Anton_A_06

Risø National Laboratory Roskilde Denmark

January 2006

Risø-I-2440

Battery Lifetime Modelling

Simulation model for improved battery lifetime for renewable based energy systems for rural areas.

Anton Andersson

Department of Applied Physics and Electronics Umeå University Sweden

Author: Anton Andersson Title: Battery Lifetime Modelling Department: VEA, VES

Risø-I-2440 January 2006

Abstract: In a renewable energy hybrid system, different power sources are combined to generate electricity for remote off-grid applications. To ensure stability and power quality, renewable energy is often compensated with a storage unit. The lead acid battery is, by far, the most common storage method since it is a well-studied science, available at a low cost and has a high availability. However, lead acid batteries are commonly agreed to be a weak link in the system since the lifetime is unsure and the investment cost significant. In order to be able to predict performance and lifetimes of batteries in hybrid systems, research cooperation has been performed between institutes in Europe that has resulted in a battery simulation tool. This report details the work that has been done to improve and tune the model for more accurate lifetime and performance estimations. The model predicts that active mass degradation is the main ageing mechanism that contributes to the reduction in capacity in hybrid systems. In the model, a new theory is implemented to simulate the effect of high and low discharge currents. The new approach is to simulate the changes in PbSO4 crystal structure at the electrodes. The improved battery model will be implemented in IPSYS, a newly developed simulation tool for hybrid systems.

Risø National Laboratory Information Service Department P.O.Box 49 DK-4000 Roskilde Denmark Telephone +45 46774004 [email protected] Fax +45 46774013 www.risoe.dk

Acknowledgements

Thanks to:

Henrik Bindner for encouraging supervision, Oliver Gehrke for sharing your excellent computer skills,

Tom Cronin for a splendid helpdesk, Julia Schiffer for your admirable battery and computer skills,

Risø for providing a suburb research environment, Anders Lundin for supervision,

My parents for never ending support and to my Eleonora!

TABLE OF CONTENTS

1 BACKGROUND..................................................................................................................... 1

1.1 Introduction .............................................................................................................................. 1 1.1.1 Hybrid systems..................................................................................................................................... 2 1.1.2 Photovoltaic panels............................................................................................................................... 2 1.1.3 Wind turbines....................................................................................................................................... 3 1.1.4 Diesel generator ................................................................................................................................... 4 1.1.5 Energy storage...................................................................................................................................... 4 1.1.6 Inverter ................................................................................................................................................ 4 1.1.7 Controller............................................................................................................................................. 4

1.2 IPSYS (Integrated Power Systems) .......................................................................................... 5 1.2.1 IPSYS simulation example ................................................................................................................... 5

2 AIM WITH PROJECT........................................................................................................... 7

3 LEAD ACID BATTERIES..................................................................................................... 8 3.1.1 Major stress factors of the batteries....................................................................................................... 9 3.1.2 Major damage mechanisms................................................................................................................... 9 3.1.3 Battery controller.................................................................................................................................10

3.2 Battery model ...........................................................................................................................11 3.2.1 Batteries simulated ..............................................................................................................................12 3.2.2 Battery model input .............................................................................................................................13 3.2.3 Test runs .............................................................................................................................................14

4 METHODS........................................................................................................................... 17

4.1 Improved code..........................................................................................................................17 4.1.1 Varying time steps...............................................................................................................................17 4.1.2 Improved design with object-oriented programming, OOP ...................................................................17 4.1.3 Vectorizing .........................................................................................................................................18 4.1.4 Global Variables..................................................................................................................................18

4.2 Current factor ..........................................................................................................................19 4.2.1 Definition of Current factor .................................................................................................................19 4.2.2 Current factor used in the initial model ................................................................................................20 4.2.3 New approach for the current factor in the improved model..................................................................21

4.3 Mathematics behind the model................................................................................................23 4.3.1 Parameters...........................................................................................................................................30

4.4 New test series ..........................................................................................................................31 4.4.1 Test series for model valuation ............................................................................................................31 4.4.2 Test series for validation of model response .........................................................................................32

5 RESULT AND DISCUSSION.............................................................................................. 33

5.1 Simulation validation ...............................................................................................................33 5.1.1 CRES simulations ...............................................................................................................................33 5.1.2 ISPRA simulations ..............................................................................................................................41 5.1.3 Simulation response for different control strategies ..............................................................................45

6 CONCLUSIONS................................................................................................................... 49

7 REFERENCES .................................................................................................................... 51

8 APPENDIX……………………………………………………………………………………52

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1 BACKGROUND

1.1 Introduction Technologies for stand alone energy systems in remote areas are steadily developing and expanding worldwide. Environmental issues and rising fuel costs have turned interest for integration of renewable energy towards these systems. At the same time new technologies have proven reliability for solar photovoltaic and small wind turbines. Consequently, renewable energy has steadily and beyond doubt become an acceptable energy source. Battery storage is often necessary in autonomous power supplies with continuous power demand. It often improves both economics and performance of the system. To be able to understand the operation characteristics of batteries, knowledge in several different topics are necessary. Battery science connects topics such as chemistry, physics and energy engineering in a genuine mixture. None of the topics can independently of one another explain and describe the battery fully. This makes battery simulations difficult, but at the same time very interesting and necessary. Battery operation is the key for optimal performance of small-scale off-grid hybrid systems. However, it is commonly agreed that batteries contribute to a significant investment cost and are the weakest link in the system. The aim with this project is to improve an existing battery model to be able to accurately predict lifetime and performance of the batteries used in renewable based rural energy systems. This project will in detail explain the steps made to improve the model, which has resulted in a model that performs faster calculations and more accurate simulations. The intension with the improved battery model is, in the future, to implement it into the larger hybrid simulation platform IPSYS. Experience has shown a need for complete system simulations, to be able to predict the performance and cost of energy systems.

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1.1.1 Hybrid systems

There are several different definitions of a hybrid system. The car company Toyota have a well-defined all-embracing description “A hybrid system combines different power sources to maximize each one's strengths, while compensating for each other's shortcomings” [1]. Off-grid power, locally produced electricity, supplies many people and applications around the world with electricity. For remote areas such as isolated communities, islands and developing countries, these systems are vital for survival. Examples of usage are rural health clinics, communication and water supply maintenance [2].

Numerous sites around the world are today powered by conventional diesel generators. By integrating renewable energy sources into these local systems, it is possible to produce environmentally friendly energy at a reasonable cost. Each kWh produced by renewable energy will save the same amount of energy generated from conventional fossil systems. Environmental awareness is something that has become very important during the past few years. Renewable energy hybrid systems consist of several components interacting together to meet the power demand. This interface between the mix of power generation results in a unique system of design issues that strongly depends on local conditions, such as wind speeds and solar radiation. The term “renewable penetration” is a measure of the amount of renewable energy used in the system. The following presents a short survey over the components that together build up a hybrid system [3],[4]

1.1.2 Photovoltaic panels

Solar power is one of the first things that come to people’s minds when the topic of renewable energy is discussed. A solar cell consists of semiconductors that transform sunlight directly into DC electricity. The concept with semiconductors and no moving parts make the PV panels highly reliable which leads to low maintenance cost. The photovoltaic (PV) panel consists of several individual cells that are connected in series or in parallel to produce the desired voltage. One solar cell normally produces a direct current with a voltage of 0.3-0.6 V. By connections of several cells in series the voltage is increased to 12 or 24 V. The panels are easily assembled into differently sized arrays to meet the demands of any given size load. PV panels are rated in terms of peak watts (Wp) which is a combined measurement of both panel size and panel efficiency. It states the amount of power delivered by a panel during solar radiation of 1 kW/m2 at 20°C. A Panel rated at 150 Wp will produce 150 W during these conditions [2]. The efficiency of regular silicon solar cells is about 10-15 %. Today higher efficiencies are available (>20%) but then the price is very high. New techniques such as chemical vapour deposition make it possible to construct multi-junction solar panels. Each junction is designed to absorb a specific wavelength of radiation. Consequently less energy is lost by heat radiation and higher efficiency is achievable [5]. The power from the PV panel equals the product between the current and voltage. Depending on different solar radiation intensities the power output from the PV panel can fluctuate. Consequently, the photovoltaic panels and the load will only have a single operation point at any given conditions

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since they have different current-voltage relations [6]. Smaller independent PV systems with a consumption power of less than 200 W are almost always designed as DC system with storage in lead acid batteries. For larger systems, greater than 1000 W consumer power, the system is converted by an inverter to an AC system. The combination of solar energy and wind power is considered beneficial over the year. The wind turbine complements the solar cells during winter and bad weather [7].

1.1.3 Wind turbines

Wind turbines are a promising alternative energy resource for remote locations with good wind conditions. The wind is a powerful energy source, since the kinetic energy of the wind is proportional to the cube of the wind velocity. Accordingly, the power production of a wind turbine is highly variable. To be able to ensure useful power production, wind turbines normally are combined with another energy source. The turbines used in independent systems are usually smaller then the wind turbines used in regular grids. This section will generally focus on wind turbines of the size <15 kW. A wind turbine operating performance is characterized by its power curve. An example is showed in Figure 1.

Figure 1 Power curve for the 11 kW Gaia wind turbine.

Risø National Laboratory has during the last three and a half years performed tests on a modified 11 kW wind turbine produced by the Danish company Gaia-Wind. The turbine has been modified to ensure stable operation in hybrid systems. The intensions with the tests have been to demonstrate that the wind turbines can operate in a proper way in the combined system. The Gaia turbine operates on the horizontal axis, down wind and free yawing principle. The rotor consists of two stall regulated blades that work in the range of 3.5 - 25 m/s with a fixed rotor speed of 56 rpm. The generator is of asynchronous induction type that needs reactive power for magnetization. This reactive power is provided by the diesel generator [3]. Smaller wind turbines are often designed as permanent magnet machines. These wind turbines produce an AC current with a frequency that is proportional to the rotor speed. By power electronics it is possible to convert the power into DC current. The voltage is also proportional to the rotation speed of the rotor [2].

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1.1.4 Diesel generator

For small independent grids containing wind turbines, the power production is often secured by diesel engines directly coupled to synchronous generators. The diesel generator operates also as a source of reactive power. This implies that the system always has to have at least one diesel generator operating. Many remote communities are today powered with diesels systems only. High fuel costs have turned interest to new alternatives. An attractive solution for many locations, with sufficient wind resources, is to reduce running cost by implementation of wind turbines [6]. At the test site at Risø National Laboratory a 48 kW / 60 kVA diesel generator is connected to the Gaia wind turbine. When the wind turbine’s power production falls short of the power demand the diesel generator steps in to complement and secure power. [6]

1.1.5 Energy storage

Batteries are used in renewable energy systems to store excess energy for later usage. Energy storage often improves both economics and performance of the system. The explanation of this is that the batteries can provide the extra energy to ride out load peaks or low wind or sun conditions. Power is ensured without a start of a diesel generator. By far the most common storage technique is lead acid batteries. This is due to a proven technology, low cost and high availability. The theory of lead acid batteries will be discussed in more detail in the chapter 1.3. There are many circumstances that need reconsidering when dimensioning the size of the storage. However, there are two main design philosophies; storage with the capacity to ensure power for one or two hours, to ride out power peaks, or a larger capacity to secure power for several days. The design is done by carefully considering and weighting circumstances as wind conditions, diesel prices and investment cost of the batteries. The smaller size storage is dimensioned after autonomy, which means discharge time or capacity at a continuous load [7].

1.1.6 Inverter

Several components, such as PV panels, batteries or small wind turbines, operate with DC current. An inverter converts DC into AC power. There are several different classes of AC current, examples are, square wave, modified sine wave and sine wave. The square wave current is only suitable for resistive load but it is the least expensive form. A modified sine wave inverter constructs a sine wave by combining small steps. Some electronic equipment does not operate properly with this modified wave. For these cases, a sine wave inverter is the only feasible method [2].

1.1.7 Controller

For effective usage of renewable resources and to be able to regulate the system a controller is necessary. The controller operates as the brain of the system; assesses system condition and performance, controls factors such as frequency and voltage. The complexity of the controller depends on the complexity of the system, since several components need to be measured and supervised. For this project the lifetime of the batteries is an essential question. A decent battery-charging algorithm is vital for continuous operation and a long battery lifetime. Furthermore, the controller should have a low and high voltage disconnect to protect the batteries against over discharge and overcharge.

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1.2 IPSYS (Integrated Power Systems) This chapter gives an overview of the simulation package, IPSYS [6]. Experience has shown that hybrid systems need to be carefully designed. This is not an easily performed task since the systems often consist of several components which make the configuration complex. The wind energy department, Risø National Laboratory, has recently developed a simulation and supervisory tool to perform the design and control of these autonomous power systems. IPSYS operates as a platform where different system configurations, in the evaluation phases, can be designed and evaluated. The ability to simulate energy systems opens up possibilities to investigate different mixes of energy generation and as a result provide energy at a lower cost. As a side effect it can be a push forward for exploitation of local renewable energy resources. To satisfy investors a technical and economical analysis needs to be done over the proposed system configuration. To be able to perform these calculations correctly the system needs either to be running or at least simulated. With the IPSYS concept it is possible to simulate how an existing diesel system should cooperate with different new system configurations. Performance estimations can be carried out as a result of different operating strategies. By simulating a wide range of configurations it is possible to show that the project can be an economically profitable deal. An additional key feature of IPSYS is that it can operate as a supervisory controller, to apply control signals to operate the interaction between renewable and conventional energy generations. IPSYS is designed to perform accurate calculations in terms of fuel consumptions, power flows and voltage levels. These calculations are performed under the condition that there exists an energy balance for the included modules [6].

1.2.1 IPSYS simulation example

This section will illustrate the potential of the simulation package IPSYS. At numerous places around the world there are communities that maintain their energy from conventional autonomous diesel generators. Several of these locations are directly suitable for implementation of sun or wind energy. The installation of the wind turbines triggers several questions that need to be answered. Available wind resources are modelled in IPSYS by integration with WAsP (Wind atlas and application program) [6]. The power system simulated in this example has a fluctuating load with an average value of 25 kW and a reactive power consumption of 10 kVAr. The hybrid system consists of two Gaia 11 kW wind turbines and three diesel generators. The diesel generators individually supply a power between 5-35 kW and a maximum reactive power of 60 kVAr. Figure 2 presents the simulation result. Figure 2(a) shows running status of the diesel generator. Diesel generator 1 runs continuously while the other two run at 18 % and 0 % respectively of the simulated period. Figure 2(b) presents the voltage fluctuations at the defined busbars. Figure 2(c) illustrates the different powers. The grey line indicates that some excess energy must be dumped to keep the power balance in the system. The fourth and last Figure 2(d) presents the total diesel consumption.

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By analysing the simulation results it is possible to see that an implementation of battery storage in the system configuration could save diesel runtime. In a hybrid system the diesel generators are configured with a minimum runtime. A small load peak can trigger the diesel generator to run for a defined period of time. The load peaks that trigger diesel generator 2 to start in Figure 2(c) could instead be levelled out by the capacity of the battery storage. The implementation of battery storage would then compensate diesel generator 3 fully and save fuel consumption for the remaining two diesel generators.

(a)

(b)

(c)

(d)

Figure 2 IPSYS output plot. From top to bottom: Diesel generator status, busbar voltages, active power, fuel consumption for diesel generators.

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2 AIM WITH PROJECT During this project, studies will be undertaken on lead acid batteries that operate in hybrid systems. A lead-acid battery model developed in cooperation between Fraunhofer-Institute and Risø national laboratory will be improved and used as a simulation tool to identify and map major damage mechanisms that cause battery ageing. The goal is to produce a simulation tool that can accurately predict the lifetime and performance of batteries by analyzing the current profiles from a hybrid system. The improved model will in a later project be implemented in a larger hybrid simulation platform IPSYS. The original MATLAB battery model requires further development, in order to shorten its simulation time and to improve battery lifetime predictions. The current model severely over predicts the lifetime of the simulated batteries and is in strong need of improvement. The current model does take into account several important physical properties, one among which is the effect of a low charge on the crystal distribution on the electrode’s surface. In order to achieve an improved model a complete understanding of the current model is essential. Consequently, a review of the mathematics and the functions used in the model is undertaken. The parts of the model that need improvements must be identified. Furthermore, the model requires reconstruction to yield faster calculations and enable a later translation and implementation in IPSYS. This project will follow-up the request for more validation of the battery model, stated in earlier projects. The simulation results from the improved model will be analyzed in search of control strategies that will improve the battery lifetime in a hybrid system. Both old and new current profiles will be simulated in order to get an enhanced validation and complete understanding of the model. Even though some of the tested batteries are of the same type and brand, their performances may differ. By comparing simulation results it maybe achievable to understand the spread in performance between “identical” batteries. In summary the project aims are:

• Review of model mathematics, MATLAB implementation and used parameters. • Identification of areas needing improvements and implement improvements. • Enhancing the validation of the model by analyzing old and new current profiles. • Analyzing simulation results in search of control strategies for improved battery lifetime.

