Optimized electric vehicle charging integrated in the unit commitment problem

Abstract

Displacing combustion engine vehicles with electric ones has recently emerged for reducing adverse environmental impacts and dependencies on fossil fuels. However, high electric vehicle penetration might disrupt the smooth operation of power sectors due to increased peak loads. A thorough investigation is therefore required, considering the charging of multiple electric vehicles, as flexible loads, in the Unit Commitment Problem. A novel approach for resolving this Operational Research problem is hereby presented, combining power flow and transmission constraints with various scenarios of electric vehicles’ penetration. A variant of Differential Evolution, aided by heuristic repair mechanisms, Priority Lists and advanced State-of-the-Art constraint handling techniques, is implemented to obtain feasible, near-optimal solutions. Well-established power systems including transmission constraints were used as benchmarks for testing the method proposed. The results are compared with those of a Mixed Integer-Linear algorithm based on the same formulation. They indicated that low and average demand cases might be resolved efficiently using the evolutionary approach proposed. As for large scale fleets, they might be handled by power systems at near optimal states exhibiting viable and resilient production schedules.

Appendices

Appendix A: case study for the 28 unit system (greek power system)

Initially, information was collected to set up the case study based on the Greek power system, i.e. the thermal energy production units, their technical characteristics and the energy demand of the Greek market in 2019. The values a, b, c of the quadratic cost function were calculated by data found in admie (Greece’s Independent Power Transmission Operator), desfa (Hellenic Gas Transmission System Operator S.A.) and freemeteo.gr. The values of Cold Startup Cost, Hot Startup Cost, minimum up time, minimum down time and Tcold were taken and matched from Schröder et al. (2013). Ramp up and Ramp down were taken and matched from Skea (2012). Finally, the source for Pmax and Pmin was also admie. These data are the technical characteristics of the units of a system consisting of 28 conventional thermal units. The first 14 units use lignite as fuel and the last 14 units use natural gas to generate electric power. These technical characteristics are demonstrated in Table 15.

Table 15 Input for the case study of 28 unit system

Moreover, the data concerning the energy demand of the Greek energy market were collected from admie, where detailed Excel sheets are provided for each day of the year. Those sheets contain information about the energy production of each unit for every hour. In this paper, we studied four cases:

• 1. A 24-h schedule for a day with high load demand

• 2. A 24-h schedule for a day with average load demand

• 3. A 24-h schedule for a day with low load demand

• 4. A 24-h schedule for a day with high load demand plus 25% of this load.

The day with high load demand was a 24 period (6 a.m. to 5 a.m.) from 09/07/2019 to 10/07/2019, representing a day with one of the highest power demands for Greece in the year. The day with average load demand was a 24 period (6 a.m. to 5 a.m.) from 01/02/2019 to 02/02/2019, representing a day with an average power demand in year 2019 for Greece. The day with low load demand was a 24 period (6 a.m. to 5 a.m.) from 01/05/2019 to 02/05/2019, representing a day with one of the lowest power demands in year 2019 for Greece. The day with high load demand plus 25% refers to the 24 period of high load demand. The load demand of each hour refers to the sum of the demand that thermal units and RES generated every hour. This amount was calculated from the data of the Excel sheets previously mentioned. The value of FD factor, i.e. the penetration of RES, was calculated according to the amount of energy that RES produced (Table 16).

Table 16 Demand and FD values for the different case studies

Subsequently, a driving pattern of electric vehicles was developed for a 24-h period. The data were gathered from UK National Travel Survey 2018 (Evans et al. 2019). More specifically, the Excel sheets from the same source, the website of the Department for Transportation (gov.uk), named ‘Average number of trips by trip length and main mode’ and ‘Trip purpose by trip start time’ are used in order to present this driving pattern. From sheet ‘Trip purpose by trip start time’ the percentage of trips that started each hour for car / van driver was found. In 'Average number of trips by trip length and main mode' the number of trips made by each person per year can be found. An adaption is made of this information in order to compute the daily trips per vehicle category. In addition, the percentage that corresponds to each trip’s length category depending on the kilometers traveled was calculated. It is assumed that the kilometers traveled for each category is the average value of the range shown in the Excel sheet. Thus, the following needed information is extracted:

• 1. The percentage of trips starting each hour

• 2. The percentage of trips of each vehicle category depending on the trip length

It is assumed that all the trips are two-way trips. For example, an EV covers 28.2 km with two trips, each 14.1 km. It is also assumed that there are no trips starting after 1.00 a.m. till 5.00 a.m. The EV’s parameters and data concerning starting time and EVs categories are summarized in Tables 17 and 18.

