (5) has a state space of 2n+1 dimensions and

an action space of n + 1 dimensions. Given any feasible generation level q, we can efficiently find

the optimal inventory decision s∗t+1(q, st,dt;S) following the steps in Figures 5 and 6, rather than

searching in a (n+1)-dimensional action space. This helps solve problem (11) efficiently. Moreover,

we do not need to solve (11) for every feasible q because the master problem in (10) involves convex

18

optimization and efficient algorithms can be readily applied for finding q∗t . We take advantage of

these structural results in the next section.

6. Numerical Analysis

Using the structural policy identified in the previous section, we numerically evaluate the relative

benefits of centralized and localized investment, and derive further insights on storage investment.

6.1 Numerical Settings

In the base case, there are n = 2 demand nodes. At each node i = 1, 2, during the even-numbered

periods demand is zero, di,2t = 0, while during the odd-numbered periods demand di,2t−1 takes three

possible values: 0, 30 MWh, and 60 MWh, with probability 0.4, 0.4, and 0.2, respectively. The

random demands {di,2t−1 : i = 1, 2, t = 1, 2, . . . } are independent across nodes and time. Thus, de-

mands exhibit both predictable (cyclic) and unpredictable variabilities, reflecting the characteristics

of energy demand. The odd and even periods represent peak time and off-peak time, respectively.

Off-peak demand is low, while the peak demand has considerable variability.

Although we are able to solve settings with much more complicated demand models, we decide to

keep the base case relatively simple, because we will evaluate tens of thousands of high-dimensional

dynamic programs (for both the base case and extended cases) to provide a comprehensive picture

of how storage investment decisions affect the total cost.

The minimum demand is zero in the base case, so that we can isolate the minimum-demand effect

examined in Section 4.2. To examine how the optimal investment changes with the demand levels,

we consider a small (resp. large) shift that raises the total demand by 6 (resp. 15) MWh; the shift is

the same for all demand nodes. We also increase the number of demand nodes in Section 6.3.

Other parameters of the model are set as follows: storage efficiency α = 0.9 (typical battery),

line efficiency β = 0.95 (5% line loss), storage cost is p = $400 per kWh (Bloomberg New Energy

Finance 2018b), discount factor γ = 0.99, and production cost c(q) = q2. Note that adding a linear

production cost term would shift total cost by a constant without affecting the optimal storage

operations and investment.

6.2 Optimal Storage Investment

Because V (S) is convex in S and demands are symmetric across locations, we can focus on symmetric

investment S1 = S2 without losing optimality. We evaluate the total cost V (S) + p |S| for thousands

of storage investment choices by varying S0 between 0 and 56 MWh and varying S1 = S2 between 0

and 28 MWh, i.e., the total local storage SL = 2S1 varies between 0 and 56 MWh.

19

For any given S, we compute the cost increase compared to the minimum cost, expressed as a

fraction of the minimum cost:

V (S) + p |S|

V (S∗) + p |S∗|− 1. (13)

The contours of the this quantity is shown in Figure 7 for the base case as well as cases with higher

demand levels. The optimal investment for the base case is S∗0 = 10.2 MWh and S∗

L = 45.2 MWh

(i.e., S∗1 = S∗

2 = 22.6 MWh). The total storage size is S∗0 + S∗

L = 55.4 MWh. It is important to

observe that the optimal investment leans toward localized investment even without the minimum

demand effect or high demand correlation (demands are independent across nodes during the periods

when stored energy is used).

Figure 7: Effect of demand shift on the total cost deviations: V (S)+p|S|V (S∗)+p|S∗| − 1

(a) Base case (b) demand shifted by 6 MWh (c) demand shifted by 15 MWh��

��

��

��

��

��

Figure 7 also shows that as demand rises, the optimal investment shifts toward localized storage

investment. Specifically, the optimal storage investment (S∗0 , S

∗L) is (10.2, 45.2) for the base case,

(8.1, 46.6) in Figure 7(b), and becomes localized investment (0, 52.8) in Figure 7(c). Since the

minimum demand is only 6 and 15 MWh, respectively, the shift toward localized storage investment

is not due to the minimum demand effect, but bears a similar logic: For a given amount of locally

stored energy, a higher (random) demand implies a higher probability that the stored energy can be

used locally and a lower probability that energy transfer across demand nodes is needed. Therefore,

localized storage investment becomes more favorable.

We make three additional observations. First, as demand increases, the total storage size de-

creases only slightly from 55.4 MWh in the base case to 54.7 and 52.8 MWh under higher demands.

This is because the benefit of storage is about the same for all three cases, since the magnitude of

demand fluctuations remains the same. Second, as demand increases, the contours shown in Figure 7

20

expand, meaning that the total cost function becomes flatter. This is because satisfying a higher

demand requires a higher production cost V (S∗), which reduces the cost ratio in (13). Third, the

total cost is more sensitive to the total storage size than storage locations, seen from the orientation

of the oval-shaped contours. This is not to say that storage placement is unimportant, because a

small percentage cost difference can be a substantial difference in practice given the magnitude of

the distribution systems. Furthermore, we use a relatively high distribution efficiency β = 0.95. If

β is lower (more line loss), storage locations will be a more important cost driver. Importantly, our

choice of a relatively higher β is to demonstrate that even with a small line loss, the optimal storage

investment is still leaning toward a localized investment.

6.3 Impact of the Number of Demand Nodes

In this section, we examine how the number of demand nodes affects the optimal storage investment.

As the number of demand nodes increases, to facilitate comparison, we shall either scale up the

production cost function (while keeping the same demand) or scale down the nodal demand (while

keeping the same aggregate demand level and production cost function). These two scaling methods

are equivalent in the sense that they lead to the same results as we examine the percentage cost

changes.

