# Lead-Acid Battery Sizing for a DC Auxiliary System in a Substation by the Optimization Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}, is released due to the electrolysis of water in the electrolyte when charging a LAB [17,18]. Hence, proper BR ventilation is required. The author in [19] discusses the risk of explosion in the BR and safety measures.

- Development of an optimization algorithm for the selection of LABs in stationary applications, taking into consideration technical, economic and reliability criteria, safety, and possible methods of battery maintenance;
- Optimum conditions of battery selection;
- A selection method that contributes to more reliable and more cost-efficient operation of LABs for the supply of a DC auxiliary system in substations;
- The selection of LABs impacts on the operational safety in the BR.

## 2. The Role of Lead-Acid Batteries in a Substation’s Auxiliary System Supply

_{DC}= 220 V or U

_{DC}= 110 V auxiliary system. Figure 1 presents the concept of the DC auxiliary system. The elements of an essential AC auxiliary system are, in all cases, also the rectifiers that supply the substation’s DC auxiliary system. The rectifiers do not have limited operation time, while the operation time of LABs is limited, although the minimum time is prescribed. Certain loads of the essential AC auxiliary system, such as supply of protection systems, measurement transducers, and switchgear drives, need DC voltage for their operation. A redundant supply of a DC auxiliary system with two rectifiers is used in all cases. The rectifiers are connected to the battery coupling switch panel through fuses (F) on both parallel systems with cables (C) (Figure 1).

_{d,A}and R

_{d,B}. Individual parts of the system are protected by fuses. Switching manipulations in the DC auxiliary system network are performed by adequate switching devices (S). In the majority of substations, the DC auxiliary system is also the main supply source of the UPS system (through an inverter). The operation depends on the load and availability of supply sources. In the normal operation, both systems, (A and B), are switched on, and supply all load and charge the LABs. In the case of rectifiers’ outage, the supply is taken over by the LABs.

## 3. Procedure of the Conventional Lead-Acid Battery Selection Method According to the Standard

_{DC}. The loads are categorized into six types with regard to the duration of load using the DC current I

_{DC}. A random load with a current I

_{R}with the duration of t

_{R}= 1 minute can, during the autonomous operation, appear at any time. In substations, such loads are circuit breaker triggering coils [20]. Let us observe the case when the LAB supplies DC load autonomously, as shown in Figure 1. During the entire autonomous operation the various loads are turned on and off. Figure 2 shows a generalized current diagram in the autonomous operation of a LAB [8]. For this purpose, the duration of the autonomous operation of the LAB is divided into m

_{s}periods, where new period p + 1 starts every time I

_{DC}is changed. At that time, a new cycle s + 1 covers all periods from 1 to the period p + 1, for period 1 is considered to be I

_{p}

_{− 1}= 0. For the cycle (s), the capacity of the LAB is computed using Equation (1).

_{T,p}[21] is the factor of operational capacity in the period (p) in [h]. It represents the relation between the operational capacity and the current that the lead-acid cell is capable of supplying permanently to the DC circuit in t minutes until it is discharged to the voltage U

_{c,min}. It can be determined graphically from three characteristics for three different ways of a lead-acid cell’s discharging.

_{s}. The necessary capacity is for all cycles calculated as the maximum value of individual cycles using (2).

_{T,R}is the factor that can be determined graphically from three characteristics for three different ways of a lead-acid cell’s discharging for a random load for the duration of t

_{R}= 1 min. n

_{rep}is the random load repetition factor, assuming t

_{R}= 1 min for all repetitions. I

_{R}is the total load current of a random load. This capacity is oversized, due to the selected duration [7].

_{a}is the aging factor of the lead-acid cell (irreversible chemical processes). The Standard defines it as K

_{a}= 1.25. This means that, at the start, up to 25% higher capacity of a lead-acid cell needs to be provided to ensure that it will be able to supply the load during the entire SL. K

_{tem}is the temperature factor. K

_{dm}(design margin) is a correction factor that takes into consideration insufficient maintenance and unexpected changes of load.

_{avg}at the average discharge voltage U

_{avg}(I

_{avg}= P/U

_{avg}) and the random load current I

_{R}. Cycles and periods of load currents are determined. The appropriate capacity is selected according to Equations (1)–(4). The factors in capacity selection (aging, temperature, manufacturing constraints) are generally taken conservatively as maximum values. The number of lead-acid cells connected in series for a given DC voltage level is predetermined and in accordance with the Standard [8]. In the case of a reconstruction of the DC auxiliary system, the current measurement and duration shall be measured for the existing one. A random current load, as well as the number of lead-acid cells connected in series, are also determined in accordance with the provisions of the Standard and equipment at the DC auxiliary system voltage. The rule is that a unifying number of cells connected in series is selected for a given DC auxiliary system voltage level. Based on these provisions, and in cooperation with the institutions for designing the DC auxiliary system in substations, we have created an application in MS Excel, which covers an active capacity selection table by the Standard [8].

## 4. Procedure of Lead-Acid Battery Selection Based on the Optimization Method

_{1}, minimum cost f

_{2}, highest possible operational reliability f

_{3}, and maximum operational safety of LABs in the BR f

_{4}. All elements of this optimization system are presented below.

_{T,p}and K

_{T,R}in the optimization process, an approximation from three characteristics for three different ways of a lead-acid cell’s discharging was made with the polynomial of the fourth degree. The “grabit” application was used for this purpose, where, for the diagram for each characteristic, the points T

_{i}(x

_{i},y

_{i}) were determined (more than 200 points per curve). Since these curves are plotted on a log-log graph, the final points were obtained using x’ = 10

^{x}transformation. The transformation is valid for all coordinates. The fourth-degree polynomial coefficients were sought using the approximation process in Equation (5).

