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An optimization model for electrical vehicles scheduling in a smart grid
Accepted Manuscript An optimization model for electrical vehicles scheduling in a smart grid G. Ferro, F. Laureri, R. Minciardi, M. Robba
PII: DOI: Reference:
S2352-4677(17)30275-8 https://doi.org/10.1016/j.segan.2018.04.002 SEGAN 143
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Sustainable Energy, Grids and Networks
Received date : 6 November 2017 Revised date : 23 March 2018 Accepted date : 16 April 2018 Please cite this article as: G. Ferro, F. Laureri, R. Minciardi, M. Robba, An optimization model for electrical vehicles scheduling in a smart grid, Sustainable Energy, Grids and Networks (2018), https://doi.org/10.1016/j.segan.2018.04.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1
An optimization model for electrical vehicles
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scheduling in a smart grid
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G. Ferro*, F. Laureri*, R. Minciardi*, M. Robba*
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* Department of Informatics, Bioengineering, Robotics and Systems Engineering – DIBRIS, University of Genoa,
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Via Opera Pia 13, I-16145, Genoa, Italy, email: [email protected], [email protected],
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[email protected], [email protected]
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Corresponding author
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Giulio Ferro, Ph.D. Student
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DIBRIS‐Department of Computer Science, Bioengineering, Robotics, and Systems Engineering,
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University of Genoa
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Via Opera Pia 13, 16145 Genoa
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Laboratory: 00390103532804
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Fax. 00390103532154
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Cell. 00393471527053
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c/o Campus Savona
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Lagorio Building
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Via A.Magliotto 2
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17100 Savona
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0039‐01921945139
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Email: [email protected]
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Abstract
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Energy Management Systems (EMSs) are recognized as essential tools for the optimal
28
management of smart grids. However, few of them consider, in their whole complexity, the
29
integration of electrical vehicles (EVs) in smart grids, taking into account the requirements and
30
the time specifications characterizing the service requests. In this paper, attention is focused on
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the formalization of a model for the optimal scheduling of charging of EVs in a smart grid, also
32
considering the vehicle to grid process (i.e., the possibility for the EV to inject power during the
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charging process). In the formalization of an optimization problem for a smart grid, a deferrable
34
demand is considered, which is represented by the charging demand of the set of EVs. The cost to
35
be optimized for the considered problem includes the economic cost of energy
36
production/acquisition (from the main grid) and the cost relevant to the delay in the satisfaction
37
of the customers’ demand (is represented as a tardiness cost). Also, the income coming from the
38
service provided to vehicles is taken into account. The developed model is tested and applied in
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connection with a real case study characterized by a photovoltaic plant, two batteries, power
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production plants that use natural gas as primary energy, and a charging station.
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Keywords: Optimization; Electric Vehicles; Microgrids; Scheduling
44
1. Introduction and literature review
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The integration of renewables, electrical vehicles, and microgrids is getting more and more
46
importance in the last few years, to reduce greenhouse gas emissions, to increase the interactions
47
between different systems, to improve the reliability of the distribution networks. In the recent
48
literature, several articles analyze smart grids and how to integrate intermittent and distributed
49
production and loads, often with reference to specific elements of the grid (renewables, electrical
50
vehicles (EVs), storage systems).
51
The integration of EVs is becoming increasingly attractive for the following reasons. First, it is
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supposed that shortly there will be an increase of such a technology and thus they can result in a
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significant intermittent and distributed load. As a consequence smart charging strategies are
54
necessary to reduce costs and peak loads. Second, EVs can be used as distributed storage
55
systems. An example is reported in [1] where the authors propose the coordination between EVs
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fleets to mitigate energy imbalances caused by renewables, offering an essential service to
57
electrical infrastructure by the use of the vehicle to grid (V2G) configuration. In fact, few of them
58
consider, in their whole complexity, Energy Management Systems (EMSs) [2] that integrate
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smart grids and the facilities for charging of EVs, taking into account the requirements and the
60
time specifications characterizing the service requests of the users relevant to the various EVs.
61
As regards the general theme of smart grid development and application, one can mention
62
Shamshiri et al. [3], who present an overview of smart grids features and highlight the most
63
recent developments. Optimization models are crucial to achieve a reduction in electricity costs
64
and to maximize investments. In this connection, Monteiro et al. [4] develop a management
65
architecture for the distributed micro-grid resource based on a multi-agent simulator that can
66
optimize load scheduling. Nasrolahpour et al. [5] consider a microgrid and determine the optimal
67
scheduling of each generator and some controllable loads during a day including the
68
determination of the amount of energy purchased from the main grid.
69
As one of the critical issues in developing practical approaches towards the attainment of real
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smart cities, different authors have presented detailed models and methods to properly manage
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the scheduling of the charging of EVs and have studied the integration of electric mobility in the
72
grid. Gonzalez et al. [6] review the state of the art related to ancillary service for smart grids
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provide by EVs, and two crucial examples are reported in [7] and [8]. In the former work, a real
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case study is presented and EVs’ active and reactive power optimal injections are determined,
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satisfying both electrical demands and maintaining the distribution grid stability. Instead, the
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authors of [8] discuss the impact of reactive power support performed by EV charging stations in
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a low-voltage distribution grid. Zhao et al. [9] present the results of an investigation for the
78
evaluation of different charging concepts and strategies. Delaimi et al. [10] propose a load
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management solution for coordinating the charging of multiple plug-in electric vehicles in a
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smart grid system. Allard et al. [11] compare the performance of different scenarios of smart
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charging and conventional charging applied to EVs integrated within a grid characterized by a
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high presence of renewables in particular wind energy.
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Electric vehicles are equipped with efficient electric engines, which are powered by high-density
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lithium-ion batteries, which, through chemical reactions, generate electrical current. A typical
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drive for these vehicles is composed of a DC/DC converter and an inverter that feeds the engine
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[12], which allows bi-directional power flow. In fact, when there is a torque request from the
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traction system, the power flow is from the battery to the wheels, whereas during the regenerative
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braking the battery is recharged. The battery life is influenced not only by the itinerary and the
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activity [13] but also by the charging modes. There are different types of charging stations: for
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example, in [14] stations based on PV power are taken into account. Such stations combine a PV
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system with electric vehicles that are also connected to the main grid. In particular, the authors
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present a load scheduling model using data collected in two years. There are many different type
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of charging modes: the most common in smart grids are Smart Charging and Vehicle to Grid
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(V2G). The Smart Charging mode represents a flexible way of charging vehicles in order to
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modulate the feeding of the vehicles according to the grid necessities, for example during the
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peak hours. Instead, in V2G configuration, the power flow is bidirectional (from the grid to the
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vehicle and vice versa). In this way the vehicle storage can help the microgrid to sustain the
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power requirement of the entire system, whenever necessary, or to regulate the voltage at the
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charging station node. Two examples of the study of this configuration are [15] and [16]; in the
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first work authors to investigate the cooperative evaluation of an EMS operation in a building
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considering different features like bidirectional energy flows of an EV fleet, the impact of PV
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uncertainty on EMS operation the effect of prioritization in selling energy to the main grid. The
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second one proposes a model to study the effects of power exchange between the grid and EVs
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on the power system’s demand profile several indices are calculated for various vehicle-to-grid
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power level to estimate the impact of different level of power exchange on system’s reliability.
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However, these types of charging have also some drawbacks, among which the most important is
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the possibility of giving rise to a fluctuating power demand. This can cause problems to the main
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grid such as power imbalances. For this reason, the authors in [17] develop a strategy to regulate
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the vehicle's charging process, introducing different priorities for the various objectives, like the
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achievement of maximum flexibility, the effective management of the power flows in order to
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smooth the demand curve and increase the system's power quality. In [18] attention is focused on
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the active contribution (i.e., power modulation via different charging modes) of the electric
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vehicles in the grid optimization, considering the V2G mode. Differently, in [19] a mixed integer
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linear programming problem is formulated, with the objective of scheduling the charging of the
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vehicles, minimizing the energy costs and taking into account reliability and stability issues,
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considered in connection with both the microgrid and the customers (the vehicles). Wang et al.
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[20] study the configuration of a system wherein EVs are connected to the smart grid as mobile
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distributed energy storage. Bracco et al. [21] present an overall EMS, based on a dynamic
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optimization model to minimize operating costs and carbon emissions considering different
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technologies as micro gas turbines, photovoltaic, concentrating solar systems, and storage based
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on different battery technologies.
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The choice, among different modes of charging influences not only the statement of the charging
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scheduling problems, but also the routing of electrical vehicles. For instance, in [22] a double
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layer smart charging algorithm for electrical vehicles (EVs) is presented, having the objective of
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allowing the single vehicle to reach the more suitable recharging station, limiting the transformer
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load and the total energy costs. In [23] the authors study different routing strategies to smooth
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microgrid power fluctuations, caused by intermittent renewables and load, ensuring at the same
128
time the grid power quality and the optimality of logistics services.
