Abstract Master thesis Electric Vehicle (EV) charging scheduling to minimize power system imbalance: From aggregator`s and EV owner`s perspective

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Electric Vehicle (EV) charging scheduling to minimize power

system imbalance: From aggregator`s and EV owner`s

perspective

Master thesis

By

Sumedh Kulkarni,

Student, MSc. Technology and Operations Management, University of Groningen.

Supervisor: dr. ir. Stefano Fazi

Co-Supervisor: prof. dr. Kees Jan Roodbergen

Abstract

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Acknowledgements

This master degree has been an uphill but pleasant journey, especially the thesis. I have encountered several instances where I thought it was better to leave the thesis but with the encouragement of my family, friends and professors it remained a thought. I have learned a lot about programming and mathematical modelling through this research. There are many people whom I want to thank.

First of all, I would like to thank Dr. Stefano Fazi for supervising my work. He was always available for a meeting to discuss issues related to thesis, even if it was a last-minute request. Additionally, I want to thank Dr. Fazi for improving my mathematical modelling skills. I also want to thank Dr. Kees Jan Roodbergen for advising me a better approach to tackle the real-world problem in my research. Next, I want to thank Robbert-Jan van der Burg for his help in building and understanding conceptual part of this thesis. Next, I want to thank Ankur Mishra, Arijit Das and Pulkesh Mishra for helping me with the programming skills required at the beginning of this research and for chilling with me in the low times.

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1. Introduction

Advancements in renewable energy technologies such as wind, solar or hydro power have made energy production viable and cheaper at the same time (Verzijlberg, 2017). This has resulted in large scale integration of renewable energy in the current power grids all over the world. However, due to uncertainty involved with the weather conditions, power produced by renewables may vary every minute (Verzijlberg, 2017). These variations cause (at times) substantial imbalance between the power supplied by renewable sources to the grid and power consumed by units connected to the grid. This difference between the energy produced and consumed in a grid at any moment is defined as power system imbalance or system imbalance (SI) (Ulbig and Andersson, 2015). In power systems the balance between demand and supply of power is of paramount importance. This imbalance can have detrimental effects on the grid such as energy congestion, system instability and loss of energy (Huber et al., 2014).

Electric vehicles (EVs) can be employed to reduce SI. An electric vehicle is “any vehicle in which some or all of the driving energy is supplied through electricity from a battery” (Ortega-Vasquez et al., 2013). Power is energy charged/discharged by EV batteries per a single unit of time (hour). Power is measured in kW and Energy is measured in kWh. At any instant when power available in the grid is more than power consumed, SI is positive and when power available is less than power consumed, SI is negative. When SI is positive EVs can charge the power from the grid and reduce imbalance. When SI is negative EVs can inject (discharge) the power back in to the grid and reduce the imbalance. In reality, each EV owner wants cheaper charging costs (Jin et al, 2013). In other words, owner wants to charge their battery when the electricity price is lowest. If this happens on a larger scale, there will be congestion in power system and SI will increase even more. To annul this situation, a charging schedule needs to be developed so that EVs can charge their batteries to desired amount. The charging schedule is developed by a third party called aggregator who acts as an intermediate between system operator and EV owners (Ortega-Vasquez, 2013).

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4 discharging schedule such that the contract is honoured on both sides. We refer to this simply as EV charging scheduling problem.

In recent years, the field of EV charging scheduling has been researched extensively. Nguyen et al (2015) developed an optimization problem for EV charging scheduling with renewable energy integrated power grid. Their objective is to reduce the SI. However, they do not consider the charging costs attached to the charging/discharging activities of EV batteries. The EV charging scheduling problem with charging cost minimization is proposed by He et al (2012). The EVs charging schedule follows the consumption pattern of units connected to grid. But their model does not include any measure to reduce SI. A study by Alonso et al (2014) focuses on the effects of voltage differences in the grid in their EV charging scheduling problem. However, they neither include charging cost nor the cost associated with system imbalance. In this thesis, we intend to achieve an EV charging and discharging scheme with the objective to minimize the charging cost of EVs and the cost associated with SI of the renewable energy integrated power grid. We consider the optimization problem as a day ahead, contract-based agreement between aggregator and system operator and EV owners.

