PEV Storage in Multi-Bus Scheduling Problems - IEEE Xplore

IEEE TRANSACTIONS ON SMART GRID, VOL. 5, NO. 2, MARCH 2014

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PEV Storage in Multi-Bus Scheduling Problems Ilan Momber, Student Member, IEEE, Germán Morales-España, Student Member, IEEE, Andrés Ramos, and Tomás Gómez, Senior Member, IEEE

Abstract—Modeling electricity storage to address challenges and opportunities of its applications for smart grids requires inter-temporal equalities to keep track of energy content over time. Prevalently, these constraints present crucial modeling elements as to what extent energy storage applications can enhance future electric power systems’ sustainability, reliability, and efficiency. This paper presents a novel and improved mixed-integer linear problem (MILP) formulation for energy storage of plug-in (hybrid) electric vehicles (PEVs) for reserves in power system models. It is based on insights from the field of System Dynamics, in which complex interactions between different elements are studied by means of feedback loops as well as stocks, flows and co-flows. Generalized to a multi-bus system, this formulation includes improvements in the energy balance and surpasses shortcomings in the way existing literature deals with reserve constraints. Tested on the IEEE 14-bus system with realistic PEV mobility patterns, the deterministic results show changes in the scheduling of the units, often referred to as unit commitment (UC). Index Terms—Direct load control, mixed-integer linear programming (MILP), multi-bus unit-commitment (UC), plug-in electric vehicles (PEVs), reserves.

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I. INTRODUCTION

ITHIN THE smart grid paradigm, plug-in electric vehicles (PEVs) have been envisioned to play a role in future power systems equipped with information and communications infrastructure to deal with higher shares of variable renewable in-feed especially from distributed generation sources [1]. With a massive penetration, PEVs may be one of the possible storage applications that can enhance sustainability, reliability, and efficiency. Commonly mentioned is the concept of vehicle-to-grid (V2G), in which PEVs are envisioned to provide reserves to the system. In a seminal contribution, [2] established the basic concept of making use of the power generation capacity installed by existing fleets of vehicles for transportation purposes as a resource for operating electric power systems. Different market designs to procure reserves can be studied in [3]. Along the same lines, [4] approximates revenues for PEVs participating in the German and Swedish context. Evidence is found that markets might get saturated at approximately 3%

Manuscript received February 15, 2013; revised September 06, 2013; accepted October 23, 2013. Date of publication February 10, 2014; date of current version February 14, 2014. The work of I. Momber and G. Morales-España is funded through the European Commission’s Joint Doctorate on Sustainable Energy Technologies and Strategies (SETS). Paper no. TSG-00153-2013. I. Momber and G. Morales-España are with the Institute for Research in Technology (IIT), Comillas University, 28015 Madrid, Spain, and also with the School of Electrical Engineering, Electric Power Systems, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden (e-mail: [email protected]; [email protected]). A. Ramos and T. Gómez are with the Institute for Research in Technology (IIT), Comillas University, 28015 Madrid, Spain (e-mail: {andres.ramos, tomas. gomez}@iit.upcomillas.es). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSG.2013.2290594

PEV penetration. This is the reason for other works, such as [5], to exclude reserves provision from PEV value assessment, which estimates $ 300–400 annual net social welfare benefit transferable for owners engaging in PEV night charging. Research on energy scheduling, i.e., unit commitment (UC), models for determining short term (one day to one week) generation unit’s commitment, startup and shutdown decisions, has been of great interest because it is economically significant for system operation as well as mathematically challenging (large scale, non-convex, and integer variables). These models have been extended to identify impacts of PEVs in the operation of power systems [6]–[11]. Also from a system perspective, however, not directly representing UC, [12] models PEV participation in the oligopolistic electricity market of Germany in Cournot-Nash-Equilibrium to highlight selected instances of market failure and regulation. [13] neither proposes exactly a UC problem, but shows the same problem ownership. A smart charging strategy is provided with locational marginal pricing highlighting transmission network strain in the Swiss-Italian-French interconnection, when PEV charging is uncontrolled. Modeling electricity storage to address the above named challenges and opportunities in smart grids, prevalently requires inter-temporal equalities [6], [9], [11], [14]. In addition to energy balances, models may include a representation of the storage device to provide reserves for power systems [8], [15]–[19] at each hourly scheduling period in order to maintain the system power equilibrium in real time. To overcome some shortcomings in existing literature, this paper presents a novel and improved mixed-integer linear problem (MILP) formulation for energy storage of PEVs for reserves in power system models. Insights from the field of System Dynamics [20], in which complex interactions between different elements are studied by means of feedback loops as well as stocks, flows and co-flows, are applied to energy storage models. The groundings of this related field can be of use in various types of smart grid models for UC [6]–[11], transmission and distribution system operation, self-scheduling [18], [21] or PEV aggregation for retail [17], [19] or to any other problem owner dealing with optimal scheduling of PEV battery charge and discharge [1], [4]–[6], [12], [15], [22]. Generalized to a multi-bus system, this formulation includes improvements in the PEV’s energy balance and surpasses shortcomings in the way existing literature deals with reserve constraints. Tested on the IEEE 14-bus system with realistic PEV mobility patterns between two nodes, the case study highlights the benefits of the proposed modeling approach and shows significant changes in the UC. A. Main Contributions Although there is an abundant variety of technical literature considering the integration of PEVs in power systems, there is a lack of rigorousness when it comes to algebraic representations