Table 17 Parameters of the EVs used in the paper

Table 18 Starting time of two-way trips over a 24-h period

The total number of EVs in the system EV_P100% is equal to the sum of the number of EVs performing a two-way trip EV_two in a 24-h period.

$$EV_{P100\% } = EV_{two}$$

(66)

The number of trips in the two-way trip group Trips_two is twice the number of EVs making two trips:

$$Trips_{two} = \, 2 \cdot EV_{two}$$

(67)

The total number of trips Trips_tot is equal to number of trips in the two-way trip group:

$$Trips_{tot} = Trips_{two}$$

(68)

After parameter Trips_tot has been calculated, Tables 18 and 19 are used to determine the number of trips started every hour. As the driving patterns and consumed energy of the EVs covering the trip lengths reported in Table 19 are different, the EVs in the system are split into eight distinct groups according to these trip lengths. Consequently, the whole set of EVs can be expressed as:

• E: set of all EVs, consisting of eight groups of EVs e travelling the same distance $e \in E= \left\{\mathrm{1,2},\dots ,8\right\}$.

Also, three other tables concerning electric vehicles are computed, containing the values of the constant MaxTransf_e,j, the values of the constant ConsEn_e,j and the values of the constant BFulle. The data, calculated in MW, are presented in Tables 20 and 21.

Table 19 Category of EVs according to the trip length and percentage of total trips

Table 20 Input for a million electric vehicle

Table 21 Input for 350 thousand electric vehicles

Appendix B: the linear UCP&EV formulation

In order to assess the results of the DE algorithm a linear formulation of the UCP, including the electric vehicles as a flexible load, is created.

Total Operating Cost:

$${minimize TOC}_{{P}_{i}^{j}{st}_{i}^{j}}= \sum_{j=1}^{T}\sum_{i=1}^{N}\left[FC\left({P}_{i}^{j}\right)+{GenSC}_{i}^{j} \right], \quad\forall j\in T,\quad\forall i\in N$$

(69)

The Eqs. (70)–(76) formulate the piecewise approximation that determines the generators’ fuel cost.

$FC\left({P}_{i}^{j}\right)$ in Eq. (69) is given by:

$$FC\left({P}_{i}^{j}\right)={FC}_{min}{\cdot st}_{i}^{j} + \sum_{sgm=1}^{LS}{K}_{sgm,i}^{j}\cdot {\xi }_{sgm,i}^{j}, \quad\forall i\in T,\quad\forall j\in N(B.2)$$

(70)

$${P}_{i}^{j}=\sum_{sgm=1}^{LS}{\xi }_{sgm,i}^{j} +{P}_{{min}_{i}}{\cdot st}_{i}^{j} , \quad\forall i\in T,\quad\forall j\in N(B.3)$$

(71)

$${\xi }_{1,i}^{j}\le {sl}_{1,i}^{j}-{P}_{{min}_{i}}, \quad\forall j\in T,\quad\forall i\in N (B.4)$$

(72)

$${\xi }_{sgm,i}^{j}\le {sl}_{sgm,i}^{j}-{sl}_{sgm-1,i}^{j}, \quad\forall j\in T,\quad\forall i\in N ,\quad\forall smg=2\dots \left(LS-1\right) (B.5)$$

(73)

$${\xi }_{LS,i}^{j}\le {P}_{{max}_{i}}- {sl}_{LS-1,i}^{j}, \quad\forall j\in T,\quad\forall i\in N (B.6)$$

(74)

$${\xi }_{sgm,i}^{j}\ge 0, \quad\forall j\in T,\quad\forall i\in N, \quad\forall smg=1\dots LS (B.7)$$

(75)

${FC}_{min}$ in Eq. 70 is given by:

$${FC}_{min}= {a}_{i}+ {b}_{i}\cdot {P}_{{min}_{i}}^{j}+ {c}_{i}\cdot {\left({P}_{{min}_{i}}^{j}\right)}^{2}, \quad\forall j\in T,\quad\forall i\in N, \quad\forall smg=1\dots LS (B.8)$$

(76)

The start-up cost step function is formulated in the Eqs. (77) and (78) with a total of 15 intervals (TP = 15).

$${GenSC}_{i}^{j}\ge {SC}_{i}^{tp}\cdot \left[\left.{st}_{i}^{j}-\sum_{n=1}^{tp}{st}_{i}^{(j-n)}\right], \quad\forall j\in T,\quad\forall i\in N ,\quad\forall tp=1\dots TP (B.9)\right.$$

(77)

$${GenSC}_{i}^{j}\ge 0, \quad\forall j\in T,\quad\forall i\in N (B.10)$$

(78)

The ramp up and ramp down constraints are formulated as follows; The Eqs. (79) and (80) formulate the Generation limits of the system and Eqs. (81)-(83) formulate the ramp rates of the system (${{p}_{max}}_{i}^{j}$ is given by Eq. 92):

$${P}_{{min}_{i}}\cdot {st}_{i}^{j}\le {P}_{i}^{j}\le {{p}_{max}}_{i}^{j}3\quad\forall j\in T,\quad\forall i\in N (B.11)$$