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3 LEAD ACID BATTERIES The unpredictable nature of renewable energy sources implies that in most autonomous cases, storage of energy is required to meet the power demand and ensure stability and power quality [8]. Lead acid batteries have been a successful solution for more than a century. The French physicist Raymond Gaston Planté performed initial research in 1860. Even though the lead acid battery suffers from a poor energy-to-weight ratio it is the most commonly used rechargeable battery today. The explanation for this is that the battery provides high-quality performance and has fairly satisfactory life characteristics. This has resulted in the battery being available in many different designs and sizes at a low price, in fact it is the least expensive storage battery and the battery type is sold in growing numbers [9]. A normal sized battery consist of six cells with a combined voltage of around 12 V and a capacity of 40-60 Ah. The overall chemical reaction that occurs is:

edisch

ech

OHPbSOSOHPbOPb

arg

arg

24422 222 +←→++

This will result in a cell voltage of 2.0 V depending on the acid concentration. Looking upon the different electrodes; at the negative electrode Pb is oxidized to Pb2+ during discharge. The Pb(II) ions then react with sulfate ions from the acid and form lead sulfate.

edisch

ech

ePbPb

arg

arg

2 2 −+ +←→

edisch

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PbSOSOPb

arg

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424

2

←→+ −+

At the positive electrode PbO2 reacts during discharge with the incoming electron and a proton from the acid to form Pb2+ and water. The Pb2+ reacts with sulfate from the acid to form lead sulfate.

edisch

ech

OHPbeHPbO

arg

arg

22_

2 224 +←→++ ++

edisch

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PbSOSOPb

arg

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424

2

←→+ −+

For charging, the reaction is reversed. As the cell approaches full capacity the electrode reactions are pushed to the left, converting the majority of PbSO4 into Pb and PbO2. The lifetime of a battery is counted in cycles. A complete cycle is equivalent to a full discharge and charge, and varies widely depending on brand and model. A SLI (Starting, Lightning, Ignition) battery is designed for shallow cycles and is not designed to survive deep discharges. A typical lifetime for the shallow cycle cell design is about 300 cycles while a battery design for deep discharges can withstand 1000-2000 cycles to an 80 % depth of discharge [2]. There are several different lead acid batteries on the market. Examples are flooded, SLA (sealed lead-acid) and VRLA (valve-regulated lead-acid) types. Flooded batteries are the most widespread type, with a liquid electrolyte, for example used in most vehicles. Market needs have driven the development of two new kinds of lead acid systems SLA and VRLA. The difference from flooded batteries is that the electrolyte is absorbed in a gel which makes it possible for the battery to operate in different orientations without spillage [8].

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3.1.1 Major stress factors of the batteries

There are several factors that influence the performance characteristics of a battery. One such factor is the discharge rate which states the current that has been applied and duration. Other examples are the amount of time the battery has been at low state of charge, the time between full charges, and the battery temperature. The temperature is a very important parameter since it strongly affects the chemical activity in the battery.

3.1.2 Major damage mechanisms

Corrosion of positive grid At the positive electrode, oxidation transforms Pb into PbO and PbO2. A thin layer with lower conductivity starts to grow around the positive electrodes as the battery ages. This layer with lower conductivity will increase electrical resistance, as the layer grows in thickness. It will also develop mechanical stresses within the structure. The battery will suffer from capacity losses through an increase in internal resistance. There are three main factors that affect corrosion: battery voltage, acid concentration and temperature. For least damage, the battery voltage should be kept at float voltage, for definition see appendix 1. Both low and high voltages increase the corrosion speed. Increased acid concentration will also increase the corrosion speed. Different acid concentrations have different densities and will react differently to gravity. Stratification will appear where several different concentration layers are formed. High concentration layers are formed close to the bottom and intensify the corrosion. The temperature also affects the corrosion speed. Lead acid batteries have an optimum operation temperature at 25°C. A temperature of 8°C would decrease the battery lifetime by half [8]. Hard/irreversible sulphation Sulfate crystals are formed at both the positive and the negative electrode throughout discharge. To achieve a continuous cycle these crystals are dissolved during charging. However, during certain operation conditions, lead sulphate crystals can aggregate into larger crystals. These crystals may be hard to dissolve and form particularly if the battery is not operated properly. This chemical response is called hard or irreversible sulphation and occurs for instance if the battery is stored during a long time in discharged condition. Least damage is caused to the battery if it is stored under float charge, when the voltage is charged with a slightly higher voltage than the battery voltage and a small current. This procedure prevents self discharge [8]. The sulfate crystals create mechanical stresses within the electrode structure since they have a larger volume than the PbO2 or Pb atoms. Active material will be lost in the formations of the crystals, resulting in capacity losses [10]. Higher discharge currents have a positive impact on the battery lifetime compared to low discharge currents, a relation that can be surprising. The explanation lies in the crystal structure of the electrodes. Under ideal conditions a fully charged battery does not have any sulfate crystals left after recharging. This is not the case for a regular battery under normal conditions. During a discharge with a small current a small number of crystals with a large radius are formed. These crystals will gradually grow over a long time to become large and difficult to dissolve. A charging for a longer time is necessary after a discharge with a small current. A large discharging current, on the other hand, results in small crystals in large quantities with a large surface area. Experiments have showed that large crystals surfaces are easer to dissolve [11].

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Active mass degradation Degradation [8] is a loss of active material due to the reconstruction process during charging and discharging. It is a change in the mechanical structure of the electrodes that will reduce the porosity and thereby the surface area that is essential for ion transport. The loss in area will affect the chemical reaction and reduce the diffusion of the electrolyte. This process cannot be restored in an attempt to fully charge the battery. The active material can, by repeated cycles, become crystalline and finally break loose from the electrode. Shedding As a consequence of both corrosion and active mass degradation, mechanical stress can result in material detaching from the battery structure. This process is called shedding [8] and results in a loss of active material. For example overcharging can cause shedding. Electrolyte stratification It can be discussed if electrolyte stratification [8] is a separate damage factor or simply an accelerator for the corrosion discussed previously. During the chemical processes that occur during charge and discharge of the battery, the concentration of the electrolyte fluctuates. Gravity will affect the different electrolyte concentrations. Higher density electrolyte will sink towards the bottom and an electrolyte gradient will be built up. By overcharging a battery, bubbles are formed which mix the electrolyte and the problem with stratification can be overcome.

3.1.3 Battery controller

The battery controller supervises the charging algorithm. A common approach is to divide the charging into three steps. The standard charging method is called an IUIa charge [8]. During charging the battery is charged with a constant current. This current is called the “bulk current”. The charging continues until the voltage reaches the upper voltage limit point, in this case 2.4 V. At the end of the first step the voltage increases up to that level where gassing occurs. This part is called the I phase. In the next phase, the U phase, the voltage is kept constant meanwhile the current lowered. The current drops down to a constant value and the Ia phase begin. The voltage can increase until it reaches an earlier defined safety point [12].

A good quality controller should also operate as a deep discharge protection. Otherwise during deep discharges a thin PbSO4 layer is formed at the surface of the battery plates and, as mentioned earlier, this layer can be hard to dissolve resulting in degradation that reduces the capacity dramatically [12].

To prevent acid stratification [8], the controller should allow some periods of over charging. Acid stratification occurs mostly in PV systems where the charging procedure is a slow process. During overcharge hydrogen and oxygen bubbles are formed that will force the electrolyte to blend. This procedure is optimised if it is allowed to occur during 1-2 h each month. If it occurs too often then the battery lifetime will be decreased due to corrosion. The acid stratification will be discussed in more detailed later.

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3.2 Battery model The following text presents an overview of the battery model used in this project. Mathematical simulations make the development of new technology both faster and more reliable. By using the power of the computer it is possible to save both time and cost from expensive experiments. The main goals with the development of the battery model are optimization of systems and minimizing cost of storage by simulating major damage mechanisms. A good knowledge in energy storage is a key factor for a successful development. The original battery model was developed at Fraunhofer-Institute, Freiburg and used for lifetime estimations of batteries in photo-voltaic systems. In order to be able to perform estimations of current profiles from wind systems, improvements of the model have been done, partly at Risø, (European Union Benchmarking research project)[8]. Simulations of battery lifetime in hybrid systems are essential to be able to make relevant cost estimations of the system. The batteries constitute a significant part of the investment cost and it is commonly agreed that the weakest part of renewable based rural energy systems is the batteries. A large part of this project has been to review the model, with the ambition to understand how the different functions of the model interact in order to predict performance and lifetime of the simulated battery. The mathematics behind the model will be discussed in more detail in chapter 3.0. This section will provide an overview of the structure of the model. There are several different methods to develop a lifetime model. The models used for this project combine a performance model with an ageing model.

Figure 3 A strongly simplified flowchart of the FhG/Risø battery model. Figure 3 provides a simplified flowchart of the calculations in the model. [8]

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The input into the model is a current profile containing the information used by the battery model to determine how the battery ages in the system. Another input is the battery specific parameters that describing the type of battery simulated. Some of these parameters are defined by the battery manufacturer others are established through a parameter fitting methodology. The performance model is primarily constituted of two parts; charge transfer and the battery voltage. The charge transfer, SOC (state of charge), is calculated by a time integral of the current input. For this battery simulation tool, and for most other battery models, the voltage is calculated by the Shepherd Equation. The voltage is calculated with respect to the current profile, discussed in more detail in chapter 3.2.2. The voltage is used as a control parameter and it is the voltage model parameters that are used in order to simulate the ageing mechanism. These calculations are performed at every model timestep according to the current input and various weighting factors. If the battery capacity falls below 80 % of its nominal capacity1 the battery life-time expires. The main assumption in the battery model is that degradation and corrosion are assumed to be calculated separately. Thereby, the ageing model can be built by a corrosion part and an active mass degradation part. The degradation embraces damage mechanisms such as hard/irreversible sulfation, shedding, electrolyte stratification and active mass degradation. Some of these damage mechanisms are simulated indirectly by using parameters and data from earlier performed battery tests. The output from these calculations results in different factors that combined simulates the reduction in capacity. By subtracting the reduction from the nominal capacity the remaining capacity is achieved which is a measurement of battery lifetime.

3.2.1 Batteries simulated

As mentioned above the batteries, if not treated correctly, can be a weak link in the system. Testing has been performed on the two types of lead acid batteries that are used extensively in renewable energy systems. As a validation of the performance of the battery model tests were carried out in three laboratories CRES, JRC ISPRA and GENEC. The batteries are manufactured by BAE and known as OGI 50 and OPz 50. Both are of the type 6 cells and an output voltage of 12 V. The rated capacities of both batteries are 50 Ah. The OGi battery has a positive electrode made by a round-grip plate in a corrosion- resistance Pd alloy and the negative electrode made by a flat plate based on antimony alloy. The OPz battery has a positive electrode consisting of a tubular plate with a woven polyester gauntlet of Pd alloy. The negative electrode corresponds to an antimony grid equivalent to the one used for the OGi battery. [14],[15]

1 For definition see appendix 1

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3.2.2 Battery model input

The battery model uses a current input in order to calculate battery voltage output. Depending on what system the battery is connected to, the current input profile will differ. In the case of the wind charge of the battery the profile has a wide distribution range compared with a PV charge. During the development of the battery model several different batteries have been subjected to test current profiles. These tests are then used for calculations, evaluation and calibration of the performance of the battery model. The present model has been tested against two different current profiles; the wind and the photovoltaic (PV). The wind and PV profiles are constructed to represent the operating conditions for batteries used in hybrid systems. Wind profile A wind block consists of 1 h discharge to reach 90 % SOC, followed by 50 wind profiles and a capacity test. One profile has the duration of 15.75 hours and is produced by multiples of I10 to ensure comparability. [8]

Figure 4 Current profile for wind simulations. PV profile The photovoltaic profile is shown in Figure 5. The magnitude of the current is smaller than for the wind profile. Consequently, the PV profile will have a smaller effect on the lifetime of the batteries compared with those tested with the wind profile, which is also the case in reality.

Figure 5 Current profile from PV simulations. The PV blocks are constructed by an I10 discharge for two hours until a SOC of 80 % is reached. This is followed by 35 PV profile cycles, a discharge for three hour to reach a SOC of 50% continued by additional 35 PV profiles. The final part is made by a charge of a three hours charge to a SOC of 80 % continued by 35 PV profiles. [8]

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Capacity tests Each test file consists of numerous blocks followed by a capacity tests. The capacity tests consist of three capacity measurements. The first one is a residual capacity test designed to measure the capacity left in the battery after each block. The second one is a solar capacity measurement, designed to replicate the typical charge from a photovoltaic system. The third measurement is to estimate the true capacity of the battery and is followed by an IUIa charge described earlier. [8]

3.2.3 Test runs

This section presents the test results of the simulations with the initial MATLAB model [15]. From Figure 6(a) it is possible to see that the model over predicts the lifetime of the test battery, since the remaining capacity (blue) does not fit the capacity test of the test battery (red stars). Instead the simulation stops when the battery remaining capacity reaches 80 % of the rated capacity (630 days).

Figure 6 Simulation results from the initial model. OPz Battery. Wind current profile from CRES. From top to bottom: (a) Remaining capacity (blue), State of charge, SOC (F, green) and test battery capacity (red stars). (b) Modelled voltage (Ucell, blue). (c) Test current (I blue), Modelled controlled current (I batt, green), Initial discharge current (Icf, red)

(a)

(b)

(c)

15

In Figure 6(a) it can be seen that the capacities start at 150 %. This is explained by the actual capacity of the battery being larger than the nominal capacity defined by the battery manufacturer.2 The manufacturer deliberately under predicts the capacity of the battery to ensure that under all circumstances the actual capacity is greater than the nominal capacity. Figure 6(b) presents the simulated voltage calculated by the Shepherd equation, described in detail in chapter 4.3. The bottom diagram, Figure 6(c), shows the test current in blue and the modeled current in green. After 210 days the controller steps in to restrict the charging current to save the battery. The controller is currently extracted from the improved model and later simulations will not show this behavior. Figure 7 is an enlargement of the model voltage compared with test. It is possible to see that the modelled voltage does not follow the peaks of the test voltage. It needs to be mentioned that this is a lifetime model and not designed to perform accurate calculations of the voltage. Most noticeable is the over prediction of voltage swings. An additional observation is that the voltage fits better during charge than discharge.

Figure 7 Difference between the modelled voltage and the simulated voltage. The battery model was made for PV systems with currents in the range of I10 or less. By including the wind profile the currents increase. This affected the simulations and made the model over predict lifetime of the batteries. The model includes simple control strategies, such as cut off. The over voltage controller steps in at voltage of 2.45 V/cell.

2 For definition of the different capacities see appendix 1

16

Figure 8 presents the coefficients that result to a decline in capacity Initial discharge capacity coefficient (Cd, green). Capacity reduction from degradation (Cdeg, red). Capacity reduction from corrosion process (Ck, light blue ). Internal resistance (rdtotal = ρd,total, mauve).

Increase in internal resistance from the corrosion process (rk =ρk, yellow). Figure 8 shows the contribution of various capacity factors to the total capacity. The degradation (Cdeg) contributes to the largest reduction in capacity while the corrosion coefficient (Ck) contributes only to a small extent for this OPz battery. The total capacity coefficient (Cdtotal) is derived by subtracting the different reductions in capacity (Cdeg, Ck) from the initial discharge capacity coefficient (Cd). From the result of the benchmarking project there was an appeal for more test data to validate the operating mechanism in the model and a request to validate the model in full scale system testing.

17

4 METHODS

4.1 Improved code

4.1.1 Varying time steps

The original model was programmed to calculate data with a constant time step of 300 s. Such a constraint is not valid for the newly assembled datasets from ISPRA since the data was sampled at different frequencies. Modifications have been made so that the model can cope with data files with varying time steps, both in calculations and presentation of simulation result. Each time step is calculated individually by (Eq1)

The code for graphical presentation has been rebuilt to incorporate a varying time step. The improved model presents the simulation results as a function of a cumulative summation of the time steps. Through this approach it is possible to get a continuous timescale measured in days that starts at zero and continues until the battery is considered dead. Furthermore, computer memory is saved by only displaying every twentieth simulation value. This resolution still guarantees sufficient data for representative figures. To facilitate further use and analysis of simulation results all figures are auto saved with proper filenames in (jpg)-format into a user defined directory.

4.1.2 Improved design with object-oriented programming, OOP

An important step to improve the structure of the model is to evolve the OOP (object-oriented programming) technique. The OOP technique was used in the initial model but not in full extent. This section will try to provide the reader with a short survey of this programming theory. OOP is about programming abstract data types (ADT), and their relationship. By this method it is possible to break up the program into separate fractions that will result in a better organised program. The modules of OOP are called classes. A class is an expansion of the principle of struct used in the traditional C-language. Each class can be seen as a blueprint to objects, like a car factory using the blueprint to produce cars of a certain brand. An object can be described as an exclusive variable that both stores data and is able to satisfy the requests sent to it. The function that answers the requests inside the object is called a method. Each object has its own user defined methods. A good program consists of a web of interacting objects, sending information telling each other what to do. By this communication it is possible to build up system complexity but at the same time hide behind the simplicity of the object. A proper OOP structure will contribute to an easier translation into IPSYS and C++ [17]. The initial MATLAB model consisted of one single large battery object that contained all mathematical methods used for the simulation. The large battery object in turn consisted of smaller

)()1( itimevectoritimevectort −+=∆ Eq 1

18

structures such as voltage and degradation. This made foreseeable difficult but it also implied that a method could by a simple mistake easily change variables outside its scope, since it had access to all variables. To improve the structure the large battery object was divided into smaller objects. The structures used in the initial model were upgraded to objects and corresponding methods were moved from the large object into the object it belonged to. By restricting the methods to only be able to write to the variables that belong to the same objects, data were encapsulated and secured for manipulation from the outside by another object. To improve understanding, all methods were renamed with a name that expressed what purpose their functions have to fill. Furthermore, since the battery model has its origin in Germany some variables had to be translated into English. To ensure that the variables are defined and used in the right object a variable overview graph was constructed using the DOT software [18]. The overview prevented the variables from being defined in the wrong objects, which would lead to unnecessary communication between the objects, and thereby use unnecessary CPU power. The structure was optimised by moving the variables into the objects where they were utilized. The principle is shown schematically in appendix 2 where also the new variable structure is shown.