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5 We compare the costs obtained from C scheme and N scheme in four scenarios and analyse the results. We do not compare the CD scheme with the other two because CD scheme also involves discharging costs in addition to the charging costs. Thus, it is only fair to compare C and N scheme. Additionally, we compare the costs obtained in all four scenarios for the CD scheme. Four scenarios are constructed based on two practical instances: first is SI of work days and holidays and second is when EV owner`s desired energy at end is variable and constant. These are explained further in later part of the thesis. We build optimization models for CD scheme and C scheme and a simple algorithmic problem for N scheme. Unlike C scheme and CD scheme, N scheme does not need optimization because all the parameters are known. Specifically, for each EV, the charging rate is already known and is constant in N scheme whereas the charging rate is variable in C and CD scheme. Thus, charging rates are optimized in C and CD schemes to get minimized costs. The optimization model in this thesis is convex optimization problem, similar to work done by He et al (2012). They use CVX, a package to solve convex optimization problems on MATLAB (Grant and Boyd,2010). We test these schemes in an experimental format using real world dataset from Belgium`s electricity grid, ELIA and use CVX in MATLAB environment to perform optimizations. For N scheme, we use MATLAB to build an algorithm for charging EVs at a constant rate.

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2. Background theory and Literature review

In this chapter, we first shed some light on the need to minimize system imbalance. Next we show how EVs can participate in minimizing system imbalance. Next, we look at aggregator contract terms with EV owners and system operators. Lastly, we review the charging and discharging schemes that have been proposed in the literature and closest to our work.

2.1 Need to minimize system imbalance

In the current power grid structure, balance between power consumption and production is maintained by traditional power systems such as thermal power plants. The power derived from traditional systems, to some extent, can be regulated to match the consumption patterns (Tang et al, 2016). Traditional systems, however, are of polluting nature. Moreover, they are not flexible enough to respond to sudden disturbances in power consumption patterns. This has adverse effects on the power grid such as power traffic congestion, power outages, grid instability and power losses (Huber et al., 2014). In addition, increasing amount of renewable energy is being integrated in to current power grids all over the world. When renewable energy is integrated, at any instant when power available in the grid can be more than power consumed by units connected to the grid. In this case, SI is positive. Power available in the grid can also be less than power consumed. Here, SI is negative. The volatile nature arises due to unpredictability of weather conditions. For example, power generated by solar farm depends on the solar energy that is available at a moment. Similarly, wind power depends on the amount and direction of winds in that moment. On a broader scale, days with similar weather conditions have similar energy production patterns (Zhang et al, 2014). On the other side, power consumption patterns can also lead to power imbalance in a grid. The consumption of power, on a normal working day can significantly vary to the power consumed on a holiday (Makarov et al, 2009). This is another aspect that adds to the existing imbalance in the power grids. Mitigation of these imbalances is, therefore, significant.

2.2 EVs as a solution to minimize system imbalance

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7 is negative or when the power produced by renewables is less that power consumed. In this way, EV acts as a temporary storage system.

The charging and discharging activities of EVs require a coordination effort to effectively minimize the SI (Sortomme et al., 2011). Precisely, if EVs charge in an uncoordinated way, all EV owners would charge their EVs only when the electricity price is lowest that is when SI is highest. Charging the EVs when the electricity price is lowest may be beneficial for the owners but power system balance organisations (System Operators) will suffer from this setting as their primary objective of balancing the grids is not achieved (Ortega-Vaszquez et al, 2013). Instead SI will increase even more as there will be no more EVs left to charge when the prices have increased. As a result, third party called aggregator is introduced that coordinates the charging and discharging activities of EVs. The aggregator is responsible to schedule EVs such that the owners are satisfied with their charging requirements and system operators are satisfied with the relatively more balanced power grid (Jin et al, 2014). Thus, aggregator acts as a link between EV owners and system operators.

2.3 Aggregator contract terms

The aggregator coordinates EV charging using a contract-based agreement with EV owners and system operators (SO) (Ortega-Vaszquez et al ,2013). The contract is designed such that all parties involved in the transaction (EV owners, aggregator and system operators) will benefit from it. From the owner`s perspective, this contract specifies their charging requirements that need to be fulfilled. From the system operator`s perspective, power imbalance needs to be minimized (Huber et al, 2014). Aggregator`s earnings are governed by efficiency of the EV charging schedule such that EV owners and system operators are satisfied. For an SO, the imbalance must be minimized regardless of any time of the year. Be it a holiday or a working day, aggregator`s earnings are decided by SO based on the performance of that day. Another aspect of aggregator`s earnings are the charging costs incurred by EVs that are coordinated. Higher charging costs negatively affects the decision of EV owners to participate in the contract. Therefore, more the charging costs, less would be aggregator`s profit. In order to profit from EV charging scheduling activity, it is necessary for aggregator to minimize the charging costs.