129
In the literature, there are also examples of stochastic approaches to model the integration
130
between electric vehicles and power networks. For example in [24], a real case study is described
131
(namely, the distribution network in Zaragoza, Spain). In particular and the authors formulate a
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stochastic problem for energy resource scheduling including uncertainties of renewable sources,
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electric vehicles, and market prices. In
134
objective of maximizing the profit by charging the plug-in electric vehicles within the lowest
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price time intervals, considering the participation in the ancillary services market as an additional
136
source of income. A stochastic problem formulation for V2G technology is also present in [26],
137
where also uncertainties in the power flow are taken into account. Instead, in [27] attention is
138
focused on energy aggregators that can help the management of multiple connected devices . In
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particular, the authors develop a stochastic model for one-day-ahead energy resource scheduling,
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integrated with the dynamic electricity pricing for electrical vehicles, to cope with the challenges
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related to the demand and renewable sources uncertainties. In [28] the authors propose the use of
142
three metaheuristic optimization techniques, in order to solve the plug-in electric vehicle charging
143
coordination problem in electrical distribution systems. Similar approaches are adopted in [29]
144
and [30] using genetic algorithms and particle swarm optimization.
[25] a scheduling problem is considered, with the
145
In the present paper, attention is focused on the formalization of a model for the optimal
146
scheduling of charging of EVs in a smart grid. Concerning the existing literature, the original
147
contribution of this paper is the formalization of an optimization problem for a smart grid with a
148
deferrable demand, represented by the charging demand of the set of electrical vehicles. The cost
149
to be optimized for the considered problem includes the economic cost of energy
150
production/acquisition (from the main grid) and the cost relevant to the delay in the satisfaction
151
of the customers' demand. The latter cost is represented as a tardiness cost. Besides, the income
152
deriving from the service provided to the vehicle is taken into account. Note that [31], where the
153
integration of plug-in EVs and renewable production plants is considered. However, the model
154
proposed in this paper goes beyond the one in [31] as the scheduling problem of the vehicle’s
155
charging is explicty considered.
156
To be more specific, the system considered in this paper is characterized by the presence of a
157
renewable energy source (photovoltaic plant), energy storage facilities, power production plants
158
that use natural gas as primary energy, and a recharging station for EVs. The decision variables
159
include the active powers exchanged in the whole system, binary variables that describe the
160
interaction between the EV and the charging station and the recharging unit price of each
161
vehicle.In particular,the last decision variables represent one of the novelties introduced in this
162
paper.
163
The rest of the paper is organized as follows. In the next section, the considered system is
164
described in detail, highlighting its specifying features and introducing the optimization problem
165
to be solved. In the third section, the optimization problem is formalized, and in section 4 the
166
application of the proposed approach to a case study is presented. Some concluding remarks are
167
finally provided in the last section.
168
2. The considered physical system
169 170
The decision model presented in this paper is based on the fundamental concepts and tools that
171
can be found in the literature for microgrids’ operational management (like in [32]). Even though
172
a new formalization is presented for the objective function and the constraints (including a
173
demand-response model for the electrical vehicles charging). In the overall formalization, it is
174
supposed that power losses can be neglected, and thus the microgrid is considered as a single
175
node. Each component of the system is represented as an active power injection in the total power
176
balance, following the active sign convention (the sign of the power is positive when itis
177
generated).
178
The considered system consists of a microgrid composed of the following elements (see Figure
179
1):
180
a) One or more renewable energy sources (considered as a unique entity in the following, for
181
the sake of simplicity); it is assumed that this renewable energy source is intermittent and
182
not controllable and that the energy is generated without cost;
183
b) One or more non-renewable high-efficiency energy sources, whose energy production is
184
costly; in particular, two internal combustion engines are considered, with different
185
maximum rated power;
186 187 188 189
c) One or more electrical storage elements; in particular, the presence of two storage systems is assumed; d) A connection (PCC–Point of Common Coupling) with the main grid, which guarantees the bidirectional power flow between the microgrid and the main gird;
190
e) A single facility (charging station) for the electrical charge of vehicles; it is assumed that
191
the power flow between the microgrid and the charging vehicle may be bidirectional. The
192
charging station may serve a single vehicle at a time, and the charging process is not pre-
193
emptive, that is, a given vehicle cannot be disconnected from the charging station and
194
reconnected later;
195
f) An electric load characterized by a given pattern over the considered time horizon.
\
196
197 198 199
Figure 11. Microgrid d design.
200 201
The objeective functtion that will w be connsidered in the decisio on problem m statementt includes
202
different terms. Suchh terms incclude the coost due to the t energy production from non-renewable
203
sources annd the energgy bought/ssold from/too the main grid. g Also, a tardiness tterm is conssidered, as
204
well as annother term referring to o the incomee due to thee service pro ovided to veehicles.
205
206
3. Formalization of the optimization problem
207
In this section, a mathematical programming problem will be formalized to represent the overall
208
decision objective and the constraints to be taken into account. The formalization of the decision
209
problem will be carried out within a discrete-time setting. The formalization of the problem is
210
provided adopting the viewpoint and selecting the utility functions of a unique decision maker,
211
that is, the manager of the microgrid. However, an attempt is made of modeling the customer’s
212
utility by introducing an elastic demand function. In other terms, the energy request by the
213
vehicles is a function of the charging prices. (which are considered as decision variables).
214 215
3.1 The considered variables
216
The state variables.
217
The state of the considered system is made of the following information:
218
-
battery and
219 220
the state of charge of the two storage systems [kWh] (i.e.,
-
related to a lithium
for a sodium battery, both with the capacity of 40[kWh]);
the state of charge of vehicle kWh
,
for i=1,…, N.
221
Each vehicle is identified by an integer number i, which may be assigned, for instance, according
222
to the arrival order. Note that all variables are considered in discrete time instants
223
The length of the discrete time interval will be denoted as ∆ , while the overall optimization
224
horizon is
,
, k=0, 1, 2,…
and consists of T time intervals.
225 226
Variables and data relevant to the set of vehicles.
227
The charging process of a vehicle has an integer duration, namely
228
-
,
is the time interval in which vehicle i begins the charging process;
229
-
,
is the time interval in which vehicle i concludes the charging process.
230
In Figure 2 an example, concerning the charging of two different vehicles is represented.
Charging of the j‐th vehicle
Charging of the i‐th vehicle
9 10 11 12 13 14 15 16 17 18 19 20 k
ci=9 231
fi=10
cj=16
fj=18
Figure 2. Charging of two different vehicles.
232 233 234
Each vehicle i, i=1, ..., N, is characterized by the following apriori information: -
i; this is the maximum amount of the energy requested by the vehicle;
235 236
,
-
is the release time interval (this may be thought as corresponding to the
time interval in which the vehicle is made available for the charging state process);
237 238
[kWh] is the aspiration level of the amount of energy to be acquired by the vehicle
,
,
-
, that is, the time interval at which the charging process of the vehicle should
239
be concluded. In particular,
240
with respect to a due date induces the payment of a penalty cost;
241
-
is the due-date of the charging of vehicle i. A tardiness
α [€/h] is the unit (tardiness) penalty cost.
242
The optimization problem considered in this paper is relevant to the sequencing and timing of the
243
charging process of the vehicles.
244
In the formalization of the problem, it is taken account the possible reduction of the actual energy
245
demand by vehicle i (from the value
246
service.
247
More specifically, it is assumed that the actual demand for vehicle i, namely Di, is elastic. This
248
means that Di is determined by the unit charging
249
demand function represented in Figure 3.
,
down to zero) owing to the prices for the charging
[€/kWh] through the application of the
Di
Dmax,i
G1 250 251
G2
Grec,i
Figure 3. The elastic demand function. The function expressing the demand is given by the following expression , ,
252
G
0 and
1, . . . ,
(1)
253
where
254
the actual number of vehicles requiring the charging service is reduced, concerning the initial
255
number of N, as some of them require zero demand.
256
The charging price
257
determined as the product between the unit cost for purchasing energy from the main grid (that is
258
given and known) in the time interval
259
are given constants, assumed independent from i. Naturallyt may turn out that
for vehicle i is considered as a decision variable, whose value is
∙
,
and the value of a function ( 1, … ,
(1)
260
Indeed, the values of the decision variables (
261
decision variables (determined by solving the optimization problem) whose values have to be
262
found taking into account the overall objective function of the decision maker (i. e., the microgrid
263
manager). For instance, higher values of such variables for some time intervals may have the
264
effect of lowering the actual service demand by vehicles arriving at the service station in those
265
intervals. This lowering, in turn, may be convenient in order to achieve a globally better solution
266
over the entire time horizon.
267
The following notation is adopted:
268 269
), for any possible (discrete) value
), namely
, are the
is the unit purchase cost (price) from the main grid, in time interval ( ,
);
270
is the unit selling price to the main grid, in time interval ( , and
0,1, … ,
1 are considered as given, whereas
271
The sequences
272
the sequence
273
Further decision variables of the problem.
274
In the problem formulation, we admit the possibility that the decision maker (that is, the
275
microgrid manager) recognizes the impossibility of providing service to one or more vehicles.
276
This possibility is modeled through the introduction of the binary decision variables , where
,
,
).
0,1, … ,
1 consists of decision variables..
277
= 1 means that a service is provided to vehicle i, whereas in case
278
served. Besides, , to ensure the consistency of the model, we impose that
279
is, when the actual demand of the customers corresponding to vehicle i becomes equal to zero,
280
owing to a value
281
Also, for each time interval
282
value of the following variables [kW]:
283
284
= 0 this vehicle is not = 0 when
=0, that
. ,
, k=0,1,2,…T-1, the decision maker has to find the optimal
, the power flow from the microgrid to the i-th vehicle, unrestricted in sign,
→ ,
i=1,2,3,…N;
285
286
→
, the power flow from the lithium storage to the microgrid, unrestricted in sign;
287
→
, the power flow from the sodium storage to the microgrid, unrestricted in sign;
288
,
→
, the power flows from the two engines to the microgrid;
, the power flow from the main grid to the microgrid, unrestricted in sign.