(79)

$$0\le {{p}_{max}}_{i}^{j}\le {P}_{{max}_{i}}\cdot {st}_{i}^{j}, \quad\forall j\in T,\quad\forall i\in N (B.12)$$

(80)

$${P}_{i}^{j}-{P}_{i}^{j-1}\le {RUp}_{i}\cdot {st}_{i}^{j-1}+{Sul}_{i}\cdot \left[\left.{st}_{i}^{j}-{st}_{i}^{j-1}\right]\right.+{P}_{{max}_{i}}\cdot \left[\left.1-{st}_{i}^{j}\right], \right.\quad\forall j\in T,\quad\forall i\in N (B.13)$$

(81)

$${{p}_{max}}_{i}^{j}\le {P}_{{max}_{i}}\cdot {st}_{i}^{j+1}+{Sdl}_{i}\cdot \left[\left.{st}_{i}^{j}-{st}_{i}^{j+1}\right]\right., \quad\forall j\in 1\dots (T-1),\quad\forall i\in N (B.14)$$

(82)

$${P}_{i}^{j-1}-{P}_{i}^{j}\le {RDown}_{i}\cdot {st}_{i}^{j}+{Sdl}_{i}\cdot \left[\left.{st}_{i}^{j-1}-{st}_{i}^{j}\right]\right.+{P}_{{max}_{i}}\cdot \left[\left.1-{st}_{i}^{j-1}\right], \right.\quad\forall i\in N,\quad\forall j\in T (B.15)$$

(83)

The formulation of the minimum up time and the minimum down time limits are defined by the Eqs. (84)–(87) for the minimum up time and the Eqs. (88)–(91) for the minimum down time.

$$\sum_{n=1}^{{Z}_{i}}\left[\left.1-{st}_{i}^{n}\right]\right.=0, \quad\forall j\in T ,\quad\forall i\in N(B.16)$$

(84)

$$\sum_{n=j}^{j+{Ton}_{i}-1}{st}_{i}^{n} \ge {Ton}_{i}\left[\left.{st}_{i}^{j}-{st}_{i}^{j-1}\right]\right., \quad\forall i\in N \quad\forall j=\left({Z}_{i}+1\right)\dots \left(T-{Ton}_{i}+1\right) (B.17)$$

(85)

$$\sum_{n=j}^{T}\left[{st}_{i}^{n}\right. \left.-\left[\left.{st}_{i}^{j}-{st}_{i}^{j-1}\right]\right.\right] \ge 0, \quad\forall i\in N, \quad\forall j=(T-{Ton}_{i}+2)\dots T (B.18)$$

(86)

$${Z}_{i}=Min\left\{\left.T,\left[\left.{Ton}_{i}-{T}_{inistateUP,i}\right]{\cdot st}_{i}^{0}\right.\right\}\right. (B.19)$$

(87)

$$\sum_{n=1}^{{Y}_{i}}\left[\left.{st}_{i}^{n}\right]\right.=0, \quad\forall j\in T,\quad\forall i\in N (B.20)$$

(88)

$$\sum_{n=j}^{j+{Toff}_{i}-1}\left[\left.{1-st}_{i}^{n} \right]\right. \ge {Toff}_{i}\left[\left.{st}_{i}^{j-1}-{st}_{i}^{j}\right]\right. \quad\forall j\in N \quad\forall i=\left({Y}_{i}+1\right)\dots \left(T-{Toff}_{i}+1\right) (B.21)$$

(89)

$$\sum_{n=j}^{T}\left[1-{st}_{i}^{n}\right. \left.-\left[\left.{st}_{i}^{j-1}-{st}_{i}^{j}\right]\right.\right] \ge 0, \quad\forall i\in N, \quad\forall j=(T-{Toff}_{i}+2)\dots T (B.22)$$

(90)

$${Y}_{i}=Min\left\{\left.T,\left[\left.{Toff}_{i}-{T}_{inistateDOWN,i}\right]\cdot \left[1-\left.{st}_{i}^{0}\right]\right.\right.\right\}\right. (B.23)$$

(91)

In the Eqs. (14) and Eq. (19) the grid-to-vehicle optimized charging of the electric vehicles is formulated.

Equation (25) describes the transmission constraints. The nodal injection can include variables such as the power production and the electric vehicle load.

Finally Eqs. (6) and Eq. (7) formulate the system’s balance and spinning reserve respectively.

${p}_{{max}_{i}}(j)$ is defined as follows:

$${{p}_{max}}_{i}^{j}={st}_{i}^{j}\cdot {P}_{{max}_{i}}, \quad\forall i\in N,\quad\forall j\in T (B.24)$$

(92)

Shut down limit and Start up limit are set to Pmax.