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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

Stochastic ﬂexible transmission operation for coordinated integration of plug-in electric vehicles and renewable energy sources

T

⁎

Ahmad Nikoobakhta, Jamshid Aghaeib,c, , Roohallah Khatamid, Esmaeel Mahboubi-Moghaddame, Masood Parvaniad a

Higher Education Center of Eghlid, Eghlid, Iran Department of Electrical and Electronics Engineering, Shiraz University of Technology, Shiraz, Iran c Department of Electric Power Engineering, Norwegian University of Science and Technology (NTNU), Trondheim NO-7491, Norway d Department of Electrical and Computer Engineering, University of Utah, Salt Lake City, UT 84112, USA e Department of Electrical Engineering, Quchan University of Technology, Quchan, Iran b

H I GH L IG H T S

G R A P H I C A L A B S T R A C T

transmission technologies are • Flexible used to increase wind energy and EVs. transformers are modeled • Adjustable as ﬂexibility tools to transfer energy. paper deals with day-ahead elec• The tricity energy markets. is proposed to model stochastic • MILP optimization framework.

A R T I C LE I N FO

A B S T R A C T

Keywords: Plug-in electric vehicles Renewable energy sources Stochastic security constrained day-ahead scheduling On-load tap changing transformers Phase shifting transformers

In this paper, the large-scale integration of plug-in electric vehicles (PEVs) with Vehicle-to-Grid (V2G) technology is taken as a promising way to promote the integration of renewable energy sources (RESs). But, it is a challenging task for the independent system operator (ISO) to manage the power ﬂows in its system, especially in the large-scale integration of PEVs and RESs into the transmission system. The coordination between PEV, as distributed storage devices, and variable RESs create variable power ﬂows and loop ﬂows and quests for utilization of controllable devices to manage these ﬂows. The On-load tap changing (OLTC) transformers and phase shifting transformers (PSTs) are two ﬂexible transmission technologies that can be solved this challenge through controlling the power ﬂows. In this paper, a novel model is developed for optimal operation of ﬂexible transmission technologies for coordinated integration of PEVs and RESs in power transmission networks. The proposed model has been implemented on the six-bus and the IEEE 118-bus test systems. The simulation results imply that: (i) The V2G technology of PEV battery can ease the impact of variability of RES on power system operations and reduce operation cost and wind power spillage; (ii) substantial economic savings can be achieved through utilization of both OLTC transformer and PSTs beyond the independent capabilities of each technology.

⁎

Corresponding author. E-mail addresses: [email protected] (A. Nikoobakht), [email protected] (J. Aghaei).

https://doi.org/10.1016/j.apenergy.2018.12.089 Received 30 April 2018; Received in revised form 26 November 2018; Accepted 30 December 2018 0306-2619/ © 2019 Elsevier Ltd. All rights reserved.

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ψ¯ / ψ̲

Nomenclature

upper/lower bound for the optimal value of objective function at iteration i (·) (·) (·) (·) (·) (·) ξ(·) , ς(·) , v(·) , γ(·) , ζ (·) , η(·) dual variables

Indices g/w n, m k t ω v

Binary variables

index for thermal units/wind farms indices for system buses index for transmission lines index for time periods index for scenario index for stationary EVs

commitment status-of-unit g at period t ugt uc v,(·)/ ud v,(·) indicator of stationary EVs v in charging/discharging modes Parameters

Continuous variables (·) δnm

φk Tk Vnt(·) Pgt(·)/ qgt(·)

the forecast value of wind energy generation of wind farm w Dn,(·) / Qn,(·) active/reactive load at bus n RgU / RgD up/down corrective action limit of unit g RvU / RvD up/down corrective action limit of stationary EVs ratio of the number of EVs in a stationary in scenario ω to NEv, tω the number of base case EVs gk / bk conductance/susceptance of line k Cg / Cv cost of thermal unit g and stationary EVs v πω probability of scenario ω value of wind power spillage ρw α the percentage of stationary EVs that are operating in storage mode γnt power factor of load at bus n round-trip eﬃciency of stationary EVs v ηv CgtSU startup cost of thermal unit g Pgmax / Pgmin max/min active power generation of unit g qgmax / qgmin max/min reactive power generation of unit g.

