Operations and Investment of Energy Storage in a Tree Network

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

*Ross School of Business, University of Michigan, Ann Arbor, MI, kapuscin@umich.edu

**Kelley School of Business, Indiana University, Bloomington, IN, owenwu@indiana.edu

***McKinsey & Company, New York, NY, santhosh.suresh1@gmail.com

June 30, 2020

Energy storage has become an indispensable part of electricity networks, and the investment in energy

storage is expected to grow rapidly. We consider the problem of energy storage siting (where to invest

storage facilities) and sizing (how much capacity to invest), with the objective of minimizing the total

investment and generation costs, in a tree network model that captures the critical features of electricity

systems, including convex costs, stochastic demand, storage efficiency, and line losses. We derive the

structures of the optimal storage operating policy, which provide insights on how storage efficiency and

line losses affect storage operations. On storage siting and sizing, our theory recommends that storage

should be first placed at the leaf nodes with positive minimum demand, and further storage investment

should consider relative benefits of distributed versus central storage. Through numerical analysis, we

find that, while both aspects are important, determining the total amount of storage investment (sizing)

is typically more critical than allocating storage capacity across the central and leaf nodes (siting).

Key words: Energy storage, inventory management, tree network, distributed storage investment

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.

Storage investment problem in a power network is very complicated due to the dependence of

storage operations on (a) multiple ways that storage can serve the network: e.g., leveling load,

buffering uncertainties (from load and renewable power generation), deferring network upgrades due

to demand growth, and (b) network topology: e.g., radial, ring, or general networks. Tree or radial

networks are almost universally used for low-voltage distribution systems (National Academies 2009,

p. 564, Csanyi 2016) and are commonly seen in North America, Europe, and many other regions

around the world (Eller and Gauntlett 2017). In this paper, we focus on radial distribution networks

and study the value of storage in mitigating both predictable and unpredictable variabilities on the

demand side, i.e., the value of storage in leveling load and buffering uncertainties in the net load.

We do not consider the storage value of deferring network upgrades due to demand growth.

Distribution loss is a critical element impacting the storage location choice (as we shall see in

Section 1.1 and throughout the paper). According to the International Energy Agency (2018), the

transmission and distribution 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%, Hong Kong

12.5%, Spain 9.6%, Canada 8.7%. In the U.S., the line losses vary from 2% to 13% across different

states (Wirfs-Brock 2015).

In this paper, we consider a distribution network with one generation node connecting to multiple

leaf nodes where demands occur. 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. We ask two related questions: First, for given

storage investment in a tree network, what is the optimal storage operating policy? Second, what

is the optimal locations and size of storage investment in a tree network? Using stochastic dynamic

1

programming, we identify the structure of the optimal storage operating policy, and then we analyze

and compare various storage investment strategies on the tree network.

1.1 An Introductory Example

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

insights, consider a tree network of one supply node (root) and three demand nodes (leaves) with

simple demand processes. Nodes 1 and 2 have independent demands fluctuating between zero and

a high level (this can be the demand from an industrial facility with an on-site generator that

occasionally needs backup), while node 3 has constant demand, as illustrated in Figure 1. The power

generator at node 0 and the distribution lines have enough capacity to meet all demands. Power

generation cost is convex in the output, and there are line losses.

Figure 1: A storage location choice problem

�

0

1 2

Demand 1 Demand 2 Demand 3

3

�

Convex generation cost

Lin

e l

oss

�

Suppose we invest in a small unit of battery storage. Where should we place this battery, allowing

for splitting the capacity among nodes? Intuitively, it is beneficial to place the battery 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 battery becomes lost along the lines, implying that part of the battery

capacity is wasted in storing line losses. Alternatively, we can place the battery at demand node 1.

However, if demand at node 2 is high while 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 tradeoffs between storage capacity loss and line loss, one may conclude

that the battery capacity should be split among nodes 0, 1, and 2, depending on the relative benefits.

Let us consider the seemingly inferior choice of placing the battery at node 3. When the demand

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

or 2 or both, the battery at node 3 releases energy to reduce the amount of energy sent to node 3,

effectively reducing the generator’s output and smoothing production without needing to incur extra

2

line losses, because energy is never sent back to node 0. Furthermore, the battery at node 3 stores

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

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

general rules for the optimal storage operations (under given storage locations and sizes) before we

can consider optimizing storage investment. This is exactly what this paper aims to achieve.

1.2 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, transmission/distribution between locations and

line losses, as well as the cost of power generation. Existing literature on energy systems 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.

One of the pioneering research papers on storage siting and sizing is by Denholm and Sioshansi

(2009). They study the tradeoff between co-locating storage with remote wind power (thus reducing

the transmission capacity needed to deliver wind power to market) and siting the storage closer to

demand (thus storage can better shift load).

With the development of storage technologies and smart grid infrastructures in recent years,

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,

3

which complements the above body of numerical methods by analyzing the storage operating policy

and deriving general lessons about investment of storage systems in distribution networks.

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

because such problems are theoretically challenging: For every possible 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 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 distribution networks

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

correlated, it is optimal to place storage near the leaves of the network. In this paper, we aim to

derive further insights on storage location choices and sizing decisions for a tree network.

Our work is related to several streams of research in operations management. The cost of energy

generation is convex, as power generators with low marginal cost are typically dispatched before high-

marginal-cost generators are dispatched to meet the demand. Inventory management with convex

purchasing cost is first analyzed by Karlin (1960). A few papers consider special cases of convex

production costs, mostly in piecewise linear format. Henig et al. (1997) consider designing contracts

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

cost, including the special case of limited production capacity. These papers, however, do not analyze

transshipment or losses due to it or storage conversion losses typical for energy storage.

For electric grids, energy storage size is typically limited due to its cost. Inventory management

with limited storage space, also known as the warehouse problem, is first 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 (if the firm acts, it either fills up the storage or sells all inventory).

Secomandi (2010) extends this work to incorporate storing/releasing speed limits. Zhou et al. (2015)

further extend the analysis to include negative prices. Our model involves storage investment in a

network and, thus, we also consider multilocation storage size constraints.

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

ilar to the transshipment operations in other industries. Multiperiod multilocation inventory man-

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

4

others. Hu, Duenyas, and Kapuscinski (2008) provide a detailed review of this area.

Despite the similarities, electricity systems have unique features that require different models.

First, transporting energy incurs line losses, and charging/discharging energy storage involves storage

losses. This feature is modeled in Zhou et al. (2015, 2019). While our focus is different, we use the

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

1.3 Contributions

This paper develops the theory on energy storage operations and investment in a tree network.

Energy can be stored centrally (at the root) or locally (at the leaves), used locally or transferred to

other nodes, but storage and transfers incur losses. We aim to minimize the total storage investment

and energy generation costs. We summarize the contributions of this paper as follows.

First, we characterize the structures of the optimal operating policy under stochastic demand

and given storage siting and sizing decisions in a tree network. The optimal policies are structurally

different for different levels of storage losses and line losses, but we derive two principles that govern

the optimal storage operations: First, balanced leaf storage levels are desirable; second, centrally

stored energy offers more operational flexibility. These principles allow us to interpret, for all cases,

storage operating policies in intuitive terms.

Second, our work advances the theory on distributed storage investment. We find that storage

should be first placed at the leaf nodes with positive minimum demand, even if the demand variability

at those nodes is very low (refer to Section 1.1). After the storage capacity reaches the minimum

demand level, further storage investment should consider relative benefits of leaf versus central

storage. Leaf storage continues to be optimal when the demands are correlated across leaf nodes.

On the other hand, central storage may improve storage usage frequency and reduce line losses.

Third, this paper improves the understanding of the relative importance of storage siting and

sizing decisions. We simulate many storage investment choices and find that determining the total

amount of storage investment is more critical than allocating storage capacity between the central

and leaf nodes. This new insight is robust with respect to various demand settings and the number

of leaf nodes. We also find that as the number of leaf nodes increases, the average value of storage

(counter-intuitively) increases.

5

2. Model

2.1 Network and Storage

We consider a utility company’s storage investment problem on a tree network that consists of central

power generation capacity at the root (referred to as node 0 or the central node thereafter) and n

demand (leaf) nodes Ldef= {1, 2, . . . , n}. Figure 2 shows this network and illustrates three storage

siting strategies. A localized (resp. centralized) investment installs storage at the leaf nodes (resp.

central node) only, and a mixed investment installs storage at both the central and leaf nodes.