289
In the above notation, the orientation of the arrow in the subscript relevant to the variables
290
representing power flows identifies the direction in which the power flow (if unrestricted in sign)
291
is considered as positive.
292
Further, it is necessary to define the set of the binary variables
293
N, where
294
0 otherwise.
295
Summing up, the decision variables of the problem are:
, k=0,1,2,…,T-1, i=1,2,…,
1 if vehicle i is connected to the charging station in time interval
296
297
298
299
0,1, … ,
1
300
0,1, … ,
1
,
→ ,
,
,
→
,
,
→
→
0,1, . . ,
i=1,2,…,N ,
i=1,2,…,N i=1,2,…,N
,
and
1
301 302
3.2 The constraints affecting the system behavior
303
Constraints representing the system state evolution. The following storage state equations (for the two batteries and the EVs, respectively) have to
304 305
be taken into account in the statement of the optimization problem:
306 307
→
308
→
309
,
∙ ∆
0, … ,
1
(2)
∙ ∆
0, . . . ,
1
(3)
→ ,
∙∆
1, . . . ,
0, . . . ,
1
(4)
310
Constraints affecting the vehicle charging process.
311
Next, we consider the constraints relevant to the dynamics of the charging process. First, it is
312
necessary to prevent that more than a single vehicle is connected to the charging station, in the
313
same time interval, namely
314
∑
315
Moreover, we must impose that the time interval at which the charging starts should be greater
316
than or equal to the release time interval:
1
1, … ,
0, . . . ,
1
(5)
1, . . . ,
317
(6)
318
Besides, we have to impose that the charging process (if any) must finish before the end of the
319
time horizon 1
320
1, … ,
(7)
321
A further constraint has the function of relating the state of the charging station with the time
322
interval during which the charging of a vehicle i takes place:
323
0 0, … ,
,
0
324
1, … ,
1
325
The above constraint allows the connection of a vehicle only when the demand expressed by the
326
vehicle is not zero, and the microgrid manager agrees to the service. In this case, the vehicle is
327
connected only during time interval
328
Another constraint prevents the service with no demand, that is
(8)
,
.
0 when
329
=0, that is, when
1, . . . ,
(9)
330
It is necessary to introduce a constraint that prevents that a vehicle departs from the station before
331
reaching the desired state of charge (that is, without having received an amount of energy equal
332
to its demand
333
,
,
,
1, . . . ,
334 335
. To this end, it is sufficient to enforce
At time instant
336
,
1
1, … ,
(10)
, the state of charge has to be equal to the given initial value, that is 1, … ,
,
0, . . ,
1
(11)
337
Constraints expressing upper and lower bounds.
338
The evolution of the system state variables is constrained by upper and lower bounds, namely 1, . . . ,
1, . . . ,
(12)
339
340
where
341
charge of vehicle i.
342
Similarly, constraints limiting the value of the state of charge of the storage elements have to be
343
taken into account:
,
,
,
and
, ,
are lower and upper bounds, respectively, for the state of
344
1, … ,
(13a)
345
1, … ,
(13b)
346
where symbols have an obvious meaning.
347
In addition, a series of constraints have to be considered in order to limit the values of the power
348
flows which are considered as decision variables in the model: → ,
0, … ,
349 350
→ ,
→ ,
1, … ,
1 (14)
→
→
→
0, … ,
1
(15)
351
0
0, … ,
1
(16)
352
0
0, … ,
1
(17)
0, … ,
1
(18)
353
→
→
→
354
→
→
→
0, . . . ,
1
(19)
355
where all symbols have a straightforward meaning. Note that we have taken into account that
356
some of the power flows are bi-directional.
357
Finally, the power balance constraint has to be fulfilled for any time instant, namely
358
→
0, … ,
359
∑
→
0
→
1
(20)
is the non-deferrable (external) power demand that has, in any case, to be
360
where
361
satisfied in time interval ( ,
362
sources in the same interval.
363
Constraints limiting the variations of the power flow to/from storage elements.
364
It is necessary to introduce constraints in order to prevent too high variations in the power flows
365
from/to the storage elements (including the storage of the vehicles). Such variations may
366
accelerate the degradation of these elements.
367
|
→
→
1
|
∆
368
|
→
→
1
|
∆
369
→ ,
∆
→ ,
is the forecasted power flow from renewable
) and
0, … ,
→
2
0, … ,
→
0, … ,
→ ,
(21) 2
2
(22) 1, … ,
(23)
370
3.3 The overall optimization problem
371
The overall cost to be minimized is composed of the sum of economic production costs plus net
372
energy buying costs (from the grid) and the overall weighted tardiness cost.
373
Thus, the optimization problem consists in the following minimization under constraints from (1)
374
to (23). (24)
375 376
where :
377
∑
∑
∆ and
∆ are the energy overall
378
production costs at the two engines, where γ is the unit cost for producing power from
379
engines (this cost is assume to be equal for the two engines);
380
→
, 0 ∆ is the cost paid to buy energy
from the main grid;
381 382
∑
∑
→
, 0 ∆ is the income due to the sale of energy
to the main grid;
383 384
Ctard ∑Ni 1 εi α i
385
∑ 1
actually provided to the customers.
386 387
t f i ‐t ddi ,0 is the total tardiness cost;
is the income due to the charging services
388
4. Application to a case study
389
For all cases, the optimization problem presented in the previous section has been solved by using
390
Lingo [33] invoking the global solver, on a PC Intel i7, 16 GB RAM. Runtime is about 30
391
minutes for Scenario 1 and 2, while for Scenario 3 is about 90 minutes. Real data have been taken
392
from a real case study located in the Savona (Italy) municipality.
393
First, let us provide, in Table 1, the values of the parameters of the elements of the microgrid of
394
the considered case study Parameter
Value 30 kW 65 kW
→
40 kW
→
40 kW 30 kW
→ ,
1000 kW
→
0,237 €/kWh
γ
∆
→
10 kW
∆
→
10 kW
∆
→ ,
7 kW
395
Table 1. Power flow data.
396
Scenarios 1 and 2 refer to the case in which the distribution of demand over time is certain and
397
relatively light and thus the optimization of the vehicles’ charging schedule can be carried out
398
over a time horizon of 24 hours.
399
Instead, Scenario 3 considers a more heavy service request (i.e. a higher number of vehicles
400
within a shorter time). In this case, the prior knowledge the distribution of service request over
401
time is quite difficult. For this reason, the application of the so called “receding horizon” scheme
402
can be reasonably chosen. That is, optimization is carried out only with respect to a limited time
403
horizon (over which the distribution of the service demand is relatively certain). Then, after the
404
applicatioon of the firrst decision ns, a new innstance of the t optimizaation probleem is consiidered and
405
solved aggain, takingg into accou unt possiblee previously y unsatisfieed service rrequests and d possible
406
new serviice requirem ments.
407
For Scenaarios 1 and 2 we consider an optiimization horizon h conssisting of 966 time interrvals, each
408
ones of 15 minutes. Thus Δt =0 0.25[h] and on the who ole, the horiizon length is equal to 24 hours.
409
S thee optimizatiion horizon is restricted d to 2 hourss with a disccretization Instead, inn the third Scenario
410
interval oof 5 minutess.
411
In Figuree 4, the pattterns (obtain ned by reall measurem ments) of thee known noot deferrablle demand
412
and of thhe renewabble energy power aree representted, over a
413
(
414
equal to
∙ ) and benefit b (
0.2
€
0.26€
415
0.15
€
0
whole dday . The unit cost
∙ ) for th he power exchanged wiith the exterrnal grid aree are taken
36
81
95
37
80
for any t
416
417
Figurre 4. Not deeferrable eleectrical dem mand, PV po ower produuction.
418
Scenario 1.
419
In this casse, we consider 3 differrent vehicl es. All dataa are summaarized in Tab able 2. Vehicle data
V Vehicle 1
Vehicle 2
Vehicle 3
40
35
65
Dmaxx,i [kWh] ,,
[kWh]
[€/h] G1, G2 [€/kWh]
65
59
79
30
30
30
5
5
5
0.1
0.3
0.1
0.5, 0.6
0.5, 0 0.6
0.5, 0.6
420
Table T 2. Daata for the fiirst scenario o.
421
The resullts obtained are shown in Figure 55, as regard ds the evoluttion of the state of chaarge of the
422
vehicles.
423
424
F Figure 5. The Th evolution n of the statte of chargee of the three vehicles (S (Scenario 1)).
425
It can be noted that vehicle 2 iss charged, but never discharged, whereas thhe other two vehicles
426
nd the EVss serve as temporary are somettimes dischharged (i.e., vehicle too grid is peerformed an
427
energy soources), but eventually y charged. T The values of o the other important ddecision vaariables, as
428
determineed in the sollution of thee problem, aare reported d in Table 3. Vehiccle output
V Vehicle 1
Vehicle 2
Vehicle 3
2
2
2
56
44
80
Di [kWh] [
61
54
96
30
30
30
1
1
1
429
Table 3. F First scenarrio results.
430
We can nnote from thhese results that in this scenario no o vehicle red duces its deemand with respect to
431
Dmax,i, andd that the microgrid m maanager satissfies all serv vice requests.