4.1.3 Vectorizing

Calculation speed can be increased by replacing traditional if and for statements with inbuilt vector functions that MATLAB offers. One example of this principle was implemented in the part of the code that calculates the dynamic timestep. The initial code was: for i=1:data_length % Main program loop. Each iteration is one timestep. if (i<data_length) dt=(data(i+1,1)-data(i,1))*24; % calculates timestep from days to hours. end if (i==data_length) dt=0.0001; end end This was replaced with the single statement: dt_series=[diff((data(:,1)-data(1,1))*24);0.0001]; Vectorizing followed by removal of unnecessary statements improved calculation speed dramatically and as a side effect improved the readability of the code.

4.1.4 Global Variables The initial model uses global variables, which are variables initial declared and accessed from any part of the program. In the improved model this element is replaced. The definitions of variables are instead moved into the objects where they are used. Furthermore, these variables are now write protected to secure their value.

19

4.2 Current factor

4.2.1 Definition of Current factor

The first battery model in MATLAB made poor predictions since it underestimated the battery lifetime dramatically. Especially for the wind profiles, where higher currents are present, it could be seen that the deep discharge factor, a factor implemented in an attempt to simulate the effects that result by deep discharging a battery, had a strong impact that reduced the predicted lifetime. This had not been seen earlier since the original model from Fraunhofer-Institute used smaller PV-currents. In reality it is the opposite. A high discharge current has a positive impact on battery lifetime; this will be explained in further detail later. A factor called the current factor (CF) had been implemented in the initial model to be able to deal with high currents that otherwise would have had an unbalanced effect on the lifetime. The factor was designed to reduce the rate of capacity degradation at high currents. For the calculations an average period of 36 hours and a tuning constant C was used, see chapter 4.2.2 for mathematic definition. These parameters are arbitrary numbers and do not have a physical basis. After some consulting it was decided that the current factor needed to be redefined and a new approach needed for the improved model. During discharge, Pb at the negative electrode and PbO2 at the positive electrode will oxidise to Pb2+. A large discharge current will result in that a large number of Pb2+ ions are formed. The opposite relation pertains to a small discharge current. If a seed PbSO4 crystal is present the reaction Pb2+ + SO4

2- → PbSO4 will begin and thereby lower the concentration of Pb2+. If a seed crystal is not present the ion concentration of Pb2+ must be high compared with the saturation concentration in order for the reaction to take place. As soon as a crystal is present the PbSO4 molecules will aggregate into larger crystals. This reaction will decrease the Pb2+ ion concentration and the production of PbSO4 will cease. Consequently, new PbSO4 molecules are formed at the initial state of the discharge. A small discharge current will result in a smaller number of PbSO4 molecules. Furthermore, a small discharge current implies a long period of discharge, ensuring that the crystals will have time to arrange into large structures. The opposite occurs for a large discharge current. A large number of crystals are formed, but since the discharge time is reduced by the battery capacity only small PbSO4 structures are able to build up [11]. A comparison of the crystals surface area can be made. The high discharge current with its large number of small crystals, will have a larger surface area than the crystals from a lower discharge current. Experiments have shown that when the battery is charged a larger surface area is easier to dissolve than a smaller surface area. Consequently a large discharge current is optimal for a battery. A small discharge current will result in longer charges to dissolve the large crystals. Additionally there is a risk that crystals are not dissolved and that the battery would age by sulfation. This theory can be used to define a new current factor. Poor recharging affects the size and number of crystals present on the electrodes of the batteries. If a battery is not properly charged the surface of a single crystal increases but the total number of crystals decreases. The result is that the total crystal surface area decreases, which makes later charges more difficult. In an attempt to simulate the influences of poor charging a factor was implemented that keep track on the number of poor charges. A bad charge is defined to occur when the SOC does not reach 99 % [11],[19].

20

4.2.2 Current factor used in the initial model For calculations of the current-factor, in the initial model [8], a running discharge current average is required. It was calculated by summing the discharge Ah-throughput over the past 36 h, Q36h in (Eq 2), and dividing by the sum of the time steps as these discharges occurred, T36h in (Eq 3). The average discharge current is expressed by (Eq 4).

This average discharge current was then used in (Eq 5) to achieve the current factor that gets the value 1 for small currents and 0.1 for large current values. CN is the nominal battery capacity.

In the above expression a constant C is introduced. By changing the magnitude of C the current factor can be tuned. A value of C equal to 0.1 will result in a current factor of size 1 for a current of I10. All higher currents will result in a current factor < 1. Accordingly, if C is set to 0.01 the current factor will equal 1 for a current of I100 and all higher currents will result in smaller values. This will reduce the impact on the aging mechanism and the simulated battery lifetime will be increased. This is something that is not physically explained and needs to be reworked in the improved model.

h

hedischaverage T

QI

36

36arg =

∑∆=t

hpasth tT

3636

∑ ∆⋅=t

hpastedischh tIQ

36arg36

edischaverage

N

I

CCtorcurrentfac

arg

⋅=

Eq 4

Eq 2

Eq 5

Eq 3

21

4.2.3 New approach for the current factor in the improved model

With the assumption that all lead sulphate crystals have the same radius it is possible to calculate the number of crystals per cm3 by dividing the total surface of all crystals with the surface of one crystal with radius r as in (Eq 6) below [19].

The number of crystals depends only of the initial discharge current and will not depend on DOD (depth of discharge) [11]. Figure 9 shows the number of crystals plotted against a normalized discharge current for 50 % and 100 % DOD. According to the discussion above the two plots results in one single curve.

Figure 9 Number of crystals visualized as function of the discharging current, I1-100. The values DOD 100% is laying exactly beneath the values for DOD 50%. By using the MATLAB toolkit polynomial regression a function that represents the number of crystals in the battery can be found by using (Eq 7).

CN is the nominal capacity of the battery and I is the initial discharge current. The constants were determined to be: C0 = 2.5 1011

π, C1 = 4.4 cm-3 , C2 = 1.5 h and C3 = 1.2 h.

24 r

Az

⋅=

π Eq 6

Eq 7

⋅+

−⋅=

N3

2

N10 exp

C

IC

IC

CCCz cf

cf

22

The number of crystals can be found by multiplying z with a reference volume V0 (Eq 8):

The radius for one crystal can be calculated by calculating the radius for a sphere in (Eq 9). The volume V of one lead sulphate crystal.

The cumulative surface area is then, (Eq 10).

We now define the current factor (CF) as the ratio between a reference surface area and the present surface area (Eq 11). Simulations have shown that I50 is an appropriate reference current, Iref.

The radius in expression in (Eq 9) now can be substituted into (Eq 11) which gives (Eq 12):

(Eq 8) then is used to express CF in terms of z. V0 can be assumed to be constant and will thereby be cancelled. Furthermore the CF is weighted with a factor that keeps track of the influences of insufficient charge Cbad as in (Eq 13).

The derived current factor gets a low value for high discharging currents and a high value for lower currents. From Figure 10 it can be seen that I10 results in a current factor of 0.5.

0Vzn ⋅=

3 refref

2

3

ref n

n

n

n

n

nCF =⋅

=

3

4

3

n

Vr

⋅⋅=

π

nrA ⋅⋅= 24π

nr

nr

A

ACF

⋅⋅

==2

ref2

refref

3 ref

z

zCCF bad=

Figure 10 Current factor vs. the discharging current, 20 represent I20 and 100 I100.

Eq 8

Eq 9

Eq 10

Eq 11

Eq 12

Eq 13

23

4.3 Mathematics behind the model In order to describe and provide the reader with some knowledge of how the improved MATLAB model is structured and operates the following chapter presents a short overview. The different functions are presented with the same name as they appear in the syntax code. Furthermore, the functions are listed together with the corresponding object. The objects contains the both the functions and the parameters (variables) used by the functions. For definition of parameters see appendix 11.

CalculateVoltage() Voltage object The voltage-current characteristic is in the model described by the Shepherd-model [8]. The model offers high precision, especially for PV-systems, and requires relatively few parameters compared to other models. The parameters can be obtained by analysis of experimental data. The Shepherd equation is expressed in (Eq 14) and consists of four terms; U0c represents the open circuit voltage which is in equilibrium with the cell voltage achieved after full charge followed by a period of rest. The second term is associated with the state of charge. Then a term follows, which represents ohmic losses that are assumed to be proportional with the current. The fourth and final term models the charge factor over-voltage which increases when the battery is either close to being empty or fully charged.

Ucell [V] terminal voltage of a battery cell F normalised SOC (Q(t)/CN, F<=1) I [A] discharge current (I<=0), charge current (I>0) U0c, U0d [V] open circuit equilibrium cell voltage gc,gd [V] proportionality constant over the electrolyte ρc, ρd [ΩAh] parameter for internal resistance CN [Ah] nominal capacity defined by battery manufacturer Mc charge transfer over-voltage coefficient Cc,Cd normalized capacity coefficient

CalculateSOC() PbBattery object The state of charge SOC, expressed by the variable F, is calculated by integrating the effective current and then dividing it with the nominal capacity (Eq 15 below). The effective current is strongly dependent on the cell voltage calculated by (Eq 14) [20].

( ) ( )FC

F

C

IM

C

IFgUUI

cN

cc

Nccccellbatt −

++−−=> ...

10 0

ρρ

( ) ( ) ( )( )FC

F

C

IM

C

IFgUUI

dN

dd

Ndddcellbatt −−

−++−−=≤1

1.

..100

ρρ

Eq 14

24

CN [Ah] nominal capacity defined by battery manufacturer Finit Previous state of charge I [A] Test current IGas [A] Gassing current At too high voltages gassing occurs that will lower the effective current.

CalculateGassingCurrent() Gassing object The gassing current is expressed in (Eq 16) [20]:

Here IG0 is the gassing current under normal conditions for a 100 Ah battery calculated by (Eq 17). Ucell is the cell-voltage (V) and UN is the cell voltage at nominal conditions (2.23 V). cU and cT are voltage and temperature coefficients respectively. TN is a reference temperature defined as 25 C and Tbatt is the cell-temperature.

In (Eq 17) ρk,t and ρk,limit are introduced for the first time. Both are variables concerning the internal resistance of the battery and are in a liner relation to describe how the gassing increases with ageing.

CalculateCorrosionVoltage() Voltage object With a modification of the Shepherd equation it is possible to determine the corrosion voltage (Uk). This voltage is a key factor in the investigations of how the corrosion layer grows at the positive grid. The equation has been equipped with a constant (0.5) to determine the voltage distribution between the positive and negative electrode, Eq 18) [20].

U0

k,0 [V] corrosion voltage per cell at full charge gk [V] electrolyte coefficient voltage ρc, ρd [ΩAh] parameter for internal resistance Mc charge transfer over-voltage coefficient

Gaseffectiv

t N

effectivinit IIIdt

C

IFF −=+= ∫

=0

( ) ( )( )NbattTNcellu TTcUUcG

NGas eI

Ah

CI −+−= 0100

itk

tkGI

lim,

,0 06.002.0

ρρ

+=

( ) ( )FC

F

C

IM

C

IFgUUI

cN

battcc

Ntckkkbatt −

++−−=> 0,,

00, 5.05.01

13

100

ρρ

( ) ( ) ( )( )FC

F

C

IM

C

IFgUUI

cN

battcd

Ntdkkkbatt −−

−++−−=≤1

15.05.01

13

100 0,

,0

0,

ρρ

Eq 15

Eq 16

Eq 17

Eq 18

25

CalculateCorrosionSpeed() Corrosion object The corrosion speed parameter (ks), is assumed to be related to the corrosion voltage. The initial studies of corrosion speed were done by J. Lander [21] and the relation is shown in Figure 11.

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Cor

rosi

on S

peed

Par

amet

er

Corrosion Voltage (V)

Lander Corrosion Speed Curve

Figure 11 Relation between Corrosion voltage and corrosion speed parameter ks.

In the function calculateCorrosionSpeed() this relation is used to derive (ks).

CalculateCorrLayerDepth() Corrosion object When the corrosion voltage and corrosion speed have been found they can be used to calculate the thickness of the corrosion layer (∆Wt), (Eq 19) [20].

( ) tk

WtwheretkWU

s

tstk ∆+

∆=⋅=∆< −6.0

1

16.0 ''74.1

( ) tkWWU sttk ∆⋅+∆=∆≥ −174.1 Eq 19

26

CalculateInternalResistance Voltage object The internal resistance, (ρd,t), of the battery is assumed to consist of an initial resistance, (ρd,0), and a contribution from corrosion, (ρk,t). For each time step the internal resistance will have a specific value that can be calculated by Eq 20 [20]:

CalculateCorrosionResCap() Corrosion object The contribution from corrosion discussed in the previous function, (ρk,t), is assumed to be exponentially proportional to the end corrosion layer thickness (∆Wlimit) as in (Eq 21) below [20]. The same exponential relation is assumed to be valid for the decrease in capacity due to corrosion (Eq 22) [20]:

ρk,limit increase in ρd,0 due to corrosion at the end of lifetime

Ck,limit decrease in Cd due to corrosion at the end of lifetime

CalculateFinalClThikness() Corrosion object (Eq 21) and (Eq 22) use the expression (∆Wlimit) as calculated below (Eq 23). (∆Wlimit) is the corrosion layer at the end of battery lifetime. It is determined by using the declared lifetime of the battery, L80%. L80% is a battery parameter in years and is determined by the battery manufacturer and states the lifetime of the battery until it reaches a capacity of 80%.

Here (ks) is the term for corrosion speed determined during float charge, (Ifloat and Uk,float).

tkdtd ,0,, ρρρ +=

−⋅=∆

∆ tit

WW

itktk eCC lim

2

1ln

,lim, 12

sit kLW ⋅⋅⋅=∆ %80lim 24365

−⋅=∆

∆ tit

WW

itktk e lim

2

1ln

,lim, 12ρρ

Eq 20

Eq 21

Eq 22

Eq 23

27

CalculateCorrIncreaceLimit() Corrosion object The proportional constant (ρk, limit), in (Eq 21) is determined by considering (Eq 20). At the end of the battery lifetime the internal resistance, (ρd,t), is equivalent to the final internal resistance, (ρd,t = ρd,end). Furthermore, at the end of the battery’s life the exponential ratio in (Eq 21) is 1 which implies that (ρk,t = ρklimit). By substitution and reorganisation of Eq 20 it is possible to derive (ρk,limit)

by (Eq 24)[8]:

The term (ρd,end) is possible to calculate with a reorganisation of the Shepard equation ( Eq 25). In this expression (Ucell) has been substituted with, (Uempty), the voltage for a fully discharge cell to simulate the end of life.

The decrease in discharge capacity due to corrosion, (Ck,limit), is assumed to be proportional to the discharge capacity coefficient by (Eq 26)[20]:

The constant 0.16 is an assumption that 16 % of the total capacity loss is lost due to corrosion.

It is assumed that the capacity reduction due to degradation can be expressed as the difference between initial capacity (Cd,0) and the reduction in capacity up until that time (Cdeg,t),(Eq 27) [20]:

CalculateCapacityLossDeg() Degradation object An exponential approach is chosen to describe (Cdeg,t) the degradation capacity decreases (Eq 28) [8]:

Cdeg,limit degradation in capacity at the end of lifetime CZN coefficient of exponential function (CZN = 5) ZgN weighted standardized charge throughput Z life cycle without corrosion, determined by multiplying the life cycle from the battery manufacturers data sheet with 1.6 (Eq 29):

0,,lim, dendditk ρρρ −=

( )

−⋅−+−=

HC

HMHgUU

I

C

d

dddodempty

batt

Nendd

0,

0,, 84.0

ρρ

0,lim, 16.0 ditk CC ⋅=

tdtd CCC deg,0,, −=

−−

⋅= Z

Zc

itt

gN

NZ

eCC1

limdeg,deg,

%806.1 ZZ ⋅=

Eq 24

Eq 25

Eq 26

Eq 27

Eq 28

Eq 29

28

CalculateAcidStratification() Acid Stratification object Acid stratification is an electrolyte concentration gradient that increases during repeated cycles of charging and discharging a battery. The acid stratification factor (fs,t) is modelled by two variables (fplus) and (fminus) to describe the build up and breakdown of the acid stratification. The current factor (CF) is implemented to adapt (fs,t) for higher currents (Eq 30) [8]:

CalculateDeepDischargeFactor() Deep Discharge object The deep discharge factor (fq) was implemented in an attempt to simulate the effects that result by deep discharging a battery. The deep discharge factor has the value 1 when the battery is fully charged and increases with elapsed time until next full charged. The growth is proportional to the time elapsed since last full charge (tF) and to the lowest state of charge reached during that period (Fmin). The current factor (CF) is implemented to adapt (fq) for higher currents, ( Eq 31) [20].