Figure 2: Location choices for storage investment

(b) Centralized investment

0

1 �

0

1 �

0

1 �

(a) Localized investment (c) Mixed investment

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

The storage technology we consider can be lithium-ion battery or flow battery. The discharging

time and recharging time of these technologies are typically one to several hours, which is suitable

for load shifting and operating reserves (Akhil et al. 2015, KEMA Inc. 2012).

We consider a planning horizon Tdef= {1, 2, . . . , T}. Each period t ∈ T represents several hours

in which the storage can be fully charged or discharged, and T represents the storage lifetime. Prior

to the first period, storage investment is decided at each node and storage facilities are built. Once

built, the storage size is fixed throughout T . The utility’s objective is to minimize the sum of storage

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

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

measured in units of energy (e.g., kWh). 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 . These and

following key variables are illustrated in Figure 3.

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

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

1The assumption of the same efficiency in both ways brings notational and analytical convenience, but does notcause 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.

6

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.

As discussed in Section 1, we do not consider the storage value in deferring upgrades of line

capacity. Thus, we assume that the lines in the tree network are not capacity-constrained, i.e., all

demand 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 and also depends on the temperature.

The linear approximation stated in Assumption 1 simplifies the analysis. The linear approximation

for line loss is often used in the literature, 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, i.e., 1 − β is the fraction of energy

lost in any line. Let ui,t denote the line flow measured at leaf node i ∈ L in period t, with ui,t > 0

for the energy sent from node 0 to leaf node i and ui,t < 0 for the reverse flow (from leaf node i to

Figure 3: Network model and key variables

Energy flow

direction when the

associated term > 0

�� ��

��

0

1

Storage level in period �:

begin: ��, end: ��,�

Demand ��, �

Generation

Storage level in period �:

begin: ��, end: ��,�

��(∆��,)

��, = ��, + ��(∆��,)

��(��,)

��(∆��,)

Storage level in period �:

begin: ��, end: ��,�

��(��,)

Demand ��,

��(∆��,)

��, = ��, + ��(∆��,)

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

7

node 0). The corresponding line flow measured at the central node 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 when storage at leaf node i is discharged to meet demand at another node.

2.2 Balancing Demand and Supply

A central decision-maker for the network system manages power generation and storage operations to

meet the demand for electricity. Let dt = (d1,t, . . . , dn,t), where di,t is the net demand at leaf node i

in period t. We assume the net demand process {dt : t ∈ T } is Markovian, and dt is realized at the

beginning of period t and must be satisfied in period t. To avoid the exposition being unnecessarily

technical, we assume dt ≥ 0 in deriving the optimal storage operations; the results can be extended

to allow negative net demand. The investment analysis in Section 4 does not assume dt ≥ 0.

In each period, observing the period-starting storage level st and demand dt, the decision-maker

decides the period-ending storage level st+1. The inventory change and the demand determine the

energy flows and required generation as below. At the leaf nodes, the 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 3):

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.

2.3 Problem Formulation

The storage investment decision trades off between the upfront investment cost and the ongoing

operating cost. To evaluate a storage investment decision S, let Vt(st,dt;S) denote the minimum

expected discounted operating 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

8

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.

The storage facilities are installed prior to period 1; no additional investment or divestment can

be made during the operating horizon T . The total investment of |S|def=

n∑i=0

Si units of storage

capacity 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: s1 = S.

Thus, the decision-maker’s objective is to minimize the expected total cost:

minS≥0

{p |S| + V (S)

}, (7)

where V (S)def= EV1(S,d1;S) and 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.

In Section 3 that follows, we derive the optimal operating policy for given storage and present

the results in a concise manner, with the objective of enabling the analysis in Sections 4 and 5. In

Section 4, we identify the key lessons related to storage investment, which are the focal point of this

paper, and Section 5 presents numerical comparisons, which lead to further insights.

3. Optimal Operating Policy for Given Storage Investment

This section derives the structural properties of the optimal operating policy under given storage

investment S ≥ 0. Because S is fixed in this section, we shorten Vt(st,dt;S) as Vt(st,dt) and express

the optimal decision for (5)-(6) as s∗t+1(st,dt).

To analyze 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 level qt:

Vt(st,dt) = minqt

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

}, (8)

and the subproblem finds the optimal use of qt by deciding the inventory levels:

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

{Et

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

]: st+1 ∈ A(qt)

}, qt ∈ Q(st,dt), (9)

9

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

}, (10)

and qt is chosen from Q(st,dt)def=

[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. (Throughout the paper, x+ = max{x, 0}.) For brevity of notations,

we do not explicitly express the dependence of A(qt), qt, and qt on (st,dt).

Let s∗t+1(qt, st,dt) denote an optimal solution to the subproblem (9). Solving (9) gives the mini-

mum expected cost-to-go functionWt(qt, st,dt), which is decreasing and convex in qt (see Lemma 1).

The master problem (8) decides the optimal production by trading off the production cost c(qt)

and the minimum expected cost-to-go Wt(qt, st,dt). Because Wt(qt, st,dt) and c(qt) are convex in

qt (Lemma 1 and Assumption 2), the master problem (8) is a one-dimensional convex optimization

problem that is relatively straightforward to solve. Therefore, the rest of this section is devoted to

analyzing the structure of s∗t+1(qt, st,dt), the solution to the subproblem (9).

3.1 Optimal Policy Overview and Intuitions

To help with readability, before we present the detailed analysis, we intuitively describe the structures

of the optimal solutions. Demand in any given period can be met by two sources of energy: stored

energy (generated in the previous periods) and current energy (generated in the current period). How

these two sources combine to meet demand and how the storage levels should be adjusted depend

on the relative magnitude of storage efficiency (α) and line efficiency (β). The following two cases

have noticeably different operating policies.

Case of α ≤ β. Since storing energy incurs more energy loss than sending energy from the central

node to the leaf nodes, it is preferred 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;

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 4. Second,

if the central storage is full, charge the leaf storage to levels as “balanced” as possible, illustrated by

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

because centrally stored energy provides more flexibility in meeting future demand than locally stored

10

Figure 4: Optimal storage level s∗t+1(q, st,dt) for q ≥ qot and α ≤ β, symmetric leaf nodes

��,���,�

��,�

��,�

�� �� ��

��

Storage

level

0

① ②

③

④

Central

storage 0

Leaf

storage 1

Leaf

storage 2

Leaf

storage 3

③

④ ④

energy. When charging leaf storage, in the case of symmetric leaf nodes (defined in (14)), equalizing

the storage levels across the leaf nodes minimizes the expected future cost. (In general, balanced leaf

storage levels are not necessarily equal.)

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. To characterize relative distance between storage and demand, for any given leaf node i, we

refer to the storage at node i as local storage and the storage at any other leaf node j 6= i as remote

storage. To reduce line losses, storage should be discharged in the sequence of local storage first,

central storage next, and remote storage last, as depicted in Figure 5. Panel (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 < qt =∑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.

In short, given any feasible generation level q, we can efficiently find the optimal inventory decision

s∗t+1(q, st,dt) following the steps in Figures 4 and 5, rather than searching in a (n + 1)-dimensional

space. This structural result helps solve problem (9) efficiently. Moreover, we do not need to solve

(9) for every feasible q because the master problem in (8) involves convex optimization and efficient

algorithms can be readily applied for finding q∗t .

11

Figure 5: Optimal storage level s∗t+1(q, st,dt) for q ≤ qot and α ≤ β, symmetric leaf 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

Leaf

storage 1

Leaf

storage 2

Leaf

storage 3

��,�

0

Storage

level

①

②

③

④

⑤

②

③

④

③ ��,�

⑦

⑧

⑥

⑧

��,� � �,� = ��,� − ��,�/�

Storage

level

��,�

� �� ��

Central

storage 0

Leaf

storage 1

Leaf

storage 2

Leaf

storage 3

0

��

Case of α > β. If q < qt, i.e., the current generation is so low that the central or remote storage

has to be discharged to meet the demand, the optimal path illustrated in Figure 5(b) is still optimal.

If q ≥ qt, storage operations are different from the case of α ≤ β, but two insights continue to

apply in this case: (a) Balanced leaf storage levels are desirable, and (b) centrally stored energy

offers more operational flexibility.

Below we describe the key differences. In the case of α ≤ β, the storage inefficiency renders it

undesirable to release energy from one location and simultaneously store at another node. When

storage is more efficient (α > β), however, the strategy of releasing energy at one location and

simultaneously storing energy at another location may be part of the optimal policy because it

helps balancing leaf storage levels and shifting stored energy to the central node, consistent with the

aforementioned insights (a) and (b).