432
The plot oof the state of charge of the two sttorage systems is provided in Figuure 6.
433 434 435
Fiigure 6. Thee evolution of o the state of charge of the two microgrid stoorage system ms. Finally, thhe various power p flowss in the mic rogrid are represented r in Figure 7 .
436 437
Figure 77. Power flow ws in the microgrid (Sccenario 1). Ps is the su um of
438
Pvehicles is the sum oof power flo ows
439
Figure 8 sshows the pattern p of
440
vehicles
→ ,
→
and
→
.
for all i..
over tiime This alllows to iden ntifythe servvice sequen nce of the
441 442
Figure 8. Evoluttion of
in Scenario 1.
443
Scenario 2.
444
In the seccond scenarrio, the num mber of vehiicles is increased to 4. Table 4 repports the datta for each
445
vehicle. Vehicle data a
Dmax,i[kWh h] ,,
[kW Wh]
Vehiclee 1
Vehicle 2
Vehicle 3
Vehicle 4
20
40
35
65
35
65
59
79
30
30
30
30
5
5
5
5
[€/h] G1, G2 [€/kW Wh]
446 447
0.2
0.1
0.3
0.1
0.5,0.66
0.5, 0.6 6
0.5, 0.6
0.5, 0.6
Table 4. 4 Vehicles data for thee second sccenario. In Figure 9, the patteerns of the state of charrge of the fo our vehicles are reporteed.
448
449
Figure 9. The T evolutio on of the staate of chargee of the fourr vehicles (SScenario 2).
450
The micrro grid mannager accep pts to chargge every veehicle. Note that in thhis case veehicle 1 is
451
initially ooverchargedd, with resp pect to its ddemand, in order o to usee it later ass a temporaary storage
452
(V2G).
453
The valuees of the moost relevant decision vaariables are reported in Table 5. Vehicle outp put
Di[kWh]]
454
Vehiclee 1
Vehicle 2
Vehicle 3
Vehicle 4
2
2
2
2
20
48
54
74
35
53
60
95
30
30
30
30
1
1
1
1
Table 5. Seecond scena ario results.
455
Scenario 3.
456
In this sceenario a recceding horizzon approacch is applieed, considerring a higheer number of o vehicles
457
within a sshorter timee horizon (2 hours). At the compleetion of the service of thhe last vehiicle served
458
(i.e., acceepted)a new w instance of the optim mization prob blem, again n over a tim me horizon of o 2 hours,
459
is definedd and solvedd. Within th his instancee, the arrivass of new vehicles are ttaken into account. a In
460
addition, also the vehicles that have not beeen accepteed and whosse due-datess are greateer than the
461
initial tim me instant of o the new optimizatioon horizon. Obviously this proceddure can bee repeated
462
indefiniteely. Here wee show only y two steps oof this proccedure. Tablle 6 reports vehicles’ data d for the
463
first step Vehicle daata
Dmax,i[kW Wh] ,,
[kkWh]
[€/h h] G1, G2 [€/k kWh]
Vehiccle 1
Vehiclle 2
Vehiclee 3
Vehicle 4
Vehicle 5
Vehicle 6
Vehicle 7
Vehicle 8
0
5.5
3.5
2
7.5
8.5
18
15
9.5
10.5 5
10
28.5
13
15.5
30
19
155
8
13
12
10
15
17
19
5
5
5
5
5
5
5
5
0.22
0.1
0.3
0.1
0.2
0.1
0.3
0.1
0.5,00.6
0.5, 0.6 0
0.5, 0..6
0.5, 0.6 6
0.5,0.6
0.5, 0.6
0.5, 0.6
0.5, 0.6
464
Table 6. Vehicles data secon nd scenario Step 1.
465
We reporrt in Figure 10 the evo olution of
466
for the moost relevantt decision vaariables
over time, t and in n Table 7 thhe values determined d
467 468
Figure 10. Evolutionn of Vehicle ou utput
Vehiccle 1
Vehiclle 2
Vehiclee 3
in Scenario o 3 Step 1. Vehicle 4
Vehicle 5
Vehicle 6
Vehicle 7
Vehicle 8
Di[kWh]
2
0
2
2
0
2
0
2
0
-
6
21
-
15
-
10
5
-
9
24
-
10
-
15
10
-
8
7
-
10
-
14
1
0
1
1
0
1
0
1
469
Table 7. Third scenario Output step 1.
470
In the solution of the first optimization step, vehicle 2, 5 and 7 are not serviced. Note that the
471
service sequence in the solution is not in accordance with the order of release times. In fact, it is
472
possible to say that the presented approach jointly optimizes timetabling and sequencing.
473
When the second optimization step is defined, two of the above three vehicles (namely, 2 and 5)
474
are definitively discarded, since both their due-dates(10,5 and 13 discretization steps,
475
corresponding to 52,5 and 65 minutes, respectively) are overcome by the initial discretization
476
step (24, that is 2 hours), that represents the starting time of the second optimization step. Instead,
477
vehicle 7 is considered in the statement of the problem (as its due date is 30, that is 2,5 hours),
478
along with 4 new vehicles arriving within the new optimization interval. Table 8 reports the data
479
for the second optimization step. Note that time values are referred to a new “zero”,
480
corresponding to the initial time instant of the optimization interval in the second step. Vehicle data
Dmax,i[kWh] ,,
[kWh]
[€/h] G1, G2 [€/kWh]
Vehicle 7
Vehicle 9
Vehicle 10
Vehicle 11
Vehicle 12
0
10
12
7
18
6
20
19
21
32
17
22
13
20
10
5
5
5
5
5
0.2
0.1
0.3
0.1
0.2
0.5,0.6
0.5, 0.6
0.5, 0.6
0.5, 0.6
0.5,0.6
481
Table 8. Vehicles data second scenario Step 2.
482
Figure 11 shows the sequence of services of step 2. In this case the microgrid manager does not
483
accept vehicle 12, but this one can be considered in the next step..
484 485 486
Figure 11. Evolutionn of
in Scenario o 3 Step 2.
Table 9 shhows the ouutput variab bles for this second step p. Vehicle output
Di[kkWh]
Veehicle 7
Veehicle 9
Veh hicle 10
Veh hicle 11
Vehhicle 12
2
2
2
2
0
1
17
13
7
-
6
23
16
12
-
12
17
7
15
-
1
1
1
1
0
487
Ta able 9. Third rd scenario Output step p 2.
488
Finally, it is important to note that the ruun time turrns out to be b acceptabble for each h Scenario
489
considereed. Namely, For Scen narios 1 annd 2 the maximum ru un time is 1 hour, wh hereas for
490
Scenario 3 (both stepps) is lower than 3 minuutes.
491 492
493
5. Conclusions and future developments
494
A new optimization model is here formalized to include electrical vehicles in a smart grid. On the
495
basis of the requests and preferences of users of EVs, the grid’s manager decides the optimal
496
schedule of its generation plants and storage systems, taking into account the non deferrable
497
internal demand and renewable availability. The developed model is applied by using real data,
498
and different scenarios related to the demand of EVs’ charging have been considered.
499
In the proposed model, an attempt has been made to model the behavior of different decision
500
makers. Namely, the vehicles’ owners may reduce their service request on the basis of an elastic
501
demand function (assumed to be the same for vehicles, for the sake of simplicity). Instead, the
502
service station manager has different degrees of freedom, as he/she can vary the unit price for
503
service, and may even decide against servicing one or more vehicles. Besides, also the time
504
intervals during which the various services take place are determined through the solution of the
505
optimization problem. These services are imposed to be non preemptive and their order does not
506
necessarily reproduces the order of the arrivals of the vehicles. Thus, is possible to say that even
507
the service sequence is determined by the solution of the optimization problem. Of course,
508
different interaction schemes among the various decision makers can be conceived, and this is
509
matter of current research.
510
The problem has been formalized in a discrete-time setting. Obviously, this choice is not the
511
unique possible. In fact, also a discrete-event formalization of the model is possible, in which the
512
only time instants represented are those corresponding to decisions to be taken. The development
513
of a decision model based on a discrete-event representation of the dynamics of the system is
514
actually in progress.
515
516
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613
List of figures and tables
614
Figures
615
Figure 1. Microgrid design.
616
Figure 2. Charging of two different vehicles
617
Figure 3. The elastic demand function
618
Figure 4. Electrical demand, PV power production.
619
Figure 5. The evolution of the state of charge of the three vehicles (Scenario 1).
620
Figure 6. The evolution of the state of charge of the two microgrid storage systems.
621
Figure 7. Power flows in the microgrid (Scenario 1). Ps is the sum of
622
Pvehicles is the sum of power flows
623
Figure 8. Evolution of
624
Figure 9. The evolution of the state of charge of the four vehicles (Scenario 2).
625
Figure 10. Evolution of δ t
626
Figure 11. Evolution of δ t in Scenario 3 Step 2.
627
Tables
628
Table 1. Power flow data.
629
Table 2. Data for the first scenario.
630
Table 3. First scenario output.
631
Table 4. Vehicle data Second scenario.
632
Table 5. Second scenario results
633
Table 6. Vehicles data second scenario Step 1.
634
Table 7. Third scenario Output step 1.
635
Table 8. Vehicles data second scenario Step 2.
636
Table 9. Third scenario Output step 2.
→ ,
→
for all i.
in Scenario 1.
in Scenario 3 Step 1.
and
→
.