The constants (c,f0) and (cf,min) are used to model impact of State of charge (Fmin).

CalculateCurrentFactor() CurrentFactor object The current-factor described earlier is defined by (Eq 32)[19].

Where (z’) and (z’ ref) represent the number of crystals. Cbad represent the status of the charge. A bad charge is defined as the state of charge (SOC) that never reaches above 99 % during charging.

CalculateChargeTroughPut() DeepDischarge object The charge throughput (Qct) is calculated in (Eq 33) by summation of the product of effective discharge currents (Ieffective) and the time period which the discharge appeared ∆t. The effective discharge current is equivalent to (Ibatt - Igas ).

CalculateWeightedChargeTroughPut() Deep Discharge object The weighting takes into account length of cycles, deep discharges, and state of charge. Extreme deep discharges lead to an additional burden for the simulated battery.

( ) CFtffff uspluststs ⋅∆−+= − min1,,

( ) CFtFccf Fffq ⋅⋅⋅−+= minmin,0,1

3 ref

z

zCCF bad ′

′=

∑ ∆⋅=t

effectivect tIQ0

Eq 30

Eq 31

Eq 32

Eq 33

29

In the function below (Eq 34) the weighted standardized charge throughput (Zgn) is calculated. Zgn is equivalent with the number of cycles that would cause the same damage to the battery as without acid stratification and deep discharges.

fq deep discharge factor fs acid stratification factor Cdeg,t decrease in discharge capacity to simulate accelerated degradation CN [Ah] nominal capacity defined by battery manufacturer

CalculateLife BattLifetime object The number of times the nominal capacity has passed through the battery is called nominal cycles (ZN). It is calculated by adding the discharge throughputs and dividing by the nominal capacity as in Eq 35. The discharge throughput is equivalent to (Ibatt - Igas )∆t when Ibatt<0 [20].

CalculateDegIncreaceLimit() Degradation object At the end of the battery lifetime the variable (Cdeg,limit) is calculated in (Eq 36) by reorganization of (Eq 27). The capacity coefficient (Cd,t) in (Eq 27) is substituted with a reorganized Shepherd equation. At the end of the battery life (Cdeg,t) in (Eq 27) is equivalent with (Cdeg,limit) since the exponent in (Eq 28) is 1 [20].

CalculateTotalCapacity() Voltage object The results from the different aging mechanisms, Eq 22 and Eq 28, are combined to form a general expression over the reduction in capacity in Eq 37 below. Cd,total is then used in the Shepherd equation to determine a voltage and to determine the remaining capacity.

( ) Nt

ctsqgngn CC

QffZZ

deg,1 1−

+= −

∑ ∆⋅=t

effectiveN

tN tIC

Z0

,

1

( )

( )

+−+−

−+−=

Ntdddcell

ddN

dit

C

IFgUU

FMC

I

HCC

,,0

0,

0,limdeg,

1

1

ρ

ρ

ttkdtotald CCCC deg,,0,, −−=

Eq 34

Eq 35

Eq 36

Eq 37

30

CalculateMaxChargingcurrent() Voltage object By the above described reorganising of the Shepherd equation it is possible to calculate the maximum permitted charging current (Eq 38). This is done by considering the voltage at full charge, Umax[20].

4.3.1 Parameters

SetVariant() PbBattery object The parameters for the Shepherd equation for the initial battery model were found by a least square fit of a parameter test profile. For a better fit some values were tuned by hand. The values used in the current model are expressed for the battery models OGi in table 1 and OPz in table 2.

Other parameters that are not related to the Shepherd equation but battery specific and used in the improved model are given in table 3-4.

( )

⋅+

−+−=

FC

FM

CFgUUI

ccctc

Ncocb

0,,

maxmax 1ρρ

U0d 2.18 V U0c 2.20 V gd 0.17 V gc 0.12413 V ρ0d 0.69902 ΩAh ρ0c 0.42 ΩAh CN 54 Ah Md 0.04642 Mc 0.88761 Cd 1.75 Cc 1.001

U0d 2.10 V U0c 2.20 V gd 0.096541 V gc 0.13071 V ρ0d 0.37885 ΩAh ρ0c 0.50 ΩAh CN 50 Ah Md 0.28957 Mc 0.36488 Cd 1.642 Cc 1.001

Table 1 OGi parameters Table 2 OPz Parameters

Lt 14 Zd 1400 Iref -100 ks 0.0435

Lt 10 Zd 600 Iref -10 ks 0.0035

Table 3 OGi specific parameters Table 4 OPz specific parameters

Eq 38

31

4.4 New test series

4.4.1 Test series for model valuation In the benchmarking project (2004) the initial MATLAB battery model was used to analyse current profiles used for testing batteries at the CRES laboratory. The results from the simulation showing the voltage behaviour, capacity and battery lifetime could then be compared with the authentic values from actual batteries tested with the same current profile. One of the results from this project was an appeal to validate the model on other test data. Several other battery tests have been carried out in the laboratories of JRC-ISPRA and GENEC [8] but the results had not been analysed or tested by the initial model. Current profiles from GENEC A current profile from GENEC consists of a (.xls) file with several sheets of data. Large efforts were put into trying to paste these sheets together into a continuous test file, but without any result. The problem was that the data sampler at GENEC had not recorded the tests properly. The time series were discontinuous, with large time gaps. This would not have been a problem if the test battery would have been dormant during the period, but this was not the case since it could be seen that the battery had been subjected to both discharges and charges during this period. An additional problem was that the sample rate differed throughout the series, shifting from several hours to zero. A zero sample rate implied that the same measurement appeared several times. The analysis of the test data lead to the conclusion that it was impossible to make anything useful out of the data sets from GENEC. Current profiles from ISPRA One current profile from ISPRA consisted of 46 (.csv) files named by the battery type (OGi1, OGi2, OPzs1, OPzs2) - Series (S) Phase(P) - date (d-m-y) and the corresponding C10 capacity tests are (OGi1, OGi2, OPzs1, OPzs2)- C10 (a, b, or c) - date (d-m-y). This awkward way of naming the files caused problems as it was impossible for a computer to sort the files into a continuous order according to the filename. Sorting by hand was not an option since there were 4*46 files. A solution was found by constructing a script that opened the files and read in the header-rows inside the files. The headers consisted of the date and the sample start time for that specific file. These values were transformed by the MATLAB operator (datenum) that converts a date vector specified by year, month, day, hour, minute, and second, into a serial date number. A serial date number can be used in a MATLAB context to generate a timescale that offers significant performance improvements to the code. 1-Jan-0000 is used as a reference point and elapsed days from this point are summarised. The following example illustrates the assignment of serial date numbers: Consider a point defined in time, as an example; 25-aug-2005 13:54:05. By the operator (datenum) it is possible to transform the time vector into a serial date number of 7.325495792245370*105. It would take considerable computer strength to perform calculations with the date vector. However, with the serial date number the calculations are preformed easily. A transformation into a serial date number enabled the sorting of the (.cvs) files into a continuous order by a sorting algorithm. The script then loads the ordered files and compounds them into a large test file. The ISPRA data have large jumps in time, but it could be seen that the battery had been dormant during these periods. Large jumps would damage several functions in the model that uses the time steps as an input. A too large time step would for example severely unbalance the weighting factors. Two alternative solutions were discussed: to fill the gaps with an artificial timescale and put the

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current for these points to zero, or simply cut out periods where a large time jump occurred. The first alternative was chosen since it would not be correct to manipulate the timescale. A program was constructed that looked up large time steps and filled them with artificial time regular sized steps. During these periods, when the battery is at rest, the script sets the current to zero.

4.4.2 Test series for validation of model response

A battery operating in a hybrid system is exposed to a wide range of different current profiles, not only the PV and wind profiles discussed in the previous sections. To be able to validate the simulation model it is necessary that the model is tested against various current profiles. Then the results have to be compared with results found in battery literature. Analysing simulation responses from a valid battery model can give hints to how a regular battery would respond to different current profiles. This can result in that the improved model may provide information that can be used to develop control strategies for enhancing battery lifetime. High and low discharge currents How will the improved model react to high or low discharge currents? According to Professor D.U. Sauer [11] a high discharge current will have a positive effect on battery lifetime compared with a low discharge current.

Figure 12 shows the current profile where the discharge current is adjusted with a factor. The discharge current is decreased with a factor of 0.8 and to get the same charge throughout the discharge time is lengthened by a division of 0.8. The effects of the manipulation can be seen by the test file is longer and the discharge current less. A high discharge test file was generated in a similar way with a factor of 1.2. The file is shown in Figure 13. From the figure it is possible to see that the test file is shorter and the discharging current higher.

Figure 13 Wind profile with high discharge current Figure 12 Wind profile with low discharge current

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5 RESULT AND DISCUSSION

5.1 Simulation validation In this chapter a selection of the results of the simulations from the improved battery model are presented. As described earlier two different batteries are simulated, OGi and OPz, and subjected to either a PV or a wind current profile. The current profiles come from two different laboratories, CRES and ISPRA. First to be tested is CRES. The battery is considered dead if it is not possible to charge to more than 80 % of its nominal capacity.

5.1.1 CRES simulations

The complete simulation results with the current profiles from CRES are shown in appendix 3-6. Figure 14 presents simulations done with the improved battery model. The battery simulated is an OGi subjected to a wind current profile from CRES. In Figure 14(a) it can be seen that the model slightly over-predicts the lifetime of the OGi test battery and the remaining capacity does not follow the test battery capacity (red stars). It can be seen that the capacity starts at 150 % and this is explained by the actual capacity being higher than the nominal capacity defined by the battery manufacturer. The over-voltage controller can be seen in operation in Figure 14(b), limiting the charging current and thereby restricting the voltage to the upper limit 2.45 V.

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Figure 14: Simulation results for OGi battery. Wind current profile from CRES. From top to bottom: (a) Remaining capacity (blue), State of charge, SOC (F, green) and test battery capacity (red stars). (b) Modelled voltage (Ucell, blue). (c) Test current (I blue), Modelled controlled current (I batt, green), Initial discharge current (Icf, red) (d) Initial discharge capacity coefficient (Cd, green). Capacity reduction from degradation (Cdeg, red). Capacity reduction from corrosion process (Ck, light blue ). Internal resistance (rdtotal = ρd,total, mauve).

Increase in internal resistance from the corrosion process (rk =ρk, yellow).

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Figure 14(c) shows the input current profile (test current) and the modeled current but the difference between them is not noticeable. At the beginning of the current profile an initialization sequence can be seen during the first 25 days. Under this period full charges and discharges are made to reset the test battery before the current profile testing starts. Figure 14(d) visualizes the different factors that contribute to the capacity reduction. It can be seen that it is the degradation (Cdeg) that gives the greatest loss in capacity, rapidly increasing at the end of the batteries life. The increase in internal resistance ρk contributes only to a small part to the aging of the OGi battery. Figure 15 presents the different factors modeled in the simulation. The weighted and unweighted standardized charge throughput (Zgn) and (ZN) is shown in Figure 15(a). The angle of the (Zgn) line illustrates the effects of the weighting factors; (fq), (fs) and (CF). If the battery is aged fast, the angle is steep. Notice that in Figure 15(b) the implemented current factor increases with increasing number of poor charges. As a consequence, the (fq) SOC weighting factor increases. Further investigation is needed to explain the peaks of (fq) SOC weighting factor that appears after 60 and 100 days.

Figure 15 OGi factor plot. From top to bottom: (a) Weighted standardised charge throughput (Zgn= dQdf, blue). Weighted standardised charge throughput (ZN = dQd, green). (b) Deep discharge weighting factor (fq, blue). The current factor (CF, green). The cumulative number of poor charges (Cbad, red). (c) Acid stratification factor (f s, blue). The contribution to acid stratification (fplus, red) and reduction in acid stratification (fminus, green)

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Figure 15(c) shows the build up of acid stratification (fs). The model uses the two variables, (fminus) and (fplus), to describe the breakdown and built up of the acid stratification. Both variables are used as weighting factors to describe the acid stratification factor (fs). By analysing Figure 14(c) and Figure 15(c) it can be seen that the stratification is being built during the current blocks and is removed when the battery is fully charged during the capacity tests. During full charges, especially at overcharges, gas bubbles are formed which stir the electrolyte and the concentration layers disappear. A conclusion from this behaviour is that full charges are important to break down acid stratification. Figure 16 shows the results from the OGi battery subjected to a PV current profile. The model correctly predicts that the PV profiles are less harsh and thereby that the battery lasts longer. Estimated remaining capacity Figure 16(a) fits well with the capacity from the test battery, with only a slight over prediction. The lower charging current for the PV profile, Figure 16(c), will by the Shepherd equation, result in a lower battery voltage, Figure 16(d), than is the case with the wind profile, Figure 14(c).

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Figure 16 Simulation results for OGi Battery. PV current profile from CRES. From top to bottom: (a) Remaining capacity (blue), State of charge, SOC (F, green) and test battery capacity (red stars). (b) Modelled voltage (Ucell, blue). (c) Test current (I blue), Modelled controlled current (I batt, green), Initial discharge current (Icf, red) (d) Initial discharge capacity coefficient (Cd, green). Capacity reduction from degradation (Cdeg, red). Capacity reduction from corrosion process (Ck, light blue ). Internal resistance (rdtotal = ρd,total, mauve).

Increase in internal resistance from the corrosion process (rk =ρk, yellow). In Figure 16(d) it can once again be seen that it is the degradation (Cdeg) that reduces the capacity. Corrosion (Ck) contributes only to a small part. Figure 17 presents the simulation results from the OPz battery subjected to a wind current profile. Once again the model predicts an accurate lifetime, Figure 17(a). The higher charging currents

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Figure 17 (c), results in a high voltage, Figure 17(b). The voltage controller has to step in to reduce the charging current and thereby limiting the voltage to 2.45 V. The OPz battery has only a small increase in internal resistance (ρk constant), Figure 17 (d). Thereby it is only the degradation (Cdeg) that causes the OPz battery to fail. Figure 17 Simulation results for OPz Battery. Wind current profile from CRES. From top to bottom: (a) Remaining capacity (blue), State of charge, SOC (F, green) and test battery capacity (red stars). (b) Modelled voltage (Ucell, blue). (c) Test current (I blue), Modelled controlled current (I batt, green), Initial discharge current (Icf, red) (d) Initial discharge capacity coefficient (Cd, green). Capacity reduction from degradation (Cdeg, red). Capacity reduction from corrosion process (Ck, light blue ). Internal resistance (rdtotal = ρd,total, mauve).

Increase in internal resistance from the corrosion process (rk =ρk, yellow). Figure 18 shows the factor plot of the OPz battery. After approximately 180 days the battery cannot be completely charged which is shown in Figure 18(b) by the increase in number of poor charges. As a consequence, the increasing number of poor charges will influence the current-factor and thereby increase ageing by degradation. The same relation is seen for the OPz battery as for the OGi

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battery, the acid stratification builds up during the different current blocks and is removed when the battery is fully charged between the blocks, Figure 18(c).

Figure 18 Simultaion results for OPz Battery. Wind profile from CRES. From top to bottom: (a) Weighted standardised charge throughput (Zgn= dQdf, blue). Weighted standardised charge throughput (ZN = dQd, green). (b) Deep discharge weighting factor (fq, blue). The current factor (CF, green). The cumulative number of bad charges (Cbad, red). (c) Acid stratification factor (f s, blue). The contribution to acid stratification (fplus, red) and reduction in acid stratification (fminus, green) Figure 19 shows results from the OPz battery subjected to a PV current profile. In Figure 19(a), the test battery capacity (stars) indicates that the test battery had not failed under the period of testing, which makes truthful comparison for longer periods than 350 days difficult. Notice how the model loops the current profile, after approximately 350 days, Figure 19(c), when it has reached the end of file. This concept of recycling the test file, occurs only in the case of an OPz battery subjected to a PV current profile since the OPz battery ages slow by PV current profile. The functions that calculate the state of charge (SOC), illustrated in Figure 19(a), and the model voltage in Figure 19(b) seems to correctly simulate the recycling.

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Figure 19 Simulation results for OPz Battery. PV current profile from CRES. From top to bottom: (a) Remaining capacity (blue), State of charge, SOC (F, green) and test battery capacity (red stars). (b) Modelled voltage (Ucell, blue). (c) Test current (I blue), Modelled controlled current (I batt, green), Initial discharge current (Icf, red) (d) Initial discharge capacity coefficient (Cd, green). Capacity reduction from degradation (Cdeg, red). Capacity reduction from corrosion process (Ck, light blue ). Internal resistance (rdtotal = ρd,total, mauve).

Increase in internal resistance from the corrosion process (rk =ρk, yellow).

The OPz battery is not aged by corrosion (Ck constant). Once again it is the degradation (Cdeg) that causes the battery to fail after 610 days, shown in Figure 19(d).