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in optimization problems with aggregated batteries. Hence, the principal contributions of this paper are as follows: 1) The presented model of PEV energy storage relies on two inventories, which proves to have several advantages over existing representations of PEVs in UC. 2) The copperplate formulation is extended with the intention to generalize for systems with more than one bus, in which PEVs can be represented connecting at different nodes. An example case with two-bus mobility connected to the IEEE 14-bus test system is presented. 3) The deviation of electric energy consumption for PEV driving is represented as an endogenous variable, which co-optimizes power system operations and PEV-charge scheduling. 4) Finally, a very restrictive yet complete model of PEV storage for reserves is proposed. In this deterministic UC model the capacity and energy constraints are represented with due detail in order to guarantee that any amount of reserve can be provided. B. Paper Organization The subsequent sections of this paper are organized as follows: Section II provides the mathematical formulation of the proposed UC problem. Section III defines the case study in terms of the IEEE 14-bus power system with PEV-mobility inputs for a single-bus as well as a two-bus mobility case. Section IV presents the numerical results of the case study and finally Section V provides a concluding discussion. In addition, the Appendix presents a more compact reformulation for the PEV reserve constraints. II. PROBLEM FORMULATION This section details the proposed formulation. It breaks down into introducing a tight and compact UC formulation in Section II-A, emphasizing the contribution of this work by pointing to shortcomings in existing formulations of the PEV energy balance in Section II-B, accounting for insights from the field of System Dynamics in Section II-C, discussing multi-bus PEV mobility representations in Section II-D and proposing improved PEV reserve constraints in Section II-E.

(1)

Similarly to [14], (1) assumes that the system operator co-optimizes the power system operation costs and PEV-charge scheduling by including the minimization of gasoline fuel cost used for PEV mobility. Hourly time intervals are considered, but it should be noted that the formulation can be easily adapted to handle shorter time periods. The power system requirements include the supply-load balance (2), provision of upwards (3) and downwards (4) secondary reserves, as well as line flow limits (5) [23]. (2) (3) (4)

(5) where energy flows from the grid to the vehicle are indicated by and vice versa . Bold letters refer to symbols in the matrix/vector notation. The constraints for thermal units are based on those proposed in [24] and presented in (7)–(12). This formulation has been proved to be computationally efficient because it is simultaneously tight and compact. The tighter characteristic reduces the search space and the more compact characteristic increases the searching speed with which solvers explore that reduced space. (6) (7)

A. Tight and Compact MILP Formulation for the Thermal UC

(8)

The thermal UC-based market clearing problem seeks to minimize the power system operation costs represented by bids, which are the sum of i) the production cost, ii) start-up cost, iii) shut-down cost and iv) capacity reserve cost. In (1), the cost parameters pertaining to PEVs use are included in the form of adequate compensation above the reservation price (accounting for battery degradation) for the discharged energy v) as well as capacity payments for providing reserves vi) and gasoline fuel costs vii):

