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Distributional Locational Marginal Pricing Based Optimization for Electric Vehicle Charging Management

Electric power generation, transmission, and distribution systems are attracting a large amount of interest from researchers with the development of the smart grid technologies. A smart grid aims at effective control and conditioning of the distribution of electricity. Pricing signal based distribution system are seen as one of the novel ways to achieve control in a smart grid. In our work, we propose to use a pricing signal modeled after the locational marginal price in the transmission system to locally provide price data to the users. The formulation and implementation of the distributional locational marginal price (DLMP) are achieved to develop a fair pricing model. The work is further practically implemented to a grid with Electric Vehicles in addition to the conventional load. The increasing popularity of EVs because of their ability to reduce greenhouse gas (GHG) emissions will pose a greater challenge in terms of demand on the power grid. We propose to use the DLMP modeling to alleviate congestion in the power grid and develop an optimal charging schedule for EVs.


Smart Grids
The most important challenge of a power system is to operate itself optimally at the true demand-supply schedule in order to minimise its losses. Historically, power systems have utilised highly controlled supply systems usually from generation grids to manage deviations in the load [1]. However, owing to the recent awareness about climate and environment issues and the spurge in integration of information technology with power systems, "Smart Grid" technologies have started to enter the market. Electric power generation, transmission and distribution systems are attracting a large amount of interest from researchers as well as corporates alike with the development of the smart grid technologies. Smart Grids aim to utilise the extensive information from users to make power systems reliable, efficient and much more environment friendly [2].Smart Grids aim to act as inter-faces between generation system and consumers to coordinate the needs of the users with the capabilities of the generators and optimise the utilisation of resources [1,2].
The envisioned smart grid aims to have the following features: • Integration of Renewable Resources:Although alternate sources of energy have started penetrating in certain areas, complete implementation of these resources will require a more flexible management system [2]. Smart Grids utilise effective Demand-Side Management(DSM), more efficient use of transmission and intelligent storage to allow for the use of renewable generation in distribution systems. Also, the pricing infrastructure developed is such that it encourages the utilisation of renewable energy over conventional sources. [3] • Demand-Side ManagementThe problems associated with the integration of distributed renewable energy resources is dealt by a lot of researchers but an alternate approach to solving the problem is the involvement of The last decade of the 20th century has seen growing concerns been raised regarding energy independence and climate change. Governments and other institutional organisations are working on developing policies to address this issue. The major focus is the consumption of fossil fuels and vehicular emission of green-house gases. In the U.S., the fuel economy standards on auto-mobiles have been set higher numerous times in the past ten years [8]. These concerns over carbon footprints have promoted the development of electric vehicles (EVs) as an alternative to petrol and diesel powered transport [9].Another major attraction for electric vehicles is the large scale integration of renewable energy into the power system. EVs are an attractive option to balance the uncertainties introduced in the system due to renewable energy [10].The number of electric vehicles on the road increased from zero in 1991 to two million in 2010 according to an Energy Foundation study [8]. Consequently, transportation economists have projected that EVs will comprise 64 to 86% of light vehicle sales in the US by 2030 [11].
Although it has immense benefits, the large-scale integration of EVs into the power grid poses serious threats towards the stability of the grid. As suggested by Rahman and Shrestha, large scale deployment of EVs will eventually mean increased demand, especially at peak hours, impacting the reliability of the system. Demand patterns inducing flows that exceed the design limits to lead to congestion in the grid. Congestion from EVs can be observed at the medium voltage (MV) level, as demonstrated by a number of studies. [12,13,14] Hence, large scale uncoordinated charging of EVs can significantly affect network security [15]. Power quality degradation is also expected from the irregular demands] [16]. These factors make it necessary to establish a coordinated standard for charging of EVs. The immense potential for smart grid technology lies in the various opportunities for control and coordination in power systems at the consumer end that have yet not been explored at all. Although a lot of research has been done recently in this regard, most of the research focuses on electronic controllers such as distribution class electronic switches and Solid State Transformers (SST) [17,18].
In our work, we have developed a pricing infrastructure which can be used as a control feature of the smart grid. These pricing signals in turn will act as control signals to affect demand response, renewable energy penetration and congestion management. We propose to use a pricing signal modelled after the locational marginal price in transmission system to locally provide price data to the users. On a practical scale, we aim to utilise the model developed in Part-1 to develop a price based modelling approach to alleviate congestion induced by electric vehicles. Price based congestion management will ideally operate by the mechanism of generation re dispatch in response to the price signals.