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8 their requirements such as start time of their EV availability for charging/discharging activities, end time of their EV availability, energy in their EV battery at the start time, desired energy in their EV battery at the end time. These requirements are fundamental to design a charging schedule simply because owners decide whether to participate in charging/discharging activity or not. Each owner can have distinct set of charging requirements. Specifically, the desired energy at the end can vary for each EV owner. For instance, EV taxi owners may not want to charge their battery to its capacity at all times. On the other hand, EV owners who work in a office might need EV charged at all times. Both instances are possible and the owner -aggregator contract must satisfy these requirements.

2.4 EV charging schemes and literature

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9 for charging are going to increase as time proceeds (Ortega-Vasquez, 2013). Thus, in future, EV charging and discharging scheme will be the best practical option to minimize the system imbalance. Thus, each scheme has valid (future) practical applications.

The methodology and objective of the aforementioned papers, however, are different. For instance, Alonso et al (2014) develop an optimal charging schedule for EVs with an objective to minimize the peaks in power consumption. They use genetic algorithm to develop the optimal schedule while considering technical power grid constraints such as voltage control. However, EV charging costs and cost of power imbalance are not considered in their objective function. Nguyen et al (2015) propose an optimization model to schedule EVs for charging/discharging the power from a grid with high penetration of renewables. In the objective function, they minimize the system imbalance, which is similar to our work. However, unlike us they do not consider EV charging costs in their problem.

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3. Problem formulation and Mathematical model

In this chapter, we first describe the parameters used for building the mathematical model. Following this we look at cost function formulations and then problem formulation where we also describe the mathematical model.

We consider a day of charging and discharging activities by EVs evenly divided in interval i of 1 hour each, denoted by interval set M. Length of the interval is τ. The set of EVs is denoted by N. Each EV n (n ϵ N) has a start time of availability, 𝑡𝑠𝑛and end time of availability, 𝑡𝑒𝑛 . At

𝑡_𝑠𝑛_{the EV starts charging and at 𝑡}

𝑒𝑛, EV should stop charging. At the start time, each EV has

𝐸_𝑠𝑛_{amount of energy in its battery and requires the desired}_𝐸

𝑒𝑛 amount of energy at the end

time. Β is the EV battery efficiency. The maximum power charging rate of an EV 𝑃𝑚𝑎𝑥. The

system imbalance is SI is difference between total energy available in a grid and total energy consumed from the grid for a day.  is the end energy factor which determines the distinct end energy value desired by each EV owner. In table 1, table 2 and table 3, the parameters, sets and variable used in the model are illustrated.

Table 1 Parameters

i The time interval of a day

 Length of the interval

n EV number

𝑡𝑠𝑛 Start time of EV n`s availability 𝑡𝑒𝑛 End time of EV n`s availability 𝐸𝑠𝑛 EV n battery energy at start time

𝐸𝑒𝑛 Desired EV n battery energy at the end time 𝑃𝑚𝑎𝑥 Maximum charging power rate

−𝑃𝑚𝑎𝑥 Maximum discharging power rate  𝑓1 𝑓2 𝑓3 𝑆𝐼𝑖 β 𝐴_𝑖𝑛

End energy factor Charging cost factor 1 Charging cost factor 2 System imbalance cost factor System imbalance in interval i. EV battery efficiency

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Table 2 Sets

N Set of EVs, 1 to N

M Set of Intervals, 1 to M

Table 3 Variables

𝒙_𝒊𝒏 EV n`s charging rate in interval i. −𝒙𝒊𝒏 EV n`s discharging rate in interval i.

3.1 Overall cost function

Here we describe the two parts of costs that an aggregator has to minimize while making a charging scheduling scheme. The first is charging cost 𝐶_𝑖1 that is linearly proportional to the of amount of energy charged by each EV, as considered by He et al (2012). The second cost 𝐶_𝑖2 that aggregator must bear is in the form of a system imbalance price at each interval. We briefly explain these costs below.