The optimal storage levels for the case of fully efficient storage (α = 1) are illustrated in Figure 6.

Figure 6(a) shows that when q ∈ [qt, qt + S0 − s0,t], we use local storage to meet as much demand as

possible (step 1©) and store q − qt in the central storage (step 2©). This strategy effectively shifts

stored energy from the leaf nodes to the central node.

When q > qt + S0 − s0,t, the central storage is fully charged and then the leaf storage levels are

kept as balanced as possible, as shown in steps 3©- 5© in Figure 6(b). Note that any period-ending

storage level on the path of 3©- 5© is reached from period-starting storage level st = (s1,t, s2,t, s3,t)

12

Figure 6: Optimal storage level s∗t+1(q, st,dt) for q ≥ qt and α = 1 > β, symmetric leaf nodes

(a) Use local storage completely,charge central storage

(b) Fully charge central storage,charge or discharge local storage

②

�� �� ��

��

Central

storage 0

Leaf

storage 1

Leaf

storage 2

Leaf

storage 3

��,�

0

Storage

level

��,�

��,�

��,�

�� �� ��

��

Central

storage 0

Leaf

storage 1

Leaf

storage 2

Leaf

storage 3

��,�

0

④

③

④

⑤ ⑤⑤

� ,� = �,� − ,�/��

�,� � ,� = �,� − ,�/��

�,�Storage

level

①

①①

directly, i.e., it is not necessary to discharge storage as in step 1©. Different from Figures 4 and 5,

energy may be released at some nodes while stored at some other nodes, which occurs, for example,

in Figure 6(b) when current generation q leads to s1,t+1 > s1,t and s3,t+1 < s3,t.

Finally, we note that if 1 > α > β, the presence of storage loss will attenuate the magnitude of

charging and discharging in the same period, and the optimal storage levels are in between the case

of α = 1 > β and the case of α ≤ β.

3.2 Structure of the Optimal Inventory Policy

While Section 3.1 illustrates the general principles of optimal storage operations, we now formally

define and prove the structures of the optimal storage policy for general settings.

We first derive the basic properties of the operating cost function Vt(st,dt). Intuitively, stored

energy has an operating-cost reduction effect. Lemma 1 confirms this intuition and further shows

that this effect declines when the storage level increases. (Throughout this paper, monotone and

convex properties are not in strict sense, unless otherwise noted.)

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

(ii) Wt(qt, st,dt) is decreasing and convex in qt for any given (st,dt) and t ∈ T .

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 online appendix.

13

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

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

losses. Both lemmas are building blocks for the structural properties of the optimal policy.

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 leaf 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.

A key construct for defining the optimal inventory decision is the constrained balanced inventory,

sb(x,y, z,dt) ∈ A, where x,y ∈ A, x ≤ y, x0 = y0, and z ∈[∑

i∈L xi,∑

i∈L yi]. A constrained

balanced inventory sb(x,y, z,dt) is constrained between x and y (thus sb0 = x0 = y0) and allocates

total leaf inventory z among the leaf nodes so that the future expected operating cost is minimized:

sb(x,y, z,dt) ∈ argmins

{Et

[Vt+1(s,dt+1)

]: x ≤ s ≤ y,

∑i∈L

si = z}. (11)

Figure 7 illustrates the constrained balanced inventory in a general setting. As the total leaf

storage level, z, increases, leaf storage levels are adjusted to minimize the expected operating cost-

to-go. Figure 7 also shows that the balanced leaf inventory levels are not necessarily equal, since

they are balanced with respect to the future demands to minimize the expected cost-to-go.

Figure 7: An example of constrained balanced inventory sb(x,y, z,dt), x ≤ y, and x0 = y0

Note: sb(x,y, z,dt) = x if z = x1 + x2 + x3. When z increases, sb(x,y, z,dt) followsthe path from 1© to 5©. When z reaches y1 + y2 + y3, s

b(x,y, z,dt) = y.

�� =��

Central

storage 0

0

Storage

level

Leaf

storage 1

Leaf

storage 2

Leaf

storage 3

��

��

��

①

②

③

④

⑤

②

③

④

③

����

��

����

Using the constrained balanced inventory defined in (11), we next define and prove the structures

of the optimal decisions for problem (9). Two cases are analyzed in sequence.

14

3.2.1 Case with α ≤ β

Recall that qot =∑i∈L

di,t/β is the production that exactly meets the demand. To define the general

structure of the optimal policy, we need the following critical inventory and production levels:

qt = qot + (S0 − s0,t)/α, st = (S0, si,t, i ∈ L),

qt =∑i∈L

(di,t − αsi,t)+/β, st =

(s0,t, (si,t − di,t/α)

+, i ∈ L),

qt =(qt − αs0,t

)+, st =

([s0,t − qt/α

]+,(si,t − di,t/α

)+, i ∈ L

).

(12a)

(12b)

(12c)

If producing qt in (12a), the excess energy qt − qot can exactly fill the central storage, leading to

inventory st. The quantities in (12b) have been introduced in Section 3.1: (di,t − si,tα)+ is the

remaining demand at leaf node i after being served by local storage and, thus, the production

needed to serve this remaining demand is qt and the remaining storage level is st, where the single

under-bar represents that only local storage is used to satisfy demand. When using both local and

central storage to serve the demand, the remaining storage level is st and the required production

is qt in (12c). When using local, central, and remote storage to serve the demand, the required

production is qt defined after (10).

Using the critical levels in (12) and the constrained balanced inventory defined in (11), we describe

the structure of the optimal policy for the case of α ≤ β in Proposition 1, in which we use L, R, and

C to abbreviate local storage, remote storage, and central storage, respectively.

Proposition 1 When α ≤ β, given state (st,dt) and feasible production quantity qt ∈ Q(st,dt), an

optimal inventory decision s∗t+1 can be expressed as follows:

s∗t+1(qt, st,dt) =

[store to full at C, then at leaves]:

sb(st, S,

∑i∈L

si,t + (qt − qt)βα, dt

), if qt ∈

[qt, qt

],

[store at C]:

st +(α(qt − qot ), 0, . . . , 0

), if qt ∈

[qot , qt

),

[release from L]:

sb(st, st,

∑i∈L

si,t − (qot − qt)β/α, dt

), if qt ∈

(qt, q

ot

),

[release fully from L, then from C]:

st −((qt − qt)/α, 0, . . . , 0

), if qt ∈

(qt, qt

],

[release fully from L,C, then from R]:

sb(0, st,

∑i∈L

si,t − (qt − qt)/(αβ), dt

), if qt ∈

[qt, qt

].

(13a)

(13b)

(13c)

(13d)

(13e)

The following properties are true for s∗t+1(qt, st,dt) for any qt, and thus also true for s∗t+1(st,dt),

15

the solution for (5)-(6):

• In each period t, either store energy and end up (weakly) above the period-starting inventory st,

or release energy and end up (weakly) below st.

• When storing energy, first store at the central storage until full (moving from st to st, as in (13b)),

and then store at the leaves, keeping inventory balanced (following sb(st,S, z,dt) in (13a)).

• When releasing stored energy to meet the demand, release first from local storage (following

sb(st, st, z,dt) in (13c)), then from central storage (moving from st to st in (13d)), and finally

from remote storage (following sb(0, st, z,dt) in (13e)).

In all cases, we try to use the current supply, qt, to satisfy the demand, and resolve the supply-

demand mismatch by using storage. Importantly, the policy structure presented in Proposition 1

holds for any storage investment and general demand distributions.

In the special case when the leaf nodes are symmetric in period t, i.e., the leaf nodes have the

same storage size and the demand distributions for future periods satisfy

Pr{dτ ≤ δ | dt} = Pr{dτ ≤ δ | dt}, ∀ τ > T, ∀ δ ∈ Rn, and δ is any permutation of δ, (14)

allocating inventory across the leaf nodes as evenly as possible minimizes the expected cost-to-go.

Lemma 3 If the leaf nodes are symmetric (S1 = S2 = · · · = Sn and (14) holds) in period t, then

there exists a constrained balanced inventory that is independent of dt and can be expressed as

sb(x,y, z) = argmins

{(maxi∈L

si)−

(mini∈L

si): x ≤ s ≤ y,

∑i∈L

si = z}, (15)

where x,y ∈ A, x ≤ y, x0 = y0, and z ∈[∑i∈L

xi,∑i∈L

yi

]. In particular, if x ≤

(x0, z/n, . . . , z/n

)≤ y,

then sb(x,y, z) =(x0, z/n, . . . , z/n

).