Literature Review
Before we move towards developing the model to generate pricing signals, we shall look into the structure of the day-ahead and real-time electricity market and develop the theoretical concept of DLMP.

Electricity Market
The fact that electricity is a commodity that cannot be stored economically allows for the development of an electricity market place. To facilitate trade among power producers and consumers, an e-commerce marketplace called pool is developed and a framework is made to allow different types of contracts [19]. The market allows short-term trades that deliver electricity for the next day by utilising day-ahead pricing and intra-day markets that deliver electricity in the next 15 minutes [20].

Real Time Pricing
Throughout a day, there are certain hours that show low demand whereas we observe huge spikes in some places. Real Time price is an electricity rate that fluctuates according to the varying costs and amounts of generation and the demand from the market. The main aim is to ensure that these real time prices reflect actual wholesale prices at different times of the day and the users are incentivised to shift loads to (o -peak) hours and hence ensure that the (peak-to-average ratio) is reduced [2,21,22]. An auction is carried out in the market every hour to calculate the market clearing price and consumption bids are taken from all end users.The concept of spot pricing for utilities was suggested by William Vickrey [23] in 1971.His idea was further developed by Bhon, Caramanis, and Schweppe [24], in 1984. They proposed a model which considered the marginal price of energy, losses, and congestion to determine real time prices. The dynamics of day ahead market is essential as it acts as the base price for the calculation of the real time prices. In a day-ahead market, different prices are quoted for each hour of the next day. The average is then taken to be the average of the day. These prices are considered to be the clearing price for the amount bid at that time. Any deviations from this bid can be bought or sold by the consumer at the real time market clearing price the next day [19,20].

Locational Marginal Pricing
The Locational Marginal Price (LMP) represents the marginal cost to supply the next megawatt hour increment in power at the transmission bus without violating any security limits [4]. The LMP is calculated in the transmission system at every node and is calculated every 5 minutes on a real time market. According to Schweppe [25] the locational marginal price represents the truest cost of electricity. The major factors that affect the LMP at a bus are the generator bid prices, the amount of congestion experienced by the transmission system elements and the losses in the system [25]. Unlike the transmission system where nodal pricing is regularly used, distribution systems still make use of at pricing system leading to market inefficiency. Distribution Locational Marginal Price (DLMP) is defined as the real-time cost of providing one extra megawatt of energy at any point on the distribution system [2]. The implementation of such a model in the distribution system would lower total electricity costs for the end-user, mitigate the need to increase line load limits and also act as a perfect support mechanism to allow the utilisation of renewable generation [26,4].

Factors Contributing To DLMP
The DLMP at any node is calculated by taking into account the transmission LMP and the distribution objectives and costs. Equation 1 is one of the formulations of DLMP.
The DLMP can be broken down into four major components. These include: • Marginal Energy Cost (MEC): This refers to the cost of energy at each substation and is calculated by the LM Pi at each substation and the contribution i of each substation.
• Marginal Loss Cost (MLC): The 2 nd nd term refers to the cost of I 2 R losses in the branch k in the power system over an hour.
• Renewable Encouragement Factor: The 3 rd term allows easy implementation of renewable energy and P denotes the cost of renewable generation over one hour.
• Marginal Congestion Cost (MCC): The 4 th applies the penalty for the congestion in the line and penalises for high current loading

Demand Side Management in EV Charging
Demand Side Management is the modification of consumer demand through various methods such as financial implications to improve service and reduce consumption. Coordinated EV charging can be broadly classified on the basis of the decision making entity into centralized and distributed methods [27]. In case of centralized decision making, there exists large aggregators who manage multiple EVs. The EVs send their availability schedule and battery specifications to the aggregator who, on the basis of its profits [28,29,30], EV power costs [?], [31], [32] and reducing peak-to-average ratios [33]. On the other hand, in the distributed system, the flow of information is from the aggregator to the EVs who utilise their own understanding of the market to develop its charging plans. A mutual agreement needs to reached among all the parties as each party tries to minimise its individual costs [34] .Various optimisation techniques have been previously utilised in DSM literature to solve energy consumption scheduling problems. These include mixed-integer linear programming as done by Zhang et al. [35], quadratic programming for EV scheduling as done by Vandel et al. [36] and mixedinteger quadratic programming to allow for decentralised demand side management [37].
Most studies concentrate on mitigating the various impacts of EV charging load such as increases in peak load, power losses and decreased voltage.