Table 6 presents the predicted and true lifetime of the batteries subjected to current profiles from CRES. Life Initial model represents the lifetime predicted by the initial simulation model at the beginning of this project. Life Improved model represents the lifetime predicted by the improved model developed under this project and presented above. Tested lifetime is the measured lifetime of

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the test battery at CRES. The lifetime of the OPz is long since the batteries had not failed when the measurements ended. An accurate comparison between the simulated lifetime and the tested lifetime is thereby impossible to make. As can be seen from the results, the improved model makes more accurate predictions for the OGi battery than the initial. Also, lifetime predictions are more accurate for PV current profiles than for wind current profiles. This is explained by the nature of the current profile for the wind cases (recall Figure 4 and Figure 5). A strongly fluctuating current applies strain to the dynamics of the model since it implies large changes in the different variables, while a fairly steady current adjusts the variables only slightly. Table 6 Summary of simulation results Battery & Profile Life Initial model (days) Life Improved model (days) Tested Lifetime (days) OGi Wind 425 200 174 OGi PV 660 235 239 OPz Wind 633 475 400+3 OPz PV 830 610 600+3

5.1.2 ISPRA simulations

This section presents the simulation results from the improved model subjected to the newly assembled ISPRA profiles. The complete results from the simulations on ISPRA data are shown in appendix 7-8. In common for all ISPRA input current profiles, the test battery has been exposed to periods when the current has been zero. An example of this is the long initial sequence (see Figure 20). This initial period needs to be subtracted from the estimated lifetime to give comparable result to the CRES data simulations. Figure 20 visualises the simulation results for an OGi battery subjected to a PV current profile. In Figure 20(a) it can be seen that the improved model just slightly underestimates the capacity of the of the OGi battery. The periods with zero current, discussed above, are shown in Figure 20(c). The improved model has no problem dealing with periods of zero current, which can be seen in Figure 20(a,b,d). The capacity factor plot, Figure 20(d), reveals that the OGi battery is aged mostly by degradation (Cdeg) but also, to a less extent, by corrosion (Ck).

3 Not complete capacity tests from CRES. Approximated lifetime.

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Figure 20 Simulation results for OGi Battery. PV current profile from ISPRA. From top to bottom: (a) Remaining capacity (blue), State of charge, SOC (F, green) and test battery capacity (red stars). (b) Modelled voltage (Ucell, blue). (c) Test current (I blue), Modelled controlled current (I batt, green), Initial discharge current (Icf, red) (d) Initial discharge capacity coefficient (Cd, green). Capacity reduction from degradation (Cdeg, red). Capacity reduction from corrosion process (Ck, light blue ). Internal resistance (rdtotal = ρd,total, mauve).

Increase in internal resistance from the corrosion process (rk =ρk, yellow). In the simulations of the OPz battery subjected to a PV current profile two rest periods appear since the input current profile is recycled (see Figure 21). For this case two initialisation periods need to be subtracted to give a comparable result with the simulations on CRES data. In Figure 21(a) it can be seen that the second capacity test (second red star) is not correct. By a further investigation it can be seen that the current profile is slightly deformed at this point. The model seems to handle this well since no disturbance can be seen in the result. The capacity test for the test battery was aborted after 520 days so it is difficult to make a truthful comparison for a longer period than that. The estimated lifetime is similar to the corresponding CRES data simulation, when the sequences at zero current have been subtracted from the ISPRA simulation (620 days). The periods of zero current can be seen breaking up the current blocks in Figure 21(c). The OPz battery is again only aged by degradation (Cdeg) shown in Figure 21(d).

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Figure 21 Simulation results for OPz Battery. PV current profile from ISPRA. From top to bottom: (a) Remaining capacity (blue), State of charge, SOC (F, green) and test battery capacity (red stars). (b) Modelled voltage (Ucell, blue). (c) Test current (I blue), Modelled controlled current (I batt, green), Initial discharge current (Icf, red) (d) Initial discharge capacity coefficient (Cd, green). Capacity reduction from degradation (Cdeg, red). Capacity reduction from corrosion process (Ck, light blue ). Internal resistance (rdtotal = ρd,total, mauve).

Increase in internal resistance from the corrosion process (rk =ρk, yellow). The simulation of the ISPRA current profiles shows that the improved model can operate with a varying time step. However, the graphs have a rougher appearance than in the simulations on CRES data and this is an effect of the periods where the current is zero. In general, it can be seen from the simulation results that batteries of same type react similarly to one and other. There is no visible spread between “identical” batteries.

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Improvements in calculation time The improved model is equipped with a function that keeps track of the time it takes for the computer to simulate the battery lifetime. For the CRES current profiles only small improvements in speed are made since the calculations were fast from the beginning due to small test files and a fixed time step. Most noticeable is the simulation speed gained under simulation of the larger ISPRA current profiles. The reasons for improved speed are the reorganization of the code and vectorization, both described in section 4.1. Table 5 presents the difference in simulation time between the initial model equipped to operate with a varying time step and the improved model. Table 5 Time for the computer to perform the simulations. Battery & Profile Time Initial model (min) Time Improved model (min) OGi PV 786.0 53.7 OPz PV 822.0 117.0

A reduced calculation time is essential for the later translation into the greater system simulation model. A slow battery model would slow down the entire hybrid simulation tool dramatically.

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5.1.3 Simulation response for different control strategies A comparison between the simulation results from high and low discharge current, 4.4.2, is complicated since the change in discharge current influences several parameters. The complete results from the high and low current simulations are shown in appendix 9-10. Figure 22 and Figure 23 present a selected part of the simulated results with a high and low discharge current respectively. In order to ease the comparison the figures have been enlarged and adjusted so that similar parts of the simulations are shown. Note that for the two figures we have different timelines, which is a result of the manipulation of the time steps done in the input current profile. The Shepherd equation gives correctly a slightly lower voltage for the higher discharge current, shown by a comparison between Figure 22(b) for a high discharge current and Figure 23(b) for a low discharge current. The relationship between discharge current and voltage is discussed in appendix 1.

The lower voltage will result in a lower gassing current than for the high current discharge and thereby affecting the state of charge (SOC) in Figure 22(a). Consequently, the number of poor charges will be reduced and thereby lowering the current factor in Figure 24(b).

Figure 22 Simulation results from a high current discharge. From top to bottom: (a) Remaining capacity (blue), State of charge, SOC (F, green). (b) Modelled voltage (Ucell, blue). (c) Test current (I blue), Modelled controlled current (I batt, green), Initial discharge current (Icf, red)

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Furthermore, a manipulation of the current profile to a higher discharge current implies a high initial discharge current, shown in Figure 22(c), which in turn will results in a lower current factor, Figure 24(b).

The increased number of poor charges will contribute to a higher current-factor. Conversely, a low discharge implies a lower initial discharge current as shown in Figure 23(c) that in its turn will result in a higher current factor, Figure 25(b). The change in the current factor influences the weighting factors (fq, fs), shown in Figure 25(b,c). As a result of the increased weighting factors the battery will age faster by degradation. A smaller discharge current will result in a higher gassing current that in its turn will influence the state of charge (SOC) in Figure 23(a). The change in (SOC) affects the counter that keeps track of poor charges, consequently there will be an increased number of poor charges, shown in Figure 23(b).

Figure 23 Simulation results from a low current discharge. From top to bottom: (a) Remaining capacity (blue), State of charge, SOC (F, green) and test battery capacity (red stars). (b) Modelled voltage (Ucell, blue). (c) Test current (I blue), Modelled controlled current (I batt, green), Initial discharge current (Icf, red)

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A lower current factor (higher discharge current) reduces the contribution of degradation to the total capacity loss; consequently, the battery will last longer before being considered dead. This is shown for a high discharge current by the number of nominal cycles (also known as unweighted standardised charge throughput (ZN, dQd)) in Figure 24(a). For the high current discharge case, 833 nominal cycles (ZN) will pass through the battery.

Figure 24 Simulation factors for a high current discharge. From top to bottom: (a) Weighted standardised charge throughput (Zgn= dQdf, blue). Weighted standardised charge throughput (ZN = dQd, green). (b) Deep discharge weighting factor (fq, blue). The current factor (CF, green). The cumulative number of bad charges (Cbad, red). (c) Acid stratification factor (f s, blue).

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For the low discharge current case, 680 nominal cycles (ZN) will pass through the battery, shown in Figure 25(a). The model thereby predicts that the lower discharge current is harsher for the battery than the high current discharge current, something that is in line with the earlier discussed crystal theory.

Figure 25 Simulation factors for a low current discharge. From top to bottom: (a) Weighted standardised charge throughput (Zgn= dQdf, blue). Weighted standardised charge throughput (ZN = dQd, green). (b) Deep discharge weighting factor (fq, blue). The current factor (CF, green). The cumulative number of bad charges (Cbad, red). (c) Acid stratification factor (f s, blue).

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6 CONCLUSIONS Lead acid batteries are a complicated piece of technology and good simulation tools are essential to reduce the calculation time and to improve understanding of the batteries’ aging mechanisms. The improved model, introduced in this work, is a small contribution to the understanding of battery aging mechanisms when combined in a hybrid system. Its full potential will be realized imminently by an implementation in the hybrid simulation platform IPSYS. By restructuring the code, the simulation time has been reduced to approximately 14 % of the earlier value. At the same time, the reorganization will ease the translation into C++. The improved model performs more accurate lifetime predictions than the earlier model, on the tested batteries OGi and OPz manufactured by BAE. According to the simulation results, the OPz battery shows good characteristics and long lifetime while the OGi battery is aged quickly by both degradation and corrosion. The explanation lays probably in the different construction of the batteries, recall 3.2.1. The model shows that degradation is the main damage mechanism for lead acid batteries in hybrid systems. Degradation is a change in mechanical structure of the electrodes caused by continuous charge and discharge processes. One way to prevent degradation is to lower the number charges and discharges. For long battery lifetime it is important to find a middle way between effective uses of the batteries on the one hand and on the other a low number of charges and discharges. In general, it can be seen from the test results that batteries operating in a wind energy system are aged faster than the batteries operating in a photovoltaic system. The explanation for the wind current profile (input to the model) causes a larger number of charge and discharge cycles as well as a higher Ah throughput. This is something that is also observed in reality since wind is much more fluctuating than solar resources. The model simulates that acid stratification is built up during the current profile blocks, as described in 3.2.2, and is only eliminated when the battery is completely charged during the capacity tests. For an increased battery lifetime it is thereby important to take the batteries out of operation and fully charge them. This should be done repeatedly after a defined period of time. This time period is strongly dependant on the system configuration. The simulations of the newly assembled ISPRA current profiles show that the improved model can perform calculations with a varying time step. This is something that is important for the coming translation into the simulation platform IPSYS. A new crystal theory, developed by Professor D. U. Sauer [11], has been implemented in the model that replaces the current-factor used in the earlier model. The new approach is to simulate the changes in PbSO4 crystal structure at the electrode. A high initial discharge current will result in many small crystals that together form a large surface area. A small initial discharge current form few large crystals that have time to grow into irreversible structures that will complicate the recharging process.

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Experiments have shown that a high discharge current will result in less aging than a small discharge current, something that the model simulation also shows. From this behavior a new control strategy could be formed. Recommendations for future development are to perform real tests on how lead acid batteries react to a higher initial discharging current. If the crystal theory is assumed to be correct, a higher initial discharge current would increase the lifetime of the battery. This would then be a new control strategy that has not previously been used. Energy storage is something that will be seen more frequently in the future. It is necessary in stand-alone renewable energy systems where the energy generation is fluctuating. One new interesting futuristic approach is to integrate the future electrical vehicle fleets’ storage capacity into the electrical grid. Excess energy during the night will be stored in the batteries of the car and used when it is needed. This concept is called “vehicle two grid” (V2G) and studies are being performed at the University of Delaware [22].

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7 REFERENCES [1] http://www.toyota.co.jp/en/tech/environment/hsd/ (22/9 2005) [2] http://www.nrel.gov/docs/legosti/fy98/25233.pdf (5/9 2005) [3] Henrik Bindner, Per Nørgård, “11kW Gaia Wind Turbine Connected to a Diesel Grid” Risø

National Laboratory, (2004) [4] Manwell J.G. McGowan A.L. Rogers, “Wind energy explained J.F.” (2002) [5] Anton Andersson, “Overview of High-efficiency Multi-junction Solar Cells”, Umeå

Universitet, (2005) [6] Henrik Bindner, Oliver Gehrke, Per Lundsager, Jens Carsten Hansen, Tom Cronin, “IPSYS-

a tool for performance assessment and supervisory controller development of integrated power systems with distributed renewable energy” Risø National Laboratory, (2004)

[7] Bengt Peres, “Komponenter i solcellssystem”, Umeå Universitet, (2004) [8] Henrik Bindner, Tom Cronin, Per Lundsager ”Lifetime modelling of Lead Acid Batteries”

Risø National Laboratory, (2005) [9] David Linden, “Handbook of Batteries, second edition”, (1994) [10] http://www.batteryuniversity.com (8/10 2005) [11] D. U. Sauer, “Optimierung des Einsatzes von Blei-Säure-Akkumulatoren in PV-Hybrid-

Systemen unter spezieller Berücksichtigung der Batteriealterung“, PhD Thesis, University of Ulm, 2003

[12] http://www.batterychargers.it/eng/iuia.htm (10/10 2005) [13] R.Kaiser, D.U. Sauer. A. Armbruster, ”New Concept for System Design and Operation

Control of Photovoltaic Systems”, Fraunhofer Institute for Solar Energy Systems ISE. [14] http://www.bae-berlin.de/produkte/Seiten_e/OGiVblock_e.PDF (1/11 2005) [15] http://www.enersafe.fr/PRODUITS/BATTERIES/OPZSBLOCS.pdf (1/11 2005) [16] MATLAB, Mathworks, www.mathworks.com [17] http://www.zib.de/Visual/people/mueller/Course/Tutorial/tutorial.html (17/11 2005) [18] DOT, Graphviz, http://www.graphviz.org/ [19] Julia Schiffer, “Internal rapport concerning implementation of currentfactor”, Risø National

Laboratory (2005) [20] H.-G. Puls, “Evolutionsstrategien zur Optmierung autonomer Photovoltaik-Systeme”,

diploma thesis, Albert-Ludwigs-University of Freiburg, (1997) [21] Lander J.: “Further Studies on the Anodic Corrosion pf Lead in H2SO4 Solution”, J

Electorochem. Soc., Vol. 103, No.1 [22] http://www.udel.edu/V2G/ (28/12 2005)

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Appendix 1

Battery parameters

In order to provide a good illustration of how a battery functions, some terms and definitions need to be explained. These parameters are used later in the battery model (chapter 1.4). The term capacity is often used in a battery context but can be a source of confusion. The term has the definition of the total number of Ah, (1Ah = 3600 C), that can be obtained from a charged battery under certain conditions of discharge. The capacity rated by the manufacturer is called nominal or rated capacity. This capacity is obtained during performance tests of normally 10 h durations, at constant rate of discharge, end voltage and temperature [1].

Figure 1 Tubular positive lead-acid cell response to various discharge currents. The size of the discharge affects the efficiency of the batteries and thereby the capacity. This special rating system is needed in order to compare different kinds of batteries. As the current increases, the battery’s capacity at that rate decreases, shown in Figure 1. For example, if the capacity C = 50 Ah, C/10 is the current that will fully discharge the battery in 10 h which is equivalent with I10 = 5 A. Compared with the capacity for the discharging current 3C = 120 A [120 (A) * 0.1 (h) = 12 Ah] is only 24 % of the nominal capacity [2]. In battery literature it is possible to find definitions such as I5 and I10. These are charge or discharge currents. I5, for example, is the current that discharges a fully charged battery in 5 hours. If we have a battery with a capacity of 50 Ah, the I5 in this case is 10 A. Note that I5 > I10. The rated or nominal capacity should not be confused with the available or actual capacity. Nominal capacity is a measurement of the capacity specified by the manufacture, typically a discharge at constant current. Charge throughput is another term frequently used in a battery context. It is defined as the number of discharge Ah that flows throw the battery.

52

SOC, (State of charge) is a unit that expresses, in percentage figures, the amount of capacity left in the battery. SOC is obtained by the relation between available capacity and the rated nominal capacity. Since the nominal capacity is used as a reference point in the model, the SOC can assume negative values. This has to do with that the nominal capacity in general being lower than the actual capacity of the battery [3]. Float charge Float charging is a low-rate charge used for storing a battery in a fully charged condition. The main purpose with the charging method is to compensate for self-discharging. Especially for hybrid systems, where a current can be generated at a relative low cost by wind or by the sun, it can be profitable to keep the batteries under float charge in order to maximize battery lifetime [4]. Gassing Gassing occurs when small bubbles of gas are formed at the surface of the electrodes. The bubbles grow bigger until the buoyancy force gets too strong and they rise to the surface of the electrolyte. Some gassing is normal, excessive gassing indicates that the battery is overcharged. Gassing can be used to reduce acid stratification but is also a stress factor for the battery [4]. References [1] http://en.wikipedia.org/wiki/Battery_%28electricity%29#Battery_capacity (12/10 2005) [2] http://www.corrosion-doctors.org/Batteries/nominal.htm (12/10 2005) [3] http://www.mpoweruk.com/soc.htm (17/11 2005) [4] Henrik Bindner, Tom Cronin, Per Lundsager ”Lifetime modelling of Lead Acid Batteries”

Risø National Laboratory, (2005)

53

Appendix 2 Variable Overview made by the DOT software, http://www.graphviz.org/. Figure 2 Initial model variable relation. The red arrow points at a variable that is in the wrong object, seen from the corresponding calls (large number of lines). The different colours refer to the different objects.

