Abstract
This paper presents a mixed-integer linear programming (MILP) formulation for sizing and siting of battery energy storage systems (BESSs). The problem formulation seeks to minimize both operation costs and BESS investment. The proposed model includes restrictions of the conventional security-constrained unit commitment problem, a piece-wise linear approximation to model power losses, and a linear model of hydro generation units. The proposed model is tested in a 6-bus test system and a 15-bus system representing the Colombian power system. For the two studied systems, simulation results show that the reduction of operation costs due to the installation of BESSs compensates the investments, under some of the considered technical cost cases. Additionally, results show that adequate sizing and siting of BESSs reduce renewable energy curtailment in the Colombian power system with high penetration of fluctuating renewable generation.
A. Avendaño Peña, D. Romero-Quete, and C. Cortes (corresponding author) are with the Department of Electrical and Electronics Engineering, Universidad Nacional de Colombia, Bogotá, Colombia (e-mail: aaavendanop@unal.edu.co; dfromeroq@unal.edu.co; caacortesgu@unal.edu.co).
BATTERY energy storage systems (BESSs) are increasingly used in electric power system applications, as quickly as their costs are decreasing. For instance, the Tranquillity project, with a capacity of 72 MW/288 MWh, was commissioned in California, USA as a support for the Tranquillity Solar Facility, as well as a support for future integration of new renewable sources [
In Colombia, for instance, only until 2018 did the CREG (the system regulator in Colombia) consider BESSs as an option to mitigate the delays in the expansion of the transmission system [
As a way to optimize different objectives, sizing and siting of BESSs play an important role in determining future investments in new elements of electrical networks. For instance, in [
In [
In [
Considering different stages of the traditional electricity supply chain and their unique features, models can represent the features of power systems with a great degree of accuracy. For instance, two different methods were proposed in [
In this paper, an MILP formulation is used as a tool to determine the sizing and siting of BESSs to minimize investment and operation costs in a system with an energy mix heavily dominated by hydro power or non-conventional renewable sources in the near future, such as the Colombian power system [
The rest of the paper is organized as follows. Section II describes the mathematical formulation of the optimization problem. Section III explains case studies that are used to test the mathematical model, and the results are also discussed in this section. Section IV presents the main conclusions.
The optimization problem aims to find the optimal sizing and siting of BESS units to minimize the overall operation cost in a power system. The mathematical formulation is partially based on [
The objective function of the model is to minimize both operation cost and BESS investment, which is formulated as:
(1) |
Term a is related to energy storage installation cost; term b is related to the operation cost of thermal generation units; and term c is related to the operation cost of hydro generation units. The functions and are used to account for the operation costs of thermal and hydro generation units, respectively. A piecewise linear upper approximation of the convex cost curve is used for the 6-bus test system as in [
The following constraint models the power balance in each node of the system, based on Kirchhoff’s current law.
(2) |
Adequate scheduling of generation units is very important to reduce the operation cost of the power system. To carry out this scheduling, it is necessary to consider operation times, ramp rate values, and initial states of generation units. Those requirements can be modeled as:
(3) |
(4) |
(5) |
(6) |
(7) |
(8) |
(9) |
(10) |
Constraint (3) maintains thermal generation units between technical limits. Constraints (4) and (5) limit ramp-up and ramp-down values of generation units. Constraint (6) relates operation statuses of generation units with binary UC variables. Constraint (7) guarantees that each generation unit is not turned on and off simultaneously. Ultimately, constraints (8)-(10) ensure the minimum up-time and down-time of generation units.
In power systems with high participation of hydro generation units, it is important to adequately represent their behavior. A linear approximation of the behavior of hydro generation units is given as:
(11) |
(12) |
(13) |
(14) |
(15) |
(16) |
Constraint (11) maintains hydro generation units between technical limits. Constraint (12) keeps water flow through the turbine between technical limits. Constraint (13) maintains water volume in the reservoir associated with the hydro generation unit between allowed limits. Constraint (14) limits spillage in the reservoir. Constraint (15) guarantees water conservation between two consecutive time periods. Finally, the electric power generated by hydro generation units is calculated with constraint (16).
Traditionally, the calculation of power flows implies non-linearity due to the use of trigonometric functions. DC power flow approximation is a common way to find power flows without non-linearities. The constraints that model this approximation are described as:
(17) |
(18) |
Power flow in each line is calculated by (17), while the maximum power flow for each line is limited by (18).
(19) |
(20) |
(21) |
(22) |
Calculation of power losses in (19) includes a non-linearity given by the quadratic expression in the angular difference. To solve this, [