Optimal EV Charging Management
Ruoyang Li et al [38] have theoretically developed an aggregator based EV charging management system to mitigate the negative impacts of EV loading on the grid. It considers the EV aggregators to be pro t seeking entities providing charging solutions to the EVs. The EV aggregators are price-takers and do not control the market. It thus proposes to forecast EV demands on the basis of all aggregators minimizing their individual costs whereas the conventional demand will be forecasted according to day-ahead prices. Figure 1 refers to the model developed to alleviate congestion caused to obtain the optimal solution for each EV aggregator. However, it can be easily proved that the efficient allocation of EV charging x i,t * of the DSO problem is optimal for each EV aggregator under the DLMPs as well. As a corollary, the efficient allocation of the DSO problem x i,t * , c i,t * can be achieved in a decentralized system under the DLMPs.

Karush-Kuhn-Tucker Conditions
The modelling and problem formulation utilises the Karush-Kuhn-Tucker (KKT) conditions (also known as the KuhnTucker conditions) to achieve optimization solution. First published in 1951 [41], the KKT conditions are first order necessary conditions for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied. KKT is essential in mathematical optimization as it generalises the method of Lagrange multi-pliers by allowing inequality constraints. In general, many optimization algorithms can be interpreted as methods for numerically solving the KKT system of equations. Through literature review, we can see the importance of LMP signals in distribution system and its use as a control signal in power systems. We have analysed the factors affecting the DLMP calculations and developed a formulation that takes care of all objectives and costs. Moreover, a detailed analysis of the demand side management techniques that have been previously used in EV charging specifically discussing different methods of decision making to decide EV charging schedule, has been done. A theoretical algorithm for coordinated EV charging from a centralized as well as decentralized perspective is discussed. In the subsequent chapters, we shall develop an optimisation problem to calculate DLMP and use it as a congestion alleviation tool.

Problem Formulation
In this section we'll formulate the problem that we have solved to determine the DLMP in a system. The cost objective function, constraints and it's relationship with DLMP is shown in the upcoming sections. We have utilised the AC-OPF (Optimal Power Flow) model to derive the DLMPs in Part I and DC-OPF model in Part II. The DCOPF is widely used and is considered to be sufficient for LMP calculation due to its computational efficiency and approximation accuracy [39].

Optimization Problem
In part I, AC-OPF optimization problem has an objective function which is to be minimized, whereas in part II, DC-OPF optimization-problem has an objective function which is to be maximized. These minimization and maximization are subject to constraints -both equality and inequality. To summarize we can write that, such that x: vector of dependent variables 3.2 Part 1

Objective Function
As discussed in Section 3.1, F (x, u) is the objective function to be minimized. It is the total cost of energy to supply the demand in the system, given by, where, G : number of grids in the system DG: number of renewable energy sources in the system P Gi : power generated by grid i P DGi : power generated by renewable generator i LM P i : price taken by grid i to deliver P Gi C DGi : price taken by renewable generator to deliever P DGi

Equality Constraints
As discussed in Section 3.1, the optimization problem has certain equality constraints, also known as Power Balance Constraints. They are obtained by balancing the real and reactive power at all the buses in the system, shown as below, where, v i : voltage at bus i i: angle at bus i G k : set of grids connected to bus k DG k : set of renewable generators connected to bus k P k : real power load at bus k Q k : reactive power load at bus k f pk : real power owing out of bus k f qk : reactive power owing out of bus k

Inequality Constraints
The network inequality constraints for an AC-OPF optimization problem can be divided in general in three parts: Line flow limits There is a constraint in the power owing any two buses i and j i.e S ij , given by where,

