# Optimal Investment Planning of Bulk Energy Storage Systems

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Knowledge Gap

#### 1.3. Modeling Methodology

#### 1.4. Contribution

- First, the investment planning includes other sources of the flexibility of the system (hydro power and consumption flexibility) which can create competition for energy storage systems and affect the revenue stream.
- Second, energy storage investments are made along with renewable generation expansion and takes into account renewable generation targets present nowadays in Europe and USA.
- Third, the decentralized planning model in addition to investment return constraints includes payback period constraint which make the simulation of investment decision on energy storage closer to real life investment planning. Moreover, a solution to the bilevel problem has been suggested.
- Fourth, the paper presents a comparative analysis based on several case studies of systems with different generation mix and different levels of congestion. The results contribute to an understanding of the benefits of energy storage under different planning strategies and dependency of existing flexibility and type of flexibility on the profitability of the energy storage and possible effect on system congestion.

#### 1.5. Structure of the Paper

## 2. Energy Storage Investment Decision and Allocation Problem

- The energy storage investor can choose between energy storage modules of different technologies, where each module has fixed energy capacity, power capability and other technical parameters such as self-discharge and efficiency.
- The energy storage charge and discharge efficiency as well as the self-discharge rate are fixed parameters and do not vary based on the charge/discharge output level or the energy level of the storage.

- The centralized player is responsible for generation expansion investments into renewable energy.
- An independent investor will only make investments which will reach break-even within a given time. For example, typical expected payback period in long term investment planning is five years.
- An independent investor has a lower limit on minimum investments returns.
- The financial benefit of the energy storage is obtained through energy arbitrage.
- The energy storage utility can exchange information with the centralized player which is in charge of the optimal dispatch of the generation, flexible load and energy storage in the system. The centralized player receives information from energy storage owner about invested and available energy storage energy capacity and power capability of each unit. On the other hand the centralized player provides information about dispatch of each energy storage unit and electricity prices.
- The power system is represented by a DC load flow model.

#### 2.1. Centralized Energy Storage Investment Decision and Allocation Problem

#### 2.2. Independent Investment Planning. MPEC Model

#### 2.3. Strong Duality Condition

#### 2.4. Reformulation of the Objective Function

#### 2.5. One-Level Problem Formulation

## 3. Case Study

#### 3.1. System Description

#### 3.2. Results and Discussions

## 4. Conclusions

- First, energy storage can be beneficial to the whole system by reducing spillage of renewable generation and relieving congestion of transmission capacity under both centralized and decentralized planning approaches. However, there are still a big gap between centralized and decentralized planning approaches. More investments are made under centralized planning and the cost and the average price reduction under centralized planning is much higher.
- Second, if treated as a market asset (decentralized planning) energy storage can profit from strategically placing energy storage units and contribute on increase to transmission congestion of power system and additional wind spillage.
- Third, negative impact of strategic behavior of energy storage can be reduced if renewable generation decisions are taken simultaneously.
- Fourth, the case studies demonstrate that decentralized unregulated allocation planning for energy storage potentially may cause congestion in the system. Thus, additional studies on proper regulation for energy storage is necessary.

- First, proposed decentralized model considers monopoly on energy storage investments and does not take into account additional competition from investments made on other flexibility sources such as hydro, flexible demand or flexible generators. Thus, an EPEC model could be developed to coordinate the investment and evaluate the dependency.
- Second, the models consider only one revenue stream which comes from providing energy arbitrage, however additional revenue streams such as provision of balancing services should be also considered to further evaluate the profitability of energy storage.
- Third, the initial formulation of the decentralized planning model is presented as a mixed integer non-linear bilevel model and later reformulated as a mixed integer linear one-level problem. The suggested technique for reformulation and linearization reduces the complexity of the model and makes it possible to find an optimal solution with reasonable computational time. However, the linearized model is still complex and a higher number of nodes and decision variables will increase the computational time. In order to apply the models to larger systems, it could be beneficial to investigate decomposition techniques (ex. Benders decomposition).
- Fourth, the choice of the number of days and operational hours also affects the computational time and the energy storage evaluation require rather large operational period to observe the charge and discharge cycles. Thus, the selection of the critical operational periods for energy storage evaluation is also a subject for future research.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

Indices | |

d | Demand; |

e | Energy storage systems; |

h | Hydro based generation; |

j | Thermal generation; |

k | Operation period (seasons); |

l | Operation period (hours); |

$n,m$ | Nodes of the system; |

p | Superset for $l,k,t$; |

s | Scenarios; |

t,${t}^{*}$ | Planning period (years); |

w | Wind based generation; |

Binary Variables | |

${y}_{e}$ | Energy storage investment decision variable |

Continuous Variables | |

$\uparrow \Delta {d}_{d},\downarrow \Delta {d}_{d}$ | Up and down regulated flexible load, ($MW$); |