PII: DOI: Reference:
S2352-4677(17)30275-8 https://doi.org/10.1016/j.segan.2018.04.002 SEGAN 143
To appear in:
Sustainable Energy, Grids and Networks
Received date : 6 November 2017 Revised date : 23 March 2018 Accepted date : 16 April 2018 Please cite this article as: G. Ferro, F. Laureri, R. Minciardi, M. Robba, An optimization model for electrical vehicles scheduling in a smart grid, Sustainable Energy, Grids and Networks (2018), https://doi.org/10.1016/j.segan.2018.04.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1
An optimization model for electrical vehicles
2
scheduling in a smart grid
3
G. Ferro*, F. Laureri*, R. Minciardi*, M. Robba*
4 5 6
* Department of Informatics, Bioengineering, Robotics and Systems Engineering – DIBRIS, University of Genoa,
7
Via Opera Pia 13, I-16145, Genoa, Italy, email: [email protected], [email protected],
8
[email protected], [email protected]
9 10
Corresponding author
11
Giulio Ferro, Ph.D. Student
12
DIBRIS‐Department of Computer Science, Bioengineering, Robotics, and Systems Engineering,
13
University of Genoa
14
Via Opera Pia 13, 16145 Genoa
15
Laboratory: 00390103532804
16
Fax. 00390103532154
17
Cell. 00393471527053
18
c/o Campus Savona
19
Lagorio Building
20
Via A.Magliotto 2
21
17100 Savona
22
0039‐01921945139
23
24
Email: [email protected]
25
26
Abstract
27
Energy Management Systems (EMSs) are recognized as essential tools for the optimal
28
management of smart grids. However, few of them consider, in their whole complexity, the
29
integration of electrical vehicles (EVs) in smart grids, taking into account the requirements and
30
the time specifications characterizing the service requests. In this paper, attention is focused on
31
the formalization of a model for the optimal scheduling of charging of EVs in a smart grid, also
32
considering the vehicle to grid process (i.e., the possibility for the EV to inject power during the
33
charging process). In the formalization of an optimization problem for a smart grid, a deferrable
34
demand is considered, which is represented by the charging demand of the set of EVs. The cost to
35
be optimized for the considered problem includes the economic cost of energy
36
production/acquisition (from the main grid) and the cost relevant to the delay in the satisfaction
37
of the customers’ demand (is represented as a tardiness cost). Also, the income coming from the
38
service provided to vehicles is taken into account. The developed model is tested and applied in
39
connection with a real case study characterized by a photovoltaic plant, two batteries, power
40
production plants that use natural gas as primary energy, and a charging station.
41 42 43
Keywords: Optimization; Electric Vehicles; Microgrids; Scheduling
44
1. Introduction and literature review
45
The integration of renewables, electrical vehicles, and microgrids is getting more and more
46
importance in the last few years, to reduce greenhouse gas emissions, to increase the interactions
47
between different systems, to improve the reliability of the distribution networks. In the recent
48
literature, several articles analyze smart grids and how to integrate intermittent and distributed
49
production and loads, often with reference to specific elements of the grid (renewables, electrical
50
vehicles (EVs), storage systems).
51
The integration of EVs is becoming increasingly attractive for the following reasons. First, it is
52
supposed that shortly there will be an increase of such a technology and thus they can result in a
53
significant intermittent and distributed load. As a consequence smart charging strategies are
54
necessary to reduce costs and peak loads. Second, EVs can be used as distributed storage
55
systems. An example is reported in [1] where the authors propose the coordination between EVs
56
fleets to mitigate energy imbalances caused by renewables, offering an essential service to
57
electrical infrastructure by the use of the vehicle to grid (V2G) configuration. In fact, few of them
58
consider, in their whole complexity, Energy Management Systems (EMSs) [2] that integrate
59
smart grids and the facilities for charging of EVs, taking into account the requirements and the
60
time specifications characterizing the service requests of the users relevant to the various EVs.
61
As regards the general theme of smart grid development and application, one can mention
62
Shamshiri et al. [3], who present an overview of smart grids features and highlight the most
63
recent developments. Optimization models are crucial to achieve a reduction in electricity costs
64
and to maximize investments. In this connection, Monteiro et al. [4] develop a management
65
architecture for the distributed micro-grid resource based on a multi-agent simulator that can
66
optimize load scheduling. Nasrolahpour et al. [5] consider a microgrid and determine the optimal
67
scheduling of each generator and some controllable loads during a day including the
68
determination of the amount of energy purchased from the main grid.
69
As one of the critical issues in developing practical approaches towards the attainment of real
70
smart cities, different authors have presented detailed models and methods to properly manage
71
the scheduling of the charging of EVs and have studied the integration of electric mobility in the
72
grid. Gonzalez et al. [6] review the state of the art related to ancillary service for smart grids
73
provide by EVs, and two crucial examples are reported in [7] and [8]. In the former work, a real
74
case study is presented and EVs’ active and reactive power optimal injections are determined,
75
satisfying both electrical demands and maintaining the distribution grid stability. Instead, the
76
authors of [8] discuss the impact of reactive power support performed by EV charging stations in
77
a low-voltage distribution grid. Zhao et al. [9] present the results of an investigation for the
78
evaluation of different charging concepts and strategies. Delaimi et al. [10] propose a load
79
management solution for coordinating the charging of multiple plug-in electric vehicles in a
80
smart grid system. Allard et al. [11] compare the performance of different scenarios of smart
81
charging and conventional charging applied to EVs integrated within a grid characterized by a
82
high presence of renewables in particular wind energy.
83
Electric vehicles are equipped with efficient electric engines, which are powered by high-density
84
lithium-ion batteries, which, through chemical reactions, generate electrical current. A typical
85
drive for these vehicles is composed of a DC/DC converter and an inverter that feeds the engine
86
[12], which allows bi-directional power flow. In fact, when there is a torque request from the
87
traction system, the power flow is from the battery to the wheels, whereas during the regenerative
88
braking the battery is recharged. The battery life is influenced not only by the itinerary and the
89
activity [13] but also by the charging modes. There are different types of charging stations: for
90
example, in [14] stations based on PV power are taken into account. Such stations combine a PV
91
system with electric vehicles that are also connected to the main grid. In particular, the authors
92
present a load scheduling model using data collected in two years. There are many different type
93
of charging modes: the most common in smart grids are Smart Charging and Vehicle to Grid
94
(V2G). The Smart Charging mode represents a flexible way of charging vehicles in order to
95
modulate the feeding of the vehicles according to the grid necessities, for example during the
96
peak hours. Instead, in V2G configuration, the power flow is bidirectional (from the grid to the
97
vehicle and vice versa). In this way the vehicle storage can help the microgrid to sustain the
98
power requirement of the entire system, whenever necessary, or to regulate the voltage at the
99
charging station node. Two examples of the study of this configuration are [15] and [16]; in the
100
first work authors to investigate the cooperative evaluation of an EMS operation in a building
101
considering different features like bidirectional energy flows of an EV fleet, the impact of PV
102
uncertainty on EMS operation the effect of prioritization in selling energy to the main grid. The
103
second one proposes a model to study the effects of power exchange between the grid and EVs
104
on the power system’s demand profile several indices are calculated for various vehicle-to-grid
105
power level to estimate the impact of different level of power exchange on system’s reliability.
106
However, these types of charging have also some drawbacks, among which the most important is
107
the possibility of giving rise to a fluctuating power demand. This can cause problems to the main
108
grid such as power imbalances. For this reason, the authors in [17] develop a strategy to regulate
109
the vehicle's charging process, introducing different priorities for the various objectives, like the
110
achievement of maximum flexibility, the effective management of the power flows in order to
111
smooth the demand curve and increase the system's power quality. In [18] attention is focused on
112
the active contribution (i.e., power modulation via different charging modes) of the electric
113
vehicles in the grid optimization, considering the V2G mode. Differently, in [19] a mixed integer
114
linear programming problem is formulated, with the objective of scheduling the charging of the
115
vehicles, minimizing the energy costs and taking into account reliability and stability issues,
116
considered in connection with both the microgrid and the customers (the vehicles). Wang et al.
117
[20] study the configuration of a system wherein EVs are connected to the smart grid as mobile
118
distributed energy storage. Bracco et al. [21] present an overall EMS, based on a dynamic
119
optimization model to minimize operating costs and carbon emissions considering different
120
technologies as micro gas turbines, photovoltaic, concentrating solar systems, and storage based
121
on different battery technologies.
122
The choice, among different modes of charging influences not only the statement of the charging
123
scheduling problems, but also the routing of electrical vehicles. For instance, in [22] a double
124
layer smart charging algorithm for electrical vehicles (EVs) is presented, having the objective of
125
allowing the single vehicle to reach the more suitable recharging station, limiting the transformer
126
load and the total energy costs. In [23] the authors study different routing strategies to smooth
127
microgrid power fluctuations, caused by intermittent renewables and load, ensuring at the same
128
time the grid power quality and the optimality of logistics services.