We keep the same production cost function and set the demand levels as follows: For a system

with n demand nodes, di,2t = 0 for all i ∈ L, and in the odd-numbered periods, demands {di,2t−1}

are independent across nodes and take values 0, 60/n MWh, and 120/n MWh with probability 0.4,

0.4, and 0.2, respectively. Note that, as n increases, the total expected demand remains the same,

and the coefficient of variability (standard deviation divided by mean) of the demand at each node

remains the same, whereas the aggregate demand variability decreases. As with the previous analysis,

we also consider different minimum demand levels by shifting the total demand by 6 and 15 MWh.

Figures 8 and 9 illustrate the results for three and four demand nodes, in parallel to the case of n = 2

in Figure 7.

Let us first compare the cases with zero minimum demand, i.e., compare Figures 7(a), 8(a), and

9(a). The pooling of n demand nodes affects the storage investment in two distinct ways: First, as

n increases, the contours (especially the innermost contour that forms the red region) shift towards

more centralized storage investment. This implies that, consistent with our intuition, the benefit

of pooling that we examined in Section 4.4 is more prominent under more demand nodes, and thus

centralized storage becomes more advantageous over localized storage when n increases. Second, as

n increases, the optimal total storage investment decreases (note that the axes of the contour plots

21

Figure 8: Total cost deviations and optimal storage investment: Case of n = 3 demand nodes

(a) zero minimum demand (b) demand shifted by 6 MWh (c) demand shifted by 15 MWh��

��

��

��

��

��

Figure 9: Total cost deviations and optimal storage investment: Case of n = 4 demand nodes

(a) zero minimum demand (b) demand shifted by 6 MWh (c) demand shifted by 15 MWh��

��

��

��

��

��

for different n have different scales). The optimal total storage investment is also shown in Table 1.

Interestingly, as the number of demand nodes n increases, the average value of storage capacity

actually increases, as revealed in Table 1 (see line 8). This is because the total amount of storage

investment (line 1) decreases in n faster than the net value of storage (line 6). In Table 1, the average

value of storage capacity increases by about 10% when n increases from 2 to 3, and further increases

by about 5% when n increases from 3 to 4. Table 1 also shows that the percentage cost change

(line 7) decreases slowly as n increases, which confirms the result that storage capacity is actually

more valuable on average when there are more demand nodes.

Comparing the cases with positive minimum demand in Figures 7, 8, and 9 and Table 1 reassures

the effects of the number of demand nodes. In addition, we observe that the average value of storage

space (line 8) barely decreases as demand increases, and the average value consistently exceeds $1,000

per kWh, while the cost of storage space is $400 per kWh.

22

Table 1: Optimal storage size and the value of storage

Number of demand nodes: 2 3 4

(a) Zero minimum demand

1. Optimal total storage size (MWh) 55.4 43.5 38

2. Optimal production cost (million $) 99.447 94.225 91.032

3. Investment cost (million $): line 1 × 0.4 M$/MWh 22.16 17.4 15.2

4. Optimal total cost (million $): line 2 + line 3 121.607 111.625 106.232

5. Total cost without storage (million $) 184.412 165.703 156.349

6. Net value of storage (million $): line 5 – line 4 62.805 54.078 50.117

7. Cost increase if without storage: line 6/line 4 51.6% 48.4% 47.2%

8. Average value of storage space ($ per kWh): line 6/line 1 1,133.7 1,243.2 1,318.9

(b) Demand shifted by 6 MWh



3. Investment cost (million $) 21.88 17.04 15.2

4. Optimal total cost (million $) 160.142 150.098 144.660

5. Total cost without storage million ($) 220.473 201.764 192.410

6. Net value of storage (million $) 60.330 51.666 47.750

7. Cost increase if without storage 37.7% 34.4% 33.0%

8. Average value of storage space ($ per kWh of storage) 1,102.9 1,212.8 1,256.6

(c) Demand shifted by 15 MWh



3. Investment cost (million $) 21.12 16.44 14.4

4. Optimal total cost (million $) 232.742 222.613 217.120

5. Total cost without storage (million $) 289.522 270.813 261.460

6. Net value of storage (million $) 56.780 48.201 44.339

7. Cost increase if without storage 24.4% 21.7% 20.4%

8. Average value of storage space ($ per kWh of storage) 1,075.4 1,172.8 1,231.7

7. Conclusions

The goal of this paper is to improve the theoretical understanding of centralized versus decentralized

investment of energy storage. Our prior expectation is that the pooling investment centrally would

be a preferred choice in many circumstances. However, we find a strong effect due to the minimum

demand: If the minimum demand is positive at any demand node, storage capacity should first

be placed at that demand node. Only after storage capacity reaches the minimum demand at all

demand nodes, should the trade-off between centralized versus localized investment become active.

23

Due to the minimum-demand effect, the location choice of the storage can be counter-intuitive:

central location and locations with a high demand variability are not necessarily the best places for

storage investment while locations with a low demand variability can well be the optimal choice for

storage investment.

The centrally located storage benefits the system by increasing storage usage and reducing the

need for energy transfer across demand nodes. On the other hand, when storage is placed at the

demand nodes, the energy released from the storage can serve the demand without incurring line

losses. In general, localized storage investment tends to be preferred over centralized storage when

(a) the minimum demand is high, or (b) demands are positively correlated across demand nodes, or

(c) the storage cost becomes very low and demand can be negative.

Under a given capacity investment decision, we also consider the corresponding operational de-

cisions. We have confirmed that inventory pooling strategy is indeed optimal at the operations

level and provided structures of the optimal storage operating policy, which facilitates numerical

evaluation of storage investment.

Through numerical analysis, we find that even under a small line loss (5%), the optimal storage

investment distributes storage capacity across all nodes, with majority of capacity invested at the

demand nodes, rather than pooled at the central node. Localized investment tends to be preferred

when there is a high chance that the stored energy is needed in the peak periods and can be used to

meet local demands. As the number of demand nodes increases, the investment shifts toward more

centralized storage capacity. We also find that, although more demand nodes dampens the overall

demand variability, requiring less storage capacity, the average value of storage capacity actually

increases.