Overall cost, 𝐶_𝑖 = 𝐶_𝑖1+ 𝐶_𝑖2

3.1.1 Charging cost

Based on the energy charged by EVs in interval i, we define the charging costs. The charging cost can be expressed as a linear function of the energy consumed by the EVs in that moment (Veldman and Verzijlberg, 2015). For one EV, at any instant, this charging cost can be formulated as follows (He et al, 2012):

𝑦(𝑥𝑡) = 𝑓1+ 𝑓2𝑥𝑡 , (4)

where 𝑓₁ and 𝑓₂are the intercept and slope for the linear cost equation, respectively. 𝑓₁ represents the cost of charged energy and 𝑓2 represents the cost corresponding to

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12 𝐶_𝑖1 = ∫ (𝑓₁+ 𝑓₂∑ 𝑥_𝑖𝑛 𝑛∈𝑁 𝐴_𝑖𝑛)𝜕𝑥_𝑡 ∑ 𝑥_𝑖𝑛.𝐴_𝑖𝑛 0 = (𝑓₁∑_𝑛∈𝑁𝑥_𝑖𝑛𝐴_𝑖𝑛+𝑓2 2 (∑ 𝑥𝑖 𝑛 𝑛∈𝑁 𝐴𝑖𝑛)2 ) (5)

Where, 𝐶_𝑖1 is the total cost of charging EV in interval, i (∀𝑖 ∈ 𝑀). ∑𝑛∈𝑁𝑥_𝑖𝑛𝐴𝑛𝑖is the amount of

energy charged/discharged by n EVs at interval i. 𝐴_𝑖𝑛 is the EV availability matrix. It will be discussed in the problem formulation section.

3.1.2 SI cost

We define the cost for system imbalance based on the amount of imbalance present in the grid when EVs perform charging/discharging activities. The SI cost are considered linear in relation to the difference between amount of SI and energy charged by EVs at interval i. This difference can be positive or negative depending on the SI value at that interval. It is positive when the SI is positive and negative when the SI is negative, as the amount of energy charged needs to be less than SI at all intervals. Regardless of being positive or negative, aggregator has to bear these costs for reducing the imbalance (van der Veen, 2014). This is why we take absolute value of the cost in our formulation. Similar function is used by Nguyen et al, 2015. See below for the equation:

𝐶_𝑖2 = 𝑓₃𝑎𝑏𝑠(𝑆𝐼_𝑖 − ∑_𝑛∈𝑁𝑥_𝑖𝑛𝐴_𝑖𝑛) , 𝑖 ∈ 𝑀 (6) Where 𝐶_𝑖2 denotes SI cost. 𝑓₃ is the average imbalance price (Euro/kWh) that is derived from ELIA`s official website. It is averaged because the SI price varies as per the value of SI for that hour. We consider the variable SI but not the price/kW. 𝑆𝐼_𝑖 denotes the system imbalance at interval i. ∑𝑛∈𝑁𝑥_𝑖𝑛𝐴𝑖𝑛 is the amount of energy charged/discharged by n EVs at interval i. 𝐴𝑖𝑛 is

the EV availability matrix, which will be discussed in the next section.

3.2 Problem formulation

In this section we formulate the three EV charging scheduling problems. First is the EV charging and discharging (CD) scheme. Second is the EV charging only (C) scheme and third is the Naive (N) charging scheme. The first and second schemes are similar to the global scheme of the model by He et al (2012).

3.2.1 Charging and discharging scheme (CD scheme)

In this scheme, we plan scheduling of EVs for a complete day. It is assumed that EV owners provide all their charging requirements (𝑡_𝑠𝑛_{, 𝑡}

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13 contract. The aggregator, then develops a charging and discharging schedule for the day. At the start of interval i, the availability matrix of EVs in that interval is defined by a matrix 𝐴_𝑖𝑛, where:

𝐴_𝑖𝑛 = { 1 , if 𝑡𝑒𝑛 ≥ 𝑖 ≥ 𝑡𝑠𝑛 , 𝑛 ∈ 𝑁 and 𝑖 ∈ 𝑀

0 .

The aim is to minimize the overall cost of EVs for a day by optimizing power charged by EVs, 𝑥 , 𝑥 ∈R. After the optimization we get a matrix of 𝑥 values for each EV for each interval.