_{1}, a

_{2}, a

_{3}, a

_{4}in a

_{5}are fourth-degree polynomial coefficients and t

_{aut}is the duration of the autonomous supply expressed in minutes. Table 1 shows the coefficients for three ways of lead-acid cells’ discharge at different U

_{c,min}.

_{tem}= 1.3) and θ = 48.9 °C (K

_{tem}= 0.86). This is also the ambient temperature in the stationary operating conditions. In the optimization process, these boundary conditions are given in a tabular form. For the intermediate temperatures, the factor K

_{tem}is defined by the linear interpolation. K

_{dm}(design margin) is a correction factor that takes into consideration insufficient maintenance and unexpected changes of load. Only the influence of insufficient maintenance is taken into account in the optimization process. The boundary values of this factor lie between K

_{dm,min}= 1.0 and K

_{dm,max}= 1.15. In the subsequent parts of the paper, this factor is addressed to maintenance factor K

_{m}= K

_{dm}.

_{DC,min}is the DC network voltage that enables uninterrupted operation of the supplied load. The maximum DC network voltage U

_{DC,max}is the maximum permitted operational voltage of the supplied load. The DC network voltage and, thus, the LAB voltage, depend on the necessary amount of energy stored in the battery. The user has to maintain the LAB and comply with the minimum maintenance requirements defined by the manufacturer. These requirements may be even higher to provide the required level of reliability. An LAB has to undergo inspections, revisions, and capacity tests to maintain its SL. The LAB is not disconnected from the DC network during the inspections. Checked are the general condition of the battery, electrolyte level, and condition of contacts between the lead-acid cells (using a thermovision camera), the sealing, and ventilation of the BR. The battery is switched off during the revisions. This process includes a thorough inspection of contacts, with measurement of contact resistance, battery connections, electrolyte density, and voltage of individual cells. The capacity test is performed in accordance with the IEC 60896-11 Standard [22], and in accordance with the manufacturer’s instructions. A detailed inspection of the battery is performed. It is loaded with a constant current through the discharging resistor. Voltage is measured until the battery is discharged to 80% of its initial capacity. The intention is to find malfunctioning lead-acid cells. During this process, the electrolyte density is also measured constantly.

_{ins}. The revision cost, on the other hand, is denoted as C

_{rev}, while the cost of the capacity test is C

_{ct}. The SL of the battery is designated as SL.

_{m}and the maintenance costs C

_{m}. On the basis of the lead-acid cells’ maintenance method and the manufacturer’s recommendations, five maintenance modes are possible, as described below. Maintenance mode 1: Twice a year, a capacity test is performed; once a year, a revision; and twice a year, an inspection. K

_{m,mode1}= 1. The cost of maintenance mode 1 is expressed as C

_{m,mode1}= (2 · C

_{ct}+ C

_{rev}+ 2 · C

_{ins}) · SL. Maintenance mode 2: Once a year, a capacity test; once a year, a revision; and twice a year, an inspection. K

_{m,mode2}= 1.05. The cost of this mode is defined by C

_{m,mode2}= (C

_{ct}+ C

_{rev}+ 2 · C

_{ins}) · SL. Maintenance mode 3: A capacity test is performed every two years. During the years of the service life, when the capacity test is not performed, a revision is carried out, and, twice a year, an inspection. The cost of this mode is C

_{m,mode3}= (0.5 · (C

_{ct}+ C

_{rev}) + 2 · C

_{ins}) · SL. K

_{m,mode3}= 1.075. Service mode 4: A capacity test is performed every five years. During the years of the service life, when the capacity test is not performed, a revision is carried out, and, twice a year, an inspection. C

_{m,mode4}= (0.2 · C

_{ct}+ 0.8 · C

_{rev}+ 2 · C

_{ins}) · SL. K

_{m,mode4}= 1.1. Service mode 5: A capacity test is performed once in the middle of the SL. During years of the service life, when the capacity test is not performed, a revision is carried out, and, twice a year, an inspection. K

_{m,mode5}= 1.15. C

_{m,mode5}= C

_{ct}+ (SL − 1)⋅C

_{rev}+ 2 · C

_{ins}· SL. The vectors K

_{m}= [1,1.05,1.075,1.1,1.15] and

**C**

_{m}= [C

_{m,mode1}, C

_{m,mode2}, C

_{m,mode3}, C

_{m,mode4}, C

_{m,mode5}] are formed with the currency unit [CU]. We are interested in the annual maintenance costs. The vector of annual maintenance costs is calculated using

**C**

_{m,a}= (1/SL)

**· C**

_{m}, where SL is the lead-acid cells’ SL. The maintenance of LABs is linked closely with their reliability of operation. A LAB is composed of a large number of cells connected in series. The capacity test gives us an indication of the weak links in this series, and the instruction which cells need to be replaced with new ones. Each weak lead-acid cell can prevent the battery from ensuring it required autonomy time during its operation [12]. The calculation of reliability depends mainly on the capacity test. A basic level of reliability is ensured by regular revisions if a capacity test is not carried out during the years of service life and inspections twice a year. For mode 1, it was found out that a failure may occur in the last half year of the SL. The reliability of operation is, thus, R

_{b,mode1}= 1 − 0.5 · SL

^{−1}. For mode 2, the reliability of operation is R

_{b,mode2}= 1 − SL

^{−1}. For mode 3, the reliability of operation is R

_{b,mode3}= 1 − 2 · SL

^{−1}. For mode 4, the reliability of operation is R

_{b,mode4}= 1 − 5 · SL

^{−1}. For maintenance mode 5, the reliability of operation is expressed as R

_{b,mode5}= 1 − 10 · SL

^{−1}. The reliability of operation vector

**R**

_{b}= [R

_{b,mode1}, R

_{b,mode2}, R

_{b,mode3}, R

_{b,mode4}, R

_{b,mode5}] is formed for the LAB.