Finally, we note that if the demand grows over time and may exceed the line capacity, then

storage can help defer or eliminate the need for line capacity expansion. To achieve this, the storage

must be localized at the demand nodes to dampen the peak demand that exceeds the line capacity.

References

Akhil, A. A., G. Huff, A. B. Currier, B. C. Kaun, D. M. Rastler, S. B. Chen, A. L. Cotter, D. T.

Bradshaw, W. D. Gauntlett 2015. DOE/EPRI Electricity Storage Handbook in Collaboration with

NRECA. Sandia National Laboratories. SAND2015-1002.

Axsater, S. 1990. Modelling emergency lateral transshipments in inventory systems. Management

Science 36(11) 1329–1338.

Benjaafar, S., W. L. Cooper, J.-S. Kim 2005. On the benefits of pooling in production-inventory

systems. Management Science 51(4) 548–565.

24

Berman, O., D. Krass, M. Mahdi Tajbakhsh 2011. On the benefits of risk pooling in inventory

management. Production and Operations Management 20(1) 57–71.

Bloomberg New Energy Finance 2018a. Energy storage is a $620 billion investment opportunity to

2040. https://about.bnef.com/blog/energy-storage-620-billion-investment-opportunity-2040.

Bloomberg New Energy Finance 2018b. New Energy Outlook 2018. Available at

https://about.bnef.com/new-energy-outlook.

Cahn, A. S. 1948. The warehouse problem. Bulletin of the American Mathematical Society 54 1073.

Carpinelli, G., G. Celli, S. Mocci, F. Mottola, F. Pilo, D. Proto 2013. Optimal integration of dis-

tributed energy storage devices in smart grids. IEEE Transactions on Smart Grid 4(2) 985–995.

Chamorro, J. M., L. M. Abadie, R. de Neufville, M. Ilic 2012. Market-based valuation of transmission

network expansion: A heuristic application in GB. Energy 44(1) 302–320.

Charnes, A., J. Drze, M. Miller 1966. Decision and horizon rules for stochastic planning problems:

A linear example. Econometrica 34(2) 307–330.

Chen, C., S. Duan, T. Cai, B. Liu, G. Hu 2011. Optimal allocation and economic analysis of energy

storage system in microgrids. IEEE Transactions on Power Electronics 26(10) 2762–2773.

Chen, L. G., S. Gavirneni 2010. Using scheduled ordering to improve the performance of distribution

supply chains. Management Science 56(9) 1615–1632.

Corbett, C. J., K. Rajaram 2006. A generalization of the inventory pooling effect to nonnormal

dependent demand. Manufacturing & Service Operations Management 8(4) 351–358.

Dada, M. 1992. A two-echelon inventory system with priority shipments. Management Science

38(8) 1140–1153.

Das, C. K., O. Bass, G. Kothapalli, T. S. Mahmoud, D. Habibi 2018. Overview of energy storage

systems in distribution networks: Placement, sizing, operation, and power quality. Renewable

and Sustainable Energy Reviews 91 1205–1230.

Denholm, P., R. Sioshansi 2009. The value of compressed air energy storage with wind in

transmission-constrained electric power systems. Energy Policy 37(8) 3149–3158.

Energy Information Administration 2020. Annual Energy Outlook 2020. www.eia.gov/outlooks/aeo.

Eppen, G. D. 1979. Effects of centralization on expected costs in a multi-location newsboy problem.

Management Science 25(5) 498–501.

Federgruen, A., P. Zipkin 1984. Approximations of dynamic, multilocation production and inventory

problems. Management Science 30(1) 69–84.

Fortenbacher, P., A. Ulbig, G. Andersson 2018. Optimal placement and sizing of distributed battery

storage in low voltage grids using receding horizon control strategies. IEEE Transactions on

Power Systems 33(3) 2383–2394.

Gerchak, Y. 2016. Manufacturing newsvendors and inventory pooling with nonlinear production

costs. International Journal of Inventory Research 3(1) 70–80.

Gerchak, Y., D. Mossman 1992. On the effect of demand randomness on inventories and costs.

Operations Research 40(4) 804–807.

25

Ghofrani, M., A. Arabali, M. Etezadi-Amoli, M. S. Fadali 2013. A framework for optimal placement

of energy storage units within a power system with high wind penetration. IEEE Transactions

on Sustainable Energy 4(2) 434–442.

Henig, M., Y. Gerchak, R. Ernst, D. F. Pyke 1997. An inventory model embedded in designing a

supply contract. Management Science 43(2) 184–189.

Heyman, D. P., M. J. Sobel 1984. Stochastic Models in Operations Research. Vol. 2. McGraw-Hill.

Hoffman, M. G., A. Sadovsky, M. C. Kintner-Meyer, J. G. DeSteese 2010. Analysis tools for sizing

and placement of energy storage for grid applications: A literature review. Pacific Northwest

National Laboratory, PNNL-19703.

Hu, X., I. Duenyas, R. Kapuscinski 2008. Optimal joint inventory and transshipment control under

uncertain capacity. Operations Research 56(4) 881–897.

International Energy Agency 2018. Electric power transmission and distribution losses. IEA Statis-

tics, https://data.worldbank.org/indicator/EG.ELC.LOSS.ZS?view=map.

Karlin, S. 1960. Dynamic inventory policy with varying stochastic demands. Management Science

6(3) 231–258.

Karmarkar, U. S. 1987. The multilocation multiperiod inventory problem: Bounds and approxima-

tions. Management Science 33(1) 86–94.

Lu, Y., M. Song 2014. Inventory control with a fixed cost and a piecewise linear convex cost. Pro-

duction and Operations Management 23(11) 1966–1984.

Nick, M., R. Cherkaoui, M. Paolone 2014. Optimal allocation of dispersed energy storage systems in

active distribution networks for energy balance and grid support. IEEE Transactions on Power

System 29(5) 2300–2310.

Ozer, O. 2003. Replenishment strategies for distribution systems under advance demand information.

Management Science 49(3) 255–272.