${f}_{n,m}$ | Power flow between node n and m, ($MW$); |

${g}_{e}^{ch}$, ${g}_{e}^{dch}$ | Charge and discharge of energy storage, ($MW$); |

${g}_{j}$,${g}_{h}$,${g}_{w}$ | Output of thermal, hydro and wind generation units, ($MW$); |

${G}_{w}^{max}$ | Expanded wind generation capacity, ($MW$); |

${s}_{h}$ | Spillage of a hydro unit h; |

$SO{C}_{e}$ | State of charge of an energy storage, ($MWh$); |

${u}_{h}$ | Hydro discharge of a hydro unit h |

${v}_{h}$ | Reservoir level of a hydro unit h; |

${\theta}_{n}$ | Voltage angles at node n, ($p.u$); |

${\lambda}_{n}$ | Price at node n, ($\$/MW$); |

${\lambda}_{e}^{SOC}$ | Lagrange multipliers, ($\$/MW$), for energy balance constraint for ES; |

${\lambda}_{n,m}^{Line}$ | Lagrange multipliers, ($\$/MW$), for power flow constraints; |

${\lambda}_{d}^{D}$ | Lagrange multipliers, ($\$/MW$), for demand response constraints; |

${\lambda}_{h,p,s}^{Gen}$ | red Lagrange multipliers,($\$/MW$), for hydro power generation constraint |

${\lambda}_{h}^{res}$ | Lagrange multipliers, ($\$/MW$), for hydrological balance constraints; |

$\tau {0}_{w},{\tau}_{w}$ | Lagrange multipliers ($\$/MW$) for generation investment constraints; |

${\underline{\xi}}_{e}^{ch},{\overline{\xi}}_{e}^{ch}$ | Lagrange multipliers, ($\$/MW$), for energy storage charge constraints; |

${\underline{\xi}}_{e}^{dch},{\overline{\xi}}_{e}^{dch}$ | Lagrange multipliers, ($\$/MW$), for energy storage discharge constraints; |

${\underline{\mu}}_{n,m},{\overline{\mu}}_{n,m}$ | Lagrange multipliers, ($\$/MW$), for line constraints; |

${\overline{\sigma}}_{h},{\underline{\sigma}}_{h}$ | Lagrange multipliers, ($\$/MW$), for water reservoir volume constraints; |

${\underline{\nu}}_{j},{\overline{\nu}}_{j}$ | Lagrange multipliers, ($\$/MW$), for generator j constraints; |

${\underline{\nu}}_{h},{\overline{\nu}}_{h}$ | Lagrange multipliers, ($\$/MW$), for generator h constraints; |

${\underline{\nu}}_{w},{\overline{\nu}}_{w}$ | Lagrange multipliers, ($\$/MW$), for generator w constraints; |

${\underline{\kappa}}_{j},{\overline{\kappa}}_{j}$ | Lagrange multipliers, ($\$/MW$), for generator j constraints; |

${\underline{\kappa}}_{h},{\overline{\kappa}}_{h}$ | Lagrange multipliers, ($\$/MW$), for generator h constraints; |

${\underline{\rho}}_{n},{\overline{\rho}}_{n},\rho $ | Lagrange multipliers, ($\$/MW$), for voltage angle constraints; |

$\uparrow {\underline{\omega}}_{d},\uparrow {\overline{\omega}}_{d}$ | Lagrange multipliers, ($\$/MW$), for demand d constraints; |

$\downarrow {\underline{\omega}}_{d},\downarrow {\overline{\omega}}_{d}$ | Lagrange multipliers, ($\$/MW$), for demand d constraints; |

${\underline{\gamma}}_{e},{\overline{\gamma}}_{e}$ | Lagrange multipliers, ($\$/MW$) for energy storage $SOC$ constraints; |

${\underline{\vartheta}}_{h},{\overline{\vartheta}}_{h}$ | Lagrange multipliers, ($\$/MW$), for spillage constraints; |

${\underline{\theta}}_{h},{\overline{\theta}}_{h}$ | Lagrange multipliers, ($/MW), for water flow constraints; |