𝑀𝑖𝑛 ∑ (𝑓1𝑎𝑏𝑠(∑𝑛∈𝑁𝑥𝑖𝑛𝐴𝑖𝑛) + 𝑓2 2(∑ 𝑥𝑖 𝑛 𝑛∈𝑁 𝐴𝑖𝑛)2+ 𝑓3𝑎𝑏𝑠(𝑆𝐼𝑖− ∑𝑛∈𝑁𝑥𝑖𝑛𝐴𝑖𝑛)) 𝑖∈𝑀 (8) Subject to 𝛾. 𝐸_𝑒𝑛 ≤ 𝐸_𝑠𝑛 + ∑_𝑖∈𝑀 𝑥_𝑖𝑛. 𝐴𝑛_𝑖 ≤ 𝐸_𝑒𝑛, 𝑖 ∈ 𝑀 and 𝑛 ∈ 𝑁 (8a) −𝑃𝑚𝑎𝑥, ≤ 𝑥𝑖𝑛 ≤ 𝑃𝑚𝑎𝑥, 𝑖 ∈ 𝑀 and 𝑛 ∈ 𝑁 (8b) 𝑆𝐼𝑖 − ∑ 𝑥𝑖𝑛. 𝐴𝑖𝑛 ≥ 0 , 𝑖 ∈ 𝑀 (8c)

The objective of function (8) minimizes overall charging cost for EVs in set 𝐵𝑖 that are charging in interval 𝑖 ∈ 𝑀 , where 𝑀 represents the set for number of intervals in a day. (8a) constrains the total energy charged by an EV n in its charging period to be at least equal to 𝛾. 𝐸_𝑒𝑛_{. (8b)}

states that the charging rate of EV n in the interval i should not exceed its maximum power rate, 𝑃𝑚𝑎𝑥 and discharging rate should not exceed −𝑃𝑚𝑎𝑥, . Constraint (8c) ensures that the

sum of energy charged by available EVs in interval i should not be more than SI at that interval. In other words, this constraint ensures EVs do not over charge in an interval.

3.2.2 Charging only scheme (C scheme)

The problem formulation for this scheme has almost the same objective function as the CD scheme. The only difference in objective function is that there is no need of absolute value for charging rates in the charging cost term simply because EVs do not discharge their batteries. In the SI cost however, we consider absolute value as the cost of imbalance must be paid by aggregator. The objective function becomes:

𝑀𝑖𝑛 ∑ (𝑓1∑𝑛∈𝑁𝑥𝑖𝑛𝐴𝑖𝑛+ 𝑓2 2 (∑ 𝑥𝑖 𝑛 𝑛∈𝑁 𝐴𝑖𝑛)2+ 𝑓3𝑎𝑏𝑠(𝑆𝐼𝑖 − ∑𝑛∈𝑁𝑥𝑖𝑛𝐴𝑖𝑛)) 𝑖∈𝑀 (9)

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14 constraint −𝑃_𝑚𝑎𝑥 is no longer required. Instead we put the lower bound for this constraint as 0. Thus, for this problem constraint 8b becomes:

0 ≤ 𝑥_𝑖𝑛 ≤ 𝑃𝑚𝑎𝑥, 𝑖 ∈ 𝑀 and 𝑛 ∈ 𝑁

For constraint (8c), again, as we do not intend to discharge EVs, the total energy charged by EVs in any interval is non-negative. Thus, at any interval i, when 𝑆𝐼_𝑖 > 0, constraint (8c) is same as in CD scheme which is:

𝑆𝐼𝑖 − ∑ 𝑥𝑖𝑛. 𝐴𝑖𝑛 ≥ 0 , 𝑖 ∈ 𝑀 and 𝑛 ∈ 𝑁

At any interval i, when 𝑆𝐼𝑖 ≤ 0, the C scheme directs each EV n to stop charging when 𝑆𝐼𝑖

reaches 0 and also to not proceed with discharging when 𝑆𝐼_𝑖 becomes negative. In other words, when 𝑆𝐼_𝑖 ≤ 0:

∑ 𝑥_𝑖𝑛. 𝐴𝑛_𝑖 = 0,𝑖 ∈ 𝑀 and 𝑛 ∈ 𝑁.

3.2.3 Naïve charging scheme (N scheme)

In this problem the EVs are charged at a constant rate till it satisfies the charging requirements. The parameters used for this scheme are same as for the C scheme. However, we do not need to perform an optimization because every parameter is known to the aggregator. Unlike the other two schemes, the charging powers of EVs for this scheme are constant and are based on the charging requirements of EV owners. Thus, an algorithmic procedure is required to schedule the EVs with constant charging power rates. These charging power rates are used to calculate the cost using the overall cost function (9). Similar to C scheme, EVs are only charged here. The objective function is same as in C scheme and constraints are also almost the same as it’s the only way to fairly compare these schemes.