Pereira, A. J., J. T. Saraiva 2011. Generation expansion planning (GEP)—A long-term approach

using system dynamics and genetic algorithms (GAs). Energy 36(8) 5180–5199.

Robinson, L. W. 1990. Optimal and approximate policies in multiperiod, multilocation inventory

models with transshipments. Operations Research 38(2) 278–295.

Saboori, H., R. Hemmati, S. M. S. Ghiasi, S. Dehghan 2017. Energy storage planning in electric

power distribution networks–a state-of-the-art review. Renewable and Sustainable Energy Reviews

79 1108–1121.

Sadegheih, A. 2009. Optimization of network planning by the novel hybrid algorithms of intelligent

optimization techniques. Energy 34(10) 1539–1551.

Sardi, J., N. Mithulananthan, M. Gallagher, D. Q. Hung 2017. Multiple community energy storage

planning in distribution networks using a cost-benefit analysis. Applied Energy 190 453–463.

Secomandi, N. 2010. Optimal commodity trading with a capacitated storage asset. Management

Science 56(3) 449–467.

Suresh, S. 2014. A Collection of Essays on Electric Grid Operations: Optimizing Energy Storage and

26

Enhancing the Effect of Social Comparisons. PhD thesis. University of Michigan.

Tang, Y., S. H. Low 2017. Optimal placement of energy storage in distribution networks. IEEE

Transactions on Smart Grid 8(6) 3094–3103.

Thrampoulidis, C., S. Bose, B. Hassibi 2016. Optimal placement of distributed energy storage in

power networks. IEEE Transactions on Automatic Control 61(2) 416–429.

Wee, K. E., M. Dada 2005. Optimal policies for transshipping inventory in a retail network. Man-

agement Science 51(10) 1519–1533.

Wirfs-Brock, J. 2015. Lost in transmission: How much electricity disappears between a power plant

and your plug? Inside Energy, http://insideenergy.org/2015/11/06/.

Yang, C., Z. Hu, S. X. Zhou 2020. Multilocation newsvendor problem: Centralization and inventory

pooling. Management Science. Forthcoming.

Zhao, H., J. K. Ryan, V. Deshpande 2008. Optimal dynamic production and inventory transshipment

policies for a two-location make-to-stock system. Operations Research 56(2) 400–410.

Zhou, Y., A. Scheller-Wolf, N. Secomandi, S. Smith 2015. Electricity trading and negative prices:

storage vs. disposal. Management Science 62(3) 880–898.

Zhou, Y., A. Scheller-Wolf, N. Secomandi, S. Smith 2019. Managing wind-based electricity generation

in the presence of storage and transmission capacity. Production and Operations Management

28(4) 970–989.

Zidar, M., P. S. Georgilakis, N. D. Hatziargyriou, T. Capuder, D. Skrlec 2016. Review of energy

storage allocation in power distribution networks: applications, methods and future research.

IET Generation, Transmission & Distribution 10(3) 645–652.

27

Online Appendix: Proofs

Proof of Lemma 1: (i) The statement holds in the last period because VT+1(·, ·) = 0. For a given

t ∈ T , suppose Vt+1(st+1,dt+1;S) is decreasing and convex in (st+1,S) for any dt+1.

The constraint q(st+1 − st,dt) ≥ 0 in (6) defines a non-convex feasible set, which is difficult for

analysis. Thus, we introduce an auxiliary objective function defined on a convex set and show that

Vt(st,dt;S) = minst+1∈A

ft(st+1, st,dt;S), (A.1)

where ft(st+1, st,dt;S)def= c

([q(st+1 − st,dt)]

+)+ γEt

[Vt+1(st+1,dt+1;S)

], for (st+1, st) ∈ A×A.

To prove (A.1), consider a state st+1 such that q(st+1− st,dt) < 0. Because q(S− st,dt) ≥ 0 and

q(·, ·) is a continuous function, we can apply the intermediate value theorem and find st+1 such that

st+1 ≤ st+1 ≤ S and q(st+1 − st,dt) = 0. The objective value at st+1 is lower than at st+1 because

ft(st+1, st,dt;S) = c(0) + γEt

[Vt+1(st+1,dt+1;S)

]

≥ c(0) + γEt

[Vt+1(st+1,dt+1;S)

]= ft(st+1, st,dt;S),

where the inequality follows from the induction hypothesis that Vt+1(st+1,dt+1;S) decreases in st+1.

Therefore, in (A.1), when minimizing ft(st+1, st,dt;S) over st+1 ∈ A, we can restrict our attention

to the states satisfying q(st+1− st,dt) ≥ 0, which is equivalent to the original problem (5)-(6).

For any given (st,dt,S), let s∗t+1 be an optimal decision found by (A.1). For any (st, S) ≥ (st,S),

Vt(st,dt;S) = ft(s∗t+1, st,dt;S) ≥ ft(s

∗t+1, st,dt; S) ≥ Vt(st,dt; S),

where the first inequality follows from c([q(∆s,dt)]

+)increasing in ∆s and the induction hypothesis

that Vt+1(s∗t+1,dt+1;S) ≥ Vt+1(s

∗t+1,dt+1; S). Thus, Vt(st,dt;S) decreases in (st,S).

To prove the convexity of Vt(st,dt;S) in (st,S), note that c([q(∆s,dt)]

+)is convex in ∆s due

to the composition of convex increasing functions, and Et

[Vt+1(st+1,dt+1;S)

]is convex in (st+1,S)

by the induction hypothesis. Therefore, ft(st+1, st,dt;S) is jointly convex in (st+1, st,S) on a closed

convex set: {(st+1, st,S) : 0 ≤ st+1≤ S, 0 ≤ st≤ S, 0 ≤ S≤ S}, where S is some large vector. Using

the theorem on convexity preservation under minimization from Heyman and Sobel (1984, p. 525),

we conclude that Vt(st,dt;S) as minimized in (A.1) is convex in (st,S).

(ii) Part (i) implies that V1(S,d1;S) is decreasing and convex in S. Thus, V (S) = EV1(S,d1;S) is

decreasing and convex in S.