(9) (10) (11) (12) The relation between commitment, start-up and shutdown statuses is presented in (6). The sum of the deployed generation and reserves must be between the unit’s maximum (7)–(12) and minimum (9) power output. Constraints (10) and (11) limit the reserve provision by the unit’s capabilities for supplying reserves. Equation (12) ensures that the unit operates within its ramp limits. The formulation used for the case studies also includes the minimum uptime and downtime constraints as well as the variable startup cost (depending on how long the unit has been offline); however, for the sake of brevity, they are not

MOMBER et al.: PEV STORAGE IN MULTI-BUS SCHEDULING PROBLEMS

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shown here. The reader is referred to [24] for a more complete UC formulation. B. Existing, Deficient PEV Energy Balances For PEV impact assessment, many sources rely on generic or qualitative mobility patterns, in which average connected and available capacities range from 80 to 90% at all times. For the energy balances of UC problems, some sources do not even explicitly take mobility patterns into account [11]. In much of the existing literature, if mobility patterns are represented at all, the energy balance or inventory for PEV batteries is modeled as shown in (13), or in a very similar way [9]. Rarely is an explicit energy balance omitted [8]. [6] and [14] have a constraint limiting the use of PEVs when driving, i.e., when being disconnected. In the common formulation, the total energy inventory, for plugged and unplugged PEVs, in period , is: (13) is the total consumption by all PEVs while supposwhere edly being disconnected and driving. The formulation presented in (13) has several shortcomings: above all, there is no clear distinction between connected and disconnected PEVs. To highlight the implications, three extreme but possible instances are given, in which (13) clearly fails: • Suppose connected PEVs were fully charged and the disconnected PEVs were near to being empty, because their batteries had been almost fully depleted in driving. Equation (13) would allow the combined inventory of connected and disconnected PEVs to continue to charge until the aggregation of both independent inventories is at its upper limit. In other words, disconnected PEVs in movement could be charging at the same time as they should be unavailable to the system. • Accordingly, suppose both the connected and disconnected PEV storage was empty. Equation (13) would not detect the infeasibility of continued electric driving, as the decision variables to charge the connected batteries are directly related to the parameter for energy consumption of PEV driving. In short, the model would allow for simultaneous driving as much as the charging capacities permit. • Finally, assume equal numbers of PEVs are connected and disconnected, and the optimal solution for a subset of consecutive hours does not yield any charging or discharging for the connected PEVs, however there is driving by the disconnected vehicles. In this case, (13) would not accurately track the differing energy levels of connected and disconnected vehicles, with all their implications on available reserve capacity and energy for the periods to follow. C. Insights for PEV Storage From System Dynamics To overcome the above named shortcomings in [9], [11], [16], [17], a more rigorous formulation is proposed, in which inventories for connected and disconnected PEVs are clearly distinguished from each other. It is ensured that only PEVs connected to the power system are charged or discharged from the system. On the other hand, it is also guaranteed that those PEVs that are disconnected from the power system are the only PEVs that can consume energy for driving.

Fig. 1. Capacity and vehicle co-flows of dis-/charging and driving as well as disaggregated energy inventories of dis-/connected PEVs.

These two inventories are modeled considering the energy transfers between them, i.e., moving from the state of connection to the state of disconnection and vice versa. [20, pp. 497–509] describes the modeling for system dynamics, where the interaction between different inventories through their flows (co-flows) is discussed in detail. Applied to the specific case of modeling the availability of an aggregated PEV fleet, it is important to highlight that connected and disconnected vehicles from the system transfer energy stored in the vehicle from one inventory to another. The following equations track the total aggregated energy inventory for connected and disconnected PEVs, respectively:

(14) (15) and are the share of vehicles arwhere riving (connecting) and leaving (disconnecting from) the power system, respectively. Though with different signs, the and two common terms in (14) and (15) are which represent the transfer of energy between the two inventories (co-flows). The former represents the total quantity of energy brought to the power system by arriving and connecting PEVs at the end of period ; and the latter term represents the total quantity of energy taken by leaving and disconnecting PEVs at the end of period . The expression stands for the total PEV consumption while disconnected and driving during period , thus interpreting as the energy needed for distance driven non-electrically. Note that following the theory of System Dynamics, (14) and (15) are discretized differential equations that, if summed up, equal the currently common formulation in literature (13). However, disaggregating the two inventories, as in (14) and (15), ensures that if the flows between them is zero, then there is no erroneous interaction leading to potentially unrealistic results. A System Dynamics style stock-flow diagram is provided in Fig. 1.It emphasizes the exogenous (mobility inputs) and endogenous (decision variables related to energy) model elements and their interactions. D. Energy Balance in a Multi-Bus System Constraints (14) and (15) are valid for a single bus copperplate system, in which a PEV can only assume either one of the two states: connected XOR disconnected. With the intention of a generalization to a system with more than one bus and PEVs connecting at different nodes, consider (16) and (17).