_{2}during battery charging [18]. For this purpose, a heating, ventilation, and air conditioning (HVAC) device is installed in the BR, which also keeps the room temperature at the desired level. The question is, what are the costs related to the operation of the HVAC device to maintain the desired temperature in the BR? For this purpose, the IEC 12831 Standard [23] is used, which defines the method of calculating the projected thermal load for buildings. To use this Standard properly, one has to know the properties of the walls, floor, ceilings, windows, and doors of both the building and the BR, as well as the surface areas of all elements. Heat losses due to natural ventilation of the BR and heat transfer through heat bridges can be neglected. It is assumed that the BR is located in a heated building with the temperature of the adjacent rooms amounting to Θ

_{np}. It is also assumed that all the adjacent heated/cooled rooms have the same temperature Θ

_{np}. The heat is transmitted between the building elements with the adjacent rooms, and the outer walls with the exterior. A stationary state of heat transmission is observed. The total heat transmission Φ

_{BR}, expressed in [W], between the BR and the adjacent rooms and the exterior, is defined by Equation (6).

_{T,BRnp}is the coefficient of transmission heat loss between the BR and the adjacent rooms in [W/K]. H

_{T,BRe}is the coefficient of transmission heat loss between the BR and the building exterior. Θ

_{BR}is the temperature inside the BR in [°C], while Θ

_{e}is the exterior temperature. The intention is to know what the levels of heat transmission from the BR are at various outside temperatures. It is assumed that the annual hourly data are known on the exterior temperature around the building. A vector Θ

_{e,a}= [Θ

_{e,a,1},…,Θ

_{e,a,i},…,Θ

_{e,a,8760}] is formed. The counter of hours i lies between 1 and m

_{h}= 8760. For a certain BR temperature Θ

_{BR}, a vector of heat transmission from the BR in all hours of the year Φ

_{BR,T}can be defined using Equation (6). This vector is defined as Φ

_{BR,}

_{Θ}= [Φ

_{BR,}

_{Θ}

_{,1},…,Φ

_{BR,}

_{Θ}

_{,i},…,Φ

_{BR,}

_{Θ}

_{,8760}]. If the heat in a certain hour of the year (i) is transferred from the BR, the element Φ

_{BR,}

_{Θ}

_{,I}> 0 is heating, and if it is transferred to it, Φ

_{BR,}

_{Θ}

_{,I}< 0 it is cooling. The c device covers both ways of heat transmission, and is able to maintain the desired temperature in the BR. To maintain this temperature, the necessary electric power in the hour (i) in a certain year is calculated using Equation (7).

_{HC}is the average heating and cooling number of the HVAC device. It is assumed that in the hour (i), the necessary electrical power for air conditioning in the Equation (7) is average. The quantity of electrical energy in this hour is calculated using Equation (8).

**W**

_{e,}

_{Θ}can be formed for maintaining the desired temperature in the BR. At the annual level, the hourly contributions of energy are summed up to get the annual electrical energy consumption of the air conditioning device for a certain temperature W

_{e,}

_{Θ}

_{,a}using Equation (9).

_{BR}by using Equation (10).

_{e,}

_{Θ}= f(Θ

_{BR}). This procedure is used in the optimization process.

_{aut,max}. This time is defined on the basis of experience. In this task, it is necessary to take into consideration the worst possible case, i.e., when the other battery in the DC network is empty. In the case of a rectifier’s outage, the other network should charge through the mobile emergency generator (Figure 1). The minimum time needed to recharge the other LAB is, for the optimization process, the highest possible autonomy time of the DC network with a LAB, t

_{aut,max}. The user also defines the minimum autonomy of supply time t

_{aut,min}, required to remove any problems in most cases and re-establish supply in the other supply network.

_{c}= [Q

_{c,1},…,Q

_{c,k},…,Q

_{c,m}] is obtained. An important factor in this process is the prices of different types of cells contained in the vector C

_{c}= [C

_{c,1},…,C

_{c,k},…,C

_{c,m}]. Since we deal with the annual costs, the vector] is formed of annual procurement costs C

_{c,a}= (1/SL)

**⋅**C

_{c}in a [CU/year]. The manufacturer also gives the technical SL of cells’ SL. Characteristic data are also the maximum lead-acid cell’s voltage U

_{c,max}in [V/cell] and minimum operating voltage that enables its normal operation, U

_{c,min}in [V/cell]. The manufacturer also specifies the recommended temperature range, limited by Θ

_{BR,r,min}and Θ

_{BR,r,max}. This temperature range is usually narrower than the one defined by the Standard.

_{c,max,int}is assigned to the maximum number of cells using Equation (12).

_{c,min}is defined by Equation (13).

_{c,min,int}is determined using Equation (14).

_{c}= [n

_{c,min,int},…,n

_{c,l},…,n

_{c,max,int}] can be formed as long as the rule n

_{c,max,int}> n

_{c,min,int}is in force. The dimension of this vector n is defined by Equation (15).