Fig. 1 Piecewise linear loss function.
(23) |
(24) |
(25) |
(26) |
(27) |
(28) |
(29) |
Constraint (23) guarantees that BESS units are not charging and discharging at the same time. The relation of energy stored at two consecutive time steps is expressed in (24). In constraint (25), the energy stored in a BESS unit at time is limited by the size found for that BESS unit. Finally, constraints (26)-(29) restrict the power size of BESS units. Since BESS units are characterized by a high ramp rate [
The model described in Section II is tested in a 6-bus test system and in a simplified 15-bus Colombian power system. A comparison of the results with and without consideration of power losses, deactivating the term of power losses in constraints (2) and (18) and constraints (19)-(22), is also presented for the 15-bus Colombian power system. A time horizon of 720 hours (a month) with 1-hour intervals is used for all the simulations. A BESS charging/discharging efficiency of 95%, and a self-discharging efficiency of 0.0000625% are considered. To study the effect of capital recovery times and eventual technical cost drops in investment decisions, 6 cases with different equivalent costs per month are considered, as shown in
In these cases, the one closest to BESS costs in 2021 is case F, according to the information from Lazard in [
(30) |
The MILP problem is implemented in Pyomo (Python) and solved with the CPLEX solver by means of NEOS server [
The 6-bus test system proposed in [

Fig. 2 6-bus test system.

Fig. 3 Wind generation profile curve.
The sizing and siting results of BESS units for the 6-bus test system are shown in

Fig. 4 Operation characteristics of load and BESS unit at bus N5.
In case A (without BESS unit), the most expensive generator G2 produces the capacity of 12.502 GWh throughout the simulation horizon (720 hours); while for case F (with BESS unit), this generator only produces the capacity of 10.924 GWh. These results show that, due to the installation of BESS units, the generators with lower operation costs (G1 and G3) are dispatched more efficiently, thus reducing the power dispatched by G2, which has the highest operation costs, as shown in

Fig. 5 Comparison of dispatch solutions in cases A and F for 6-bus test system. (a) G1. (b) G2. (c) G3.
Finally,
The power system shown in

Fig. 6 Simplified Colombian power system.

Fig. 7 Load profile curves in both 2020 and 2030 scenarios.
Current installed capacity of energy mix in Colombia is strongly dominated by hydro generation, with 68.3% of participation, followed by thermal generation, with 30.6% of participation. The remaining 1.1% is composed of wind generation, solar generation, and co-generation [

Fig. 8 Solar generation profile curve.
The sizing and siting results of BESS units for cases A-F in 2020 scenario obtained by the formulation presented in Section II, with and without consideration of power losses, are shown in

Fig. 9 Operation characteristics of load and BESS units at bus BOL in 2020 scenario.
In this scenario, 19 thermal generation units, 23 hydro generation units, 4 wind farms (with a total of 2154 MW of installed power), 9 solar farms (with a total of 2917 MW of installed power), and 25 transmission lines are considered, based on the information presented in [
The evaluation results of operation costs, BESS investments, and total costs with and without consideration of power losses for different cases in 2030 scenario is shown in
It can be observed that a more efficient dispatch of generators, resulting in a lower operation cost, compensates the cost associated with the higher power losses presented in cases where a bigger capacity of BESS units is installed, e.g., case F. As in the 2020 scenario, BESS investment increases when power losses are not considered.