Voltage Limits
The voltage at all the buses can only vary between a minimum and a maximum value. Hence for any bus k, Power Supply Limits The power supplied by the grid and the renewable generators can also vary only in a given range. Hence for any grid or renewable generator i we can write,

Calculating DLMP
The optimization problem described in the Section 3.1 can be solved using the Lagrangian multipliers The DLMP at each bus k is simply the Lagrange multiplier associated with the real power balance constraint for that bus. [43] 3.3 Part II

Objective Function
As discussed in Section 3.1, F (x; u) is the objective function to be minimized. We are solving the problem from the DSO side and the objective is economic allocation for household demand and EV charging energy. Hence, the DSO objective is to maximise the social surplus, which can be mathematically presented as where, N c : Number of demand nodes T : Number of hours for which system is analysed T i,t :Demand variable at time t and node i P i,t (τ i , t)dτ i,t : Benefits of using demand : System LMP at time t for the node feeding the grid q g,t : Power supplied by the generator to the grid at t c i,t :Conventional household demand at time t at node i The objective function consists of two terms. The first one represents the socialvalue of meeting the conventional demand, given by the area under the demand functions. The bene t of EV demand is not used as it is constant as long as the EV demand is met within the day [40]. The second term rep-resents the total cost of satisfying both the conventional as well as the EV demand. The optimization function can be further decomposed as follows: where, is the social welfare corresponding to the conventional demand.
is the EV charging cost.

Equality Constraints
Energy Balance Constraints The total energy coming into the grid at time t should be same as the energy supplied by the generator. Net active power import/ export at time t for all nodes should be zero. where, here, r i,t is the net active power import/ export at time t at node i

Generation Node Balance Constraint
In this constraint the r i,t of the generator is made equal to the power supplied by the generator to the grid. Mathematically,

Inequality Constraints
Transmission Balance Constraint There is a limit on the power that can flow in each line and hence the constraint is represented as, where, K l : MVA capacity of line l D l,i : Power Transfer Distribution Factor (PTDF) coefficient of line l with respect to a unit injected at node i PTDF can be calculated as follows, here, A: branch-to-node incidence matrix X: diagonal reactance matrix A r : reduced incidence matrix

Charging Energy Limit Constraints
There is a limit on the maximum energy that a EV can take at one time from a charging station where, E i,t : EV battery size

Driving Requirement Constraints
The SOC (State of Charge) of EV batteries at time t for EV i is the sum of its initial SOC and total charging energy up to time period t-1 minus the total driving energy requirement up to time t.
where, D i,t : distance travelled at time t by

Calculating DLMP
The DLMP for all the nodes in the system can be calculated by using the KKT multipliers found by solving the optimization problem defined in 3.4.1 using the constraints defined in 3.4.2 and 3.4.2.
We have theoretically formulated the optimization problem with the objective function and the constraints which were solved to gives us the DLMP at each bus. In the next chapter we will show how this theoretical approach can be modelled in software and can be utilised for calculation of DLMP in practical systems.

Introduction
In the previous chapter in Section 3.2 we theoretically formed the optimization problem that needs to be solved to obtain the DLMP. This chapter deals with creating a model in MATLAB that solves the optimization problem and calculates the DLMP.

Non-Linear Programming Function MATLAB
We have used the MATLAB function f mincon to solve the optimisation problem. f mincon is a non-linear programming solver which gives the minimum of a function with respect to constraints.

Cost Function
We have used a separate function myf un which returns the total cost of the energy summing the cost of energy supplied by each generator. We have used cost functions of two types in our analysis:

Polynomial Cost Function
The fuel cost function for grids or conventional thermal generators is represented by a quadratic polynomial function [44] given as,

Linear Cost Function
The cost for renewable energy is taken to be linear. It is assumed to contain the investment costs of equipments and the operation and management cost of generated energy. The cost function for renewable energy is hence given as Where, m i is the slope of the assumed linear cost function.

Constraints
There are two types of constraints, equality and inequality. These constraints can be further divided as linear or non linear.

Linear Constraints
The voltage limits and power generation limits are linear.

Non-Linear Constraints
The power balance equations and Line flow limits are non linear constraints.
Having summarised the method in which we have modelled the DLMP in the software MATLAB, in the coming two chapters we'll consider a sample system of 12 buses, define its specifications and discuss the results of various cases on it.