129
In the literature, there are also examples of stochastic approaches to model the integration
130
between electric vehicles and power networks. For example in [24], a real case study is described
131
(namely, the distribution network in Zaragoza, Spain). In particular and the authors formulate a
132
stochastic problem for energy resource scheduling including uncertainties of renewable sources,
133
electric vehicles, and market prices. In
134
objective of maximizing the profit by charging the plug-in electric vehicles within the lowest
135
price time intervals, considering the participation in the ancillary services market as an additional
136
source of income. A stochastic problem formulation for V2G technology is also present in [26],
137
where also uncertainties in the power flow are taken into account. Instead, in [27] attention is
138
focused on energy aggregators that can help the management of multiple connected devices . In
139
particular, the authors develop a stochastic model for one-day-ahead energy resource scheduling,
140
integrated with the dynamic electricity pricing for electrical vehicles, to cope with the challenges
141
related to the demand and renewable sources uncertainties. In [28] the authors propose the use of
142
three metaheuristic optimization techniques, in order to solve the plug-in electric vehicle charging
143
coordination problem in electrical distribution systems. Similar approaches are adopted in [29]
144
and [30] using genetic algorithms and particle swarm optimization.
[25] a scheduling problem is considered, with the
145
In the present paper, attention is focused on the formalization of a model for the optimal
146
scheduling of charging of EVs in a smart grid. Concerning the existing literature, the original
147
contribution of this paper is the formalization of an optimization problem for a smart grid with a
148
deferrable demand, represented by the charging demand of the set of electrical vehicles. The cost
149
to be optimized for the considered problem includes the economic cost of energy
150
production/acquisition (from the main grid) and the cost relevant to the delay in the satisfaction
151
of the customers' demand. The latter cost is represented as a tardiness cost. Besides, the income
152
deriving from the service provided to the vehicle is taken into account. Note that [31], where the
153
integration of plug-in EVs and renewable production plants is considered. However, the model
154
proposed in this paper goes beyond the one in [31] as the scheduling problem of the vehicle’s
155
charging is explicty considered.
156
To be more specific, the system considered in this paper is characterized by the presence of a
157
renewable energy source (photovoltaic plant), energy storage facilities, power production plants
158
that use natural gas as primary energy, and a recharging station for EVs. The decision variables
159
include the active powers exchanged in the whole system, binary variables that describe the
160
interaction between the EV and the charging station and the recharging unit price of each
161
vehicle.In particular,the last decision variables represent one of the novelties introduced in this
162
paper.
163
The rest of the paper is organized as follows. In the next section, the considered system is
164
described in detail, highlighting its specifying features and introducing the optimization problem
165
to be solved. In the third section, the optimization problem is formalized, and in section 4 the
166
application of the proposed approach to a case study is presented. Some concluding remarks are
167
finally provided in the last section.
168
2. The considered physical system
169 170
The decision model presented in this paper is based on the fundamental concepts and tools that
171
can be found in the literature for microgrids’ operational management (like in [32]). Even though
172
a new formalization is presented for the objective function and the constraints (including a
173
demand-response model for the electrical vehicles charging). In the overall formalization, it is
174
supposed that power losses can be neglected, and thus the microgrid is considered as a single
175
node. Each component of the system is represented as an active power injection in the total power
176
balance, following the active sign convention (the sign of the power is positive when itis
177
generated).
178
The considered system consists of a microgrid composed of the following elements (see Figure
179
1):
180
a) One or more renewable energy sources (considered as a unique entity in the following, for
181
the sake of simplicity); it is assumed that this renewable energy source is intermittent and
182
not controllable and that the energy is generated without cost;
183
b) One or more non-renewable high-efficiency energy sources, whose energy production is
184
costly; in particular, two internal combustion engines are considered, with different
185
maximum rated power;
186 187 188 189
c) One or more electrical storage elements; in particular, the presence of two storage systems is assumed; d) A connection (PCC–Point of Common Coupling) with the main grid, which guarantees the bidirectional power flow between the microgrid and the main gird;
190
e) A single facility (charging station) for the electrical charge of vehicles; it is assumed that
191
the power flow between the microgrid and the charging vehicle may be bidirectional. The
192
charging station may serve a single vehicle at a time, and the charging process is not pre-
193
emptive, that is, a given vehicle cannot be disconnected from the charging station and
194
reconnected later;
195
f) An electric load characterized by a given pattern over the considered time horizon.
\
196
197 198 199
Figure 11. Microgrid d design.
200 201
The objeective functtion that will w be connsidered in the decisio on problem m statementt includes
202
different terms. Suchh terms incclude the coost due to the t energy production from non-renewable
203
sources annd the energgy bought/ssold from/too the main grid. g Also, a tardiness tterm is conssidered, as
204
well as annother term referring to o the incomee due to thee service pro ovided to veehicles.
205
206
3. Formalization of the optimization problem
207
In this section, a mathematical programming problem will be formalized to represent the overall
208
decision objective and the constraints to be taken into account. The formalization of the decision
209
problem will be carried out within a discrete-time setting. The formalization of the problem is
210
provided adopting the viewpoint and selecting the utility functions of a unique decision maker,
211
that is, the manager of the microgrid. However, an attempt is made of modeling the customer’s
212
utility by introducing an elastic demand function. In other terms, the energy request by the
213
vehicles is a function of the charging prices. (which are considered as decision variables).
214 215
3.1 The considered variables
216
The state variables.
217
The state of the considered system is made of the following information:
218
-
battery and
219 220
the state of charge of the two storage systems [kWh] (i.e.,
-
related to a lithium
for a sodium battery, both with the capacity of 40[kWh]);
the state of charge of vehicle kWh
,
for i=1,…, N.
221
Each vehicle is identified by an integer number i, which may be assigned, for instance, according
222
to the arrival order. Note that all variables are considered in discrete time instants
223
The length of the discrete time interval will be denoted as ∆ , while the overall optimization
224
horizon is
,
, k=0, 1, 2,…
and consists of T time intervals.
225 226
Variables and data relevant to the set of vehicles.
227
The charging process of a vehicle has an integer duration, namely
228
-
,
is the time interval in which vehicle i begins the charging process;
229
-
,
is the time interval in which vehicle i concludes the charging process.
230
In Figure 2 an example, concerning the charging of two different vehicles is represented.
Charging of the j‐th vehicle
Charging of the i‐th vehicle
9 10 11 12 13 14 15 16 17 18 19 20 k
ci=9 231
fi=10
cj=16
fj=18
Figure 2. Charging of two different vehicles.
232 233 234
Each vehicle i, i=1, ..., N, is characterized by the following apriori information: -
i; this is the maximum amount of the energy requested by the vehicle;
235 236
,
-
is the release time interval (this may be thought as corresponding to the
time interval in which the vehicle is made available for the charging state process);
237 238
[kWh] is the aspiration level of the amount of energy to be acquired by the vehicle
,
,
-
, that is, the time interval at which the charging process of the vehicle should
239
be concluded. In particular,
240
with respect to a due date induces the payment of a penalty cost;
241
-
is the due-date of the charging of vehicle i. A tardiness
α [€/h] is the unit (tardiness) penalty cost.
242
The optimization problem considered in this paper is relevant to the sequencing and timing of the
243
charging process of the vehicles.
244
In the formalization of the problem, it is taken account the possible reduction of the actual energy
245
demand by vehicle i (from the value
246
service.
247
More specifically, it is assumed that the actual demand for vehicle i, namely Di, is elastic. This
248
means that Di is determined by the unit charging
249
demand function represented in Figure 3.
,
down to zero) owing to the prices for the charging
[€/kWh] through the application of the
Di
Dmax,i
G1 250 251
G2
Grec,i
Figure 3. The elastic demand function. The function expressing the demand is given by the following expression , ,
252
G
0 and
1, . . . ,
(1)
253
where
254
the actual number of vehicles requiring the charging service is reduced, concerning the initial
255
number of N, as some of them require zero demand.
256
The charging price
257
determined as the product between the unit cost for purchasing energy from the main grid (that is
258
given and known) in the time interval
259
are given constants, assumed independent from i. Naturallyt may turn out that
for vehicle i is considered as a decision variable, whose value is
∙
,
and the value of a function ( 1, … ,
(1)
260
Indeed, the values of the decision variables (
261
decision variables (determined by solving the optimization problem) whose values have to be
262
found taking into account the overall objective function of the decision maker (i. e., the microgrid
263
manager). For instance, higher values of such variables for some time intervals may have the
264
effect of lowering the actual service demand by vehicles arriving at the service station in those
265
intervals. This lowering, in turn, may be convenient in order to achieve a globally better solution
266
over the entire time horizon.
267
The following notation is adopted:
268 269
), for any possible (discrete) value
), namely
, are the
is the unit purchase cost (price) from the main grid, in time interval ( ,
);
270
is the unit selling price to the main grid, in time interval ( , and
0,1, … ,
1 are considered as given, whereas
271
The sequences
272
the sequence
273
Further decision variables of the problem.
274
In the problem formulation, we admit the possibility that the decision maker (that is, the
275
microgrid manager) recognizes the impossibility of providing service to one or more vehicles.
276
This possibility is modeled through the introduction of the binary decision variables , where
,
,
).
0,1, … ,
1 consists of decision variables..
277
= 1 means that a service is provided to vehicle i, whereas in case
278
served. Besides, , to ensure the consistency of the model, we impose that
279
is, when the actual demand of the customers corresponding to vehicle i becomes equal to zero,
280
owing to a value
281
Also, for each time interval
282
value of the following variables [kW]:
283
284
= 0 this vehicle is not = 0 when
=0, that
. ,
, k=0,1,2,…T-1, the decision maker has to find the optimal
, the power flow from the microgrid to the i-th vehicle, unrestricted in sign,
→ ,
i=1,2,3,…N;
285
286
→
, the power flow from the lithium storage to the microgrid, unrestricted in sign;
287
→
, the power flow from the sodium storage to the microgrid, unrestricted in sign;
288
,
→
, the power flows from the two engines to the microgrid;
, the power flow from the main grid to the microgrid, unrestricted in sign.