Proposition 1 and Lemma 3 confirm the optimality of the inventory decisions in Figures 4 and 5.

3.2.2 Case with α > β

As intuitively discussed in Section 3.1, when the transfer loss rate, 1 − β, is more than the storage

loss rate, 1 − α, it may be desirable to store energy in one location and release energy in another.

Such a policy is formalized in Proposition 2. The formalization requires the following critical values:

qt = qt + (S0 − s0,t)/α, st =(S0, (si,t − di,t/α)

+, i ∈ L), (16)

where qt is defined in (12b), and qt is the production required to fill the central storage while satisfying

the demand after the local storage is used to meet as much local demand as possible. Note that st

and st differ only in the central storage level.

16

Proposition 2 When α > β, given state (st,dt) and feasible production quantity qt ∈ Q(st,dt), an

optimal inventory decision s∗t+1 can be expressed as follows.

(i) If storage operations are perfectly efficient (β < α = 1),

s∗t+1(qt, st,dt) =

[store to full at C, store or release at leaves]:

sb(st, S,

∑i∈L

si,t + (qt − qt)β,dt

), if qt ∈

(qt, qt

],

[release fully from L, store at C]:(s0,t + qt − qt, (si,t − di,t)

+, i ∈ L), if qt ∈

(qt, qt

],

[same as (13e) and (13d)] if qt ∈[qt, qt

].

(17a)

(17b)

(17c)

(ii) If storage operations are not perfectly efficient (β < α < 1),

s∗t+1(qt, st,dt) =

[store to full at C, store or release at leaves]:

s∗t+1 ∈ Et, if qt ∈(qt, qt

],

[release or store at L, store at C]:

s∗t+1 ∈ Ft ∪ Et, if qt ∈(qt, qt

],

[release from L, store at C]:

s∗t+1 ∈ Ft, if qt ∈(qt, qt

],

[same as (13e) and (13d)] if qt ∈[qt, qt

],

(18a)

(18b)

(18c)

(18d)

where the face Ft and edges Et are defined as

Ft ={st+1 ∈ A(qt) : s0,t+1 ≥ s0,t, si,t+1∈

[(si,t − di,t/α)

+, si,t], i ∈ L

},

Et ={st+1 ∈ A(qt) : s0,t+1 = S0, si,t+1∈

[(si,t − di,t/α)

+, Si], i ∈ L

}.

In contrast with Proposition 1, when α > β, the current-period demand is not always satisfied

from current-period generation to the extent possible, and it may be desirable to release energy at

one location and simultaneously storing energy at another location.

Despite the differences, the same insights remain useful. Intuitively, when the storage is fully

efficient (α = 1), the only loss in the system is the line loss. Part of the line loss is unavoidable

because all energy generated at node 0 will eventually be sent to the leaf nodes. The other part of

the line loss is incurred when releasing energy from one leaf node to meet the demand at another.

To minimize this loss, it is desirable to store energy at the central node whenever possible and keep

the leaf storage levels balanced at the same time.

17

4. Optimal Investment Decisions

Building on the analysis in Section 3, we consider in this section the storage investment problem

stated in (7): minS≥0

V (S) + p |S|, where V (S) = EV1(S,d1;S), p > 0 is the cost per unit of storage

capacity, and |S| =n∑

i=0Si. Our goal is to understand the trade-off between localizing and centralizing

storage investment (refer to Figure 2). The analysis in this section does not assume dt ≥ 0.

We first present the basic properties of the expected operating cost function V (S).

Lemma 4 The expected operating cost V (S) is decreasing and convex in S.

Lemma 5 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).

In Lemma 4, the multidimensional convexity in S implies diminishing marginal returns on storage

investment. Lemma 5 states that, for any given investment S, there exist a centralized investment Sc

and a localized investment Sl constructed as in the lemma that yield a lower expected operating cost

than S. However, because Sc and Sl require a higher investment cost than S (note that |Sc| ≥ |S|

since β ∈ (0, 1), and that |Sl| ≥ |S| if β ≥ 1/n, which holds for most practical situations), the

preference is not obvious among the location choices depicted in Figure 2.

Before presenting detailed analysis, let us intuitively consider how line losses affect the economic

value of storage in the tree network: One unit of energy released from a local storage can serve one

unit of local demand, or only β2 units of demand at other leaf nodes; one unit of energy released

from the central storage can serve β units of demand. Thus, the economic value of storage is affected

by the fact that some storage space is wasted in storing line losses.

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

}, (19)

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

}. (20)

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

}may coincide with Sc∗ or Sl∗, or may

be a mixed investment.

4.1 Demand Correlation and Storage Placement

Leaf-to-leaf energy transfer reduces economic value of locally stored energy, but such transfer is rarely

needed if demands are highly positively correlated across nodes. Indeed, with perfectly correlated

net demand, localized investment decision is proven to be optimal, as stated below.

18

Proposition 3 If for all i ∈ L and t ∈ T , di,t = ki d1,t for some constant ki > 0, then S∗ = Sl∗.

Note that electricity systems are different from other logistic systems in that producing energy to

meet demand involves zero lead-time. Thus, the result in Proposition 3 is not driven by lead-times

and safety stocks as in the classic inventory theory. Instead, the driving force is the line losses, as

explained below. Multiple leaf nodes with perfectly correlated demands can be treated as a single

demand node. With only one demand node connected to node 0, investing at node 0 is never optimal,

because a smaller investment of βS0 at the demand node provides the same operational benefit as

investing S0 at node 0.

4.2 Impact of Minimum Demand on Storage Investment

When demands are not highly correlated, leaf-to-leaf energy transfer still will not be needed if the

minimum net demand is sufficiently high so that locally stored energy can always be used locally.

Specifically, if dmini > αSl∗

i , where dmini is the minimum net demand at node i ∈ L and Sl∗

i is the

optimal localized investment given by (20), then stored energy can always serve the local demand.

In fact, when dmini > αSl∗

i , it can be verified that qt = qt = qt. Thus, the case of using remote storage

in Propositions 1 and 2 does not occur. The result is formalized in Lemma 6 and Proposition 4.

Lemma 6 Suppose dminj > 0 for a given leaf 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 6(i) states that we can maintain the same operating cost, while reducing the total storage

size by replacing δ units of central storage capacity by βδ units of local storage capacity, as long as

the increased local storage is below the minimum net demand. Lemma 6(ii) suggests that we can

maintain the same storage size while reducing the operating cost by shifting storage capacity among

the leaf nodes, as long as the increased local storage is below the minimum net demand.

Proposition 4 (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 4(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 leaf node. Proposition 4(ii) implies that positive minimum net demand precludes

19

centralized investment from being optimal. In other words, if dminj > 0 for some j ∈ L, then the

optimal storage investment is either mixed or localized.

The practical implication of Proposition 4 is that for a tree network in practice, we can consider

leaf nodes with high minimum demand as good potential sites for building storage with capacity at

or below the minimum demand. Intuitively, energy stored in these storage facilities can always be

used locally, avoiding leaf-to-leaf energy transfer.

4.3 Benefits of Centralized Storage Investment

Next, we identify the benefits of centralizing storage investment. The first and expected benefit is

that the energy stored at node 0 can serve demand at either leaf node without incurring leaf-to-leaf

line losses. Second, storage at node 0 may avoid investing in storage capacity dedicated to each leaf

node. Both of these benefits are illustrated in a simple example below. For illustrative purpose, the

parameters are chosen so that the centralized investment is optimal in this example.

Example 1 Suppose the tree network has two leaf nodes. In even-numbered periods, d2t = 0, while

in odd-numbered periods, d2t−1 = (0, 10) or (10, 0) with equal probability (t = 1, 2, ...).

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

optimal localized investment Sl∗(p). We find that under relatively high p, the optimal policy for

operating Sl∗(p) is to fully charge storage in period 2t when demand is zero, and empty all storage in

period 2t− 1 to meet the demand at one node, incurring leaf-to-leaf transfer losses. The centralized

investment increases the operational flexibility by sending energy from node 0 to only the node with

high demand and avoiding leaf-to-leaf energy transfer.

Under relatively low p, the optimal policy for operating Sl∗(p) is to fully charge storage in

period 2t, but discharge only local storage to serve the demand in period 2t−1, i.e., storage capacity

is dedicated to the local demand. Dedicated storage investment for each leaf node results in over-

investment of storage capacity. Although leaf-to-leaf transfer is avoided, the storage capacity is

under-utilized. A centralized investment is more economical and optimal in this case.