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Fig. 2. Capacity and vehicle co-flows in a multi-bus system.

The energy balances for connected PEVs are now described on a per node basis, in which the energy connected in hour at node is obtained by:

(16) where is the share of vehicles arriving at node from ; and is the share of vehicles the trajectory leaving from node for trajectory . These shares at the end of period are defined as: if otherwise

(19)

if

(20)

otherwise Analogously, the energy balances for disconnected PEVs are now described on a trajectory basis. This means that there is one inventory for every possible mobility trajectory from one node to another per period :

(17) where

output capacity. For example, a unit could not produce at the maximum (minimum) output in order to have some power capacity left to provide upwards (downwards) spinning reserves [10], [18], [21]. However, when modeling PEV batteries for providing reserves, apart from these capacity constraints (i.e., the aggregated battery size), the available stored energy is also a limiting parameter. For a correct PEV storage model, energy content must hence also be constrained. To give a straightforward example, if downwards reserve capacity is offered, the batteries cannot be fully charged, otherwise they are not capable of absorbing the energy to provide the reserve being offered and vice versa. There is a general lack of both capacity and energy reserve constraints in the existing literature. References [6] and [14] include reserve energy constraints: a two hour sustained discharge bound accounting for a potential use of reserves in a very short-term. Even in models that include a representation of hydro-storage, these drawbacks have been detected. In short, the work in [10], [18], [21] does not guarantee that the stored energy is sufficient to provide a sustained discharge for spinning reserve. Summarizing, the formulation presented here, proposes a very restrictive yet complete model for PEV storage for reserves, where the capacity and energy constraints are represented with due detail in order to guarantee that any amount of the offered reserve can be provided. 1) Reserve Equality: We define the bids in reserve markets for upwards (19) and downward (20) reserve as follows:

is the electricity consumption in driving trajectory and it is defined by (18)

must Note that the consumption in driving by gasoline always be lower than or equal tp , because both variables and are defined as positive. In the other case where that pure electric vehicles are modeled, (18) must be rewritten as . Fig. 2 gives a graphical illustration of the resulting capacity and vehicle co-flows. E. PEV Reserve Constraints PEVs could potentially provide upward and downward reserves in two different ways depending on whether vehicles are charging or discharging. In existing UC formulations, constraints for reserves are given by a generating unit’s power

where both bids split up in a part that is achieved through energy flows from the grid to the vehicle and vice versa . 2) Charging and Discharging Limits: Equation (21) and (22) ensure that the charges and discharges respectively remain within the boundaries, given by e.g., connection capacity. Note that charging (discharging) can be achieved through energy from day-ahead energy markets, or from downward (upward) reserve markets. However the sum of both energy flows should not exceed the imposed limits. (21) (22) allows where the binary variable decision of discharging either discharging or charging, but never both simultaneously. Similarly, (23) and (24) ensure that the downward (upward) reserve bid cannot exceed the upstream energy flows: (23) (24) With the intention to increase clarity of (23) and (24), the relationship between different reserve types and directions of charge is given in Fig. 3. Assuming that PEV batteries are fast enough in operation, it visualizes how, from a given operation point , down (up) reserves can be provided until the maximum discharge (charge) rate limit . Furthermore, the detailed formulation provided in constraints (19)–(24) can be


Fig. 3. Providing reserves from different operation points

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.

reformulated in a concise way as shown in (28)– (31) of the appendix. 3) Energy Capacity Limits: Equation (25) ensures that the states of charge of disconnected PEVs are kept within the battery’s capacity limits for operation: (25)

Fig. 4. IEEE 14-bus system—PEVs connect to bus 9 and 10.