_{c}complies with the technical requirements of the Standard. The size of the vector n

_{c}is defined primarily to enable setting of the maximum charging voltage U

_{c,max}. If it is higher, the number of possible cells connected in series is lower. The variables and procedures described so far are input data for the optimization process. In addition to the input data, they are also influenced by the optimization parameters. The output from the DEA is a set of optimization parameter values. They are limited directly in the algorithm. The parameters were not normalized in the concept described in the paper. The limitations are of a physical nature. The optimization algorithm deals with five optimization parameters: Electrolyte temperature (BR) (p

_{1}), maintenance mode of the LAB (p

_{2}), selection of cells’ capacity from the set of possible ones (p

_{3}), selection of the number of lead-acid cells connected in series (p

_{4}), and duration of autonomy of supply with the LAB (p

_{5}). The first parameter in the optimization process is electrolyte temperature. In stationary operating conditions, this is also the BR temperature. The room temperature takes the value of parameter p

_{1}, as Θ

_{BR,sel}= p

_{1}; p

_{1}∈ [Θ

_{BR,r,min},Θ

_{BR,r,max}]; p

_{1}∈ ℝ. With the selection of this parameter, the temperature factor K

_{tem,sel}= f(Θ

_{BR,sel}) can be determined using linear interpolation, as well as the annual cost of maintaining this temperature with the air conditioning device C

_{e,}

_{Θ},

_{sel}using the procedure defined by the set of Equations (6)–(10). The parameter p

_{2}is connected with the battery maintenance. It is an element of the set of natural numbers, p

_{2}∈ {1,2,3,4,5}; p

_{2}∈ N. Since the optimization process yields the parameter p

_{2}as a real number, it should be transformed to a natural number using the function round(). This parameter actually represents a counter of possible maintenance modes j = p

_{2}. The parameter p

_{2}is used to obtain the maintenance factor K

_{m,sel}= K

_{m,j}from the vector of maintenance factors K

_{m}, reliability of operation of LABs R

_{b,sel}= R

_{b,j}from the vector of reliability of operation R

_{b}, and maintenance-related cost C

_{m,sel}= C

_{m,j}from the vector of maintenance costs

**C**

_{m}. The parameter p

_{3}is related to the selection of lead-acid cells’ capacity, defined by the manufacturer. It represents a counter of selection of cell capacities from the set of products k offered by the manufacturer, k = p

_{3}; p

_{3}∈ {1,2,…,m}; p

_{3}∈ N. With this parameter, the lead-acid cells’ capacity Q

_{c,sel}= Q

_{c,k}is selected from the vector Q

_{c}, as well as the procurement costs distributed by the years C

_{c,a,sel}= C

_{c,a,k}from the vector of annual costs C

_{c,a}. The parameter p

_{4}represents the counter of selection of the number of cells connected in series. This is the counter l in the vector n

_{c}, l = p

_{4}; p

_{4}∈ {1,2,…,n}; p

_{4}∈ N. The number of cells connected in series n

_{c,sel}= n

_{c,l}is selected from the vector

**n**

_{c}with this parameter,. The parameter p

_{5}represents the duration of battery autonomy in supplying energy to the DC auxiliary network. The duration of autonomy takes the value of the parameter p

_{5}as t

_{aut,sel}= p

_{5}; p

_{5}∈ [t

_{aut,min},t

_{aut,max}]; p

_{5}∈ R. This parameter enables calculation of the factor K

_{T,p,sel}= f(t

_{aut,sel}), which makes possible calculation of the needed lead-acid cell’s capacity Q

_{c,need}.

_{tem,sel}, C

_{e,}

_{Θ}

_{,sel}, K

_{m,sel}, C

_{m,sel}, R

_{b,sel}, Q

_{c,sel}, C

_{c,a,sel}, n

_{c,sel}, K

_{T,p,sel}), it is possible to calculate the needed capacity Q

_{c,need}using Equation (4). If Q

_{c,sel}≥ Q

_{c,need}, then the capacity was selected properly, and complies with the technical criteria defined by the Standard. If Q

_{c,sel}< Q

_{c,need}, then the counter k is set at the last place, k = m. The capacity Q

_{c,sel}and cost C

_{c,a,sel}are selected once again. After this, we may proceed to the computation of the Objective Function.

_{1}, f

_{2}, f

_{3}, and f

_{4}are normalized functions for autonomy of supply, costs, reliability of operation, and the operational safety of LABs in a BR. α

_{1}, α

_{2}, α

_{3}, and α

_{4}are weighting factors of the objective function. The normalized function f

_{1}is a normalized objective function for the criterion of autonomy of supply. This function is normalized through the linear transformation in Equation (17).

_{5}. The most cost efficient selection of a LAB is sought. This means that the minimum of the objective function f

_{2}is sought. Normalized function f

_{2}is the normalized function for the criterion of cost. Normalization is performed using linear transformation in Equation (18). Since the minimum of this function is sought, it can only be increasing linearly.

_{sel}is defined by Equation (19).

_{max}is defined by the inequality in Equation (20).

_{min}is defined by the inequality in Equation (21).

_{b}has distributed five probabilities discretely, and each of them has reliability mostly above R

_{b}> 0.5. The normalization of the reliability criterion with linear transformation is, in this case, not adequate. A Gaussian curve is used instead for the normalization of the objective function, assuming that the objective is to ensure the highest possible reliability of operation of the LAB. The function f

_{3}is obtained by Equation (22).

_{l}is the “shape factor” on the left side of the Gaussian curve, calculated for our normalization case using the equation m

_{l}= ln(0.1)/(0.82 − 1)

^{2}.

_{4}is the operational safety criterion of LABs in the BR. It refers to the amount of released hydrogen, H

_{2,}into the BR while charging LABs. The Standard [17] deals with the safety requirements for secondary batteries and installations. An important part of this Standard is explosion hazards and protection against them in the BR. During the charging process of the LAB at U

_{c}> 2,23 V/cell, a chemical reaction of electrolysis of water is initiated in the electrolyte. The concentration of hydrogen in the air in the BR must not exceed 4%

_{vol}(a potentially explosive mixture). The battery for a DC auxiliary system supply in the substation is recharged all the time. This means the H

_{2}concentration is increasing regularly. The solution is adequate forced ventilation of the BR. The amount of replaced air in hours is defined by Equation (23).