${\beta}_{t}$ | Lagrange multipliers for renewable target constraint; |

Parameters | |

${C}_{w}$ | Capital cost of wind gen. expansion, ($\$/MW$); |

${C}_{e}$ | Capital cost of energy storage block, ($\$/block$); |

${D}_{d}$ | Non-dispatchable load, ($MW$); |

${D}_{d}^{max}$, ${D}_{d}^{min}$ | Limits of flexible load, ($MW$); |

$dis$ | Discount rate; |

$FC$ | Expected future cost of electricity for period k ; |

$f{l}_{h}$ | Inflow of hydro unit h; |

${\sigma}_{h}$ | Efficiency of hydro unit h; |

${\tau}_{{h}^{*}}$ | Hydro discharge time delay; |

${G}_{j}^{max}$,${G}_{h}^{max}$,${G}_{e}^{max}$ | Upper generation limits, ($MW$); |

${G}_{j}^{min}$,${G}_{h}^{min}$,${G}_{e}^{min}$ | Lower generation limits, ($MW$); |

${G}_{w}$ | Existing capacity of wind power generation (MW); |

$inf$ | Annual inflation rate; |

${I}_{n,j}$,${I}_{n,h}$,${I}_{n,w}$ | Incidence matrix for thermal, hydro and wind generation units; |

${I}_{n,e}$ | Incidence matrix for energy storage units; |

${I}_{n,d}$ | Incidence matrix for flexible demand units; |

${I}_{n,m}$ | Incidence matrix for transmission; |

$IR$ | Investments return coefficient; |

L | Operation time period; |

M | Big-M parameter,sufficiently large number; |

$PBP$ | Payback period, (years) ; |

$P{w}_{t}$ | Present worth factor ; |

$R{G}^{min}$ | Renewable generation penetration target ; |

$R\_Total$ | Total expected ramping capability of a system ; |

$R{U}_{j}^{max}$,$R{U}_{h}^{max}$ | Ramp-up hourly limits, ($MW$); |

$R{D}_{j}^{max}$,$R{D}_{h}^{max}$ | Ramp-down hourly limits, ($MWh$); |

${S}_{h}^{max}$ | Maximum spillage of hydro units; |

$SO{C}_{e}^{max}$ | Storage capacity, ($MWh$); |

${T}_{n,m}^{max}$ | Transmission line capacity, ($MW$); |

T | Investment planning period; |

$RTY$ | Target year for renewable generation penetration; |

${U}_{h}^{max}$ | Maximum flow of hydro units; |

${V}_{h}^{max}$ | Maximum reservoir of hydro units; |

$w{p}_{p,s}$ | Wind power output for each scenario as percentage of capacity; |

${\u03f5}_{e}$ | Energy conversion efficiency; |

${\varphi}_{e}$ | Self discharge of energy storage; |

${\pi}_{s}$ | Scenario probability; |

$m{c}_{e},m{c}_{j},m{c}_{d}$ | marginal costs of energy storage units,thermal units |

and flexible demand units | |

$\psi $ | Scaling factor for operation and investment values |

${\rm Y}(*)$ | 1 if * is true and 0 otherwise; |

## Appendix A. Stationary Conditions

## Appendix B. Complementary Slackness Conditions for Lower Level Problem

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Thermal | Hydro-Thermal | |||
---|---|---|---|---|