Equation (25) and (27) restrict the amount of reserve that can be used for system imbalance management: (26) (27) This is a very restrictive formulation and can be interpreted as zero risk-taking since it requires taking into account all the potential states of charges in the connected battery of the previous hours. That is, the constraint becomes more restrictive the higher the number of periods. However, (26) and (27) ensure, that the capacity in the upward and downward reserve markets can actually be fulfilled by the load flow, even in the very unlikely but possible worst case scenario, in which all capacity bids are needed and transformed into energy calls by the system operator. III. CASE STUDY DESCRIPTION This section provides all the input data that describes the case study, to which the above model formulation is applied. First the power system related data and later on the PEV parameters together with the mobility inputs are given. A. Power System Data The IEEE 14-bus system is depicted in Fig. 4. to which the detailed data is provided in [25]. This system has 5 generating units, 20 branches and 11 loads. Simple approximation of geographical distances between the nodes are based on reactance. These lead to placements of the PEV loads in buses 9 and 10. The start-up and shut-down capabilities are assumed equal to the unit’s minimum output . The initial power production of units 1 – 5, prior to the first period of the time span, are (85,50,15,20,10) MW, respectively.

Fig. 5. Inputs: single-bus (9): arrivals, departures [1], PEVs dis-, connected.

B. PEV Parameters and Mobility Input Universal, i.e., non-case-dependent, PEV parameter settings are indicated in Table I. The cost of discharging is deliberately set very high to account for battery degradation. Also, the reserve cost bids are slightly less than those of the most expensive generator. The mobility profiles and corresponding consumption patterns used are generically designed based on intuition. With the intention to make the benefits of this paper’s PEV storage modeling compared to existing formulations most apparent, two cases are distinguished in the following. The mobility inputs are summarized in Figs. 5 and 6. Please note that the figures contain information for total consumption in energy terms (bars with the scale on the left hand y-axis) and fleet sizes in vehicle terms (curves with the scale on the right hand y-axis). 1. Single-Bus Mobility case: A fleet of PEVs are either connected or disconnected to bus 9. The following assumptions are made: on average, trips have a length of 12.18 km, 2.2 kWh of Energy are consumed on such a trip, it takes less than one hour to reconnect, and vehicle inventories are initialized with . Veand arrivals represented hicle flows for departures by a circle markers, follow exactly the probability distribution for starting a trip on a weekday other than Monday or Friday

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TABLE II CASE STUDY SOLVING SUMMARY

Fig. 6. Inputs: PEVs dis-/connected for multi-bus (9,10) mobility. TABLE I PEV PARAMETERS IN CASE STUDY

in Germany as provided in [1]. The remaining parameters are derived by simple arithmetic. Therefore, the amount of vehicles connected marked by squares, and disconnected marked by crosses stays the same, while hourly varying numbers of vehicles always equally disconnect and reconnect as depicted in Fig. 5. This corresponds with , the target consumption of the entire fleet, marked by a bar chart in the background. The two inventories for energy in connected and disconnected batteries are initialized with . 2. Multi-Bus Mobility: The fleet has twice the size of the one in the single-bus case; however, now PEVs are accounted for in the inventories for being connected at either one of the buses and , as well as in any possible trajectory between them:

Each stock has an inflow (circumscribed as “arriving” for and “leaving” for ) as well as an outflow (vice versa: “leaving” for and “arriving” for ). However, since no PEV can leave the system, not all of the flows are distinguishable, or in short, each outflow of one stock represents an inflow to another. Hence for the two-node mobility there are 6 stocks and 12 (6 distinguishable) flows. The reactance based approximation of distance and hence consumption spent on trajectories between different nodes are 30 km and 5.4 kWh on average. The average trip performed and corresponding consumption spent when reconnecting to the same node remains 12.18 km and 2.2 kWh as in the single bus case. Hence, again consumption is proportional to movement with a factor of 0.18 kWh/km. In terms of total vehicle fleet shares, initialization of the inventories are The multi-bus mobility case is depicted in Fig. 6, where the respective data for stocks as well as flows are illustrated. The mobility, split up for commuting between nodes and reconnecting at the same node, is constructed quasi-symmetrically, with again the