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4. Numerical analysis

For numerical analysis, we first compare the performance of naive (N) scheme and charging (C) scheme in terms of the overall cost and difference between SI and charged energy. Next, we show the results of the optimization performed in the charging and discharging (CD) scheme for all scenarios. We also examine whether aggregator`s contract terms with the EV owners and System operators are met in all three schemes. The objective functions (8) and (9) are of quadratic and linear form and constraints are all linear. As explained by He et al (2012), these forms of optimization problems are the type of convex optimization, that can be solved by MATLAB based CVX (Grant and Boyd,2010), a solver specifically designed for convex optimization problem. Thus, for CD and C schemes, we use CVX. For N scheme, we perform an algorithmic coding in MATLAB.

Based on the insights from literature, we consider four scenarios to compare the performance of these charging schemes. The scenarios are as follows:

1. Scenario 1:

The system imbalance (SI) here is of a holiday and each EV owner requires unique desired energy at the end time of its availability to charge/discharge. The SI here not only depends on the production of renewables but also the consumption pattern of units connected to grid. For instance, most offices are closed on holidays. Thus, the peaks and lows in energy consumption on this day will vary significantly when compared to a working day.

2. Scenario 2:

Here, the SI is of a holiday but each EV owner requires same desired energy at the end time of its availability. We consider that each EV must charge the battery to its capacity. Thus, final energy charged by EVs would be more here.

3. Scenario 3:

The SI is of a working day and each EV owner requires unique desired energy at the end. On this day as well, the peaks and lows in energy consumption pattern can vary significantly within the day.

4. Scenario 4:

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16 We consider energy system imbalance of Belgium`s electric grid, ELIA. The period of 24 hours is divided in 24 intervals of length 1 hour each. System imbalance SI can be positive or negative where positive SI indicates energy production surplus and negative SI indicates energy consumption surplus. The SI values are scaled down from megawatt (MW) to kilowatt (kW) with a factor of [1/1000]. The charging cost factors are set as: 𝑓₁=20−3_{EUR/kWh and}

𝑓₂=15−3_{EUR/kW/kWh and} _𝑓

3=75−3EUR/kW. Starting time 𝑡𝑠𝑛 is randomly distributed

between [1,20]. The charging period of EVs is randomly set between [4 ,12] hours. Thus, end time 𝑡_𝑒𝑛 is randomly distributed between [𝑡_𝑠𝑛+ 4, 𝑡_𝑠𝑛 + 12] hours. Energy at start 𝐸_𝑠𝑛 for each EV is randomly distributed between [16 x 0.1,16 x 0.7] kWh. We consider the EV battery`s charging/discharging efficiency, β as 90%. This means that any EV`s new battery capacity becomes 16 x β = 14.40 kWh. Desired energy at end 𝐸𝑒𝑛 for each EV varies per scenario. See

next section for details. Maximum charging rate (Pmax) is 5 kW and maximum discharging rate

(- Pmax) is -5 kW.  is randomly distributed between [0.9,1.0]. Thus, for each EV, the desired

energy at the end becomes [0.9 x 14.40, 1.0 x 14.40] kWh. 0 ≥ 𝛾 ≥ 1. Number of EVs as per default setting is 200.

4.1 Sensitivity Analysis

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17 Fig 1. System imbalance on a working day Fig 2. System imbalance on a holiday

4.1.1 C scheme vs N scheme for Scenario 1

In the first scenario the desired energy at the end for each EV is considered to be different. For each EV we randomly pick ɣ from [0.9,1.0]. Thus, 𝐸𝑒𝑛 for each EV is distributed randomly

between [0.9×14.40, 1×14.40] kWh. SI for this scenario is that of a holiday (see fig. 2).

Fig. 3. Cost comparison in C scheme vs N scheme scenario 1

From the above figure, we can observe the sensitivity of all charging schemes` costs to the number of EVs. When the number of EVs are 100, 200, 300 and 400, Charging only (C) scheme outperforms Naïve (N) scheme by 18.02%,36.72 %, 43.49% and 49.08% respectively. For 100, 200, 300 and 400 EVs, in each scheme, the total energy charged by EVs remains the same. In other words, energy charged by each EV remains the same. This is illustrated in table 1 of Appendix I. 0 100 200 300 400 500 600 100 200 300 400 Ov era ll cos t (€) Number of EVs

Scenario 1 overall cost comparison

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In this scenario the desired energy at the end for each EV is considered to be its capacity of 14.40 kWh or simply, for each EV, ɣ is set as 1. SI for this scenario is also of a holiday (see fig. 2).