Proof of Lemma 2: (i) The statement in part (i) holds in period T + 1 as VT+1(·, ·) = 0. Suppose

the statement holds in t + 1. For period t, we consider states (st,dt) and (st,dt) that satisfy the

conditions in part (i). Let s∗t+1 be the optimal decision for state (st,dt). Denote ∆s∗t = s∗t+1− st and

q∗t = q(∆s∗t ,dt). We now construct a feasible decision for state (st,dt). Consider two cases:

1

Case 1: st+∆s∗t ∈ A. In this case, a feasible decision for state (st,dt) is to produce q∗t and change

inventory to st+1 = st +∆s∗t . Then, s∗j,t+1 = sj,t+1 − δ, s∗k,t+1 = sk,t+1 + β2δ, and s∗i,t+1 = si,t+1 for

all i 6= j, k. The induction hypothesis implies that Vt+1(st+1,dt+1) ≤ Vt+1(s∗t+1,dt+1) for any dt+1,

leading to Vt(st,dt) ≤ c(q∗t ) + γEt

[Vt+1(st+1,dt+1)

]≤ c(q∗t ) + γEt

[Vt+1(s

∗t+1,dt+1)

]= Vt(st,dt).

Case 2: st + ∆s∗t 6∈ A, i.e., s∗j,t+1 + δ > Sj or s∗k,t+1 − β2δ < 0 or both inequalities hold. Let

δ ≡ min{Sj − s∗j,t+1, s

∗k,t+1/β

2}. By definition, δ ∈ [0, δ). For state (st,dt), consider a candidate

inventory decision st+1 ∈ A satisfying sj,t+1 = s∗j,t+1+ δ, sk,t+1 = s∗k,t+1−β2δ, and si,t+1 = s∗i,t+1 for

all i 6= j, k. Then, the induction hypothesis implies Vt+1(st+1,dt+1) ≤ Vt+1(s∗t+1,dt+1) for any dt+1.

Define ∆st = st+1− st and qt = q(∆st,dt). If we can show qt ≤ q∗t , then we have the intended result:

Vt(st,dt) ≤ c([qt]+) + γEt

[Vt+1(st+1,dt+1)

]≤ c(q∗t ) + γEt

[Vt+1(s

∗t+1,dt+1)

]= Vt(st,dt), (A.2)

where we used the relation Vt(st,dt) = minst+1∈A

ft(st+1, st,dt) given in (A.1).

The rest of the proof shows qt ≤ q∗t . The choice of δ gives sj,t+1 = Sj or sk,t+1 = 0, which implies

∆sj,t = sj,t+1 − sj,t ≥ 0 or ∆sk,t = sk,t+1 − sk,t ≤ 0. (A.3)

Let ε = δ − δ > 0. Then, by definitions, we have ∆s∗j,t = ∆sj,t + ε, ∆s∗k,t = ∆sk,t − β2ε, and

∆s∗0,t = ∆s0,t. Using the definition in (4), we have

q∗t − qt = ψβ

(dj,t+ ψα(∆s

∗j,t)

)− ψβ

(dj,t+ ψα(∆sj,t)

)−

[ψβ

(dk,t+ ψα(∆sk,t)

)− ψβ

(dk,t+ ψα(∆s

∗k,t)

)]

≥ β[ψα(∆sj,t + ε)− ψα(∆sj,t)

]− β−1

[ψα(∆sk,t)− ψα(∆sk,t − β2ε)

]≡ Γ, (A.4)

where the inequality is because ψβ(u) increases in u with a slope of either β or β−1. Now consider

the cases under the two conditions derived in (A.3):

• If ∆sj,t ≥ 0, then Γ = βα−1ε− β−1[ψα(∆sk,t)−ψα(∆sk,t − β2ε)

]≥ βα−1ε− β−1α−1β2ε = 0.

• If ∆sk,t ≤ 0, then Γ = β[ψα(∆sj,t + ε)− ψα(∆sj,t)

]− β−1αβ2ε ≥ βαε− βαε = 0.

Hence, qt ≤ q∗t and the result in (A.2) holds.

(ii) For the case of k = 0, the proof follows the same lines as in part (i), except that s0,t exceeds s0,t

by βδ instead of β2δ. The case of j = 0 can be proved similarly.

Proof of Lemma 3: In (5)-(6), for a given S, we express an optimal decision rule in period t

as s∗t+1(st,dt;S) and abbreviate it as s∗t+1. Let {s∗t : t ∈ T } denote an optimal policy, and let

u∗i,t = di,t + ψα(∆s∗i,t), i ∈ L, be the corresponding energy flows according to (3).

2

(i) Under investment Sc =(S0 + β−1

∑i∈L Si, 0, . . . , 0

), we construct a policy {st : t ∈ T }:

s0,t = s∗0,t + β−1gt, si,t = 0, i ∈ L, ∀ t ∈ T . (A.5)

g1 =∑i∈L

Si, (A.6)

gt+1 = min{ ∑

i∈LSi, gt +

( ∑i∈L

∆s∗i,t)− 1−β2

α

∑i∈L

min{u∗i,t, 0}}. (A.7)

At t = 1, (A.5) and (A.6) imply that s0,1 = s∗0,1 + β−1∑

i∈L Si. Since s∗1 = S, we have s1 = Sc.