total of arrivals and departures following exactly the probability distribution as provided in [1]. Thus, as in the single-bus case the total amount of vehicles, i.e., the sum of connected ( with hollow square markers and with filled square markers) stays the same, while hourly varying numbers of vehicles disconnect and reconnect. The resulting consumption is once again marked by a bar chart in the background, although here different shades distinguish between the four possible trajectories. Furthermore, the two inventories for energy in connected batteries at bus 9 and 10 are initialized at their maximum and minimum capacity, i.e., with and , respectively. Trajectory specific stocks of energy in unplugged vehicles are all initialized at their minimum capacity . IV. RESULTS A. Model Run Procedures Highlighting the functionality of the UC model with PEV formulation, three runs are performed. First the UC is solved, excluding all variables and constraints with regard to PEVs. Then the cases described in Section III, with single-bus mobility and multi-bus mobility are solved. For comparison of the indications regarding solving time, all calculations were performed running GAMS BUILD 24.0.1 with the CPLEX 12.5.0 solver on a 64-bit MS Windows 7 machine with 8.00 GB RAM and an Intel© Core™ i7–3770 CPU clocked at 3.4 GHz. Table II reports the summary of all runs and parts of the solutions. It can be seen that the proposed formulation is rather efficient, i.e., solutions to all runs are found in less than one and a half seconds. Including PEVs and increasing the amount of buses to which the vehicles connect lets the problem size augment, but remains manageable. In the following, the results of the cases which include PEVs are discussed in more detail with regard to energy and vehicle inventories, driving consumption and reserve modeling of the proposed formulation. B. Single-Bus Mobility Case This subsection emphasizes how the model tracks energy quantities and via (14) and (15), as well as shows how avoids infeasibilities the variable for gasoline consumption from transportation demand. Please see Fig. 7. for graphical illustration of these effects. Both connected as well as disconnected batteries, both marked as dashed line without markers and tracked on the left hand y-axis of Fig. 7, have been initialized close to minimum capacity . Therefore, the cheapest means to provide energy for mobility is through charging the connected stock via and transfer the energy, which is precisely what letting flows


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Fig. 8. Outputs: multi-bus connected energies and reserve constraint. Fig. 7. Outputs: single-bus mobility, energy total and per vehicle.

the program finds as the optimal solution. Total energy quantities of connected and disconnected vehicles support this: and disconeven though the number of vehicles connected nected remains the same for all hours, first increases with higher mobility and then decreases accordingly, while the unplugged, traveling energy remains more or less constant. kW for Charging occurs at rates around hours , which is not graphically illustrated here. Vehicles transferring from the two states and tend to equal out the differences in average SOC between the stocks and . This behavior can be observed in the inventories of connected and disconnected vehicles marked by solid marked by solid line with cross, line with square and on a per vehicle basis, tracked on the right hand y-axis of Fig. 7. The average energy connected follows the same profile as the total energy . However, the average disconnected provides more insights. The conenergy per vehicle sumption from driving , as indicated by the bars in Fig. 5, causes a decrease of until a minimum is met at hour . In these hours the flows between the stocks and are not sufficient to compensate for the sharp rise in energy consumption . , not The variable for non-electric energy consumption graphically illustrated here, is greater than zero for exactly those hours , when the average PEV needs to turn to the second fuel, gasoline consumption for driving. It is interesting to note that this happens even though it is heavily penalized in the objective function. Without the variable , the program would obviously be infeasible, even though the exact origin of the infeasibility would be difficult to detect. C. Multi-Bus Mobility Case This subsection emphasizes how the model tracks energy quantities and via (16) and (17), as well as shows how energy can be moved between buses and how the reserve constraints (19) and (20) affect the optimal outcome. Please see Fig. 8. for graphical illustration. The most important take away is that the size of the aggregated battery connected at each of the buses , can vary largely because energy is not only being depleted through driving but even transferred according to the mobility. This fact is illustrated by Fig. 8, which shows how the battery model accurately tracks the connected energies on a per vehicle basis. Furthermore, it shows how the reserve constraints (26) and (27), here indicating the lower and upper bounds by dotted lines with facing triangles, over the course of the day become ever more restrictive to the SOC.