289
In the above notation, the orientation of the arrow in the subscript relevant to the variables
290
representing power flows identifies the direction in which the power flow (if unrestricted in sign)
291
is considered as positive.
292
Further, it is necessary to define the set of the binary variables
293
N, where
294
0 otherwise.
295
Summing up, the decision variables of the problem are:
, k=0,1,2,…,T-1, i=1,2,…,
1 if vehicle i is connected to the charging station in time interval
296
297
298
299
0,1, … ,
1
300
0,1, … ,
1
,
→ ,
,
,
→
,
,
→
→
0,1, . . ,
i=1,2,…,N ,
i=1,2,…,N i=1,2,…,N
,
and
1
301 302
3.2 The constraints affecting the system behavior
303
Constraints representing the system state evolution. The following storage state equations (for the two batteries and the EVs, respectively) have to
304 305
be taken into account in the statement of the optimization problem:
306 307
→
308
→
309
,
∙ ∆
0, … ,
1
(2)
∙ ∆
0, . . . ,
1
(3)
→ ,
∙∆
1, . . . ,
0, . . . ,
1
(4)
310
Constraints affecting the vehicle charging process.
311
Next, we consider the constraints relevant to the dynamics of the charging process. First, it is
312
necessary to prevent that more than a single vehicle is connected to the charging station, in the
313
same time interval, namely
314
∑
315
Moreover, we must impose that the time interval at which the charging starts should be greater
316
than or equal to the release time interval:
1
1, … ,
0, . . . ,
1
(5)
1, . . . ,
317
(6)
318
Besides, we have to impose that the charging process (if any) must finish before the end of the
319
time horizon 1
320
1, … ,
(7)
321
A further constraint has the function of relating the state of the charging station with the time
322
interval during which the charging of a vehicle i takes place:
323
0 0, … ,
,
0
324
1, … ,
1
325
The above constraint allows the connection of a vehicle only when the demand expressed by the
326
vehicle is not zero, and the microgrid manager agrees to the service. In this case, the vehicle is
327
connected only during time interval
328
Another constraint prevents the service with no demand, that is
(8)
,
.
0 when
329
=0, that is, when
1, . . . ,
(9)
330
It is necessary to introduce a constraint that prevents that a vehicle departs from the station before
331
reaching the desired state of charge (that is, without having received an amount of energy equal
332
to its demand
333
,
,
,
1, . . . ,
334 335
. To this end, it is sufficient to enforce
At time instant
336
,
1
1, … ,
(10)
, the state of charge has to be equal to the given initial value, that is 1, … ,
,
0, . . ,
1
(11)
337
Constraints expressing upper and lower bounds.
338
The evolution of the system state variables is constrained by upper and lower bounds, namely 1, . . . ,
1, . . . ,
(12)
339
340
where
341
charge of vehicle i.
342
Similarly, constraints limiting the value of the state of charge of the storage elements have to be
343
taken into account:
,
,
,
and
, ,
are lower and upper bounds, respectively, for the state of
344
1, … ,
(13a)
345
1, … ,
(13b)
346
where symbols have an obvious meaning.
347
In addition, a series of constraints have to be considered in order to limit the values of the power
348
flows which are considered as decision variables in the model: → ,
0, … ,
349 350
→ ,
→ ,
1, … ,
1 (14)
→
→
→
0, … ,
1
(15)
351
0
0, … ,
1
(16)
352
0
0, … ,
1
(17)
0, … ,
1
(18)
353
→
→
→
354
→
→
→
0, . . . ,
1
(19)
355
where all symbols have a straightforward meaning. Note that we have taken into account that
356
some of the power flows are bi-directional.
357
Finally, the power balance constraint has to be fulfilled for any time instant, namely
358
→
0, … ,
359
∑
→
0
→
1
(20)
is the non-deferrable (external) power demand that has, in any case, to be
360
where
361
satisfied in time interval ( ,
362
sources in the same interval.
363
Constraints limiting the variations of the power flow to/from storage elements.
364
It is necessary to introduce constraints in order to prevent too high variations in the power flows
365
from/to the storage elements (including the storage of the vehicles). Such variations may
366
accelerate the degradation of these elements.
367
|
→
→
1
|
∆
368
|
→
→
1
|
∆
369
→ ,
∆
→ ,
is the forecasted power flow from renewable
) and
0, … ,
→
2
0, … ,
→
0, … ,
→ ,
(21) 2
2
(22) 1, … ,
(23)
370
3.3 The overall optimization problem
371
The overall cost to be minimized is composed of the sum of economic production costs plus net
372
energy buying costs (from the grid) and the overall weighted tardiness cost.
373
Thus, the optimization problem consists in the following minimization under constraints from (1)
374
to (23). (24)
375 376
where :
377
∑
∑
∆ and
∆ are the energy overall
378
production costs at the two engines, where γ is the unit cost for producing power from
379
engines (this cost is assume to be equal for the two engines);
380
→
, 0 ∆ is the cost paid to buy energy
from the main grid;
381 382
∑
∑
→
, 0 ∆ is the income due to the sale of energy
to the main grid;
383 384
Ctard ∑Ni 1 εi α i
385
∑ 1
actually provided to the customers.
386 387
t f i ‐t ddi ,0 is the total tardiness cost;
is the income due to the charging services
388
4. Application to a case study
389
For all cases, the optimization problem presented in the previous section has been solved by using
390
Lingo [33] invoking the global solver, on a PC Intel i7, 16 GB RAM. Runtime is about 30
391
minutes for Scenario 1 and 2, while for Scenario 3 is about 90 minutes. Real data have been taken
392
from a real case study located in the Savona (Italy) municipality.
393
First, let us provide, in Table 1, the values of the parameters of the elements of the microgrid of
394
the considered case study Parameter
Value 30 kW 65 kW
→
40 kW
→
40 kW 30 kW
→ ,
1000 kW
→
0,237 €/kWh
γ
∆
→
10 kW
∆
→
10 kW
∆
→ ,
7 kW
395
Table 1. Power flow data.
396
Scenarios 1 and 2 refer to the case in which the distribution of demand over time is certain and
397
relatively light and thus the optimization of the vehicles’ charging schedule can be carried out
398
over a time horizon of 24 hours.
399
Instead, Scenario 3 considers a more heavy service request (i.e. a higher number of vehicles
400
within a shorter time). In this case, the prior knowledge the distribution of service request over
401
time is quite difficult. For this reason, the application of the so called “receding horizon” scheme
402
can be reasonably chosen. That is, optimization is carried out only with respect to a limited time
403
horizon (over which the distribution of the service demand is relatively certain). Then, after the
404
applicatioon of the firrst decision ns, a new innstance of the t optimizaation probleem is consiidered and
405
solved aggain, takingg into accou unt possiblee previously y unsatisfieed service rrequests and d possible
406
new serviice requirem ments.
407
For Scenaarios 1 and 2 we consider an optiimization horizon h conssisting of 966 time interrvals, each
408
ones of 15 minutes. Thus Δt =0 0.25[h] and on the who ole, the horiizon length is equal to 24 hours.
409
S thee optimizatiion horizon is restricted d to 2 hourss with a disccretization Instead, inn the third Scenario
410
interval oof 5 minutess.
411
In Figuree 4, the pattterns (obtain ned by reall measurem ments) of thee known noot deferrablle demand
412
and of thhe renewabble energy power aree representted, over a
413
(
414
equal to
∙ ) and benefit b (
0.2
€
0.26€
415
0.15
€
0
whole dday . The unit cost
∙ ) for th he power exchanged wiith the exterrnal grid aree are taken
36
81
95
37
80
for any t
416
417
Figurre 4. Not deeferrable eleectrical dem mand, PV po ower produuction.
418
Scenario 1.
419
In this casse, we consider 3 differrent vehicl es. All dataa are summaarized in Tab able 2. Vehicle data
V Vehicle 1
Vehicle 2
Vehicle 3
40
35
65
Dmaxx,i [kWh] ,,
[kWh]
[€/h] G1, G2 [€/kWh]
65
59
79
30
30
30
5
5
5
0.1
0.3
0.1
0.5, 0.6
0.5, 0 0.6
0.5, 0.6
420
Table T 2. Daata for the fiirst scenario o.
421
The resullts obtained are shown in Figure 55, as regard ds the evoluttion of the state of chaarge of the
422
vehicles.
423
424
F Figure 5. The Th evolution n of the statte of chargee of the three vehicles (S (Scenario 1)).
425
It can be noted that vehicle 2 iss charged, but never discharged, whereas thhe other two vehicles
426
nd the EVss serve as temporary are somettimes dischharged (i.e., vehicle too grid is peerformed an
427
energy soources), but eventually y charged. T The values of o the other important ddecision vaariables, as
428
determineed in the sollution of thee problem, aare reported d in Table 3. Vehiccle output
V Vehicle 1
Vehicle 2
Vehicle 3
2
2
2
56
44
80
Di [kWh] [
61
54
96
30
30
30
1
1
1
429
Table 3. F First scenarrio results.
430
We can nnote from thhese results that in this scenario no o vehicle red duces its deemand with respect to
431
Dmax,i, andd that the microgrid m maanager satissfies all serv vice requests.