The definition in (A.7) implies gt ∈[∑

i∈L s∗i,t,

∑i∈L Si

]for all t ∈ T .3 Hence, 0 ≤ st ≤ Sc, thus

the constructed policy {st : t ∈ T } is feasible under Sc.4 Under Sc and the policy {st : t ∈ T }, the

production is qt = ψα(∆s0,t) + β−1∑

i∈L di,t. We now prove that

qt ≤ q∗t = ψα(∆s∗0,t) +

∑i∈L

ψβ(u∗i,t). (A.8)

1) Case of u∗i,t ≥ 0 for all i ∈ L. In this case, ψβ(u∗i,t) = β−1u∗i,t. Then,

q∗t = ψα(∆s∗0,t) +

∑i∈L

β−1(di,t + ψα(∆s∗i,t)) = ψα(∆s

∗0,t) +

∑i∈L

[β−1di,t + ψα(β

−1∆s∗i,t)]

≥ ψα

(∆s∗0,t +

∑i∈L

β−1∆s∗i,t)+ β−1

∑i∈L

di,t

≥ ψα(∆s∗0,t + β−1∆gt) + β−1 ∑

i∈L

di,t = qt,

where the first inequality utilizes the subadditivity of ψα(·), i.e., ψα(x)+ψα(y) ≥ ψα(x+y), and

the second inequality is because u∗i,t ≥ 0 and (A.7) imply that ∆gt ≡ gt+1 − gt ≤∑

i∈L∆s∗i,t.

2) Case of u∗j,t < 0 for j ∈ L− ⊂ L, i.e., some energy is transmitted from the nodes in L− to other

nodes. This immediately implies that ∆s∗t ≤ 0 because Lemma 2 states that energy should

not be released from one node only to store it in another node. These conditions imply that

∆gt =∑

k∈L\L−

∆s∗k,t + β2∑

j∈L−

∆s∗j,t +β2−1α

∑j∈L−

dj,t < 0.5 Then,

qt = α(∆s∗0,t + β−1∆gt) + β−1 ∑i∈L

di,t

= α∆s∗0,t + β∑

j∈L−

(α∆s∗j,t + dj,t) + β−1∑

k∈L\L−

(α∆s∗k,t + dk,t) = q∗t .

Note that u∗j,t < 0 for all j ∈ L is not possible because reverse flows on all lines are suboptimal by

Lemma 2. Therefore, in all cases, we have qt ≤ q∗t , implying that the policy {st : t ∈ T } achieves an

operating cost no higher than V (S). Therefore, V (Sc) ≤ V (S).

3We inductively show gt ≥ ∑i∈L s∗i,t. This is true for t = 1. Suppose gt ≥ ∑

i∈L s∗i,t for some t < T . Then,

gt +∑

i∈L ∆s∗i,t ≥ ∑i∈L(s

∗i,t + ∆s∗i,t) =

∑i∈L s∗i,t+1. This, together with β2−1

α

∑i∈L min{u∗

i,t, 0} ≥ 0, implies thatgt+1 ≥ ∑

i∈L s∗i,t+1.4We do not require st to satisfy the non-negative production constraint in (6), because if qt < 0, there exists another

inventory decision that results in qt ≥ 0 and the same objective value, which is shown in the proof of Lemma 1.5To see this, note that the last two terms in (A.7) are

(∑i∈L ∆s∗i,t

)+ β2−1

α

∑j∈L−

(dj,t + α∆s∗j,t) =(∑

k∈L\L−∆s∗k,t

)+ β2

(∑j∈L−

∆s∗j,t)+ β2−1

α

∑j∈L−

dj,t < 0.

3

(ii) Under Sl = (0, S1 + βS0, . . . , Sn + βS0), we construct a policy {st : t ∈ T }:

s0,t = 0, sj,t = s∗j,t + βgj,t, j ∈ L, ∀ t ∈ T , (A.9)

gj,1 = S0, j ∈ L, (A.10)

∆gj,t = gj,t+1 − gj,t =

max{∆s∗0,t −

∑i∈L, i<j

∆gi,t, −u∗+j,t /(αβ)

}, if ∆s∗0,t < 0,

min{∆s∗0,t −

∑i∈L, i<j

∆gi,t, S0 − gj,t

}, if ∆s∗0,t ≥ 0.

(A.11)

Using techniques similar to part (i), we can prove V (Sl) ≤ V (S).

The proof of Lemma 4 requires some properties of the optimal operating policy and the value

function when dminj > 0, as stated in the following lemma.

Lemma A.1 Suppose dminj > 0 for given j ∈ L. For given storage investment S with αSj < dmin

j ,

(i) There exists an optimal policy satisfying ∆s∗0,t ·∆s∗j,t ≥ 0 for all t ∈ T ;

(ii) In period t, suppose δ > 0 and st, st ∈ A satisfy s0,t = s0,t− δ, sj,t = sj,t+ βδ, and si,t = si,t for

all i ∈ L, i 6= j, then Vt(st,dt) = Vt(st,dt) for any dt.

Proof of Lemma A.1: The condition αSj < dminj means that the demand at node j cannot be met

solely by storage j in a period. Thus, energy is transmitted from node 0 to j in every period.

Suppose part (ii) holds for period t+1 (it clearly holds for period T +1). In period t, we consider

any given state (s,d) and any decision st+1 with inventory change ∆s ≡ st+1 − s satisfying ∆s0 > 0

and ∆sj < 0. Set δ = min{∆s0, −β−1∆sj} > 0. We now show that a strictly better decision is st+1

with s0,t+1 = s0,t+1 − δ, sj,t+1 = sj,t+1 + βδ, and si,t+1 = si,t+1 for i 6= j. This new decision satisfies

∆s0 = ∆s0 − δ ≥ 0, ∆sj = ∆sj + βδ ≤ 0, and ∆s0 ·∆sj = 0. To show the superiority of st+1, note

that Vt+1(st+1,dt+1) = Vt+1(st+1,dt+1) by the induction hypothesis and

q(∆s,d)− q(∆s,d) = β−1(dj+ α∆sj) + α−1∆s0 − β−1(dj+ α∆sj)− α−1∆s0 = αδ − α−1δ < 0.

Similarly, any decision st+1 with ∆s0 < 0 and ∆sj > 0 can also be improved. Thus, part (i) holds

for period t. We next prove part (ii) for period t.

Consider states (s,d) and (s,d) in period t, with s0,t = s0,t − δ, sj,t = sj,t + βδ for some δ > 0,

and si,t = si,t for all i ∈ L, i 6= j. Lemma 2 implies that Vt(s,d) ≤ Vt(s,d). Thus, we only need

to show Vt(s,d) ≤ Vt(s,d). Let s∗t+1 be the optimal decision for (s,d) and denote ∆s∗ = s∗t+1 − s.