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

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

of localized investment to the demand.

4.4 Impact of Storage Cost in Storage Investment

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

reduced cost, which will impact the optimal storage investment decisions. As in Section 4.3, we

20

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

Proposition 5 (i) The minimum total cost V (S∗(p)) + p |S∗(p)| increases in p;

(ii) The optimal total investment |S∗(p)| decreases in p.

Proposition 5 confirms that, as storage cost declines, more investment in storage capacity will

take place and it generates a higher net benefit. However, storage cost decline also affects the relative

values of centralized and localized storage. Centralized storage tends to store more line losses than

localized storage, but this disadvantage is less prominent when the storage cost is low. Thus, the

optimal storage investment may shift toward more centralized as storage cost declines. Therefore,

although |S∗(p)| decreases in p, the elements of S∗(p) may not be monotonic in p.

Next, we consider a more futuristic case when storage cost becomes very small. Proposition 6

below suggests that often the location of the storage does not matter any more: The localized

investment Sl∗(p) is asymptotically optimal as p → 0+. This asymptotic optimality also holds for

the centralized investment Sc∗(p), as long as the net demand is non-negative at each node.

Proposition 6 (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.

The intuition for part (i) is that at very low p, we can invest in storage dedicated to each leaf

node and its size can be large enough to eliminate the need for leaf-to-leaf distribution. For part (ii),

a very low p suggests that installing storage at node 0 costs little, even though some of the stored

energy will be lost during distribution. However, when net demand can be negative (i.e., distributed

generation exceeds local demand), localized storage avoids storing energy remotely, and centralized

investment is no longer asymptotically optimal.

In summary, localized storage investment tends to be optimal when (a) net demands are highly

positively correlated across leaf nodes, or (b) the minimum net demand is high, or (c) the storage

cost becomes very low. The location choice of the localized storage can be counter-intuitive: nodes

with high demand variability are not necessarily the best place for storage while nodes with low

demand variability can well be the optimal location for storage. We have also identified the benefits

of centralized storage in reducing leaf-to-leaf energy transfer and increasing usage of storage. In

practice, many factors simultaneously affect the value of storage, rendering the optimal investment

often mixed. In the next section, we numerically study various situations and explore additional

insights on storage investment.

21

5. Numerical Analysis

In evaluating the expected operating cost under a given storage investment, the dynamic program

in (5) has a state space of n dimensions of demand and n + 1 dimensions of storage levels and has

an action space of n+ 1 dimensions of storage levels. In Section 3, we have identified the structures

of the optimal storage operating policies, which are used in this section to significantly simplify

the computation. This section numerically studies the relative benefits of centralized and localized

investment strategies, and derive further insights on storage investment.

5.1 Base Case Settings

We start with considering a base case, in which there are only n = 2 leaf nodes. At each node i = 1, 2,

net demand is zero during the even periods: di,2t = 0. During the odd periods, net demand di,2t−1

takes three possible values: 0, 30 MWh, and 60 MWh, with probability 0.4, 0.4, and 0.2, respectively,

and the net demands are independent across nodes and time. This simple model reflects the reality

that net demand exhibits both predictable and unpredictable variabilities. The odd and even periods

represent peak time and off-peak time, respectively. Off-peak net demand is low, while the peak net

demand has considerable variability. We set the minimum net demand level to zero, so that the

minimum-demand effect (detailed in Section 4.2) does not exist in the base case. Positive minimum

demand will be studied in Section 5.3.

Other parameters of the model are set as follows: storage efficiency α = 0.9 (representative of

typical lithium-ion battery), line efficiency β = 0.95 (5% distribution loss), storage cost is p = $400

per kWh (Bloomberg New Energy Finance 2018b), production cost c(q) = q2 (adding a linear cost

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

investment), and discount factor γ = 0.99.

Because V (S) is convex in S and symmetric with respect to S1 and S2, we can focus on sym-

metric investment S1 = S2 without losing optimality. Furthermore, to gain more insights on storage

investment, we depict the shape of the total cost function: for the base case, we evaluate the total

cost for thousands of storage investment choices by varying S0 between 0 and 56 MWh and varying

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

5.2 Operational Benefit of Storage and Optimal Investment

Our prior intuition is that the placement of storage critically affects the operating cost. Unexpectedly,

our numerical results provide some counterintuitive insights, as detailed below.

Recall V (0) is the system operating cost without storage. The operational benefit of storage S can

22

be measured by the fraction of operating cost reduced by storage:(V (0)−V (S)

)/V (0). Figure 8(a)

shows the contours of this fraction as S0 and SL = 2S1 vary in a wide range. In the figure, the

contour labels correspond to the fraction of operating cost reduction. Clearly, diminishing marginal

benefit (in terms of cost reduction) can be observed. A striking observation is that all contours are

approximately 45-degree lines (the slopes vary between 43 and 46 degrees), implying that allocation

of storage capacity between the central and leaf nodes have much less effect on operating cost than

the total storage capacity. (If the contours were exactly 45-degree straight lines, allocation of storage

capacity between central and leaf nodes would not affect operating cost.)

Figure 8: Effect of storage investment on operating cost and total cost: Base case

(a) Operating cost reduction: 1− V (S)V (0) (b) Total cost increase: V (S)+p|S|

V (S∗)+p|S∗| − 1

�� ��

�� ��

Next, we find the optimal storage investment S∗, which minimizes the total cost V (S)+p |S|. 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. (21)

The contours of the this quantity is shown in Figure 8(b). The optimal investment (indicated by a

star) 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.

We make two important observations from Figure 8(b). First, the total cost function is quite flat

near the optimal decision: if the investment decision deviates from the optimal decision by 10%, the

23

cost increases by less than 0.2%. Second, the total cost is fairly robust to the allocation of storage

capacity across the central and leaf nodes, seen from the orientation of the oval-shaped contours.

In fact, holding the total storage size at 55.4 MWh (= S∗0 + S∗

L), the total cost under the localized

investment (S0 = 0 and SL = 55.4) is only 0.07% higher than the optimal cost, and the total cost

under the centralized investment (S0 = 55.4 and SL = 0) is 0.8% higher than the optimal cost.

To confirm the robustness of the operating cost with respect to storage locations, we perform

an additional analysis. For each given total storage size, we calculate the operating costs under

centralized investment, localized investment, and the constrained optimal investment for the given

total storage size. These three operating costs and the minimum total cost are shown in Figure 9.

The three operating cost curves are very close, with the largest percentage difference being only

1.1%. Thus, when the total storage size is fixed at any level, it continues to hold that the total cost

is fairly robust with respect to whether the storage is placed at the central or leaf nodes.

The key message from the above results is that it is more critical to decide the right amount

of total storage investment than to allocate storage capacity between central and leaf nodes. This

insight continues to hold for a wide range of settings, as we will see in the rest of this section.

We remark that this insight is not to say that storage placement is unimportant. The result that

centralized storage investment leads to 1% cost increase may look small percentage wise, but can be

a substantial cost in practice given the magnitude of the electrical systems.

We now provide a theoretical explanation for the above numerical finding. Consider the case

Figure 9: Cost components under various given total storage size: Base case

0

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80 100

Minimum total cost for giventotal storage size

System operating cost(central investment only)

System operating cost(distributed investment only)

Minimum system operating costfor given total storage size

Storage investment cost

Total storage size (MWh)

(million $)

24

of perfectly correlated demands, for which Proposition 3 ensures that the localized investment is

optimal. Let v(S0, SL) be the operating cost under given central storage S0 and total leaf storage

SL. As discussed after Proposition 3, under perfectly correlated demand, one unit of central storage

is operationally equivalent to β units of local storage. Thus, we have

v(S0, SL) = v(0, βS0 + SL).

This equation implies that the contours of v(S0, SL) are determined by βS0 +SL = c, where c is the

contour level. Hence, the contours are straight lines with a slope of −β in the (S0, SL) space. (For

β = 0.95, the angle is 43.5 degrees.) When the demand is not perfectly correlated, a central storage

facility brings some operational benefit, rendering the slopes of the contours closer to −1.

An efficient heuristic. Based on the robustness property discovered above, we can develop an

efficient heuristic to search for the optimal storage investment:

Step 1: Search for the optimal centralized storage investment Sc∗ defined in (19).

Step 2: Search for the optimal allocation of total storage size |S| = |Sc∗|.