Capacity | Node | Capacity | Node | |

Thermal, (MW) | 600 | 1, 2, 22, 27, 23, 13 | 300 | 1, 2, 23 |

Hydro, (MW) | - | - | 300 | 22, 27 |

Wind, (MW) | 100 | 5, 6 | 100 | 5, 6 |

Max. demand, (MW) | 600 | - | 600 | - |

Flexible demand | 10% | - | 10% | - |

Transmission limits, (MW) | 100 | - | 100 | - |

Congested transmission limits, (MW) | 70 | - | 70 | - |

Ramping capability, (MW) | 300 | - | 420 | - |

Renewable generation target | 20% | - | 20% | - |

Parameter | Value |
---|---|

Annual inflation rate, (inf) | 2% |

Discount rate, (dis) | 10% |

Technology | CAES | Battery |
---|---|---|

Energy storage capacity, $SO{C}_{e}^{max}$, (MWh) | 100 | 15 |

Power limit, ${G}_{e}^{max}$, (MW) | 20 | 6 |

Energy conversion efficiency, ($\epsilon $) | 0.75 | 0.85 |

Self discharge of energy storage, ($\eta $) | 0.78 | 0.99 |

Initial state of charge | 50% | 50% |

Capital cost, Energy, ($/kWh) | 5 | 400 |

Capital cost, Power, ($/kW) | 700 | 400 |

Maximum number of units | 5 | 10 |

Parameter | Value |
---|---|

Planning period, (T) | 10 years |

Investments return parameter, ($IR$) | 1.2 |

Payback period limit, ($PBP$) | 5 |

Short term operation period, (l) | 74 h |

Renewable penetration target year, ($RTY$) | 5th year |

Congested | Not Congested | |||||
---|---|---|---|---|---|---|

T | T+D | H-T | H-T+D | T | H-T | |

Case 1. Base case (no ES investments) | ||||||

Available capacity, (MW) | 600 | 600 | 600 | 600 | 600 | 600 |

Wind power investments, (MW) | 288 | 274 | 273 | 271 | 281 | 271 |

Ramping capability, (MWh) | 300 | 360 | 420 | 480 | 300 | 420 |

Number of congested lines | 3 | 1 | 3 | 1 | 0 | 0 |

Curtailed wind, (%) | 10 | 8 | 11 | 6 | 7 | 5 |

Std of price | 14.2 | 13.1 | 5.2 | 5.7 | 13.3 | 4.5 |

Average price, ($) | 52 | 45 | 32 | 32 | 45 | 34 |

Case 2. Centralized planning | ||||||

Wind power investments, (MW) | 241 | 235 | 232 | 238 | 248 | 236 |

Energy storage investments, (MW) | 115 | 45 | 45 | 45 | 100 | 30 |

Ramping capability, (MWh) | 390 | 390 | 455 | 510 | 384 | 447 |

Std of price | 4.3 | 3.9 | 3.4 | 3.1 | 4.3 | 3.1 |

Average price, ($) | 40 | 40 | 30 | 30 | 40 | 30 |

Number of congested lines | 1 | 0 | 1 | 0 | 0 | 0 |

Curtailed wind, (%) | ≥1 | ≥1 | ≥1 | ≥1 | ≥1 | ≥1 |

Case 3. Decentralized planning | ||||||

Wind power investments, (MW) | 257 | 241 | 273 | 271 | 270 | 271 |

Energy storage investments, (MW) | 90 | 45 | 0 | 0 | 45 | 0 |

Ramping capability, (MWh) | 330 | 390 | 420 | 480 | 390 | 420 |

Std of price | 6.2 | 5.4 | 5.2 | 5.7 | 5.2 | 4.5 |

Average price, ($) | 44 | 40 | 35 | 35 | 40 | 34 |

Number of congested lines | 2 | 1 | 2 | 1 | 0 | 0 |

Curtailed wind, (%) | 6 | 6 | 9 | 6 | 7 | 4 |

Case 4. Decentralized planning with fixed wind investments | ||||||

Energy storage investments, (MW) | 90 | 45 | 45 | 45 | 100 | 30 |

Ramping capability, (MWh) | 330 | 390 | 420 | 480 | 384 | 447 |

Std of price | 6.9 | 6.1 | 5.8 | 5.9 | 4.3 | 3.1 |

Average price, ($) | 46 | 41 | 38 | 37 | 40 | 30 |

Number of congested lines | 3 | 1 | 3 | 1 | 0 | 0 |

Curtailed wind, (%) | 6.5 | 6.8 | 9 | 6 | ≥1 | ≥1 |

Congested | Not Congested | |||||||
---|---|---|---|---|---|---|---|---|

Centralized | Decentralized | Centralized | Decentralized | |||||

CAES | Bat. | CAES | Bat. | CAES | Bat. | CAES | Bat | |

Thermal system | ||||||||

Time period, (t) | t1 | t1 | - | t5 | t1 | t1 | - | t5 |

Node, (n) | 3 | 25 | - | 4, 6, 8 | 4 | 25 | - | 4, 8, 25 |

Number of units | 1 | 6 | - | 1 | 1 | 3 | - | |

Hydro-Thermal system | ||||||||

Time period, (t) | - | t1 | - | - | - | t1 | - | - |

Node, (n) | - | 3, 25, 15 | - | - | - | 3, 25 | - | - |

Number of units | - | 3 | - | - | - | 2 | - | - |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Khastieva, D.; Dimoulkas, I.; Amelin, M.
Optimal Investment Planning of Bulk Energy Storage Systems. *Sustainability* **2018**, *10*, 610.
https://doi.org/10.3390/su10030610

**AMA Style**

Khastieva D, Dimoulkas I, Amelin M.
Optimal Investment Planning of Bulk Energy Storage Systems. *Sustainability*. 2018; 10(3):610.
https://doi.org/10.3390/su10030610

**Chicago/Turabian Style**

Khastieva, Dina, Ilias Dimoulkas, and Mikael Amelin.
2018. "Optimal Investment Planning of Bulk Energy Storage Systems" *Sustainability* 10, no. 3: 610.
https://doi.org/10.3390/su10030610