For state (s,d), we construct a decision st+1 satisfying s0,t+1 = s∗0,t+1 − δ, sj,t+1 = s∗j,t+1 + βδ, with

δ = min{δ, s∗0,t+1, β

−1(Sj − s∗j,t+1)}, and si,t+1 = s∗i,t+1 for all i ∈ L, i 6= j. We next show that

st+1 for (s,d) gives the same production cost as s∗t+1 for (s,d). First, by the induction hypothesis,

Vt+1(s∗t+1,dt+1) = Vt+1(st+1,dt+1). Second, we show the production quantities are the same. Let

∆s = st+1 − s = ∆s∗ − (−ε, 0, . . . , 0, βε, 0, . . . , 0), where ε = δ − δ. Consider two cases:

4

• Case 1: ∆s∗0 ≥ 0 and ∆s∗j ≥ 0. We have s∗0,t+1 ≥ s0 = s0 + δ ≥ δ. Thus, either δ = δ or

δ = β−1(Sj − s∗j,t+1). In either case, we can verify that ∆sj ≥ 0. Also, ∆s0 ≥ 0. Hence,

q(∆s,d)− q(∆s∗,d) = β−1(dj + α−1∆sj) + α−1∆s0 − β−1(dj + α−1∆s∗j)− α−1∆s∗0 (A.12)

= −β−1α−1βε+ α−1ε = 0.

• Case 2: ∆s∗0 ≤ 0 and ∆s∗j ≤ 0. Using similar logic, we can show ∆s0 ≤ 0, ∆sj ≤ 0, and

q(∆s,d) = q(∆s∗,d).

These are the only cases we need to consider, as indicated by part (i). Equal production and equal

future expected cost together imply that Vt(s,d) ≤ Vt(s,d), completing the proof.

Proof of Lemma 4: Under investment S, let {s∗t : t ∈ T } be an optimal policy satisfying ∆s∗0,t ·

∆s∗j,t ≥ 0, which follows from Lemma A.1(i). Under investment S, we define δt = min{δ, s∗0,t} and

construct a policy st such that s0,t = s∗0,t − δt, sj,t = s∗j,t + βδt, and si,t = s∗i,t for i ∈ L and i 6= j, for

all t ∈ T . The policy {st : t ∈ T } is feasible under S because sj,t ≥ 0, sj,t ≤ s∗j,t+βδ ≤ Sj +βδ = Sj ,

and s0,t = max{s∗0,t − δ, 0} ∈ [0, S0].

We next show that the two policies yields the same production levels. If ∆s∗j,t ≥ 0 and ∆s∗0,t ≥ 0,

we have δt+1−δt ∈ [0,∆s∗0,t], which implies ∆sj,t = ∆s∗j,t+β(δt+1−δt) ≥ 0 and ∆s0,t = ∆s∗0,t−(δt+1−

δt) ≥ 0. Then, following exactly the same logic in (A.12), q(∆st,dt) = q(∆s∗t ,dt). If ∆s∗j,t ≤ 0 and

∆s∗0,t ≤ 0, similar logic applies. Therefore, q(∆st,dt) = q(∆s∗t ,dt) for all t ∈ T , and consequently

the total production costs are the same for both policies, which implies V (S) ≤ V (S). The opposite

inequality V (S) ≥ V (S) can be proved similarly.

The proof of part (ii) is parallel, but note that V (S) ≥ V (S) may not hold because we are not

given the relationship between Sk and dmink .

Proof of Proposition 1: (i) Because C(S) ≡ p |S| + V (S) is convex in S (Lemma 1), it suffices

to show that Sl∗ achieves a local minimum. Let Sdef= Sl∗ + δ, where δ = (δ0, δ1, . . . , δn) satisfies

−Sl∗ ≤ δ < 12 (α

−1dminj − Sl∗

j )1. We aim to show C(Sl∗) ≤ C(S).

Note that δ0 ∈[0, 12(α

−1dminj − Sl∗

j )). Define another localized investment S such that S0 =

S0−δ0 = 0, Sj = Sj+βδ0, and Si = Si for i ∈ L, i 6= j. By definition, Sj = Sl∗j +δj+βδ0 < α−1dmin

j .

Then, we have

C(S)− C(Sl∗) = V (S)− V (Sl∗) + p(δ0 +

∑i∈L

δi)

≥ V (S)− V (Sl∗) + p(βδ0 +

∑i∈L

δi)

= C(S)− C(Sl∗) ≥ 0,

where the first inequality follows from Lemma 4(i) and δ0 ≥ βδ0, and the last inequality follows

from optimality of Sl∗ for the constrained investment problem (9). This proves the optimality of Sl∗.

5

Furthermore, if δ0 is set to be positive, then δ0 > βδ0 and the first inequality holds strictly, which

implies that investment with S0 > 0 is strictly dominated by Sl∗.

(ii) The statement in the proposition clearly holds when dmini = 0 for all i ∈ L. We only need to

prove the case when dminj > 0 for some j ∈ L. We prove by contradiction. Let the optimal investment

be S∗ with S∗0 > 0, and suppose S∗

j < α−1dminj . Define S such that S0 = S∗

0 − δ and Sj = S∗j + βδ,

where δ = min{S∗0 , (α

−1dminj − S∗

j )/2}. Note that Sj < α−1dmini . Then, by Lemma 4(i), we have

V (S∗) = V (S). Because |S∗| > |S|, we have C(S∗) > C(S), contradicting to the optimality of S∗.