The search in Step 1 is considerably simpler than the original problem because each dynamic

program contains only one storage state and one decision to make per period. This first step aims

to determine the total storage size, which is a critical decision. Step 2 optimally allocates the given

total storage capacity across central and leaf nodes.

Applying this heuristic to the base case, we find the optimal centralized storage investment:

Sc∗ = 56.6 MWh, which is very close to the optimal total storage size 55.4 MWh found earlier.

Then, we search along the line S0 + SL = 56.6 MWh to find the near-optimal solution. Graphically,

Step 1 essentially finds the optimal point along the S0 axis in Figure 8(b), and then Step 2 searches

along a 45-degree line passing that point. In short, utilizing the robustness property with respect to

storage allocations brings computational efficiency.

5.3 Impact of Demand Levels on Optimal Storage Investment

In the base case, we assume zero minimum demand to isolate the minimum-demand effect examined

in Section 4.2. We now examine how the optimal investment changes with the demand levels.

Specifically, we consider a small (resp. large) shift that raises the nodal demand levels by 3 (resp.

7.5) MWh; the total demand is thus raised by 6 (resp. 15) MWh.

Figure 10 shows that as demand rises, the optimal investment shifts toward localized storage

investment, consistent with the insights from Section 4.2. Specifically, the optimal storage investment

(S∗0 , S

∗L) is (10.2, 45.2) for the base case, (8.1, 46.6) in Figure 10(b), and becomes localized investment

(0, 52.8) in Figure 10(c).

25

Figure 10: 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��

��

��

��

��

��

We make two additional observations. First, as the demand increases, the optimal total storage

size decreases 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 and the marginal production cost is linear,

c′(q) = 2q. Second, as demand increases, the contours shown in Figure 10 expand. This is because

satisfying a higher demand requires a higher operating cost V (S∗), which reduces the cost ratio in

(21). The shape and orientation of the contours remain the same.

5.4 Impact of the Number of Demand Nodes

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

the number of leaf nodes increases, to facilitate comparison, we shall either scale up the production

cost function (while keeping the same nodal 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 leaf nodes, the demand in the even periods is di,2t = 0 for all i ∈ L, and in the odd periods,

demands di,2t 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 nodal

demand by 6/n and 15/n MWh. Figures 11 and 12 illustrate the results for three and four leaf nodes,

26

in parallel to the case of n = 2 in Figure 10.

Figure 11: Total cost deviations and optimal storage investment: Case of n = 3 leaf nodes

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

��

��

��

��

��

Figure 12: Total cost deviations and optimal storage investment: Case of n = 4 leaf nodes

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

��

��

��

��

��

Let us first compare the cases with zero minimum demand, i.e., compare Figures 10(a), 11(a), and

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

n increases, the contours (especially the contour at 0.1% or the red region) shift towards more central

storage investment. Intuitively, the benefit of pooling is more prominent under more leaf nodes, and

thus central storage becomes more advantageous over local storage when n increases. Second, as n

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

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

Interestingly, as the number of leaf 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

27

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 10, 11, and 12 and Table 1

reassures the effects of the number of leaf nodes. In addition, we observe that the average value of

storage (Line 8) hardly decreases as demand increases, and the average value consistently exceeds

$1,000 per kWh, while the cost of storage is $400 per kWh.

Table 1: Optimal storage size and the value of storage

Number of leaf nodes: 2 3 4

(a) Zero minimum demand

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

2. Optimal operating 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 ($ per kWh): Line 6/Line 1 1,133.7 1,243.2 1,318.9

(b) Demand shifted by 6 MWh

1. Optimal total storage size (MWh) 54.7 42.6 38

2. Optimal operating cost (million $) 138.262 133.058 129.460

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 ($ per kWh of storage) 1,102.9 1,212.8 1,256.6

(c) Demand shifted by 15 MWh

1. Optimal total storage size (MWh) 52.8 41.1 36

2. Optimal operating cost (million $) 211.622 206.173 202.720

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 ($ per kWh of storage) 1,075.4 1,172.8 1,231.7

28

6. Conclusions

The goal of this paper is to improve the theoretical understanding on the operations and distributed

investment of energy storage on a tree network. For given storage investment, we have derived

structures of the optimal operating policy under convex generation costs, stochastic demand, storage

and line losses. We also derived two principles that govern the optimal storage operations: balanced

leaf storage levels are desirable; centrally stored energy offers more operational flexibility. We then

compare the net benefits of various storage investment strategies.

Our investment analysis reveals that, for a system with positive minimum demand, pooling all

investment centrally is suboptimal, and localized investment tends to be optimal when the minimum

demand is high or the spatial correlation of demand is high. On the other hand, centrally located

storage benefits the system by increasing storage usage frequency and reducing the need for leaf-

to-leaf energy transfer. We also find that the total investment and operating cost is more sensitive

to the total storage size than to the allocation of storage between central and leaf nodes. Finally,

although more demand nodes dampens the overall demand variability, requiring less storage capacity,

the average value of storage capacity actually increases.

We discuss several limitations of our model and analysis. First, the height of the tree model is

one. It would be interesting to consider a higher tree, where storage can be placed at nodes of various

depth. Nodes of medium depth (neither root nor leaves) might blend the advantages of centralized

and localized storage. Second, although our theoretical model allows for general demand processes,

our numerical analysis uses a two-period cyclic demand model to ease computational burden. Third,

although optimizing storage operations is greatly simplified by the structural policies, the state

space is still large: the dynamic program for a tree with n leaf nodes has a state space of 2n + 1

dimensions. We leave further numerical analysis on larger networks with general demand processes

for future research.

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31

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) is decreasing and convex in st+1 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) = minst+1∈A

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

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

([q(st+1 − st,dt)]

+)+ γEt

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

], 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) = c(0) + γEt

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

]≥ c(0) + γEt

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

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

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

Therefore, when minimizing ft(st+1, st,dt) 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 state (st,dt), let s∗t+1 be an optimal decision found by (A.1). For any st ≥ st,

Vt(st,dt) = ft(s∗t+1, st,dt) ≥ ft(s

∗t+1, st,dt) ≥ Vt(st,dt),

where the first inequality is because c([q(∆s,dt)]

+)increases in ∆s. Thus, Vt(st,dt) decreases in st.

To prove the convexity of Vt(st,dt) in st, 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)

]is convex in st+1 by the

induction hypothesis. Therefore, ft(st+1, st,dt) is jointly convex in (st+1, st) on the closed convex set

A × A. Using the theorem on convexity preservation under minimization from Heyman and Sobel

(1984, p. 525), we conclude that Vt(st,dt) as minimized in (A.1) is convex in st.

(ii) Because q(st+1 − st,dt) increases in st+1 and Vt+1(st+1,dt+1) decreases in st+1 (part (i)), the

subproblem in (9) with equality constraint (10) is equivalent to the following problem with an

inequality constraint:

Wt(qt, st,dt) = minst+1∈A

{Et

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

]: q(st+1 − st,dt) ≤ qt

}. (A.2)

As qt increases, the feasible set in (A.2) expands, and thus Wt(qt, st,dt) decreases in qt.

To show convexity, note that the set Ydef= {(qt, st+1) : qt ∈ Q(st,dt), st+1 ∈ A, q(st+1 − st,dt) ≤

qt} is a closed convex set. From Lemma 1, the objective Et

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

]in (A.2) is convex in

st+1, and thus it is also convex on the set Y. Using the theorem on convexity preservation under

1

minimization from Heyman and Sobel (1984, p. 525), we conclude Wt(qt, st,dt) is convex in qt.

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:

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.3)

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.4)

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.5)

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.4):

• 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.3) 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: Symmetry in leaf nodes (S1 = S2 = · · · = Sn and (14)) implies that the cost

function is also symmetric: Et

[Vt+1

(st+1,dt+1

)]= Et

[Vt+1

(st+1,dt+1

)], where s0,t+1 = s0,t+1 and

2

(si,t+1, i ∈ L) is any permutation of (si,t+1, i ∈ L).

Suppose sb1 is a minimizer for (11) (thus feasible for (15)), but it is not a minimizer for (15).

Then, there exists sb2 that is feasible and achieves a lower objective in (15). Such sb2 can be obtained

by decreasing sb1,p and increasing sb1,q for some p, q ∈ L, i.e., sb2,p = sb1,p − ε, sb2,q = sb1,q + ε, and

sb2,i = sb1,i for i ∈ L and i 6= p, q, and 0 < ε < (sb1,p − sb1,q)/2.