_{gas}is a specific equivalent charge current producing H

_{2}in [mA/Ah]. The thermodynamics of the electrochemical reaction of the electrolysis of water in the electrolyte depends on the charge current density [18]. With increasing charging current, the amount of H

_{2}release also increases. The current is dependent on the lead-acid cell’s charge voltage U

_{c}, since the conductivity is presumed to be constant in this area of charging. In the worst case, the charge voltage of the lead-acid cell is determined as U

_{c}= U

_{DC,max}/n

_{c}. The limit currents I

_{gas}= 5 mA/Ah at the voltage U

_{c}= 2.23 V/cell and I

_{gas}= 20 mA/Ah at the voltage U

_{c}= 2,40 V/cell are specified in the Standard [17]. Based on the two boundary points, function I

_{gas}= f(U

_{c}) is determined by linear interpolation. Inserting this function into Equation (23), Equation (24) is obtained for the ventilation airflow Q

_{air,sel}.

_{c,sel}is the selected capacity from vector Q

_{c}. Equation (24) is normalized, and the objective function in Equation (25) obtained. As with the normalization for the objective function f

_{3}, the Gaussian bell curve function is also applied here.

_{l}is the “shape factor”, defined by m

_{l}= ln(0.01)/50

^{2}. By minimizing the objective function in Equation (25), the amount of H

_{2}released into the BR is reduced, along with the required airflow for forced ventilation with HVAC.

_{max}.

^{−6}. The optimization parameters become the optimum parameters. The vector p

_{opt}= [p

_{1,opt}, p

_{2,opt}, p

_{3,opt}, p

_{4,opt}, p

_{5,opt}] is obtained as a solution. If the parameters are decoded, we can obtain the optimum operational temperature Θ

_{opt}= p

_{1,opt}, optimum maintenance mode m

_{mode,opt}= p

_{2}, optimum capacity of the lead-acid cell Q

_{c,opt}, optimum number of cells connected in series n

_{c,opt}, optimum duration of autonomy t

_{aut,opt}= p

_{5,opt}, and optimum costs C

_{opt,}that can be calculated using Equation (19) if optimal costs C

_{e,}

_{Θ}

_{,opt}, C

_{m,opt}in C

_{c,a,opt}are used as parameters in this equation.

## 5. Two Examples of the Selection of Lead-Acid Batteries Using the Conventional Method and the Optimization Process

_{max}= 50, and they were determined based on previous experience for such optimization cases. The boundaries for normalization for the objective function f

_{1}are t

_{aut,min}= 1 h and t

_{aut,max}= 8 h.

_{2}, the boundary values for normalization are C

_{min}= 938 CU and C

_{max}= 4000 CU. For f

_{3}, the boundary values are 0 and 1. For the safety criterion f

_{4}, the boundary values for normalization are Q

_{air,min}= 12 m

^{3}/h and Q

_{air,max}= 100 m

^{3}/h. All four objective functions are equally weighted, α

_{1}= α

_{2}= α

_{3}= α

_{4}= 0.25.

#### 5.1. Input Data for the Selection of Lead-Acid Batteries for Small Substations

_{L1,1}= 2.42 A. The inverter’s current was also measured on the DC side of the UPS system. The permanent DC current amounted to I

_{L1,2}= 1.41 A. Both loads, according to Figure 1, represent the total permanent current I

_{L1}= 3.83 A. Since there is only one type of load with the current I

_{L1}decisive for the selection of LABs, the calculation of needed capacity Q according to Equation (1) was simplified, since the number of cycles equals n

_{s}= 1, and one period is p = 1. The calculation of the factor K

_{T,L1}was also simplified at the same time.

_{tc}= 1.63 A.

_{R}= 1.63⋅(6⋅6 + 3⋅1) = 63.57 A. During the autonomous operation of the LAB, the system can also be restored by sequential switching of all bays. We assume that the actuators of the circuit breaker have stored energy for the three switching maneuvers. The equivalent trigger time of all coils is t

_{R}= 1 min, which means the same K

_{T,R}factor for all triggers. Given these assumptions, we can define n

_{rep}= 2. By integrating the n

_{rep}repetition factor, we included in the LAB selection concept with optimization the determination of the required capacity reserve to restore the system after switching off the bays in substation. To enable their operation, the loads also define U

_{DC,max}= 250 V and minimal load voltage U

_{DC,min}= 188 V. Let us assume that 8 h are needed to charge the second battery system. This means that t

_{aut,max}= 8 h. The minimum time of autonomy of supply is set to t

_{aut,min}= 1 h. The cost of LAB inspections in the substation is known, and amounts to C

_{pr}= 100 CU, cost of revision amounts to C

_{rev}= 240 CU, while the cost of the capacity test amounts to C

_{kp}= 350 CU. The vector of maintenance costs

**C**

_{m}= [1140, 790, 495, 462, 445.5] CU, is calculated on the basis of these data. The lead-acid battery should be located in the BR. Proper BR ambient conditions are provided by the HVAC device. The room has one outer wall with H

_{T,BRe}= 10.3 W/K. The other walls, ceiling, and floor are uninsulated, with H

_{T,BRnp}= 87.7 W/K. The temperature in the neighboring rooms is constant, and amounts to Θ

_{np}= 21 °C. In the year 2018, a temperature profile of the outer side of the building was measured, comprising hourly temperature readings. The temperature profile is shown in Figure 6.

_{HC}= 3.7. The specific cost for electricity is c

_{e}= 0.124 CU/kWh. Using the procedure defined by Equations (6)–(10), the annual electricity costs were calculated to maintain the temperature between 10 °C and 25 °C in the BR. The diagram of costs for different BR temperatures is shown in Figure 7.