432
The plot oof the state of charge of the two sttorage systems is provided in Figuure 6.
433 434 435
Fiigure 6. Thee evolution of o the state of charge of the two microgrid stoorage system ms. Finally, thhe various power p flowss in the mic rogrid are represented r in Figure 7 .
436 437
Figure 77. Power flow ws in the microgrid (Sccenario 1). Ps is the su um of
438
Pvehicles is the sum oof power flo ows
439
Figure 8 sshows the pattern p of
440
vehicles
→ ,
→
and
→
.
for all i..
over tiime This alllows to iden ntifythe servvice sequen nce of the
441 442
Figure 8. Evoluttion of
in Scenario 1.
443
Scenario 2.
444
In the seccond scenarrio, the num mber of vehiicles is increased to 4. Table 4 repports the datta for each
445
vehicle. Vehicle data a
Dmax,i[kWh h] ,,
[kW Wh]
Vehiclee 1
Vehicle 2
Vehicle 3
Vehicle 4
20
40
35
65
35
65
59
79
30
30
30
30
5
5
5
5
[€/h] G1, G2 [€/kW Wh]
446 447
0.2
0.1
0.3
0.1
0.5,0.66
0.5, 0.6 6
0.5, 0.6
0.5, 0.6
Table 4. 4 Vehicles data for thee second sccenario. In Figure 9, the patteerns of the state of charrge of the fo our vehicles are reporteed.
448
449
Figure 9. The T evolutio on of the staate of chargee of the fourr vehicles (SScenario 2).
450
The micrro grid mannager accep pts to chargge every veehicle. Note that in thhis case veehicle 1 is
451
initially ooverchargedd, with resp pect to its ddemand, in order o to usee it later ass a temporaary storage
452
(V2G).
453
The valuees of the moost relevant decision vaariables are reported in Table 5. Vehicle outp put
Di[kWh]]
454
Vehiclee 1
Vehicle 2
Vehicle 3
Vehicle 4
2
2
2
2
20
48
54
74
35
53
60
95
30
30
30
30
1
1
1
1
Table 5. Seecond scena ario results.
455
Scenario 3.
456
In this sceenario a recceding horizzon approacch is applieed, considerring a higheer number of o vehicles
457
within a sshorter timee horizon (2 hours). At the compleetion of the service of thhe last vehiicle served
458
(i.e., acceepted)a new w instance of the optim mization prob blem, again n over a tim me horizon of o 2 hours,
459
is definedd and solvedd. Within th his instancee, the arrivass of new vehicles are ttaken into account. a In
460
addition, also the vehicles that have not beeen accepteed and whosse due-datess are greateer than the
461
initial tim me instant of o the new optimizatioon horizon. Obviously this proceddure can bee repeated
462
indefiniteely. Here wee show only y two steps oof this proccedure. Tablle 6 reports vehicles’ data d for the
463
first step Vehicle daata
Dmax,i[kW Wh] ,,
[kkWh]
[€/h h] G1, G2 [€/k kWh]
Vehiccle 1
Vehiclle 2
Vehiclee 3
Vehicle 4
Vehicle 5
Vehicle 6
Vehicle 7
Vehicle 8
0
5.5
3.5
2
7.5
8.5
18
15
9.5
10.5 5
10
28.5
13
15.5
30
19
155
8
13
12
10
15
17
19
5
5
5
5
5
5
5
5
0.22
0.1
0.3
0.1
0.2
0.1
0.3
0.1
0.5,00.6
0.5, 0.6 0
0.5, 0..6
0.5, 0.6 6
0.5,0.6
0.5, 0.6
0.5, 0.6
0.5, 0.6
464
Table 6. Vehicles data secon nd scenario Step 1.
465
We reporrt in Figure 10 the evo olution of
466
for the moost relevantt decision vaariables
over time, t and in n Table 7 thhe values determined d
467 468
Figure 10. Evolutionn of Vehicle ou utput
Vehiccle 1
Vehiclle 2
Vehiclee 3
in Scenario o 3 Step 1. Vehicle 4
Vehicle 5
Vehicle 6
Vehicle 7
Vehicle 8
Di[kWh]
2
0
2
2
0
2
0
2
0
-
6
21
-
15
-
10
5
-
9
24
-
10
-
15
10
-
8
7
-
10
-
14
1
0
1
1
0
1
0
1
469
Table 7. Third scenario Output step 1.
470
In the solution of the first optimization step, vehicle 2, 5 and 7 are not serviced. Note that the
471
service sequence in the solution is not in accordance with the order of release times. In fact, it is
472
possible to say that the presented approach jointly optimizes timetabling and sequencing.
473
When the second optimization step is defined, two of the above three vehicles (namely, 2 and 5)
474
are definitively discarded, since both their due-dates(10,5 and 13 discretization steps,
475
corresponding to 52,5 and 65 minutes, respectively) are overcome by the initial discretization
476
step (24, that is 2 hours), that represents the starting time of the second optimization step. Instead,
477
vehicle 7 is considered in the statement of the problem (as its due date is 30, that is 2,5 hours),
478
along with 4 new vehicles arriving within the new optimization interval. Table 8 reports the data
479
for the second optimization step. Note that time values are referred to a new “zero”,
480
corresponding to the initial time instant of the optimization interval in the second step. Vehicle data
Dmax,i[kWh] ,,
[kWh]
[€/h] G1, G2 [€/kWh]
Vehicle 7
Vehicle 9
Vehicle 10
Vehicle 11
Vehicle 12
0
10
12
7
18
6
20
19
21
32
17
22
13
20
10
5
5
5
5
5
0.2
0.1
0.3
0.1
0.2
0.5,0.6
0.5, 0.6
0.5, 0.6
0.5, 0.6
0.5,0.6
481
Table 8. Vehicles data second scenario Step 2.
482
Figure 11 shows the sequence of services of step 2. In this case the microgrid manager does not
483
accept vehicle 12, but this one can be considered in the next step..
484 485 486
Figure 11. Evolutionn of
in Scenario o 3 Step 2.
Table 9 shhows the ouutput variab bles for this second step p. Vehicle output
Di[kkWh]
Veehicle 7
Veehicle 9
Veh hicle 10
Veh hicle 11
Vehhicle 12
2
2
2
2
0
1
17
13
7
-
6
23
16
12
-
12
17
7
15
-
1
1
1
1
0
487
Ta able 9. Third rd scenario Output step p 2.
488
Finally, it is important to note that the ruun time turrns out to be b acceptabble for each h Scenario
489
considereed. Namely, For Scen narios 1 annd 2 the maximum ru un time is 1 hour, wh hereas for
490
Scenario 3 (both stepps) is lower than 3 minuutes.
491 492
493
5. Conclusions and future developments
494
A new optimization model is here formalized to include electrical vehicles in a smart grid. On the
495
basis of the requests and preferences of users of EVs, the grid’s manager decides the optimal
496
schedule of its generation plants and storage systems, taking into account the non deferrable
497
internal demand and renewable availability. The developed model is applied by using real data,
498
and different scenarios related to the demand of EVs’ charging have been considered.
499
In the proposed model, an attempt has been made to model the behavior of different decision
500
makers. Namely, the vehicles’ owners may reduce their service request on the basis of an elastic
501
demand function (assumed to be the same for vehicles, for the sake of simplicity). Instead, the
502
service station manager has different degrees of freedom, as he/she can vary the unit price for
503
service, and may even decide against servicing one or more vehicles. Besides, also the time
504
intervals during which the various services take place are determined through the solution of the
505
optimization problem. These services are imposed to be non preemptive and their order does not
506
necessarily reproduces the order of the arrivals of the vehicles. Thus, is possible to say that even
507
the service sequence is determined by the solution of the optimization problem. Of course,
508
different interaction schemes among the various decision makers can be conceived, and this is
509
matter of current research.
510
The problem has been formalized in a discrete-time setting. Obviously, this choice is not the
511
unique possible. In fact, also a discrete-event formalization of the model is possible, in which the
512
only time instants represented are those corresponding to decisions to be taken. The development
513
of a decision model based on a discrete-event representation of the dynamics of the system is
514
actually in progress.
515
516
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613
List of figures and tables
614
Figures
615
Figure 1. Microgrid design.
616
Figure 2. Charging of two different vehicles
617
Figure 3. The elastic demand function
618
Figure 4. Electrical demand, PV power production.
619
Figure 5. The evolution of the state of charge of the three vehicles (Scenario 1).
620
Figure 6. The evolution of the state of charge of the two microgrid storage systems.
621
Figure 7. Power flows in the microgrid (Scenario 1). Ps is the sum of
622
Pvehicles is the sum of power flows
623
Figure 8. Evolution of
624
Figure 9. The evolution of the state of charge of the four vehicles (Scenario 2).
625
Figure 10. Evolution of δ t
626
Figure 11. Evolution of δ t in Scenario 3 Step 2.
627
Tables
628
Table 1. Power flow data.
629
Table 2. Data for the first scenario.
630
Table 3. First scenario output.
631
Table 4. Vehicle data Second scenario.
632
Table 5. Second scenario results
633
Table 6. Vehicles data second scenario Step 1.
634
Table 7. Third scenario Output step 1.
635
Table 8. Vehicles data second scenario Step 2.
636
Table 9. Third scenario Output step 2.
→ ,
→
for all i.
in Scenario 1.
in Scenario 3 Step 1.
and
→
.