Next, swap sb1,p and sb1,q and define the new vector as sb1 . Similarly, swap sb2,p and sb2,q and define

the new vector as sb2 . Notice that sb2 and sb2 each are convex combination of sb1 and sb1 . Furthermore,

sb2 + sb2 = sb1 + sb1 . Therefore, we have

Et

[Vt+1(s

b

1 ,dt+1)]=

1

2

(Et

[Vt+1(s

b

1 ,dt+1)]+ Et

[Vt+1(s

b1 ,dt+1)

])

≥1

2

(Et

[Vt+1(s

b

2 ,dt+1)]+ Et

[Vt+1(sb2 ,dt+1)

])

= Et

[Vt+1(s

b

2 ,dt+1)].

Because sb1 is a minimizer for (11), the above inequality must hold with equality, i.e., sb2 is also a

minimizer for (11) and achieves a lower objective value in (15). Continuing this procedure, we can

identify a constrained balanced inventory that is also a minimizer for (15).

Finally, if allocating the total leaf inventory z equally across all leaf nodes is feasible, i.e., x ≤(x0, z/n, . . . , z/n

)≤ y, then by the symmetry and convexity of the objective function, such an equal

allocation minimizes the expected cost.

Proofs of Propositions 1 and 2: Overview and Preliminaries

These two propositions provide structures of the optimal solution to (9), which is equivalent to (A.2),

where we minimize a convex function over a convex set. Thus, to show that a solution is optimal,

we only need to prove that it achieves a local minimum in (9).

Using the definition from (4), the set of feasible st+1 for (9) is the iso-production hypersurface

A(q) ={st+1 ∈ A : ψα(s0,t+1 − s0,t) +

∑i∈L

ψβ

(di,t + ψα(si,t+1 − si,t)

)= q

}, (A.6)

which separates A into two parts (production < q and > q). Note that A(qt) is a piecewise linear

hypersurface in A, because ψα(s0,t+1− s0,t) is piecewise linear in s0,t+1 with slopes α and α−1 (slope

changes at s0,t+1 = s0,t), and ψβ

(di,t + ψα(si,t+1 − si,t)

)is piecewise linear in si,t+1 with slopes αβ,

αβ−1, and α−1β−1 (slope changes at si,t+1 = si,t − di,t/α and si,t+1 = si,t; if si,t − di,t/α ≤ 0, the

segment with slope αβ does not exist). For ease of exposition, we refer to any linear hypersurface of

A(q) as a face (which has n dimensions), and the intersection of any two adjacent faces as an edge

(which has n− 1 dimensions)

To prove local minimum, we show that the objective value Et

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

]in (9) increases

as st+1 deviates from the prescribed s∗t+1 (or the set containing s∗t+1). We prove this using two steps:

3

Step 1. Find all faces of A(qt) that intersect the prescribed s∗t+1 (or the set containing s∗t+1), and

identify the edges formed by these faces.

Step 2. Prove that the objective value Et

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

]increases when st+1 moves away from

s∗t+1 (or the set containing s∗t+1) in the directions parallel to any of the edges identified in Step 1.

(We in fact prove a stronger result that Vt+1(st+1,dt+1) increases for any realization of dt+1.)

Steps 1 and 2 prove local minmum because from s∗t+1 (or the set containing s∗t+1) we can reach

any point in any neighboring face identified in Step 1 by taking at most n moves parallel to the edges

of the faces; each move increases the objective value, as shown in Step 2.

Instead of repeating Step 1 for every case, we first identify all possible faces and edges of A(qt).

To find faces, we can consider A(qt) as consisting of two parts: a part with s0,t+1 > s0,t (store at

node 0) and another part with s0,t+1 < s0,t (release at node 0). The boundary of the two parts,

which contains the edges formed by faces from both parts, corresponds no change in storage level at

node 0.

Let k index the faces of A(qt). Face k satisfies (A.6), which can be expressed as a linear equation

ak· st+1 ≡∑

i∈L∪{0}

ak,isi,t+1 = bk, for st+1 ∈ face k,

where ak,0 is either α or α−1, while ak,i, i ∈ L, takes three possible values: αβ, αβ−1, or α−1β−1.

These values are exactly the slopes discussed after (A.6).

For the part of A(qt) with s0,t+1 > s0,t, we have ak,0 = α−1, while ak,i, i ∈ L, have 3n com-

binations. Thus, this part of A(qt) has up to 3n faces (the actual number of faces depends on qt).

For n = 2, the contours of the 9 possible faces are shown in Figure A.1(a), labeled from 0 to 8.

The other part of A(qt) with s0,t+1 < s0,t consists of faces with ak,0 = α. These faces are shown in

Figure A.1(b) and labeled from 0′ to 8′. Note that A(qt) cannot contain the lower-left area because

s0,t+1 < s0,t implies that∑i∈L

ψβ

(di,t + ψα(si,t+1 − si,t)

)> 0 due to (A.6).

Let {e(m)ij : m = 1, . . . , n − 1} denote a basis for the (n− 1)-dimensional vector space parallel to

the edge formed by faces i and j. Because all coefficients ak,i > 0, we can always choose the basis

such that e(m)ij contains exactly two non-zero elements, with one being −1 and the other belongs to

(0, 1]. For n = 2 and α ≤ β, these basis are shown as vectors in Figure A.1; we omit index m because

each edge has only one dimension.

We next prove a lemma on how the value function changes along the directions of these basis.

Lemma A.1 (In this lemma, ‘. . . ’ represents omitted zeros.) For any dt+1, we have

(i) Vt+1(st+1,dt+1) increases as st+1 moves along e(m)ij = (0, . . . ,−1, . . . , β2, . . . ) or (0, . . . , β2, . . . ,−1, . . . ).

(ii) Vt+1(st+1,dt+1) increases as st+1 moves along e(m)ij = (β, . . . ,−1, . . . ) or (−1, . . . , β, . . . ).

4

Figure A.1: Contours of A(qt), faces, and edges: n = 2

This figure illustrates the iso-production surface ψα(s0,t+1 − s0,t) +∑

i=1,2

ψβ

(di,t + ψα(si,t+1 − si,t)

)= q.

Each contour line represents a fixed s0,t+1 level; the lower-left contour line has the highest s0,t+1 level.

If si,t − di,t/α ≤ 0, then the faces between 0 and si,t − di,t/α do not exist.

(a) s0,t+1 > s0,t: store energy at node 0

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���� = 0, �, −1 ���� = 0, ��, −1

5

(iii) Vt+1(st+1,dt+1) increases as st+1 moves along e(m)ij = (α2β, . . . ,−1, . . . ) or (−1, . . . , α2β, . . . )

or (0, . . . ,−1, . . . α2β2, . . . ) or (0, . . . , α2β2, . . . ,−1, . . . ).

(iv) If α ≤ β, then Vt+1(st+1,dt+1) increases as st+1 moves along e(m)ij = (α2/β, . . . ,−1, . . . ) or

(0, . . . ,−1, . . . α2, . . . ) or (0, . . . , α2, . . . ,−1, . . . ).

Proof of Lemma A.1: Parts (i) and (ii) follow directly from Lemma 2(i) and (ii), respectively.

For part (iii), suppose st+1 moves in the direction (α2β, . . . ,−1, . . . ) by a small amount δ > 0.

Then,

Vt+1(st+1,dt+1) ≤ Vt+1(st+1 + (βδ, . . . ,−δ, . . . ),dt+1) ≤ Vt+1(st+1 + (α2βδ, . . . ,−δ, . . . ),dt+1),

where the first inequality is due to Lemma 2(ii) and the second inequality follows from α ≤ 1 and

the monotonicity in Lemma 1. The proof for all other directions in part (iii) are similar.

For part (iv), suppose st+1 moves in the direction (α2/β, . . . ,−1, . . . ) by a small amount δ > 0.

Then,

Vt+1(st+1,dt+1) ≤ Vt+1(st+1 + (βδ, . . . ,−δ, . . . ),dt+1) ≤ Vt+1(st+1 + (α2/β · δ, . . . ,−δ, . . . ),dt+1),

where the second inequality follows from α ≤ β (so that α2/β < β) and the monotonicity in Lemma 1.

The proof for all other directions in part (iv) are similar.

Proof of Proposition 1: For qt ∈ (qot , qt), per proposition, s∗t+1 = st +(α(qt − qot ), 0, . . . , 0

), i.e.,

serve all demand using qot and store remaining qt − qot at node 0; no operations at the leaf storage.