_{e,}

_{Θ}. The manufacturer provides a set of possible LAB cells’ capacities

**Q**

_{c}= [100,150,200,250,300,350,420,490,600,800,1000,1200] Ah and costs

**C**

_{c}= [91,104,115,130,148,163,185,204,235,325,378,431] CU. The manufacturer also specifies the SL of the lead-acid cells, SL = 20 years, and operating temperature range between Θ

_{BR,r,min}= 10 °C and Θ

_{BR,r,max}= 25 °C. The maximum charging voltage of lead-acid cells is U

_{c,max}= 2.35 V/cell. The minimum voltage, to which a cell can discharge, amounts to U

_{c,min}= 1.81 V/cell.

#### 5.2. Selection of Lead-Acid Batteries Using the Conventional Method for Small Substations

_{c,need}= 291.65 Ah. From the scale of offered capacities from the manufacturer, we have chosen Q

_{c}= 300 Ah. It is assumed that the lead battery operates at a temperature Θ

_{BR}≥ 10 °C. The number of lead-acid cells connected in series is n

_{c}= 104 cells.

#### 5.3. Optimization-Based Selection of Lead-Acid Batteries for Small Substations

_{opt}= [20.17,1,4,3,8] was obtained as a result. From this, it can be seen that the optimum temperature amounts to Θ

_{opt}= 20.17 °C, and optimum maintenance mode is m

_{mode,opt}= 1. This means that C

_{m,1,opt}= 1140 CU/year and R

_{b,opt}= 0.975 are selected. The selected maintenance mode provides for the highest operational reliability. The optimum lead-acid cell capacity selected is Q

_{c,opt}= 250 Ah. The optimum number of cells connected in series is n

_{c,opt}= 106 cells, while the optimum duration of autonomy is t

_{aut,opt}= 8 h. Optimum costs are C

_{opt}= 1851 CU/year. For the optimally selected optimization parameters according to Equation (24), we have calculated the minimum amount of exchanged air in the BR as Q

_{air}= 21.65 m

^{3}/h. With the selected temperature Θ

_{opt}= 20.17 °C, these are the necessary data for selecting an HVAC device for the BR. Figure 8a shows the results of all objective functions. It is obvious that the values of the objective functions converge. Figure 8b shows the results of all values of the objective functions.

#### 5.4. Input Data for Selection of Lead-Acid Batteries for Large Substations

_{L1}= 20.4 A. The circuit breaker drives are the same as in the small substation. When the busbar protection is activated at 110 kV, it switches off all 110 kV bays, as well as the transformer bays on both sides of the transformers. The total current in this case amounts to I

_{R,110}= 1.63⋅(9⋅6 + 2⋅3⋅3) = 117.36 A. At the 220 kV level, it stays equal, while at 400 kV, it equals I

_{R,400}= 1.63⋅(4⋅6 + 2⋅2⋅3) = 58.68 A.

_{R}= 117.36 A is used as a random current. Also, in this case, during the LABs’ autonomous operation, the system can be restored to its original state after the busbar protection is activated. We assume the random load repetition factor n

_{rep}= 2. The BR has one insulated wall with H

_{T,BRe}= 4.71 W/K. The other walls and ceiling are shared with the neighboring rooms through uninsulated walls with H

_{T,BRnp}= 100.53 W/K. The heat loss coefficient to the uninsulated floor is neglected. The temperature in the neighboring rooms is constant, and amounts to Θ

_{np}= 21 °C. The annual profile of costs for the HVAC device C

_{e,}

_{Θ}is shown in Figure 11.

_{e}were used in the optimization process. The other data are the same as in the previous example. The temperature profile is preserved, since both substations are located within a radius of 20 km. The profile of annual costs for maintaining the specified temperature was also calculated as shown in Figure 11. The optimum temperature was shifted towards 21 °C, due to the reduced impact of the outside temperature due to the insulated wall.

#### 5.5. Selection of Lead-Acid Batteries Using the Conventional Method for Large Substations

_{c,need}= 721.86 Ah. From the scale of offered capacities from the manufacturer, we have chosen Q

_{c}= 800 Ah. It is assumed that the LAB operates at a temperature Θ

_{BR}≥ 10 °C. The number of lead-acid cells connected in series is n

_{c}= 104 cells.

#### 5.6. Optimization-Based Selection of Lead-Acid Batteries for Large Substations

_{opt}= [20.65,1,9,3,8]. The optimum temperature obtained in the second computation was Θ

_{opt}= 20.65 °C. This result is slightly different than that obtained by the first computation. The optimum maintenance mode is also, in this case, m

_{mode,opt}= 1. The transmission system operator should stem for the highest possible reliability. This means that C

_{m,1,opt}= 1140 CU/year and R

_{b,opt}= 0.975 are selected. The selected maintenance mode provides the highest reliability of operation. The optimum lead-acid cell capacity is Q

_{c,opt}= 600 Ah. The selected capacity is so high due to the high constant loading and the possibility of a random current. The optimum number of cells connected in series is n

_{c,opt}= 106 cells. The optimum duration of autonomy is t

_{aut,opt}= 8 h. The optimum costs are C

_{opt}= 2397 CU/year. For the optimally selected optimization parameters according to Equation (24), we have calculated the minimum amount of exchanged air in the BR as Q

_{air}= 51.95 m

^{3}/h. These costs are only higher due to the higher annuity for purchasing the LAB. The only differences between the two computations are the capacities of the lead-acid cells, which is a result of the higher current loadings of the DC network in the large substation.