System Specifications
In the previous section we have discussed modelling of the optimization problem in MATLAB. We shall utilise the model for two different systems. Firstly, a 12 bus system is studied to calculate DLMP across the system. Secondly, a part of the bus 4 distribution system of RBTS is used to demonstrate the efficacy of DLMP signals as an effective tool to alleviate congestion from EV demand. This chapter deals with the specifications of both the systems and the discusses the data selected for each case 5.1 Part-I

Elements and Circuit Diagram
The sample system we have considered is a 12 bus system as shown in Figure 2[4].

Generators
It has two grids connected to bus 1 and 10. Four renewable generators are connected to buses 2, 5, 9 and 12. We have taken the renewable energy to be wind energy.

Loads
There are loads connected to buses 2, 3, 5, 6, 8, 9, 11 and 12, whose values are decided by the load profiles shown in Figure 6 in Section 6.2 for different cases.

Other Specifications of the Distribution System
Some other specifications of the system like base units and feeder impedance are shown in the Table 1.

Cost Funtions
As discussed in Section 4.3 the costs are of two types.

Grids
The cost of grids is taken to be quadratic polynomial and the coefficients a i , b i and c i for the same are tabulated in Table 2[40]

Renewable Wind Generators
The cost of the renewable wind generators is taken as linear passing through origin and the slope for the same is tabulated in the Table 3. [40]

Penalty in Real Time
In the day ahead market power from the grids as well as the wind generators are decided based on the forecasted load.  But since in the real time market the actual load differs from the forecasted load, some extra power is drawn from either the grid or the wind generators. The grid in real time imposes a penalty i.e. charges a high price if more than scheduled load is drawn. In such a case getting power from the wind generators become cheaper. But there is a limit on the power generated by the wind generator (Section 3.4.3), so when its maximum is reached, the system is forced to draw power from the generator leading to an increase in the cost.

System Outline
A sub part of the Bus 4 distribution network of the RBTS system is used as a case study to test the model.  Table 4 lists the different types of lines along with respective line lengths.

Customer Data
The customer data consists of peak and average loads for each load point. The inverse demand function at each bus is assumed to be linear with a price elasticity of 0.1 as shown in Figure 4. This level of demand price elasticity is consistent with empirical studies.
EV Table 6 summarises the EV data. The battery size of each EV varies according to EV driving requirements. It is assumed that the maximum charging power is 1.15 kW (based on a 5 A, 230 V connection). The energy consumption while

Generator Cost Function
The generator cost function is defined as a quadratic polynomial represented by the equation where Q is the energy demand and P is the cost Figure 5 shows the graph of the generator cost with the net energy demand.
In this section we have defined the system specifications, limits and cost functions. Using these values in the model (Section 4) we'll run different cases which will be discussed in the next Section.

Results and Discussions
This section is divided in two parts -Part I and Part II. For each part we first present the data that is used as input for the system and then we discuss the observations and conclusions for different scenarios.

Part-I
In this section we shall discuss the following scenarios: • Day ahead market To get the day-ahead pricing signal, we shall use the forecast of the load for 24 hours as shown in Figure 6 [48] and the forecast of the wind energy production, as shown in

Results
Day ahead market As discussed in Section 2.1.2, the bidding for the day ahead market happens 24 hours prior to the day when the actual requirement occurs. Since the actual load profile and wind power generation data is not available, we use the forecasted data to predict the production bills by all renewable generators as well as the grid. Figure 8 shows a comparison of the Total Load, total contribution from both the grids and the renewable sources and the value of the total cost function over the period of 24 hours. From Figure 8, it can be observed that as the total load in the system increases, the total cost as well as the contribution of wind energy increases. However the contribution of grid remains almost the same. This is beneficial for the system as the renewable sources are able to take care of load variations and we do not have to retrieve extra energy from the grid. The grid supplies up to a predefined rate at a price which is cheaper than renewable sources as discussed in 5.2.2. After that level, it is cheaper to take from the DGs, in order to promote these sources of energy.  Figure 9 shows the variation in DLMP at different hours of the day for all the buses. It is very interesting to see that the DLMP is highest at t=14 hours which is a peak load hour and lowest at t=0 hours which is an off peak hour. Hence, it will provide an added incentive to the end-user to shift his loads to o-peak hours. The DLMP at buses 1, 2, 5, 9, 10 Figure 10: Day Ahead: Variation in DLMP with load and wind generation and 12 are lower as compared to the other buses. This is because at these buses we either have a grid or a wind power generator. Since in supplying the loads at these buses the line loss component of DLMP doesn't come into picture, the DLMP is significantly less. Figure 10 is highlights the variations in DLMP with the changing load and wind generation pro les at bus 8. An increase in the DLMP is seen with the increasing load as is expected. Further, when the generation is less, an increase in the DLMP is observed.