Figure 3 Improved model variable relation. A more efficient structure since there are fewer calls between the objects.

54

Appendix 3 Simulation results of an OGi battery subjected to a wind current profile from CRES

Figure 4 OGi Wind Profile Simulation Result, CRES

Figure 5 OGi Wind Profile Simulation Result Factors, CRES

55

Figure 6 OGi Wind Profile Simulation Result Factors, CRES

Figure 7 OGi Wind Profile Simulation Result Losses, CRES

56

Appendix 4 Simulation results for an OGi battery subjected to a PV current profile from Gres.

Figure 8 OGi PV Profile Simulation Result

Figure 9 OGi PV Profile Simulation Result Factors, CRES

57

Figure 10 OGi PV Profile Simulation Result Factors, CRES

Figure 11 OGi PV Profile Simulation Result Losses, CRES

58

Appendix 5 Simulation results for an OPz subjected to a wind current profile from Gres.

Figure 12 OPz Wind Profile Simulation Result, CRES

Figure 13 OPz Wind Profile Simulation Result Factors, CRES

59

Figure 14 OPz Wind Profile Simulation Result Factor, CRES

Figure 15 OPz Wind Profile Simulation Result losses, CRES.

60

Appendix 6 Simulation results for an OPz battery subjected to a PV current profile from Gres.

Figure 16 OPz PV Profile Simulation Result, CRES.

Figure 17 OPz PV Profile Simulation Result Factors, CRES.

61

Figure 18 OPz PV Profile Simulation Result Factors, CRES

Figure 19 OPz PV Profile Simulation Result Losses, CRES

62

Appendix 7 Simulation results for an OGi battery subjected to a PV current profile from ISPRA.

Figure 20 OGi PV Profile Simulation Result, ISPRA

Figure 21 OGi PV Profile Simulation Result Factors, ISPRA

63



64

Appendix 8 Simulation results for an OPz battery subjected to a PV current profile from ISPRA.

Figure 24 OPz PV Profile Simulation Result, ISPRA

Figure 25 OPz PV Profile Simulation Result Factors, ISPRA

65

Figure 26 OPz PV Profile Simulation Result Factors, ISPRA

Figure 27 OPz PV Profile Simulation Result Losses, ISPRA

66

Appendix 9 Simulation results for an OPz battery subjected to manipulated wind current profile from CRES. Factor 0.8.

Figur 28 Wind Profile Simulation Result Factor 0.8, CRES


67



68

Appendix 10 Simulation results for an OPz battery subjected to manipulated wind current profile from CRES. Factor 1.2.



69



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Nor

mal

cha

rge

thou

ghpu

t dur

ing

deep

dis

char

ge p

hase

D2

IV@

sDee

pDis

char

ge/c

alcu

late

Wei

ghte

dCha

rgeT

hrou

ghpu

t, @

cPbB

atte

ry/r

eset

Dee

pDis

char

ge

dtdt

Tim

e st

ep[h

]3.

63IP

Use

d ex

tens

ivel

y th

roug

hout

N

ote

that

tim

este

p ha

s un

its o

f hou

rs n

ot

seco

nds.

Inpu

t

-dt

low

up1

Wei

ghte

d tim

e ne

eded

by

Um

ax to

ris

e fr

om U

low

to U

up-

IV@

cBat

tCon

trol

/set

Cha

rgeC

ontr

olU

sed

in th

e pr

oces

s fo

r de

cidi

ng o

n ex

tern

al

char

ge, e

tc.

Con

trol

t Fdt

soc

0D

urat

ion

of d

eep-

disc

harg

e ph

ase

[h] ,

Tim

e si

nce

last

full

char

ge3,

67IV

@sD

eepD

isch

arge

/cal

cula

teW

eigh

tedC

harg

eThr

ough

put,

@cP

bBat

tery

/res

et,

@sD

eepD

isch

arge

/cal

cula

teD

eepD

isch

arge

Fac

tor

, @B

attC

ontr

ol/s

etN

ewT

hres

hold

s,

@cB

attC

ontr

ol/s

etC

harg

eCon

trol

dtso

c tr

ies

to k

eep

trac

k of

the

dura

tion

of a

cy

cle.

Con

trol

, Dee

p D

isch

arge

-dt

u_re

fH

ours

nee

ded

to d

ecom

pose

DF

PLU

S a

t U_b

att=

2.5V

-IV

@sA

cidS

trat

ifica

tion/

calc

ulat

eAci

dStr

atifi

catio

nT

o do

with

aci

d st

ratif

icat

ion…

.lo

cal

∆W

t (&

∆W

t+∆

t)dW

0C

orro

sion

laye

r de

pth

[mg/

cm2 ]

3.44

IV@

cCor

rosi

on/c

alcu

late

Cor

rLay

erD

epth

, @

cCor

rosi

on/c

alcu

late

Cor

rosi

onR

esC

ap,

rese

tC

orro

sion

∆W

limit

dWlim

0C

orro

sion

laye

r th

ickn

ess

at th

e en

d of

bat

tery

's li

fesp

an o

n st

andb

y op

erat

ion

3,47

IV@

cCor

rosi

on/c

alcu

late

Fin

alC

LThi

ckne

ss,

@cC

orro

sion

/cal

cula

teC

orro

sion

Res

Cap

Cor

rosi

on

-F

ER

MIK

T0,

5C

ontr

ols

the

shap

e of

the

ferm

i dis

trib

utio

nD

2C

@sA

cidS

trat

ifica

tion/

calc

ulat

eAci

dStr

atifi

catio

nH

as s

omet

hing

to d

o w

ith a

cid

stra

tific

atio

n di

strib

utio

n.A

cidS

trat

ifica

tion

-F

Ext

Load

Sta

rtT

hres

hold

for

star

t of r

echa

rge

-IV

@cB

attC

ontr

ol/s

etN

ewT

hres

hold

s,

@cB

attC

ontr

ol/s

etC

harg

eCon

trol

Use

d in

the

proc

ess

for

deci

ding

on

exte

rnal

ch

arge

, etc

.C

ontr

ol

IP =

Inpu

t Par

amet

er, O

= O

utpu

t, IV

= In

tern

al V

aria

ble,

B =

Boo

lean

, V =

Val

ue, C

= C

onst

ant

2

Tom

Cro

nin

Julia

Sch

iffer

Ant

on A

nder

sson

Pag

e 3

Par

amet

er N

ames

231

105.

xls

Nam

e in

Pap

erN

ame

in c

od

eV

alu

e (I

nit

ial V

alu

e)

Des

crip

tio

nE

qu

atio

n i

init

ial

pap

er

Pu

rpo

seL

oca

tio

n in

Mat

lab

Mo

del

Co

mm

ents

bel

on

gs

to

clas

s

-F

Ext

Load

Sto

p0,

8T

hres

hold

for

rech

arge

sto

p-

IV@

cBat

tCon

trol

/set

New

Thr

esho

lds,

@

cBat

tCon

trol

/set

Cha

rgeC

ontr

olU

sed

in th

e pr

oces

s fo

r de

cidi

ng o

n ex

tern

al

char

ge, e

tc.

Con

trol

-F

LIM

ITD

FP

LUS

*2Li

mit

(Fer

mi d

istr

ibut

ion)

for

wei

ghtin

gD

2C

@sA

cidS

trat

ifica

tion/

calc

ulat

eAci

dStr

atifi

catio

nH

as s

omet

hing

to d

o w

ith a

cid

stra

tific

atio

n di

strib

utio

n.A

cidS

trat

ifica

tion

-F

load

She

dLo

ad s

hedd

ing

thre

shol

d of

hig

hest

prio

rity

-IV

@cB

attC

ontr

ol/s

etC

harg

eCon

trol

, res

etU

sed

in th

e pr

oces

s fo

r de

cidi

ng o

n ex

tern

al

char

ge, e

tc.

Con

trol

-F

min

00,

3In

itial

end

-of-

disc

harg

e th

resh

old

-IV

@cB

attC

ontr

ol/s

etC

harg

eCon

trol

, res

etU

sed

in th

e pr

oces

s fo

r de

cidi

ng o

n ex

tern

al

char

ge, e

tc.

Con

trol

-F

min

dQE

nd-o

f-di

scha

rge

thre

shol

d w

ith r

espe

ct to

cha

rge

thro

ughp

ut-

IV@

cBat

tCon

trol

/set

Cha

rgeC

ontr

ol, r

eset

Use

d in

the

proc

ess

for

deci

ding

on

exte

rnal

ch

arge

, etc

.C

ontr

ol

-F

min

dtE

nd-o

f-di

scha

rge

thre

shol

d w

ith r

espe

ct to

tim

e si

nce

full

char

ge-

IV@

cBat

tCon

trol

/set

Cha

rgeC

ontr

ol, r

eset

Use

d in

the

proc

ess

for

deci

ding

on

exte

rnal

ch

arge

, etc

.C

ontr

ol

f -fm

inus

Dec

reas

e in

aci

d st

ratif

icat

ion

wei

ghtin

g fa

ctor

due

to g

assi

ng3,

66IV

@sA

cidS

trat

ifica

tion/

calc

ulat

eAci

dStr

atifi

catio

nIn

fluen

ces

acid

str

atifi

catio

n di

ssol

utio

nlo

cal

f +fp

lus

Incr

ease

in w

eigh

ting

fact

or c

ause

d by

aci

d st

ratif

icat

ion

3,66

IV@

sAci

dStr

atifi

catio

n/ca

lcul

ateA

cidS

trat

ifica

tion

Influ

ence

s ac

id s

trat

ifica

tion

grow

thlo

cal

f Ffq

0W

eigh

ting

fact

or (

stat

e of

cha

rge)

3,67

IV@

sDee

pDis

char

ge/c

alcu

late

Dee

pDis

char

geF

act

or, @

Bat

tCon

trol

/set

New

Thr

esho

lds,

res

et,

@sD

eepD

isch

arge

/cal

cula

teW

eigh

tedC

harg

eThr

ough

put

Fac

tor

that

trie

s to

kee

p tr

ack

of in

fluen

ce o

f de

pth

of d

isch

arge

(or

tim

e at

sta

te o

f cha

rge)

Con

trol

, Dee

p D

isch

arge

-F

Q00

8W

eigh

ting

afte

r 3

mon

ths

with

F_m

in=

0.0

C@

sDee

pDis

char

ge/s

Dee

pDis

char

geU

sed

in c

alcu

latio

n of

FQ

MC

& F

QM

M. T

o do

w

ith d

epth

of d

isch

arge

wei

ghtin

g.lo

cal

-F

Q07

3W

eigh

ting

afte

r 3

mon

ths

with

F_m

in=

0.7

C@

sDee

pDis

char

ge/s

Dee

pDis

char

geU

sed

in c

alcu

latio

n of

FQ

MM

. To

do w

ith d

epth

of

disc

harg

e w

eigh

ting.

loca

l

-fq

Ext

Load

999

Tur

nove

r w

eigh

ting

fact

or, a

t whi

ch r

echa

rge

begi

ns-

C@

cBat

tCon

trol

/set

New

Thr

esho

lds

Use

d to

influ

ence

ext

erna

l cha

rgin

g re

gim

e.C

ontr

olc f

,0F

QM

C(F

Q00

- 1

)/ 2

160

Incr

ease

of f

F p

er h

our

at F

min=

0 (?

). C

onst

ant s

lope

for

fq.

3,67

C@

sDee

pDis

char

ge/c

alcu

late

Dee

pDis

char

geF

act

orD

eepD

isch

arge

c f,m

inF

QM

M-(

FQ

00-

FQ

7)/(

2160

*0.7

)In

fluen

ce o

f Fm

in o

n f F

. (F

min-d

epen

dent

frac

tion.

)3,

67C

@sD

eepD

isch

arge

/cal

cula

teD

eepD

isch

arge

Fac

tor

Dee

pDis

char

ge

f sfs

Aci

d st

ratif

icat

ion

wei

ghtin

g fa

ctor

3,66

IV@

sAci

dStr

atifi

catio

n/ca

lcul

ateA

cidS

trat

ifica

tion,

O

pera

tion,

@B

attC

ontr

ol/s

etN

ewT

hres

hold

s,

Cal

cula

ted

in

@sA

cidS

trat

ifica

tion/

calc

ulat

eAci

dStr

atifi

catio

nA

cidS

trat

ifica

tion

f -(t,f

s)fs

min

usD

ecre

ase

of fs

dur

ing

times

tep

D2

IV@

sAci

dStr

atifi

catio

n/ca

lcul

ateA

cidS

trat

ifica

tion

= fm

inus

* d

tA

cidS

trat

ifica

tion

f +(t

,f s)

fspl

usIn

crea

se o

f fs

durin

g tim

este

pD

2IV

@sA

cidS

trat

ifica

tion/

calc

ulat

eAci

dStr

atifi

catio

n=

fplu

s *

dtA

cidS

trat

ifica

tion

g cgc

Ele

ctro

lyte

coe

ffici

ent o

f the

cel

l vol

tage

(ch

argi

ng)

[V]

3.34

IP@

cVol

tage

/cal

cula

teV

olta

ge, S

etV

aria

nt,

@cV

olta

ge/c

alcu

late

Max

Cha

rgin

gCur

rent

, @

cVol

tage

/cal

cula

teC

orro

sion

Vol

tage

She

pher

d eq

uatio

n pa

ram

eter

Par

amet

ers

g dgd

Ele

ctro

lyte

coe

ffici

ent o

f the

cel

l vol

tage

(di

scha

rgin

g) [V

]3.

35IP

@cV

olta

ge/c

alcu

late

Vol

tage

, Set

Var

iant

, @

cCor

rosi

on/c

alcu

late

Cor

rInc

reas

eLim

it,

@sD

egra

datio

n/ca

lcul

ateR

emai

ning

Cap

acity

, @

sDeg

rada

tion/

calc

ulat

eDeg

Incr

ease

Lim

it,

@cV

olta

ge/c

alcu

late

Cor

rosi

onV

olta

ge

She

pher

d eq

uatio

n pa

ram

eter

Par

amet

ers

HH

Dep

th o

f dis

char

ge3.

48IV

@sD

egra

datio

n/ca

lcul

ateD

egIn

crea

seLi

mit,

@

cCor

rosi

on/c

alcu

late

Cor

rInc

reas

eLim

itU

sed

as lo

cal v

aria

ble

for

dod.

loca

l

-H

0A

vaila

ble

capa

city

(I =

- I 1

0 ,U

leer

=1.

8V, t

=0)

-IV

@cP

bBat

tery

/Set

Var

iant

Cap

acity

of b

atte

ry w

hen

new

Vol

tage

-H

0lim

Cd

Dec

reas

e of

Cd

due

to d

egra

datio

n-

IV@

sDeg

rada

tion/

calc

ulat

eCap

acity

Loss

Deg

Con

nect

ed to

dcd

Deg

rada

tion

-H

0lim

dQC

harg

e th

roug

hput

-IV

@sD

egra

datio

n/ca

lcul

ateC

apac

ityLo

ssD

egC

onne

cted

to d

Qdf

nD

egra

datio

n-

H0l

imm

Slo

pe (

?)-

IV@

sDeg

rada

tion/

calc

ulat

eCap

acity

Loss

Deg

Con

nect

ed to

dcd

and

ZD

egra

datio

n

IP =

Inpu

t Par

amet

er, O

= O

utpu

t, IV

= In

tern

al V

aria

ble,

B =

Boo

lean

, V =

Val

ue, C

= C

onst

ant

3

Tom

Cro

nin

Julia

Sch

iffer

Ant

on A

nder

sson

Pag

e 4

Par

amet

er N

ames

231

105.

xls

Nam

e in

Pap

erN

ame

in c

od

eV

alu

e (I

nit

ial V

alu

e)

Des

crip

tio

nE

qu

atio

n i

init

ial

pap

er

Pu

rpo

seL

oca

tio

n in

Mat

lab

Mo

del

Co

mm

ents

bel

on

gs

to

clas

s

Hm

ax,t

H0t

Rem

aini

ng c

apac

ity c

oeffi

cien

t at t

ime

t3,

6O

@sD

egra

datio

n/ca

lcul

ateR

emai

ning

Cap

acity

, @

sDeg

rada

tion/

calc

ulat

eCap

acity

Loss

Deg

, E

xcha

nge,

Get

AhC

apR

emai

ning

, @

Bat

tLife

time/

calc

ulat

eLife

, Set

Var

iant

, T

oBeE

xcha

nged

,

An

outp

ut o

f the

mod

el.C

alcu

late

d in

@

sDeg

rada

tion/

calc

ulat

eRem

aini

ngC

apac

ityV

olta

ge

-H

0tE

xcha

nge

0,8

Rem

aini

ng c

apac

ity a

t whi

ch b

atte

ry is

cha

nged

-IP

@cP

bBat

tery

/ToB

eExc

hang

edS

et b

y ar

gum

ent w

hen

crea

ting

batte

ry o

bjec

t.Li

fetim

e

I bat

tI

Cur

rent

[A]

3.34

IPU

sed

exte

nsiv

ely

thro

ugho

ut

Inpu

t

-Ic

fF

irst d

isch

arge

cur

rent

-IV

@sC

urre

ntF

acto

r/ca

lcul

ateC

urre

ntF

acto

r,

rese

tCur

rent

Fac

tor,

@cP

bBat

tery

/Set

Var

iant

Sto

res

the

first

dis

char

ge c

urre

nt v

alue

afte

r a

full

char

geC

urre

ntF

acto

r

I G0

Ig0

Gas

sing

cur

rent

und

er n

orm

al c

ondi

tions

(2.