Fig. 4. Cost comparison in C scheme vs N scheme: scenario 2

Here we can observe similar cost trends to Scenario 1. For example, for number of EVs as 100 ,200, 300 and 400, the C scheme outperforms N scheme by 6.28%, 38.40%, 44.68 % and 49.67% respectively. The energy charged by EVs in all cases are illustrated in table 2 of Appendix I. In both scenarios 1 and 2, the overall costs for CD scheme are always higher than C scheme and N scheme.

In this scenario the desired energy at the end for each EV is considered to be different. Similar to Scenario 1, we randomly pick ɣ from [0.9,1.0]. Thus, 𝐸_𝑒𝑛 for each EV is randomly distributed value between [0.9×14.40, 1×14.40] kWh. The SI for this scenario is different than scenario 1 and 2 and it represents imbalance of a working day (see fig. 1).

0 100 200 300 400 500 600 700 800 100 200 300 400 Ov era ll cos t (€) Number of EVs

Scenario 2 overall cost comparison

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19 Fig. 5 Cost comparison in C scheme vs N scheme: scenario 3

In this case for 100, 200, 300 and 400 EVs, the C scheme outperforms N scheme by 4.91%, 10.41%, 20.42% and 32.67% respectively. The total energy charged in each scheme for all EVs is still the same as that in Scenario 1 for all number of EVs (see table 1 and 3 in appendix I). Thus, charging costs remain the same in scenario 1and 3 but overall costs vary due to the influence of SI cost.

In this scenario the desired energy at the end for each EV is considered to be its capacity of 14.40 kWh or simply, for each EV, ɣ is set as 1. SI for this scenario is of a Sunday (see fig. 1).

Fig. 6 Cost comparison in C scheme vs N scheme: scenario 4 0 100 200 300 400 500 600 100 200 300 400 Ov era ll cos t (€) Number of EVs

Scenario 3 overall cost comparison

C scheme N scheme 0 100 200 300 400 500 600 700 100 200 300 400 Ov era ll cos t (€) Number of EVs

Scenario 4 overall cost comparison

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20 As we can see in table 6, for 100, 200, 300 and 400 EVs the overall cost of N scheme is more than cost of C scheme by 5.53%, 12.09%, 24.52% and 30.68% respectively. Note that the total energy charged by EVs in this scenario equals to that in scenario 2 (see Table 2 and 4 of appendix I). This implies charging cost is same for both scenarios but overall costs vary under the influence of SI cost.

4.1.5 CD scheme cost comparison for all scenarios

Here we compare the overall costs obtained by CD scheme optimization in all scenarios. The comparison is illustrated in Fig. 7.

For each scenario, when the number of EVs are increased, the overall cost increases. Even though SI cost decreases, as number of EVs increase, the charging cost significantly increases which leads to overall cost increase. Another observation is that the variation in costs when SI is of a working day and that of a holiday. Precisely, scenarios with a working day SI have significantly lesser costs than scenarios with a holiday SI.

Fig. 7 Cost comparison in CD scheme for all scenarios

4.2 Aggregator contract performance

In all the above scenarios, the model developed in this study meets the aggregator contract terms with the system operator and each EV owner. For instance, in each interval the difference between SI and energy charged by EVs is minimized by the optimization schemes. We show the SI vs charged energy comparisons for 200 EVs in Fig. 8 and Fig 9. Note that Fig. 8 shows the energy comparison for Scenario 2 and Fig. 9 shows energy comparison for Scenario 3.

0 500 1,000 1,500 100 200 300 400 Ov era ll cos t (€) Number of EVs

CD scheme cost comparison

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21 Moreover, the EV owner charging requirements 𝑡_𝑠𝑛, 𝑡_𝑒𝑛 ,𝐸_𝑠𝑛and 𝐸_𝑒𝑛 are all satisfied by the proposed scheme. For instance, in Fig. 10 the energy evolution of a randomly chosen EV can be observed. The energy (𝐸_𝑠75_{) at start time (𝑡}

𝑠75) of availability and energy (𝐸𝑒75) at the end

time (𝑡𝑒75) of availability for 75th EV is same in all schemes even though the charging profiles

in all schemes are different. In CD scheme, EV 75 charges its battery in intervals 11-14, discharges in 15-18 and again charges in 19-21 (see fig. 10). In C scheme, EV 75 does not charge the battery till 12th_{interval after which it charges from 13-15 and stops charging again}

till interval 17. From 18-21 it charges again. Lastly, in N scheme, the EV charges at constant rate till it reaches the desired end energy 𝐸𝑒75. The corresponding charging and discharging

power rates for EV 75 in these schemes are illustrated in figure 11.