In comparison to the previous case, energy can now be moved from one bus to another, whichever is favorable for the system’s UC given the network saturation. Even though the total amount of vehicles connected does not vary at all, the high (low) SOC at initialization in bus is resolved over the course of the optimization horizon. This can be seen in Fig. 8. The sub-fleet of vehicles in bus 10, marked by the solid line with a cross, is completely depleted at the beginning, but then charged to service the demand in energy for mobility and to stay at a rather high level of SOC until the end of the day. The inflow of highly charged vehicles from commuting mobility supports the rapid increase of energy in the first hours of the day. For level at bus the energy connected at bus 9, marked by the solid line with can be an asterisk, the contrary is true: no charging observed, while the batteries are merely depleted to a medium SOC , servicing demand for mobility at reconnection as indicated by the bars in Fig. 6. Not explicitly depicted in a figure are the resulting charging and discharging , as well as reserve up and down schedules. However, in short, reserves are deployed much more and almost in similar amounts upwards as downwards in bus 9 where no charging occurs, compared to bus 10 where almost all the charging occurs. V. CONCLUSION This paper has presented an optimization model for solving the multi-bus unit commitment problem with plug-in electric vehicles as storage for providing reserves. The tight and compact formulation minimizes thermal system operation cost and includes the discharge of PEV batteries, capacity payments for providing reserves and gasoline fuel costs. Insights from System Dynamics have lead to a representation of the aggregated vehicle batteries, which splits up inventories for connected and disconnected shares of the fleet by means of discretized differential equations. With further rigor, this model has formalized these inter-temporal energy balances as an extension to correctly depict PEV mobility in multi-bus power systems. In a case study of the IEEE 14-bus system with stylized mobility patterns connecting to either one or two buses, the model functionality has been demonstrated. An endogenous variable for deviations from parametrized energy consumption for mobility appropriately resolves infeasibility and can be interpreted as gasoline consumption, when energy constraints do not permit preferred electricity consumption for driving. Both capacity, as well as energy constraints have been taken into account to show a deterministic, zero-risk taking approach for using PEV batteries for reserves.

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3) PEV Parameters:

APPENDIX A REFORMULATION

Energy consumption for PEVs electric driving [kWh/Veh].

Constraints (19)– (24) can be re-formulated as: (28) (29)

PEVs arriving, leaving at the end of Veh./h].

(30)

PEVs connected, disconnected [M Veh.].

(31)

Discharge compensation, gasoline cost .

Note that the variables , , and are not required in the re-formulation (28)– (31). In order to observe that (19)–(24) and (28)–(31) are equivalent, a case where PEVs are charging is analyzed: when then (22) forces and (19) together with (24) become (30). In short, (19), (22) and (24) are equivalent to (28) and (30) when PEVs are charging . Similarly, (20), (21) and (23) are equivalent to (28) and (31) when PEVs are discharging . APPENDIX B NOMENCLATURE Upper-case letters denote input parameters and sets (calligraphic), while lower-case letters are for variables and indexes. Indexes and Sets:

PEV capacity connection incentives . Max. battery rate of discharge, charge [kW /Veh.]. Max., Min. battery state of charge [kWh/Veh.]. Efficiencies: Grid-battery and gasoline [p.u.]. Positive and Continuous Variables: 1) Thermal Units: Power output above

of unit

[MW].

Up, down reserve provided by unit

Time periods in hourly resolution spanning one day.

2) PEVs:

Generating units.

,

[MW].

PEV charging and discharging [kW /Veh.]. Average battery SOC dis-, connected [kWh/Veh.].

Nodes or system buses. Subsets of : PEV location and possible trajectories.

,

Capacity for regulation up/ down [kW /Veh.]. Capacity for regulation as a generator [kW /Veh.].

Transmission lines. Constants: 1) System Requirements: Load demand in hour

[M

Capacity for regulation as a load [kW /Veh.]. [MW].

Gasoline consumption, trajectory [kW /Veh.].

Up, down secondary reserve requirement [MW].

Electricity consumption, trajectory [kW /Veh.].

Flow limit on transmission line [MW].

Binary Variables: 1) Thermal Units and PEVs:

Shift factor vector for line [p.u.]. Bus- gen./load/PEV incidence matrix

.

2) Thermal Unit Parameters: Linear-variable, No-load cost Up, Down Reserve cost Bid Start-up, Shut-down cost

. .

.

Max., Min. power output [MW]. Max. up, down reserve contribution [MW]. Ramp-up, -down rate limit [MW/h]. Start-up, Shut-down capability [MW]. Minimum Up, Down Time [h].

Commitment decision of thermal unit . Start-up, shut-down of thermal unit Discharging decision in hour

. .