Fig. 10 Comparison of number of hourly dispatched generators for different cases in 2030 scenario.
Finally,

Fig. 11 Operation characteristics of load and BESS units at bus GCM in 2030 scenario.
Note that in
This paper describes a formulation to determine the sizing and siting of BESSs to reduce operation costs in a power system. Simulation results for different studied systems demonstrate the adequacy of the proposed formulation to reduce operation costs by optimally determining the sizing and siting results of BESSs. Moreover, it is also shown that the sizing and siting results of BESSs obtained with the proposed formulation improve the use of renewable sources by reducing power curtailment, especially in systems with high penetration of renewable sources. Furthermore, the simulations show that a deferral in new transmission line investment is possible with BESS inclusion. Additionally, the comparison with and without consideration of power losses shows the impact of transmission capacity of the systems on the sizing and siting results of BESSs to minimize the operation cost. Finally, the results shows that possible projected reductions in BESS costs will allow a further use of this technology in power systems, due to its benefits to the system.
Nomenclature
Symbol | —— | Definition |
---|---|---|
A. | —— | Indices |
—— | Buses | |
—— | Reservoir | |
—— | Hydro generation unit | |
—— | Thermal generation unit | |
—— | Renewable generation unit | |
—— | Battery energy storage system (BESS) unit | |
—— | Time period | |
B. | —— | Sets |
—— | Set of all buses | |
—— | Set of all reservoirs | |
—— | Set of all hydro generation units | |
—— | Subset of hydro generation units at bus b | |
—— | Subset of hydro generation units in reservoir d | |
—— | Set of all thermal generation units | |
—— | Subset of thermal generation units at bus b | |
—— | Set of all renewable generation units | |
—— | Subset of renewable generation units at bus b | |
—— | Set of all lines | |
—— | Set of all BESS units | |
—— | Subset of BESS units at bus b | |
—— | Set of all time periods | |
K | —— | Set of blocks |
C. | —— | Parameters |
—— | Slope of block voltage angle linearization | |
, , | —— | Coefficients of thermal generation units of quadratic cost curves |
, | —— | Charging and discharging efficiencies of BESS units |
—— | Self-discharging efficiency of BESS units | |
—— | The maximum angular difference | |
—— | Susceptance of line | |
—— | Cost reduction in power and energy elements | |
—— | Power/energy cost of BESS units | |
, | —— | Power and energy equivalent costs of BESS units |
, | —— | Start-up and shut-down costs of thermal generation units |
, | —— | Thermal and hydro generation costs |
—— | Demand forecast | |
—— | Initial status of thermal generation units | |
, | —— | The minimum up-time and down-time |
—— | Line conductance | |
—— | Conversion factor | |
—— | Forecasted water rate in reservoir d | |
, | —— | Number of time steps that thermal generation unit is required to stay on and off at the beginning of the optimization horizon |
—— | Large enough positive constant | |
—— | The maximum limit of line capacity | |
, | —— | The minimum and maximum thermal generation capacities |
, | —— | The minimum and maximum hydro generation capacities |
, | —— | The minimum and maximum water flows in turbine |
—— | Capital recovery time | |
, | —— | Ramp-up and ramp-down rate limits |
—— | The maximum spillage in reservoir d | |
—— | Time horizon of the optimization problem | |
, | —— | The minimum and maximum water volumes in reservoir d |
D. | —— | Variables |
—— | Voltage angles of buses d and r | |
—— | Size of block voltage angle linearization | |
—— | Auxiliary variables to calculate the absolute value of | |
—— | Energy stored in BESS units | |
—— | Energy capacity of BESS units | |
—— | Power losses in the line connecting buses b and r | |
, | —— | Active power generated by thermal and hydro generation units |
—— | Power flow in the line connecting buses b and r | |