Fig. 8 System imbalance vs energy charged by Fig. 9 System imbalance vs energy charged 200 EVs (scenario 2) by 200 EVs (scenario 3)

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5. Discussion

Almost all papers with similar researches show that the overall costs of their proposed charging scheme are slightly lower than the cost of naïve (N) charging scheme. In this research, charging only (C) scheme optimizes the charging power rates of EVs that are available to charge energy from grid. On the contrary, in N scheme, the charging power rates of EVs are known to aggregator and are constant. In this study, for all scenarios, the C scheme performs better than N scheme in terms of overall costs and in terms of minimizing the power system imbalance (SI). In all the scenarios performed in this study, as the number of EVs increase the percent cost difference between C scheme and N scheme increases. Therefore, our model performs better when the number of available EVs to charge increase. This is especially useful for the aggregator as number of EVs available for charging scheduling and system imbalance due to renewables are going to increase in future.

Another managerial implication is that our optimization model is robust to some changes in objective function. The SI cost can be modified as per the electricity price infrastructure in that geographical region. As the norms and tariffs of SI pricing varies per region or country, these changes can be incorporated easily. Moreover, computational effort and time is less. The maximum time required for an optimization in all the scenarios was 180.76 seconds. Hence, it is a practical, simple and efficient charging/discharging scheme.

The overall costs of charging and discharging (CD) scheme are always higher than C scheme for all scenarios. However, this was expected because in the CD problem formulation we consider the charging and discharging costs whereas in C scheme we only consider charging costs. Similarly, it N scheme performs better than CD scheme. The advantage of CD scheme is that it minimizes SI even when it is negative. Thus, as explained earlier, with future power grids this would be a viable EV charging and discharging scheduling scheme.

5.1 Limitations

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6. Conclusion

The aim of this thesis is to build a charging/discharging schedule for EVs that charge their batteries from the grid. The EV owners are, a day ahead, in contractual agreement with the aggregator. The contract states all charging requirements of EV owners such as start time of EV`s availability to charge, end time of EV`s availability, EV battery energy at start time, desired EV battery energy at the end time. System operators` contract with aggregator contains historical data of power system imbalance (SI) information. SI is the difference between the power produced by renewables and traditional sources and power consumption by the units connected to grid. With this knowledge, aggregator creates a schedule for the day to charge the contracted EVs to their desired energy level.

We develop three mathematical models for scheduling the charging of EVs. Two of these models consider only charging of EV battery from the grid (N scheme and C scheme) and other considers charging and discharging of EVs (CD scheme). The objectives of these models are to minimize charging costs and SI cost. In CD and C schemes, the EV`s charging power rates are optimized whereas in N scheme, EV`s charging power rates are constant. Four scenarios are developed. These scenarios are based on two SI patterns, one for a working day and the other for a holiday and for these two SI patterns, we consider the desired energy level by EV owners at the end time is either same or distinct for each owner. A sensitivity analysis is performed where the costs of C scheme and N scheme are compared for four scenarios. For each scenario overall costs are obtained by keeping number of EVs as 100,200,300 and 400. C scheme performs better than N scheme in all scenarios. Along with the overall costs, we shoe that C scheme also minimizes the system imbalance for the day.

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7. References

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Appendix

Appendix I

Table 1 Scenario 1 analysis Number of

EVs

CD scheme C scheme N scheme

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26 Table 2 Scenario 2 Analysis

Table 3 Scenario 3 Analysis Number of

EVs

Overall Cost Total Energy charged (kWh) Overall Cost Total Energy charged (kWh) Overall Cost Total Energy charged (kWh) 100 228,2435 842,38 61,0291 842,38 73,3644 842,38 200 585,4068 1695,1 128,8750 1695,1 178,3666 1695,1 300 888,7844 2628,4 256,5513 2628,4 371,1843 2628,4 400 1355,4 3553,4 451,7 3553,4 676,1 3553,4 Number of EVs

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27 Table 4 Scenario 4 Analysis

400 624,7709 3239.4 379,7710 3239,4 503.8352 3239,4

Number of EVs