ACKNOWLEDGMENT The authors would like to express gratitude towards all partner institutions delivering the SETS joint degree, particularly including the the Universidad Pontificia Comillas (UPCO) in Spain, the Royal Institute of Technology (KTH) in Sweden, and Delft University of Technology (TUDelft) in The Netherlands. Many thanks are directed to the very welcoming colleagues from the department of Electric Power Systems at


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[19] M. A. Ortega-Vazquez, F. Bouffard, and V. Silva, “Electric vehicle aggregator/system operator coordination for charging scheduling and services procurement,” IEEE Trans. Power Syst., vol. 28, no. 2, pp. 1806–1815, May 2013. [20] J. D. Sterman, Business Dynamics: Systems Thinking and Modeling for a Complex World. New York: McGraw-Hill Higher Education, 2000. [21] A. K. Varkani, A. Daraeepour, and H. Monsef, “A new self-scheduling strategy for integrated operation of wind and pumped-storage power plants in power markets,” Appl. Energy, vol. 88, no. 12, pp. 5002–5012, Dec. 2011. [22] T. Sousa, H. Morais, J. Soares, and Z. Vale, “Day-ahead resource scheduling in smart grids considering vehicle-to-grid and network constraints,” Appl. Energy, vol. 96, pp. 183–193, Aug. 2012. [23] M. Shahidehpour, H. Yamin, and Z. Li, Market Operations in Electric Power Systems: Forecasting, Scheduling, and Risk Management, 1st ed. New York: Wiley-IEEE Press, 2002. [24] G. Morales-España, J. M. Latorre, and A. Ramos, “Tight and compact MILP formulation for the thermal unit commitment problem,” IEEE Trans. Power Syst., vol. 28, no. 4, pp. 4897–4908, Nov. 2013. [25] A. Lotfjou, M. Shahidehpour, Y. Fu, and Z. Li, “Security-constrained unit commitment with ac/dc transmission systems,” IEEE Trans. Power Syst., vol. 25, no. 1, pp. 531–542, Feb. 2010. Ilan Momber received the degree of Business & Industrial Engineering from the Karlsruhe Institute of Technology (KIT), Germany in 2010. He is currently pursuing the SETS Joint Doctorate at Universidad Pontificia Comillas (UPCo), Madrid, Spain, and KTH, Stockholm, Sweden. Previously, he worked at the Fraunhofer Institute for Systems and Innovation Research (FhG ISI) in Karlsruhe, Germany, visited the Lawrence Berkeley National Laboratory (LBNL), USA, and the University College London (UCL), U.K. His interests include the regulation as well as technical and economic modeling of power systems with particular focus on plug-in electric vehicles and distributed generation. Germán Morales-España (S’10) received the B.Sc. degree in electrical engineering from the Universidad Industrial de Santander (UIS), Colombia, in 2007 and the M.Sc. degree from the Delft University of Technology (TUDelft), The Netherlands, in 2010. He is now pursuing the (Ph.D.) Erasmus Mundus Joint Doctorate in Sustainable Energy Technologies and Strategies (SETS) hosted by the Universidad Pontificia Comillas (Comillas), Spain; the Royal Institute of Technology, Sweden; and TUDelft, The Netherlands. He is currently an Assistant Researcher at the Institute for Research in Technology (IIT) at Comillas, and he is also a member of the Research Group on Electric Power Systems (GISEL) at the UIS. His areas of interest are power systems operation, economics and reliability, as well as power quality and protective relaying. Andres Ramos received the degree of Electrical Engineering from Universidad Pontificia Comillas, Madrid, Spain, in 1982 and the Ph.D. degree in electrical engineering from Universidad Politécnica de Madrid in 1990. He is a Research Fellow at Instituto de Investigación Tecnológica, Madrid, and a Full Professor at Comillas’ School of Engineering, Madrid, where he has been the Head of the Department of Industrial Organization. His areas of interest include the operation, planning, and economy of power systems and the application of operations research to industrial organization. Tomás Gómez (M’88–SM’09) received the Ph.D. degree in industrial engineering from the Universidad Politécnica, Madrid, Spain, in 1989. He is a Professor of electrical engineering at the Engineering School of Universidad Pontificia Comillas (UPCo), Madrid, Spain. He has broad industrial experience in joint research projects in the field of electric power systems. His areas of interest are the operation and planning of transmission and distribution systems, power quality assessment and regulation, as well as economic and regulatory issues in the electric power sector. Since May 2011 he is on extended leave of absence acting as commissioner at the Spanish regulator: National Energy Commission (CNE).