, | —— | Charging and discharging power of BESS units |
—— | Power capacity of BESS units | |
—— | Water flow rate in the turbine | |
, | —— | Start-up and shut-down decision variables |
—— | Spillage in reservoir d | |
, | —— | Binary variables to model the status of BESS units |
—— | Water volume in reservoir d | |
—— | Power output of renewable generation units | |
, | —— | Generation states of thermal and hydro generation units |
References
Southern Power Company. (2021, Mar.). Tranquility battery energy storage system. [Online]. Available: https://www.southernpowercompany.com/content/dam/southernpower/pdfs/fact-sheets/Tranquillity_Energy_Storage_factsheet.pdf [Baidu Scholar]
Aurecon. (2020, Feb.). Hornsdale power reserve: year 2 technical and market impact case study. [Online]. Available: https://westerndownsgreenpowerhub.com.au/wp-content/uploads/2021/10/Aurecon-Hornsdale-Power-Reserve-Impact-Study-year-2019.pdf [Baidu Scholar]
Comisión de Regulación de Energía y Gas CREG. (2018, Oct.). Por la cual se ordena hacer público el proyecto de resolución “Por la cual se definen los mecanismos para incorporar sistemas de almacenamiento en el Sistema Interconectado Nacional”. [Online]. Available: http://apolo.creg.gov.co/Publicac.nsf/1c09d18d2d5ffb5b05256eee00709c02/20e2dd9910530b24052583470059837c/$FILE/Creg127-2018.pdf [Baidu Scholar]
Unidad de Planeación Minero Energética UPME. (2020, May). Convocatoria pública UPME STR 01-2020 almacenamiento de energía con baterías–Atlántico. [Online]. Available: https://www1.upme.gov.co/PromocionSector/InformacionInversionistas/Paginas/UPME-STR-01-2020-Almacenamiento-de-Energ%C3%ADa-con-Baterias-Atlantico.aspx [Baidu Scholar]
Unidad de Planeación Minero Energética UPME. (2021, Jan.). Convocatoria pública UPME STR 01-2021 almacenamiento de energía con baterías–Atlantico. [Online]. Available: https://www1.upme.gov.co/PromocionSector/InformacionInversionistas/Paginas/UPME-STR-01-2021-Almacenamiento-de-Energ%C3%ADa-con-Baterias-Atlantico.aspx [Baidu Scholar]
R. A. Jabr, I. Dzafic, and B. C. Pal, “Robust optimization of storage investment on transmission networks,” IEEE Transactions on Power Systems, vol. 30, no.1, pp. 531-539, Jan. 2015. [Baidu Scholar]
J. A. Aguado, S. de la Torre, and A. Trivio, “Battery energy storage systems in transmission network expansion planning,” Electric Power Systems Research, vol. 145, pp. 63-72, Apr. 2017. [Baidu Scholar]
S. Wang, G. Geng, and Q. Jiang, “Robust co-planning of energy storage and transmission line with mixed integer recourse,” IEEE Transactions on Power Systems, vol. 34, no. 6, pp. 4728-4738, Nov. 2019. [Baidu Scholar]
L. Fiorini, G. A. Pagani, P. Pelacchi et al., “Sizing and siting of large-scale batteries in transmission grids to optimize the use of renewables,” IEEE Journal on Emerging and Selected Topics in Circuits and Systems, vol. 7, no. 2, pp. 285-294, Jun. 2017. [Baidu Scholar]
G. C. van Kooten, P. Withey, and J. Duan, “How big a battery?” Renewable Energy, vol. 146, pp. 196-204, Feb. 2020. [Baidu Scholar]
M. Motalleb, E. Reihani, and R. Ghorbani, “Optimal placement and sizing of the storage supporting transmission and distribution networks,” Renewable Energy, vol. 94, pp. 651-659, Aug. 2016. [Baidu Scholar]
A. Hassan and Y. Dvorkin, “Energy storage siting and sizing in coordinated distribution and transmission systems,” IEEE Transactions on Sustainable Energy, vol. 9, no. 4, pp. 1692-1701, Oct. 2018. [Baidu Scholar]
A. R. López, A. Krumm, L. Schattenhofer et al., “Solar PV generation in Colombia–a qualitative and quantitative approach to analyze the potential of solar energy market,” Renewable Energy, vol. 148, pp. 1266-1279, Apr. 2020. [Baidu Scholar]
A. Borghetti, C. D’Ambrosio, A. Lodi et al., “An MILP approach for short-term hydro scheduling and unit commitment with head-dependent reservoir,” IEEE Transactions on Power Systems, vol. 23, no. 3, pp. 1115-1124, Aug. 2008. [Baidu Scholar]