## 6. Comparison between the Conventional and New Selection Method Using Optimization

_{c}= 300 Ah using the conventional selection method for the intended autonomy of operation. Using the new method, the selected capacity after optimization is Q

_{c}= 250 Ah for the estimated LAB autonomy of 8 h, which is one level lower than the selection according to the conventional method. The number of lead-acid cells is n

_{c}= 106 cells according to the new method. According to the conventional method, it is n

_{c}= 104 cells. These selections satisfy all the technical requirements of the Standard [8]. Selection according to the new method is cost effective. The highest level of reliability and the lowest H

_{2}emissions in the BR were reached. According to the conventional method, the minimum amount of required air exchange in the BR is Q

_{air}= 31.73 m

^{3}/h, which is 46% more than the selection using the optimization method. In the event of a failure on the HVAC system for the BR, the gas concentration would rise earlier, with the same mode of operation. Maintenance and explosion protection requirements are part of the optimization process. However, they do not affect the selection of LABs using the conventional method.

_{c}= 800 Ah. According to the new selection method, the capacity is Q

_{c}= 600 Ah, which is two levels lower than the selection according to the conventional method. For the conventional method, this means significantly higher H

_{2}emissions and higher required airflow ventilation requirements in the BR (Q

_{air}= 84.61 m

^{3}/h for Q

_{c}= 800 Ah).

## 7. Conclusions

_{2}emissions are reduced to a minimum (up to 40% in comparison to the conventional method). Based on the comparison between the conventional and new method of selection, it can be concluded that the conventional selection method satisfies the technical criterion only. Maintenance tasks and associated reliability of operation are determined independently of selection. Also, the HVAC device selection for the BR is made independently. The proposed new method using optimization includes them in the selection process. Thus, we achieve the optimum choice of both LAB, as well as maintenance, operational safety BR, and designing of HVAC.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Parvini, Y.; Vahidi, A.; Fayazi, S.A. Heuristic Versus Optimal Charging of Supercapacitors, Lithium-Ion, and Lead-Acid Batteries: An Efficiency Point of View. IEEE Trans. Control Syst. Technol.
**2018**, 26, 167–180. [Google Scholar] [CrossRef] - Neto, P.B.L.; Saavedra, O.R.; de Souza Ribeiro, L.A. A Dual-Battery Storage Bank Configuration for Isolated Microgrids Based on Renewable Sources. IEEE Trans. Sustain. Energy
**2018**, 9, 1618–1626. [Google Scholar] [CrossRef] - Gallo, D.; Landi, C.; Luiso, M.; Morello, R. Optimization of Experimental Model Parameter Identification for Energy Storage Systems. Energies
**2013**, 6, 4572–4590. [Google Scholar] [CrossRef] - Vrettos, E.I.; Papathanassiou, S.A. Operating Policy and Optimal Sizing of a High Penetration RES-BESS System for Small Isolated Grids. IEEE Trans. Energy Convers.
**2011**, 26, 744–756. [Google Scholar] [CrossRef] - Salameh, Z.M.; Casacca, M.A.; Lynch, W.A. A mathematical model for lead-acid batteries. IEEE Trans. Energy Convers.
**1992**, 7, 93–98. [Google Scholar] [CrossRef] - Papic, I. Simulation model for discharging a lead-acid battery energy storage system for load leveling. IEEE Trans. Energy Convers.
**2006**, 21, 608–615. [Google Scholar] [CrossRef] - Jancauskas, J.R.; Shook, D.A. Optimization of station battery replacement [nuclear plant]. IEEE Trans. Nuclear Sci.
**1994**, 41, 1384–1388. [Google Scholar] [CrossRef] - IEEE. Recommended Practice for Sizing Lead-Acid Batteries for Stationary Applications. IEEE Std 485-2010 (Revis. IEEE Std 485-1997)
**2011**, 1–90. [Google Scholar] [CrossRef] - IEEE. Guide for the Design of Low-Voltage Auxiliary Systems for Electric Power Substations. IEEE Std 1818-2017
**2017**, 1–95. [Google Scholar] [CrossRef] - Sun, Y.; Jou, H.; Wu, J. Aging Estimation Method for Lead-Acid Battery. IEEE Trans. Energy Convers.
**2011**, 26, 264–271. [Google Scholar] [CrossRef] - Stevanatto, L.C.; Brusamarello, V.J.; Tairov, S. Parameter Identification and Analysis of Uncertainties in Measurements of Lead–Acid Batteries. IEEE Trans. Instrum. Meas.
**2014**, 63, 761–768. [Google Scholar] [CrossRef] - Liu, X.; Yang, Y.; He, Y.; Zhang, J.; Zheng, X.; Ma, M.; Zeng, G. A new dynamic SOH estimation of lead-acid battery for substation application. Int. J. Energy Res.
**2017**, 41, 579–592. [Google Scholar] [CrossRef] - Azzollini, I.A. Lead-Acid Battery Modeling Over Full State of Charge and Discharge Range. IEEE Trans. Power Syst.
**2018**, 33, 6422–6429. [Google Scholar] [CrossRef] - Zhao, X.; Zhang, J.; Chen, C.; Guo, L. Operation Optimization of Standalone Microgrids Considering Lifetime Characteristics of Battery Energy Storage System. IEEE Trans. Sustain. Energy
**2013**, 4, 934–943. [Google Scholar] [CrossRef] - Ribič, J.; Pihler, J.; Kitak, P. Impact of Electrode Shape on the Performance of a Gas Discharge Arrester. IEEE Trans. Power Deliv.
**2015**, 30, 463–471. [Google Scholar] [CrossRef] - Glotić, A.; Glotić, A.; Kitak, P.; Pihler, J.; Tičar, I. Parallel Self-Adaptive Differential Evolution Algorithm for Solving Short-Term Hydro Scheduling Problem. IEEE Trans. Power Syst.
**2014**, 29, 2347–2358. [Google Scholar] [CrossRef] - CENELEC—EN 50272-2—Safety Requirements for Secondary Batteries and Battery Installations Part 2: Stationary Batteries|Engineering360. Available online: https://standards.globalspec.com/std/418395/EN%2050272-2 (accessed on 18 October 2019).
- Pavlov, D. Lead-Acid Batteries: Science and Technology; Elsevier: Amsterdam, The Netherlands, 2011. [Google Scholar]
- Brzezinska, D. Ventilation System Influence on Hydrogen Explosion Hazards in Industrial Lead-Acid Battery Rooms. Energies
**2018**, 11, 2086. [Google Scholar] [CrossRef] [Green Version] - Thompson, M.J.; Wilson, D. Auxiliary DC Control Power System Design for Substations. In Proceedings of the 2007 60th Annual Conference for Protective Relay Engineers, College Station, TX, USA, 27–29 March 2007; pp. 522–533. [Google Scholar]
- Kim, D.; Cha, H. Kt Factor analysis of lead-acid battery for nuclear power plant. In Proceedings of the 2013 International Conference on Electrical Machines and Systems (ICEMS), Busan, Korea, 26–29 October 2013; pp. 526–529. [Google Scholar]
- TC/SC 21, I. IEC 60896-11 Ed. 1.0 b:2002, Stationary Lead-Acid Batteries-Part 11: Vented Types-General Requirements and Methods of Tests; IEC: Geneva, Switzerland, 2003. [Google Scholar]
- CEN/TR 12831-2:2017. Energy, Performance of Buildings - Method for Calculation of the Design Heat Load - Part 2: Explanation and Justification. EN 12831-1; Module M3-3; CEN: Brussels, Belgium, 2017. [Google Scholar]