23V

, 20^

C,

100A

h) [m

A] i

nclu

ding

age

ing

effe

ct.

3,62

IV@

cGas

sing

/cal

cula

teG

assi

ngC

urre

ntlo

cal

I G0,

min

IG0C

0,02

Gas

sing

cur

rent

und

er n

orm

al c

ondi

tions

of a

new

bat

tery

[A].

Con

stan

t par

t of I

G0

(gas

sing

cur

rent

of a

new

bat

tery

und

er

norm

al c

ondi

tions

).

3.62

C@

cGas

sing

/cal

cula

teG

assi

ngC

urre

nt, r

eset

, @

sAci

dStr

atifi

catio

n/ca

lcul

ateA

cidS

trat

ifica

tion

used

in a

cid

stra

tific

atio

n w

eigh

ting

fact

orG

assi

ng

I G0,

limit

IG0R

0,06

Incr

ease

in I G

0 fr

om 8

0% s

ervi

ce li

fe. M

axim

um IG

0 ch

ange

du

e to

of i

nter

nal r

esis

tanc

e.3.

62C

@cG

assi

ng/c

alcu

late

Gas

sing

Cur

rent

Gas

sing

-Ig

0tG

asin

g cu

rren

t at 2

.23V

and

20

deg

C (

aged

)D

2IV

@cG

assi

ng/c

alcu

late

Gas

sing

Cur

rent

, S

et to

Igo

in

@cG

assi

ng/c

alcu

late

Gas

sing

Cur

rent

, use

d in

ac

id s

trat

ifica

tion

wei

ghtin

g fa

ctor

Gas

sing

I Gas

Igas

Leak

age

curr

ent c

ause

d by

gas

sing

(i.e

. gas

sing

cur

rent

(lo

ss))

.3.

63IV

@cG

assi

ng/c

alcu

late

Gas

sing

Cur

rent

, ca

lcul

ateC

harg

eThr

ough

put,

calc

ulat

eSoc

, res

etC

alcu

late

d in

@

cGas

sing

/cal

cula

teG

assi

ngC

urre

nt. U

sed

to

redu

ce c

harg

e th

roug

hput

into

bat

tery

whi

lst

gass

ing

Gas

sing

-Im

in-0

,002

min

imum

val

ue fo

r th

e cu

rren

t to

be r

egar

ded

as a

dis

char

ge

curr

ent

-IV

@sC

urre

ntF

acto

r/ca

lcul

ateC

urre

ntF

acto

r,

rese

tCur

rent

Fac

tor,

@cP

bBat

tery

/Set

Var

iant

Use

d to

avo

id th

at a

mea

sure

men

t err

or is

re

gard

ed a

s a

disc

harg

e cu

rren

tC

urre

ntF

acto

r

-Ir

efre

fere

nce

curr

ent f

or c

urre

nt fa

ctor

cal

cula

tion

-IV

@sC

urre

ntF

acto

r/ca

lcul

ateC

urre

ntF

acto

r,

rese

tCur

rent

Fac

tor,

@cP

bBat

tery

/Set

Var

iant

Use

d fo

r ca

lcul

atio

n of

the

curr

ent f

acto

rP

aram

eter

s

k k

kks

Cor

rosi

on s

peed

par

amet

er3.

42?

IV@

cCor

rosi

on/c

alcu

late

Cor

rosi

onS

peed

loca

l

Cor

rosi

on R

ate

kks2

0,00

1S

peed

par

amet

er fo

r co

rros

ion

(>1.

74V

SH

E)

3,47

C@

cCor

rosi

on/c

alcu

late

Fin

alC

LThi

ckne

ss,

Par

amet

ers

Cor

rosi

on R

ate

KK

ST

log2

/15

Fac

tor

for

rate

of i

ncre

ase

of c

orro

sion

with

tem

pera

ture

C@

cCor

rosi

on/c

alcu

late

Cor

rosi

onS

peed

Tem

pera

ture

dep

enda

ncy

of c

orro

sion

rat

eC

orro

sion

ρk,

limit

klim

"Cal

cula

ted

cum

ulat

ive

resi

stan

ce."

3,49

IV@

cCor

rosi

on/c

alcu

late

Cor

rInc

reas

eLim

itC

alcu

late

d cu

mul

ativ

e re

sist

ance

or

is th

is

calc

ulat

ion

of r

klim

it? O

r ca

lcul

atio

n of

klim

?lo

cal

k k,1

(Uk<

1.74

0VS

HE)

ks0

Cor

rosi

on s

peed

par

amet

er3.

44IV

@cC

orro

sion

/cal

cula

teC

orrL

ayer

Dep

th,

@cC

orro

sion

/cal

cula

teC

orro

sion

Spe

ed, r

eset

Cal

cula

ted

by

@cC

orro

sion

/cal

cula

teC

orro

sion

Spe

edC

orro

sion

L 80%

,dat

Lt12

Life

of a

bat

tery

(on

con

tinuo

us fl

oat c

harg

e on

ly)

from

m

anuf

actu

rer's

dat

a sh

eets

[yea

rs]

3,47

IP@

cPbB

atte

ry/S

etV

aria

nt,

@cC

orro

sion

/cal

cula

teF

inal

CLT

hick

ness

Par

amet

ers

-Lt

80M

ean

Mea

n lif

etim

e at

H=

80%

ofH

0 -

IV@

cPbB

atte

ry/E

xcha

nge,

@

Bat

tLife

time/

calc

ulat

eLife

Life

time

-Lt

80N

owR

emai

ning

life

time

of th

e pr

esen

t bat

tery

, i.e

. unt

il H

=80

%of

H0

-IV

@cP

bBat

tery

/Exc

hang

e,

@B

attL

ifetim

e/ca

lcul

ateL

ifeLi

fetim

e

-Lt

Mea

nM

ean

lifet

ime

deriv

ed fr

om L

tNow

.-

IV@

cPbB

atte

ry/E

xcha

nge

Life

time

-Lt

Now

Pre

sent

age

of t

he p

rese

nt b

atte

ry.

-IV

@cP

bBat

tery

/Exc

hang

e,

@B

attL

ifetim

e/ca

lcul

ateL

ife, T

oBeE

xcha

nged

Life

time

Mc

Mc

Act

ivat

ion

pola

rizat

ion

volta

ge c

oeffi

cien

t (ch

argi

ng)

[V-1

]3.

34IP

@cP

bBat

tery

/Set

Var

iant

, @

cVol

tage

/cal

cula

teV

olta

ge,

@cV

olta

ge/c

alcu

late

Cor

rosi

onV

olta

ge,

@cV

olta

ge/c

alcu

late

Max

Cha

rgin

gCur

rent

Par

amet

er o

f the

She

pher

d eq

uatio

n.P

aram

eter

s

IP =

Inpu

t Par

amet

er, O

= O

utpu

t, IV

= In

tern

al V

aria

ble,

B =

Boo

lean

, V =

Val

ue, C

= C

onst

ant

4

Tom

Cro

nin

Julia

Sch

iffer

Ant

on A

nder

sson

Pag

e 5

Par

amet

er N

ames

231

105.

xls

Nam

e in

Pap

erN

ame

in c

od

eV

alu

e (I

nit

ial V

alu

e)

Des

crip

tio

nE

qu

atio

n i

init

ial

pap

er

Pu

rpo

seL

oca

tio

n in

Mat

lab

Mo

del

Co

mm

ents

bel

on

gs

to

clas

s

Md

Md

Act

ivat

ion

pola

rizat

ion

volta

ge c

oeffi

cien

t (di

scha

rgin

g) [V

-1]

3.35

IP@

cPbB

atte

ry/S

etV

aria

nt,

@cV

olta

ge/c

alcu

late

Vol

tage

, @

cVol

tage

/cal

cula

teC

orro

sion

Vol

tage

, @

sDeg

rada

tion/

calc

ulat

eDeg

Incr

ease

Lim

it,

@sD

egra

datio

n/ca

lcul

ateR

emai

ning

Cap

acity

, @

cCor

rosi

on/c

alcu

late

Cor

rInc

reas

eLim

it

Par

amet

er o

f the

She

pher

d eq

uatio

n.P

aram

eter

s

-N

omV

olta

ge2

Nom

inal

vol

tage

of a

cel

l [V

]-

IP@

cPbB

atte

ry/S

etV

aria

nt-

Par

amet

ers

-N

umbe

rN

umbe

r of

bat

terie

s us

ed (

by r

epla

cem

ent)

-O

@cP

bBat

tery

/Exc

hang

e,

@B

attL

ifetim

e/ca

lcul

ateL

ifeLi

fetim

e

-Q

_ct

Cha

rge

thro

ughp

ut d

urin

g tim

e in

terv

al d

tD

2IV

@sD

eepD

isch

arge

/cal

cula

teW

eigh

tedC

harg

eThr

ough

put,

@cP

bBat

tery

/Ope

ratio

n,

@sD

eepD

isch

arge

/cal

cula

teC

harg

eThr

ough

put

Dee

pDis

char

ge

-Q

cycl

e0

Cha

rge

thro

ughp

ut s

ince

last

det

ecte

d fu

ll ch

arge

-IV

@cB

attC

ontr

ol/s

etN

ewT

hres

hold

sC

ontr

ol-

QD

EE

PP

EN

ALT

Y20

Add

ed n

omin

al c

harg

es p

er d

ay, a

t U <

0.5

C@

sDee

pDis

char

ge/c

alcu

late

Wei

ghte

dCha

rgeT

hrou

ghpu

tU

sed

in d

eter

min

ing

the

dept

h of

dis

char

ge

wei

ghtin

g fa

ctor

fq.

Dee

pDis

char

ge

-Q

oldc

ycle

0C

harg

e th

roug

hput

at l

ast d

etec

ted

full

char

ge-

IV@

cBat

tCon

trol

/set

New

Thr

esho

lds,

@

cBat

tCon

trol

/set

Cha

rgeC

ontr

olC

ontr

ol

-Q

SO

CLI

M0,

98U

pper

bou

ndar

yD

2C

@sD

eepD

isch

arge

/cal

cula

teW

eigh

tedC

harg

eThr

ough

put,

Ope

ratio

nD

eter

min

es n

ew o

r pr

esen

t cyc

le.

Dee

pDis

char

ge

-Q

TLI

M0

Low

er ti

me

boun

dary

[h]

-C

@sD

eepD

isch

arge

/cal

cula

teD

eepD

isch

arge

Fac

tor

Use

d in

det

erm

inin

g th

e de

pth

of d

isch

arge

w

eigh

ting

fact

or fq

.D

eepD

isch

arge

ρc,

0rc

Par

amet

er fo

r in

itial

ohm

ic r

esis

tanc

e (c

harg

ing)

[ΩA

h]3.

34IP

@cP

bBat

tery

/Set

Var

iant

, @

cVol

tage

/cal

cula

teM

axC

harg

ingC

urre

nt,

@cV

olta

ge/c

alcu

late

Vol

tage

, @

cVol

tage

/cal

cula

teC

orro

sion

Vol

tage

, @

cVol

tage

/cal

cula

teIn

tern

alR

esis

tanc

e

Initi

al p

aram

eter

of t

he S

heph

erd

equa

tion.

Par

amet

ers

ρc,

trc

tota

lP

aram

eter

for

ohm

ic r

esis

tanc

e (c

harg

ing)

at t

ime

t (rc

tota

l = r

c +

rk)

3.

34IV

@cP

bBat

tery

/Set

Var

iant

, @

cVol

tage

/cal

cula

teM

axC

harg

ingC

urre

nt,

@cV

olta

ge/c

alcu

late

Vol

tage

, @

cVol

tage

/cal

cula

teC

orro

sion

Vol

tage

, @

cVol

tage

/cal

cula

teIn

tern

alR

esis

tanc

e

Vol

tage

ρd,

0rd

Par

amet

er fo

r in

itial

ohm

ic r

esis

tanc

e (d

isch

argi

ng)

at ti

me

t [Ω

Ah]

3.35

IP@

cPbB

atte

ry/S

etV

aria

nt,

@cV

olta

ge/c

alcu

late

Vol

tage

, @

cVol

tage

/cal

cula

teC

orro

sion

Vol

tage

, @

cCor

rosi

on/c

alcu

late

Cor

rInc

reas

eLim

it,

@sD

egra

datio

n/ca

lcul

ateR

emai

ning

Cap

acity

, @

sDeg

rada

tion/

calc

ulat

eDeg

Incr

ease

Lim

it,

@cV

olta

ge/c

alcu

late

Inte

rnal

Res

ista

nce

Initi

al p

aram

eter

of t

he S

heph

erd

equa

tion.

Par

amet

ers

ρd,

trd

tota

lP

aram

eter

for

ohm

ic r

esis

tanc

e (d

isch

argi

ng)

at ti

me

t (rd

tota

l =

rd

+ r

k)3.

35IV

@cP

bBat

tery

/Set

Var

iant

, @

cVol

tage

/cal

cula

teIn

tern

alR

esis

tanc

e,

@cV

olta

ge/c

alcu

late

Vol

tage

, @

cVol

tage

/cal

cula

teC

orro

sion

Vol

tage

, @

sDeg

rada

tion/

calc

ulat

eRem

aini

ngC

apac

ity

Vol

tage

ρk,

trk

Incr

ease

in in

tern

al o

hmic

res

ista

nce

due

to c

orro

sion

at t

ime

t3.

62IV

@cP

bBat

tery

/Set

Var

iant

, @

cVol

tage

/cal

cula

teIn

tern

alR

esis

tanc

e,

@cG

assi

ng/c

alcu

late

Gas

sing

Cur

rent

Cor

rosi

on

ρk,

limit

/ ∆W

limit

rkld

dWl

0rk

lim/d

Wlim

(R

atio

of ρ

k,lim

it to

∆W

limit)

-IV

@cC

orro

sion

/cal

cula

teF

inal

CLT

hick

ness

Cor

rosi

on

IP =

Inpu

t Par

amet

er, O

= O

utpu

t, IV

= In

tern

al V

aria

ble,

B =

Boo

lean

, V =

Val

ue, C

= C

onst

ant

5

Tom

Cro

nin

Julia

Sch

iffer

Ant

on A

nder

sson

Pag

e 6

Par

amet

er N

ames

231

105.

xls

Nam

e in

Pap

erN

ame

in c

od

eV

alu

e (I

nit

ial V

alu

e)

Des

crip

tio

nE

qu

atio

n i

init

ial

pap

er

Pu

rpo

seL

oca

tio

n in

Mat

lab

Mo

del

Co

mm

ents

bel

on

gs

to

clas

s

ρk,

limit

rklim

0In

crea

se in

inte

rnal

ohm

ic r

esis

tanc

e du

e to

cor

rosi

on a

t the

en

d of

ser

vice

life

. Not

sur

e th

is fi

ts in

to E

q. 3

.48?

? Is

it

corr

ect?

3.62

IV@

cPbB

atte

ry/S

etV

aria

nt,

@cC

orro

sion

/cal

cula

teF

inal

CLT

hick

ness

, @

cGas

sing

/cal

cula

teG

assi

ngC

urre

nt,

@cC

orro

sion

/cal

cula

teC

orrI

ncre

aseL

imit,

@

cCor

rosi

on/c

alcu

late

Cor

rosi

onR

esC

ap

Cor

rosi

on

-S

eria

lNum

ber

1N

umbe

r of

cel

ls in

ser

ies

-IP

Use

d m

any

times

Alw

ays

cons

ider

ed a

s a

sing

le c

ell f

or th

is m

odel

.P

aram

eter

s

FS

ocS

tate

of c

harg

e3.