Real Time Market
In the real-time market, the bidding is usually done 15 minutes before the energy is required at the load. At this moment, we have actual data relating to the load requirements at all buses and the generation by the renewable sources and hence that is used. The load usually shows a positive diversion from the forecast and is higher. It is this difference in load that has to be auctioned to the production companies. Also, the actual generation by renewable sources might be less as compared to the forecast and in case if the day-ahead bid amount is not produced on one of the buses, the concerned generator has to buy it from other sources. Figure 11 shows a comparison of the Total Load, total contribution from both the grid and the renewable sources and As the load requirements in the real time increase as compared to the day ahead case, the DLMP signals also show a spurge. This is seen in Figure 12. Hence, an added penalty is applied to the loads that divert from the initially bid amount making the process of buying electricity in the real time market expensive. This makes economic sense as well since if there is a load in urgent need of electricity, the price charged can be increased. It is also seen that the change in DLMP is proportional to the change in the load. If major diversions from the initial bid are made, the DLMP will shoot up.Also, as in the day ahead case, the DLMP shows spikes in value when the load increases on the corresponding bus. This behaviour is seen in Figure 13 for the case of bus 8 and can be a major deterrent for peak loading by different consumers.

Real time market with responsive loads
The results for the real time indicate that all the buses are at their maximum loads with Bus 6 having the highest DLMP of 5.21 as well as the maximum load of 0.23 MW. In a responsive system, the load should ideally adjust its consumption with respect to the price signal. The load at Bus 6 is made responsive to see the effect of demand response. It can adjust its load from the maximum value of 0.23MW to 0 MW. The results of the test shows clear instance of demand response. When the load at Bus 6 is made responsive, it responds by reducing its consumption to 0.20 MW. The DLMP at Bus 6 also goes down to 4.40 from 5.21 indicating adequate amount of responsivity in the demand. Hence, we can reduce the demand in peak hours as well (peak-to-average ratio) by implementing demand response in smart grid systems.

Real time market with congestion
In the real time, the power owing in branch between buses 11-12 was seen to be the maximum. Line limits on this branch are changed to 0.1 MW from 3.2 MW to induce congestion on this line. The AC OPF solution illustrates that the injection from the generator at bus 12 decreases as it will no longer be able to supply to bus 11. As a result, the DLMP at Bus 11 sees a spike and goes upto a maximum value of 5.94.Practically this is because the load at Bus 11 cannot take power either from Bus 12 due to the line congestion limit and will have to look upto Bus 0 or Bus 2 for its load requirements. Since, this is an added pressure and will also incur additional losses, DLMP shows a sharp increase in value.

Driving Data
The driving data used in the model is from the Danish National Travel Survey [13]. The Danish driving data were chosen because the driving behavior in Denmark could be representative of the EV users driving pattern. In Denmark, the average driving distance is about 40 km per day. Customers who need to drive a longer distance might not choose to use EVs. To get the profile of driving pattern for the EV customers in our model we assumed a total of 50 EVs. Then using Table A.11 we get the number of EVs travelling a distance of 0-10km, 10-20km and so on every day. Using the EV availability curve from [40], we divide the distance of each EV travels equally among the hours that it is available for travelling. We eventually combined the first 25 EVs under one EV aggregator and the last 25 EVs under the 2nd EV aggregtor. Figure 17 shows the sample EV availability for an EV. At the times which are marked black the EV is travelling and hence cannot be charged but for the times for which it is shown white it is available for charging.