**Figure 1.**Presentation of a DC auxiliary system in a substation with lead-acid battery (LAB), switching devices and DC auxiliary system loads.

**Figure 4.**Oscillogram of the measured DC current of protection and control system’s DC auxiliary system in a small substation.

**Figure 7.**Annual costs of the heating, ventilation, and air conditioning (HVAC) device in the battery room (BR) for various BR temperatures in a small substation.

**Figure 8.**The results of all objective functions and all objective functions values for small substation: (

**a**) Results of all objective functions; (

**b**) Results of all values of the objective functions.

**Figure 10.**Oscillogram of the measured DC current of protection and control system’s DC auxiliary system in a large substation.

**Figure 11.**Annual costs of the HVAC device in the BR for various BR temperatures in a large substation.

**Figure 12.**The results of all objective functions and all objective function values for large substation: (

**a**) Results of all objective functions; (

**b**) Results of all values of the objective functions.

U_{c,min} [V/Cell] | a_{1} | a_{2} | a_{3} | a_{4} | a_{5} |
---|---|---|---|---|---|

1.81 | −8.7584⋅10^{−11} | 1.0610⋅10^{−7} | −4.4639⋅10^{−5} | 22.9857⋅10^{−3} | 1.0599 |

1.75 | −1.7060⋅10^{−10} | 1.8196⋅10^{−7} | −6.5694⋅10^{−5} | 24.5160⋅10^{−3} | 0.7513 |

1.69 | −1.0067⋅10^{−10} | 1.6330⋅10^{−7} | −4.5739⋅10^{−5} | 23.6610⋅10^{−3} | 0.5792 |

Lead-Acid Batteries Capacity Sizing by Standard IEEE 85-2010, Using K_{T}; Case 1 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

p | I_{p} [A] | I_{p} – I_{p}_{− 1} [A] | M_{p} [min] | T_{p} [min] | K_{T}_{,p} [h] | Q_{1} = (I_{p} − I_{p} _{−} _{1})⋅K_{T,p} = | ||||||

Section 1—First Period | ||||||||||||

p = 1 | I_{1} = | 3.83 | I_{1}−0 = | 3.83 | M_{1} = | 480 | T_{1} = M_{1} = | 480 | 8 | 30.64 | Ah | |

S = | 1 | TOTAL | 30.64 | Ah | ||||||||

Random Equipment Load | ||||||||||||

p | I_{R} [A] | I_{R}−I_{0} [A] | M_{R} [min] | T_{R} [min] | K_{T}_{,R} [h] | Q_{R}= (I_{R} − 0)⋅K_{T}_{,R} = | ||||||

p = R | I_{R} = | 127.14 | I_{R} − 0 = | 127.14 | M_{R} = | 1 | T_{R} = M_{R} = | 1 | 1,1 | 139.85 | Ah | |

Max Section Capacity Size | 30.64 | Ah + | Random Section Q | 139.85 | Ah = | 170.49 | Ah | |||||

Uncorected Capacity Size | 170.49 | Ah | × Temp. Correction | 1.19 | × | Aging Factor | 1.25 | |||||

× | Design Margin | 1.15 | = Total needed Q of cell = | 291.65 | Ah | - | - |

© 2019 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**

Ribič, J.; Pihler, J.; Maruša, R.; Kokalj, F.; Kitak, P.
Lead-Acid Battery Sizing for a DC Auxiliary System in a Substation by the Optimization Method. *Energies* **2019**, *12*, 4400.
https://doi.org/10.3390/en12224400

**AMA Style**

Ribič J, Pihler J, Maruša R, Kokalj F, Kitak P.
Lead-Acid Battery Sizing for a DC Auxiliary System in a Substation by the Optimization Method. *Energies*. 2019; 12(22):4400.
https://doi.org/10.3390/en12224400

**Chicago/Turabian Style**

Ribič, Janez, Jože Pihler, Robert Maruša, Filip Kokalj, and Peter Kitak.
2019. "Lead-Acid Battery Sizing for a DC Auxiliary System in a Substation by the Optimization Method" *Energies* 12, no. 22: 4400.
https://doi.org/10.3390/en12224400