34O

@cP

bBat

tery

/cal

cula

teS

oc, O

pera

tion,

@

sDee

pDis

char

ge/c

alcu

late

Wei

ghte

dCha

rgeT

hrou

ghpu

t, @

Bat

tCon

trol

/set

New

Thr

esho

lds,

@

cVol

tage

/cal

cula

teM

axC

harg

ingC

urre

nt,

Set

Soc

, @cV

olta

ge/c

alcu

late

Cor

rosi

onV

olta

ge,

@cV

olta

ge/c

alcu

late

Vol

tage

An

outp

ut o

f the

mod

el b

ut a

lso

used

to fe

ed

back

into

the

She

pher

d eq

uatio

n.P

bBat

tery

Fm

inso

cmin

1M

inim

al s

tate

of c

harg

e du

ring

deep

-dis

char

ge p

hase

, Low

est

stat

e of

cha

rge

sinc

e la

st fu

ll ch

arge

3,67

IV@

sDee

pDis

char

ge/c

alcu

late

Wei

ghte

dCha

rgeT

hrou

ghpu

t, @

sDee

pDis

char

ge/c

alcu

late

Dee

pDis

char

geF

act

or, @

cBat

tCon

trol

/set

New

Thr

esho

lds,

@

cBat

tCon

trol

/cal

cula

teD

eepD

isch

arge

Fac

tor

Kee

ps tr

ack

of th

e lo

wes

t SO

C in

a c

ycle

, In

fluen

ces

the

fact

or fo

r st

ate

of c

harg

eC

ontr

ol, D

eep

Dis

char

ge

Fne

uso

cnN

orm

al s

tate

of c

harg

e fo

r a

new

bat

tery

3.63

IV@

cPbB

atte

ry/c

alcu

late

Soc

loca

l

-S

tart

Soc

1S

tart

ing

SO

C fo

r a

new

bat

tery

-IP

@cP

bBat

tery

/Set

Sta

rtS

oc, r

eset

Sta

rtin

g S

OC

of n

ew b

atte

ry s

et b

y ar

gum

ents

w

hen

crea

ting

batte

ry o

bjec

t.P

bBat

tery

-S

wba

ttcha

rged

true

TR

UE

: "B

efue

"'s fu

ll ch

arge

crit

eria

are

met

-B

@cB

attC

ontr

ol/s

etN

ewT

hres

hold

sO

nly

used

for

dete

rmin

ing

exte

rnal

cha

rgin

gC

ontr

ol-

Sw

Dyn

amic

Thr

esho

lds

Allo

ws

exte

rnal

cha

rgin

g th

resh

olds

to c

hang

e.-

B@

cBat

tCon

trol

/set

New

Thr

esho

lds,

@

cBat

tCon

trol

/set

Cha

rgeC

ontr

olO

nly

used

for

dete

rmin

ing

exte

rnal

cha

rgin

gC

ontr

ol

-S

wE

xtLo

adfa

lse

TR

UE

: Pha

se in

whi

ch a

full

char

ge is

don

e us

ing

"BS

Z"

-B

@cB

attC

ontr

ol/s

etN

ewT

hres

hold

sO

nly

used

for

dete

rmin

ing

exte

rnal

cha

rgin

gC

ontr

ol-

Tam

bien

tA

mbi

ent t

empe

ratu

re-

IV@

cPbB

atte

ry/O

pera

tion,

Cel

lTem

pera

ture

Am

bien

t tem

pera

ture

. Set

to 2

0 de

g C

whe

n ba

ttery

obj

ect c

reat

ed b

y S

tart

up.m

pro

cedu

re.

Inpu

t

-ba

ttTem

pB

atte

ry te

mpe

ratu

re [

deg

C]

-IV

@cC

orro

sion

/cal

cula

teC

orro

sion

Spe

edT

batt

is th

e lo

cal v

aria

ble

used

whe

n de

term

inin

g co

rros

ion

spee

d pa

ram

eter

.lo

cal

Tba

ttT

Cel

lC

ell t

empe

ratu

re3.

61IV

@cP

bBat

tery

/Cel

lTem

pera

ture

, @

cGas

sing

/cal

cula

teG

assi

ngC

urre

nt, O

pera

tion,

@

sAci

dStr

atifi

catio

n/ca

lcul

ateA

cidS

trat

ifica

tion,

@

cBat

tCon

trol

/UM

axC

orre

ctio

n

Tce

ll is

the

varia

ble

used

in c

alcu

latin

g th

e ga

ssin

g cu

rren

t, ac

id s

trat

ifica

tion

and

corr

ectio

n of

Um

ax.

PbB

atte

ry

-tc

harg

eT

ime

afte

r w

hich

Um

ax is

set

to U

low

[h]

-IV

@cB

attC

ontr

ol/s

etN

ewT

hres

hold

sU

sed

in d

eter

min

ing

the

exte

rnal

cha

rge.

Con

trol

-tc

harg

elow

3tv

oll a

t Um

ax =

Ulo

w [h

]-

IV@

cBat

tCon

trol

/set

New

Thr

esho

lds

Use

d in

det

erm

inin

g th

e ex

tern

al c

harg

e.C

ontr

ol-

tcha

rgeu

p3

tvol

l at U

max

= U

up [h

] -

IV@

cBat

tCon

trol

/set

New

Thr

esho

lds

Use

d in

det

erm

inin

g th

e ex

tern

al c

harg

e.C

ontr

ol-

tF0

Tim

e af

ter

full

char

ge w

ithou

t ris

e of

thre

shol

ds [h

]-

IV@

cBat

tCon

trol

/set

Cha

rgeC

ontr

olU

sed

in d

eter

min

ing

the

exte

rnal

cha

rge.

Con

trol

-tg

as0

Tim

e si

nce

the

last

gas

ing

perio

d

[h]

-IV

@cB

attC

ontr

ol/s

etC

harg

eCon

trol

Use

d in

det

erm

inin

g th

e ex

tern

al c

harg

e.C

ontr

ol-

tgas

set

1000

0T

ime

to n

ext g

asin

g cy

cle

[d]

-IV

@cB

attC

ontr

ol/s

etC

harg

eCon

trol

Use

d in

det

erm

inin

g th

e ex

tern

al c

harg

e. S

et in

cB

attC

ontr

ol.

Con

trol

-tr

emas

Min

imum

tim

e fo

r re

mov

ing

acid

str

atifi

catio

n at

2.5

VD

2IV

@sA

cidS

trat

ifica

tion/

calc

ulat

eAci

dStr

atifi

catio

nT

o do

with

aci

d st

ratif

icat

ion…

.A

cidS

trat

ifica

tion

-tr

emas

25T

ime

for

rem

ovin

g ac

id s

trat

ifica

tion

at 2

.5V

D2

IV@

sAci

dStr

atifi

catio

n/ca

lcul

ateA

cidS

trat

ifica

tion

Equ

al to

dtu

_ref

Aci

dStr

atifi

catio

n

tts

Inte

rmed

iate

var

iabl

e3.

44IV

@cC

orro

sion

/cal

cula

teC

orrL

ayer

Dep

thU

sed

in c

alcu

latio

n of

dep

th o

f cor

rosi

on la

yer.

loca

l

IP =

Inpu

t Par

amet

er, O

= O

utpu

t, IV

= In

tern

al V

aria

ble,

B =

Boo

lean

, V =

Val

ue, C

= C

onst

ant

6

Tom

Cro

nin

Julia

Sch

iffer

Ant

on A

nder

sson

Pag

e 7

Par

amet

er N

ames

231

105.

xls

Nam

e in

Pap

erN

ame

in c

od

eV

alu

e (I

nit

ial V

alu

e)

Des

crip

tio

nE

qu

atio

n i

init

ial

pap

er

Pu

rpo

seL

oca

tio

n in

Mat

lab

Mo

del

Co

mm

ents

bel

on

gs

to

clas

s

-tU

0A

fter

full

char

ge: t

ime

with

out r

ise

of U

max

[h

]-

IV@

cBat

tCon

trol

/set

Cha

rgeC

ontr

olU

sed

in d

eter

min

ing

the

exte

rnal

cha

rge.

Con

trol

-tU

max

0T

ime

(Bat

tery

vol

tage

>=

end

-of-

char

ge v

olta

ge)

[V]

-IV

@cB

attC

ontr

ol/s

etN

ewT

hres

hold

sU

sed

in d

eter

min

ing

the

exte

rnal

cha

rge.

Con

trol

U0c

U0c

Vol

tage

at f

ull S

OC

(F

=1)

for

char

ging

[V]

3.34

IP@

cPbB

atte

ry/S

etV

aria

nt,

@cV

olta

ge/c

alcu

late

Max

Cha

rgin

gCur

rent

, @

cVol

tage

/cal

cula

teV

olta

ge

She

pher

d eq

uatio

n pa

ram

eter

.P

aram

eter

s

U0d

U0d

Vol

tage

at f

ull S

OC

(F

=1)

for

disc

harg

ing

[V]

3.35

IP@

cPbB

atte

ry/S

etV

aria

nt,

@cV

olta

ge/c

alcu

late

Vol

tage

, @

cCor

rosi

on/c

alcu

late

Cor

rInc

reas

eLim

it,

@sD

egra

datio

n/ca

lcul

ateR

emai

ning

Cap

acity

, @

sDeg

rada

tion/

calc

ulat

eDeg

Incr

ease

Lim

it

She

pher

d eq

uatio

n pa

ram

eter

.P

aram

eter

s

-U

Bat

tB

atte

ry v

olta

ge-

IV@

cVol

tage

/cal

cula

teV

olta

ge, c

alcu

late

Soc

, res

etU

batt

= U

cell

* S

eria

lNum

ber

Vol

tage

-U

BR

OK

EN

0,3

Bat

tery

vol

tage

whe

n co

mpl

etel

y em

pty

-C

@cP

bBat

tery

/cal

cula

teS

oc,

@sD

eepD

isch

arge

/cal

cula

teW

eigh

tedC

harg

eThr

ough

put,

@cV

olta

ge/c

alcu

late

Vol

tage

, @

cVol

tage

/cal

cula

teC

orro

sion

Vol

tage

Vol

tage

UZ

elle

UC

ell

Cel

l vol

tage

3.61

, 3.3

4O

rese

t, @

Bat

tCon

trol

/set

New

Thr

esho

lds,

@

sAci

dStr

atifi

catio

n/ca

lcul

ateA

cidS

trat

ifica

tion,

@

sDee

pDis

char

ge/c

alcu

late

Wei

ghte

dCha

rgeT

hrou

ghpu

t, @

cGas

sing

/cal

cula

teG

assi

ngC

urre

nt,

@cV

olta

ge/c

alcu

late

Vol

tage

, cal

cula

teS

oc

Out

put f

rom

She

pher

d eq

uatio

n.V

olta

ge

Uem

pty

UE

MP

TY

1,8

Vol

tage

of a

n em

pty

batte

ry3.

60C

Set

Var

iant

, res

et,

@sD

egra

datio

n/ca

lcul

ateR

emai

ning

Cap

acity

Vol

tage

Uk

Uk

0C

orro

sion

vol

tage

at p

ositi

ve e

lect

rode

(on

e ce

ll)3.

44IV

@cC

orro

sion

/cal

cula

teC

orrL

ayer

Dep

th, r

eset

, ca

lcul

ateS

oc,

@cC

orro

sion

/cal

cula

teC

orro

sion

Vol

tage

, @

cCor

rosi

on/c

alcu

late

Cor

rosi

onS

peed

Vol

tage

Uk

ukC

orro

sion

vol

tage

per

cel

l [V

SH

E]

3.39

IV@

cVol

tage

/cal

cula

teC

orro

sion

Vol

tage

Loca

l var

iabl

e us

ed w

ithin

@

cVol

tage

/cal

cula

teC

orro

sion

Vol

tage

rou

tine.

lo

cal

U0 k,

0U

KS

01,

716

Cor

rosi

on v

olta

ge p

er c

ell [

V] w

hen

fully

cha

rged

. No-

load

co

rros

ion

volta

ge w

hen

fully

cha

rged

.3.

39C

@cV

olta

ge/c

alcu

late

Cor

rosi

onV

olta

geV

olta

ge

-U

low

2,35

Low

er li

mit

of e

nd-o

f-ch

arge

vol

tage

[V]

-IV

@cB

attC

ontr

ol/s

etN

ewT

hres

hold

s,

@cB

attC

ontr

ol/s

etC

harg

eCon

trol

Use

d fo

r de

term

inin

g ex

tern

al c

harg

ing

Con

trol

-U

max

End

-of-

char

ge v

olta

ge-

IV@

cBat

tCon

trol

/set

New

Thr

esho

lds,

@

cBat

tCon

trol

/set

Cha

rgeC

ontr

ol,

@cB

attC

ontr

ol/U

Max

Cor

rect

ion

Con

trol

-U

Max

2,45

Glo

bal e

nd-o

f-ch

arge

vol

tage

-IV

@cV

olta

ge/c

alcu

late

Max

Cha

rgin

gCur

rent

used

for

high

vol

tage

con

trol

ler

Vol

tage

-U

max

gas

2,35

End

-of-

char

ge v

olta

ge d

urin

g th

e ga

sing

per

iod

[V]

-IV

@cB

attC

ontr

ol/s

etC

harg

eCon

trol

Con

trol

-U

max

grad

T-0

,004

Tem

pera

ture

gra

dien

t for

ada

ptin

g U

max

[V/K

]-

IV@

cBat

tCon

trol

/UM

axC

orre

ctio

nC

ontr

ol-

Um

axT

End

-of-

char

ge v

olta

ge (

tem

pera

ture

cor

rect

ed)

[

V]

-IV

@cB

attC

ontr

ol/s

etN

ewT

hres

hold

s,

@cB

attC

ontr

ol/U

Max

Cor

rect

ion

Con

trol

-U

RE

F2,

5C

ell v

olta

ge fo

r w

eigh

ted

deco

mpo

sitio

n tim

e [V

] (ca

lcul

atin

g c)

.D

2C

@sA

cidS

trat

ifica

tion/

calc

ulat

eAci

dStr

atifi

catio

nS

omet

hing

to d

o w

ith a

cid

stra

tific

atio

nA

cidS

trat

ifica

tion

-U

ST

AR

T2,

3C

ell v

olta

ge a

t whi

ch d

ecom

posi

tion

begi

ns [V

] (ca

lcul

atin

g c)

.D

2C

@sA

cidS

trat

ifica

tion/

calc

ulat

eAci

dStr

atifi

catio

nS

omet

hing

to d

o w

ith a

cid

stra

tific

atio

nA

cidS

trat

ifica

tion

-U

up2,

35U

pper

lim

it of

end

-of-

char

ge v

olta

ge [V

]-

IV@

cBat

tCon

trol

/set

New

Thr

esho

lds,

@

cBat

tCon

trol

/set

Cha

rgeC

ontr

olU

sed

for

dete

rmin

ing

exte

rnal

cha

rgin

gC

ontr

ol

x 1x

Inte

rmed

iate

par

amet

er fo

r ca

lcul

atio

n3.

60IV

@sD

egra

datio

n/ca

lcul

ateR

emai

ning

Cap

acity

loca

l

IP =

Inpu

t Par

amet

er, O

= O

utpu

t, IV

= In

tern

al V

aria

ble,

B =

Boo

lean

, V =

Val

ue, C

= C

onst

ant

7

Tom

Cro

nin

Julia

Sch

iffer

Ant

on A

nder

sson

Pag

e 8

Par

amet

er N

ames

231

105.

xls

Nam

e in

Pap

erN

ame

in c

od

eV

alu

e (I

nit

ial V

alu

e)

Des

crip

tio

nE

qu

atio

n i

init

ial

pap

er

Pu

rpo

seL

oca

tio

n in

Mat

lab

Mo

del

Co

mm

ents

bel

on

gs

to

clas

s

x 2w

(M

atla

b), a

(C

++

)In

term

edia

te p

aram

eter

for

calc

ulat

ion

3.60

IV@

sDeg

rada

tion/

calc

ulat

eRem

aini

ngC

apac

itylo

cal

ZZ

Cyc

le li

fe w

ithou

t cor

rosi

on3,

55IV

@sD

egra

datio

n/ca

lcul

ateC

apac

ityLo

ssD

eglo

cal

Z80

%,d

atZ

dT

he c

ycle

life

, as

indi

cate

d by

the

man

ufac

ture

r.3,

55IP

@cP

bBat

tery

/Set

Var

iant

, @

sDeg

rada

tion/

calc

ulat

eCap

acity

Loss

Deg

Par

amet

ers

-Z

dMea

nM

ean

num

ber

of n

omin

al c

ycle

s-

IV@

sBat

tLife

time/

calc

ulat

eLife

, Exc

hang

eLi

fetim

eZ

NZ

dNow

Num

ber

of im

plem

ente

d no

min

al c

ycle

s w

ith th

e pr

esen

t ba

ttery

3,64

IV@

sBat

tLife

time/

calc

ulat

eLife

, Exc

hang

eLi

fetim

e

-z

norm

aliz

ed n

umbe

r of

sul

fat c

ryst

als

at Ic

f-

IV@

sCur

rent

Fac

tor/

calc

ulat

eCur

rent

Fac

tor

used

for

calc

ulat

ing

the

curr

ent f

acto

rlo

cal

-z0

norm

aliz

ed n

umbe

r of

sul

fat c

ryst

als

at Ir

ef-

IV@

sCur

rent

Fac

tor/

calc

ulat

eCur

rent

Fac

tor

used

for

calc

ulat

ing

the

curr

ent f

acto

rlo

cal

-I_

norm

norm

aliz

ed Ic

f-

IV@

sCur

rent

Fac

tor/

calc

ulat

eCur

rent

Fac

tor

used

for

calc

ulat

ing

the

curr

ent f

acto

rlo

cal

-I_

rel

equa

l to

-Ire

f-

IV@

sCur

rent

Fac

tor/

calc

ulat

eCur

rent

Fac

tor

used

for

calc

ulat

ing

the

curr

ent f

acto

rlo

cal

IP =

Inpu

t Par

amet

er, O

= O

utpu

t, IV

= In

tern

al V

aria

ble,

B =

Boo

lean

, V =

Val

ue, C

= C

onst

ant

8

Tom

Cro

nin

Julia

Sch

iffer

Ant

on A

nder

sson