Case-I
Case I represents the most basic case with 2 EV aggregators and the system at 100% EV penetration. Figure 17 and Figure 18 shows the availability of both EV aggregators for this case. Figure 19 depicts the distance travelled by each an initial EV charge of 25%. Figs 6.16 and 6.17 illustrate the use of DLMPs in reducing the impact of EV loads on the grid at any particular hour. Comparing Figure 21 with Figure 20, we see that EV loads are spread out across multiple hours according to the LMPs at each hour instead of charging at any one hour. This is because charging at one hour would lead to a serious spike in the value of DLMP at that hour and therefore leading to instability in the system. As seen, the EVs charge at times with minimum conventional demand so as to minimize their charging costs and maximize social welfare.

Case-II
Case II represents a case with 2 EV aggregators and the system at 100% EV penetration. Figure 17 and Figure 18 shows the availability of both EV aggregators for this case. In both cases the availability is such that it necessitates that the EVs charge during the high price hours. Figure 19 depicts the distance travelled by each EV at every hour when it is on duty. The driving schedule of each EV.The results of Case Study II are shown in Figure 20- Figure 22. The difference in Case II with respect to Case I is evident from Figure 26 and Figure 27. The EV availability schedule is such that the EVs are forced to charge in hours 12, 13 and 14 when the LMPs are generally high. However, even in this case the charging is balanced rather than concentrated at one particular hour.

Case-III
Case III represents a system which has 500% EV penetration in terms of the net EV demand. ?? and Figure 29 illustrate the results for the case. The essential point in this case is the EV loading after DLMPs are used in the system. We see an optimized distribution of EV loading across the 24 hours such as the hours that have low LMP see higher EV load as compared to hours that have high LMP. This is almost constant across time except for hours when the EVs are driving. This ultimately leads to an approximately equitable demand at each hour for the system and hence means a general stability of the system.  Figure 30 and Figure 31 represent the results achieved. As a result of the congestion in the system, there is a visible change in the power used to charge the EVs at time 4 when the systems meets the limits. Fig 6.27 depicts a comparison between the EV demand with and without congestion. The induced congestion is at time 4.As a result this EV charging at time 4 is reduced and in turn compensated at other times of the day. Figure 32 represents the variation of DLMP and LMP with time at node 11. The DLMP follows the LMP at all times except at time 4 when congestion is induced into the system. This is because the system is lossless system and the distribution price is same as the transmission price. However, at time t=4, congestion in the adjoining line causes a spurt in price of demand indicating a model that discourages users to charge at times when the load is high and shift load accordingly to other hours.

Conclusion
This work has two main conclusions. One of the main conclusion of this work is the development of a pricing signal modelled on the lines of Locational Marginal Pricing (LMP) in distribution systems as a control signal. A multi objective optimisation function has been developed for the calculation of the pricing signal. The implementation of the optimisation was developed in MATLAB and a robust model has been developed for the same. Various control objectives such as reduction in cost of energy, reduction of demand in peak hours, improvement of (peak-to-average ratio) and controlled flow in selected lines was successfully achieved. Secondly, a real-life practical implementation of the model is developed for a system with Electric Vehicle based loads in addition to the conventional household demand. The model is developed with the aim to mitigate the risks that the grid might face due to large scale integration of EVs. Again,a non-linear constrained optimization schedule is developed and tested on a part of the Bus 4 RBTS system. The model is seen to have successfully alleviated congestion and can be used as a tool for demand side management and EV charge scheduling.

Future Work
The demand response shows interesting results and can be practically used to reduce peak power demand. The modelling can be used as a suitable benchmark for the study and development of distribution systems in the future smart grid. The future work requires further development of the model, introduction of other control objectives and testing the model on real-time cases. A well developed pricing infrastructure has the potential to work as a competent control mechanism for the future smart grid. For the part II of the model, immense scope lies in adding additional feature of spatial optimization across different Figure 32: DLMPs at node 11 in Case III charging stations. We can modify the system to allow EV users to drive to a different station to specifically charge. Further, a higher degree of precision can be achieved by including practical features such as driving distance, opportunity cost of driving time and spatial difference in EV charging costs. We can also expand the system to a multitude of EV aggregators. Further, the system can be expanded to a micro level by expanding the optimization to individual EV aggregators as well.