Possible directions of deviation from this prescribed solution includes (−1, . . . , β, . . . ), i.e., move

some energy from central storage to a leaf storage, (α2/β, . . . ,−1, . . . ), i.e., use some local storage

and store more at the central, and (0, . . . ,−1, . . . α2, . . . ), i.e., use some local storage and store at

another leaf. Lemma A.1 asserts that Vt+1(st,dt) increases along these directions, which ensures the

local optimality of s∗t+1.

For qt ∈ (qt, qot ), per proposition, s∗t+1 = sb(st, st, z,dt). From the definition of sb(st, st, z,dt)

in (11), s∗t+1 minimizes Et[Vt+1(st+1,dt+1)] within a set X ={st+1 : s0,t+1 = s0,t, si,t+1 ∈

[(si,t −

di,t/α)+, si,t

],∑i∈L

si,t+1 = z}. (For n = 2, the setX is the intersection of faces 0 and 0′.) We only need

to show that Vt+1(st+1,dt+1) increases as st+1 moves away from X. Possible directions of deviation

includes (−1, . . . , β, . . . ), i.e., use some energy from central storage instead of local storage to serve

demand, (0, . . . ,−1, . . . , β2, . . . ), i.e., use remote storage, (α2/β, . . . ,−1, . . . ), i.e., store energy at

the central node instead of sending to leaves. Lemma A.1 ensures that Vt+1(st,dt) increases along

these directions.

For q = qt, per proposition, s∗t+1 = st, i.e., the set X shrinks to a point, which leads to additional

directions of deviation: (α2β, . . . ,−1, . . . ), i.e., release energy from a leaf storage and store it at the

6

central. Similarly, for q = qot , the set X shrinks to s∗t+1 = st, which leads to additional directions:

(−1, . . . , α2β, . . . ), i.e., release energy from the central and store it at a leaf. Lemma A.1 ensures

that Vt+1(st,dt) increases along these directions.

Proofs for the other cases of Proposition 1 are parallel.

Proof of Proposition 2: The proof for the case of qt ∈ [qt, qt] remains same as Proposition 1

because Vt+1(st,dt) increases along all directions for qt ∈ [qt, qt], irrespective of the relative value of

α and β.

In Proposition 1, the condition α ≤ β (thus α2/β ≤ β) is crucial for Vt+1(st+1,dt+1) to increase

along the direction (α2/β, . . . ,−1, . . . ). When α > β, however, this result may not hold. In fact,

if α = 1, this direction becomes (1/β, . . . ,−1, . . . ) or (1, . . . ,−β, . . . ). Lemma 2 suggests that

Vt+1(st+1,dt+1) decreases when st+1 moves in the direction of (1, . . . ,−β, . . . ).

Equations (17b) and (18c): Case of qt < qt ≤ qt

When α ≤ β and min{qot , qt} < qt ≤ qt, Proposition 1 prescribes st +(α(qt − qot ), 0, . . . , 0

).

However, when α = 1, Vt+1(st+1,dt+1) decreases as st+1 moves along (α2/β, . . . ,−1, . . . ), i.e., use

local storage and store energy at the central node. Moving along these directions until local storage

is fully used, we reach(s0,t + qt − qt, (si,t − di,t)

+, i ∈ L), as prescribed in (17b).

Similarly, when α ≤ β and qt < qt ≤ min{qot , qt}, from the proof of Proposition 1, s∗t+1 lies within

the set X ={st+1 : s0,t+1 = s0,t, si,t+1 ∈

[(si,t − di,t/α)

+, si,t],∑i∈L

si,t+1 = z}. From X, moving

along (α2/β, . . . ,−1, . . . ) reduces Vt+1(st+1,dt+1), until we reach(s0,t + qt − qt, (si,t − di,t)

+, i ∈ L)

in (17b).

When β < α < 1, the solution s∗t+1 lies between the solution for the case of α ≤ β and the

solution for the case of α = 1. That is, (part or all) local storage is used while some energy is stored

at the central node. Thus, the solution lies in the set Ft defined in the proposition; this proves (18c).

Equations (17a) and (18b): Case of qt < qt < qt

In this case, the set Ft contains candidate solutions that involve filling up the central storage

while partially using the local storage at some nodes. Rebalancing local storage levels can involve

moving along the direction (0, . . . ,−1, . . . α2, . . . ), which means using more local storage at some

nodes while increasing the storage level at another leaf node. These candidate solutions are part of

the set Et.

Therefore, when β < α < 1, the solution s∗t+1 belongs to Ft ∪ Et. When α = 1, s∗t+1 belongs to

Et, which becomes a linear surface. Minimizing Et[Vt+1(st+1,dt+1)] within this linear surface gives

sb(x,y, z,dt), which is the result in (17a) in Proposition 2.

Equations (17a) and (18a): Case of qt ≤ qt < qt

7

When qt > qt, Ft no longer exists. Thus, s∗t+1 ∈ Et as in (18a). Further, when α = 1, Et becomes

linear and thus s∗t+1 can be expressed using sb(x,y, z,dt).

Proof of Lemma 4: For a given S, we use s∗t+1 as short for an optimal decision rule s∗t+1(st,dt;S),

and let {s∗t : t ∈ T } be an optimal policy. For any S ≥ S, we can construct a feasible policy:

{s∗t = s∗t + S− S : t ∈ T }. The two policies yield the same inventory changes, ∆s∗t = ∆s∗t , and thus

the same expected operating cost. Therefore, V (S) ≤ V (S).

Using inductive proof similar to that for Lemma 1, we can show that Vt(st,dt;S) is convex in

(st,S) for any t ∈ T . In particular, V1(s1,d1;S) is convex in (s1,S). Hence, V (S) = EV1(S,d1;S) is

convex in S.

Proof of Lemma 5: For a given S, we use s∗t+1 as short for an optimal decision rule s∗t+1(st,dt;S).

Let {s∗t : t ∈ T } be an optimal policy, and let u∗i,t = di,t + ψα(∆s∗i,t), i ∈ L, be the corresponding

energy flows according to (3).

(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.7)

g1 =∑i∈L

Si, (A.8)

gt+1 = min{ ∑

i∈LSi, gt +

( ∑i∈L

∆s∗i,t)− 1−β2

α

∑i∈L

min{u∗i,t, 0}}. (A.9)

At t = 1, (A.7) and (A.8) imply that s0,1 = s∗0,1 + β−1∑

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

The definition in (A.9) implies gt ∈[∑

i∈L s∗i,t,

∑i∈L Si

]for all t ∈ T .2 Hence, 0 ≤ st ≤ Sc, thus

the constructed policy {st : t ∈ T } is feasible under Sc.3 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.10)

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

2We 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+1.3We 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.

8

the second inequality is because u∗i,t ≥ 0 and (A.9) 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 < 0.4 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).

(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.11)

gj,1 = S0, j ∈ L, (A.12)

∆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.13)

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

Proof of Proposition 3: 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.14)

Furthermore, (A.14) holds with equality if di,t = ki d1,t and Si = ki S1, for all i ∈ L. This can be

4To see this, note that the last two terms in (A.9) 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.

9

shown by using the optimal policy for the new system to construct a feasible policy for the original

system that yields the same operating cost. The construction maintains the leaf 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.15)

Following the reasoning after Proposition 3 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.16)

Because S is arbitrary, we conclude from (A.16) that the localized investment is optimal.

The proof of Lemma 6 requires some properties of the optimal operating policy and the value

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

Lemma A.2 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.2: 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

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st+1 for (s,d) gives the same operating 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:

• 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.17)

= −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 6: 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.2(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.17), 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 operating 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 4: (i) Because C(S) ≡ p |S| + V (S) is convex in S (Lemma 4), 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 .

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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 6(i) and δ0 ≥ βδ0, and the last inequality follows from

optimality of Sl∗ for the constrained investment problem (20). This proves the optimality of Sl∗.

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 6(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 5: 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

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.3 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.3: 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+1

an

]<

bNbN1

N1∑n=1

anbn +N∑

n=N1+1

anbn <ε

2MM +

ε

2= ε. (A.18)

Hence the limiting result holds.

Proof of Proposition 6: To prove this proposition, we first show

limp→0

p |S∗(p)| = 0. (A.19)

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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 5(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.3, 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 5(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 .

Note that S∗0 is a function of p, and lim

p→0p S∗

0(p) = 0 due to (A.19). 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 5(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.19), we have

limp→0

C(S∗(p))− C(Sc∗(p)) = 0.

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