D. Rajan and S. Takriti. (2005, Jun.). IBM research report minimum up/down polytopes of the unit commitment problem with start-up costs. [Online]. Available: https://dominoweb.draco.res.ibm.com/reports/rc23628.pdf [Baidu Scholar]
D. Romero-Quete and C. A. Canizares, “An affine arithmetic-based energy management system for isolated microgrids,” IEEE Transactions on Smart Grid, vol. 10, no. 3, pp. 2989-2998, May 2019. [Baidu Scholar]
M. Carrion and J. M. Arroyo., “A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem,” IEEE Transactions on Power Systems, vol. 21, no. 3, pp. 1371-1378, Aug. 2006. [Baidu Scholar]
N. Alguacil, A. L. Motto, and A. J. Conejo, “Transmission expansion planning: a mixed-integer LP approach,” IEEE Transactions on Power Systems, vol. 18, no. 3, pp. 1070-1077, Aug. 2003. [Baidu Scholar]
H. A. Behabtu, M. Messagie, T. Coosemans et al., “A review of energy storage technologies’ application potentials in renewable energy sources grid integration,” Sustainability, vol. 12, no. 24, pp. 1-20, Dec. 2020. [Baidu Scholar]
P. V. Brogan, R. J. Best, D. J. Morrow et al., “Effect of BESS response on frequency and RoCoF during underfrequency transients,” IEEE Transactions on Power Systems, vol. 34, no. 1, pp. 575-583, Jan. 2019. [Baidu Scholar]
V. Gevorgian, M. M. Baggu, and D. Ton. (2019, May). Interconnection requirements for renewable generation and energy storage in island systems: Puerto Rico example. [Online]. Available: https://www.nrel.gov/docs/fy19osti/73848.pdf [Baidu Scholar]
F. Nadeem, S. M. S. Hussain, P. K. Tiwari et al., “Comparative review of energy storage systems, their roles, and impacts on future power systems,” IEEE Access, vol. 7, pp. 4555-4585, Dec. 2019. [Baidu Scholar]
Lazard. (2020, Oct.). Lazard’s levelized cost of storage analysis–version 6. [Online]. Available: https://www.lazard.com/media/451565/lazards-levelized-cost-of-storage-version-60-vf2.pdf [Baidu Scholar]
R. Fu, T. Remo, and R. Margolis. (2018, Nov.). 2018 U.S. utility-scale photovoltaics-plus-energy-storage system costs benchmark. [Online]. Available: https://www.nrel.gov/docs/fy19osti/71714.pdf [Baidu Scholar]
J. Czyzyk, M. P. Mesnier, and J. J. More, “The NEOS server,” IEEE Computational Science and Engineering, vol. 5, no. 3, pp. 68-75, Jul.-Sept. 1998. [Baidu Scholar]
E. D. Dolan. (2001, May). NEOS server 4.0 administrative guide. [Online]. Available: https://publications.anl.gov/anlpubs/2001/07/39844.pdf [Baidu Scholar]
W. Gropp and. J. Moré, “Optimization environments and the NEOS server,” in Approximation Theory and Optimization. Cambridge: Cambridge University Press, 1997, pp. 167-182. [Baidu Scholar]
L. Wu, M. Shahidehpour, and T. Li, “Stochastic security-constrained unit commitment,” IEEE Transactions on Power Systems, vol. 22, no. 2, pp. 800-811, May 2007. [Baidu Scholar]
L.Wu, M. Shahidehpour, and Z. Li, “Comparison of scenario-based and interval optimization approaches to stochastic SCUC,” IEEE Transactions on Power Systems, vol. 27, no. 2, pp. 913-921, May 2012. [Baidu Scholar]
Unidad de Planeación Minero Energética UPME. (2018, Jul.). Plan de expansión de referencia generación-transmisión 2017-2031. [Online]. Available: http://www.upme.gov.co/Docs/Plan_Expansion/2017/Plan_GT_2017_2031.pdf [Baidu Scholar]
XM Compañía de Expertos en Mercados. (2021, Jul.). Demanda y fronteras–Históricos. [Online]. Available: https://sinergox.xm.com.co/dmnd/Paginas/Historicos/Historicos.aspx [Baidu Scholar]
A. Avendaño. (2021, Jul.). 15-bus Colombian power system dataset. [Online]. Available: https://github.com/aaavendanop/colombian_dataset [Baidu Scholar]
XM Compañía de Expertos en Mercados. (2021, Jul.). Generación-capacidad efectiva por tipo de generación. [Online]. Available: http://paratec.xm.com.co/paratec/SitePages/generacion.aspx?q=capacidad [Baidu Scholar]
Unidad de Planeación Minero Energética. (2015, Sept.). Integración de las energías renovables no convencionales en Colombia. [Online]. Available: https://www1.upme.gov.co/DemandayEficiencia/Doc_Hemeroteca/Estudio_integracion_energias/Integracion_energias_renovables.pdf [Baidu Scholar]