Proof of Proposition 2: For any given S ≥ 0 and the associated optimal policy {s∗t : t ∈ T }, we

construct a new system with node 0 and a single demand node. The storage size and operations at

node 0 remain the same as in the original system. The single demand node combines the demand

and storage of all n nodes in the original system: demand is dLt =∑i∈L

di,t, storage size is SL =∑i∈L

Si,

and a feasible operating policy is s0,t = s∗0,t, sLt =∑i∈L

s∗i,t, t ∈ T . Let C(S) ≡ p |S|+V (S) denote the

total cost under investment S in the original system, and let C(S0, SL) denote the total cost under

(S0, SL) for the new system. The subadditivity of ψα and ψβ implies

ψβ

(dLt + ψα(∆sLt)

)≤ ψβ

(dLt +

∑i∈L

ψα(∆s∗i,t)

)≤

∑i∈L

ψβ

(di,t + ψα(∆s

∗i,t)

), t ∈ T ,

which in turn implies that the new system produces no more than the original system. Thus,

C(S0,

∑i∈L

Si)≤ C(S0, S1, . . . , Sn). (A.13)

Furthermore, (A.13) holds with equality if di,t = ki d1,t and Si = ki S1, for all i ∈ L. This can be

shown by using the optimal policy for the new system to construct a feasible policy for the original

system that yields the same production cost. The construction maintains the local storage levels at

the ratios ki. In other words, under di,t = ki d1,t, we have

C(S0, SL) = C(S0,

k1SL∑i∈L ki

, · · · ,knSL∑i∈L ki

). (A.14)

Following the reasoning after Proposition 2 in the paper, a localized investment is optimal for the

new system with only one demand node. Denote the optimal localized investment as S∗L. Then,

under di,t = ki d1,t, we have

C(0,

k1S∗L∑

i∈L ki, · · · ,

knS∗L∑

i∈L ki

)= C(0, S∗

L) ≤ C(S0,

∑i∈L

Si)≤ C(S), (A.15)

Because S is arbitrary, we conclude from (A.15) that the localized investment is optimal.

Proof of Proposition 3: The proof for part (i) is straightforward and omitted. To prove part (ii),

consider any p1 and p2 with p1 < p2. The optimality of S∗(p1) suggests p1 |S∗(p1)| + V (S∗(p1)) ≤

p1 |S∗(p2)| + V (S∗(p2)). Similarly, p2 |S

∗(p2)| + V (S∗(p2)) ≤ p2 |S∗(p1)| + V (S∗(p1)). Combining

6

these two inequalities, we have

p1(|S∗(p1)| − |S∗(p2)|) ≤ V (S∗(p2))− V (S∗(p1)) ≤ p2(|S

∗(p1)| − |S∗(p2)|),

which implies (p1 − p2)(|S∗(p1)| − |S∗(p2)|) ≤ 0. Because p1 < p2, we have |S∗(p1)| ≥ |S∗(p2)|.

Lemma A.2 For n = 1, 2, . . . , suppose an ≥ 0, bn > 0, bn ≥ bn+1, limn→∞

bn = 0, and∞∑n=1

anbn < ∞.

Then, limn→∞

(bn

n∑i=1

ai

)= 0.

Proof of Lemma A.2: First, anbn ≥ 0 and∞∑n=1

anbn <∞ imply∞∑n=1

anbn exists. Let∞∑n=1

anbn =M .

For any ε > 0, there exists N1 such that∞∑

n=N1

anbn <ε

2. Because bn > 0 decreases in n and converges

to zero, there exists N2 > N1 such thatbN2

bN1

<ε

2M. Then, for any N > N2, we have

bNN∑

n=1an = bN

[N1∑n=1

an +N∑

n=N1+1an

]<

bNbN1

N1∑n=1

anbn +N∑

n=N1+1anbn <

ε

2MM +

ε

2= ε. (A.16)

Hence the limiting result holds.

Proof of Proposition 4: To prove this proposition, we first show

limp→0+

p |S∗(p)| = 0. (A.17)

Let {pn} be a sequence of positive prices such that pn decreases in n and converges to zero. For

simplicity, let Sn ≡ S∗(pn). Proposition 3(ii) implies that |Sn| − |Sn−1| ≥ 0.

By optimality of Sn, we have pn|Sn|+ V (Sn) ≤ pn|Sn−1|+ V (Sn−1) or

pn(|Sn| − |Sn−1|) ≤ V (Sn−1)− V (Sn).

Summing over n, we have

∞∑n=1

pn(|Sn| − |Sn−1|) ≤ V (S0)− limn→∞

V (Sn) <∞.

Applying Lemma A.2, we have limn→∞

pn(|Sn| − |S0|) = 0. Since limn→∞

pn|S0| = 0, we have

limn→∞

pn|Sn| = 0. Because {pn} is chosen arbitrarily, we have limp→0+

p |S∗(p)| = 0.

(i) Let C(S) ≡ p |S| + V (S). Given an optimal investment S∗, consider a localized investment

S = (0, S∗1 + βS∗

0 , . . . , S∗n + βS∗

0). Lemma 3(ii) suggests that V (S) ≤ V (S∗). In addition, as the

optimal localized investment is Sl∗, we have C(Sl∗) ≤ C(S). Utilizing these inequalities, we have

0 ≤ C(Sl∗)−C(S∗) ≤ C(S)− C(S∗) = V (S) + p|S| − V (S∗)− p|S∗|

≤ p|S| − p|S∗| = (nβ − 1)p S∗0 .

7

Note that S∗0 is a function of p, and lim

p→0+p S∗

0(p) = 0 due to (A.17). Hence,

limp→0+

C(S∗(p))− C(Sl∗(p)) = 0.

(ii) Consider a centralized investment S =(S∗0 + β−1

∑i∈L S

∗i , 0, . . . , 0

). Using similar logic and the

result in Lemma 3(i) (which requires non-negative demand), we have

0 ≤ C(Sc∗)− C(S∗) ≤ C(S)− C(S∗) = V (S) + p|S| − V (S∗)− p|S∗|

≤ p|S| − p|S∗| = (β−1 − 1)p∑i∈L

S∗i .

Because limp→0+

p∑i∈L

S∗i = 0 due to (A.17), we have

limp→0+

C(S∗(p))− C(Sc∗(p)) = 0.

8

storage investment dec2020 - iu

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