Pw,(·)

phase angle diﬀerence across line k(n, m) at period t phase angle of the PST k tap ratio of OLTC k voltage magnitude at bus n at period t active/reactive power output-of-thermal unit g at period t

U D rgtω / rgtω

up/down corrective action deployed by unit g in period t and scenario ω D U up/down corrective action deployed by stationary EVs v in rvtω / rvtω period t and scenario ω Pc v,(·)/ Pd v,(·) charging/discharging power of stationary EVs v 0 charging/discharging power rate of stationary EVs v ΔPv,(·) c&d Pv (·) the stationary EVs v operating in charging/discharging modes Pvc(·) the stationary EVs v operating in charging mode net discharged energy of stationary EVs v En v,(·) Ev,(·) state of charge of EVs v, as a storage energy wind power spillage WS(·) flkP,(·) / flkq,(·) active/reactive power ﬂow on line k (n, m) H &N W &S C(·) / EC(·) operation cost in (here-and-now)/ (wait-and-see) condition χ value of objective function in the wait and see subproblem κ1,(·), κ2,(·) slack variables mismatch of here-and-now realizations sub-problems Rt Ψ t(i) wait and see production cost at iteration i and period t

Evmax / Evmin max/min energy stored in stationary EVs v loss active loss in line k fl¯k max Vn / Vnmin max/min of voltage magnitude δkmax / δkmin max/min of voltage angle diﬀerence across line k φkmax / φkmin max/min phase of the PST k αnm, l/ βnm, l slope-of-the l th-piecewise-linear-block/value-of-the-linearized cos(δmn) at l

1. Introduction

storage capacity provided by PEV would be prominent. Aggregated PEVs can act as a distributed storage system and demand which can decrease the inﬂuence of uncertainties imposed on power grids by the high penetration of WPG and reduce wind power spillage (WPS) [10]. Considering the aggregators as the association between the distribution network and the bus-level transmission network, the operation of PEV would be a two-stage solution:

During the last decade, the share of variable renewable energy sources (RESs) in power systems operation has been at the center of much attention [1]. Among various types of RESs, wind power generation (WPG) plays a vital role in the most modern power systems. A major advantages of WPG are eﬃciency and economic merit [2]. Unfortunately, due to its stochastic and intermittent nature, the integration of large-scale wind power brings two main challenges for power system operation: (i) uncertainty in wind power generation may have been an important factor in the imbalance between power supply and consumption [3] and (ii) higher uncertainty in energy ﬂows due to variable injections [4]. The ﬁrst challenge can be solved by the aggregated storage capability of PEVs. From the power system operator perspective, PEVs are equipped with batteries that can also be used as energy storage systems, due to the fact that almost all PEVs stay parked for up to 96% time of a day, and this stay time is much longer than the necessary time to fully recharge the batteries [5]. During these stay times, PEVs may discharge electricity back to the grid using the V2G technology [6]. Therefore, independent system operator (ISO) can use PEVs as ﬂexible storage systems that compensate the uncertainty of WPGs [7]. That is, the PEV batteries can capture the wind power excess during oﬀ peak periods once the demand is low, and improve the reliability of the energy supply during peak hours when the demand is high and WPG is low [8,9]. As the penetration of PEV ﬂeet increases in power systems, the

(1) Once the PEVs are aggregated at the distribution level, the distribution company (DISCO) would track the mobility of individual PEVs and coordinate the PEV charging and discharging sequence at the distribution level. The DISCO would submit the information of the marginal cost, storage capacity, and the number of PEV ﬂeets in each hour to the ISO for participation in the day-ahead market [8,11]. (2) The ISO will run the security constrained unit commitment problem with stationary PEV ﬂeets to obtain the optimal PEV schedule considering the charge and discharge constraints. The ISO would optimally dispatch the PEV and send the corresponding scheduling signals back to the DISCO which would control the charging/discharging sequences of individual PEVs considering the consumer’s energy requirements in the day-ahead electricity market [8]. The iteration between the two-stages are continued until a balance point will be obtained. This paper will address the second stage of the day-scheduling at the ISO level in which the aggregated stationary PEV ﬂeets are participated in the day-ahead scheduling. Fig. 1 226

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increase transmission capacity [13,14]. Power ﬂow control technologies (PFCTs) allow for re-routing of the power, and, thus, enhance the transfer capability over the existing transmission network [15]. The ability to control power ﬂows, will become more crucial as the share of WPG uncertainty increases in the generation mix [16]. The ﬂexible transmission technologies with PFCTs, such as the onload tap changing (OLTC) transformers and phase shifting transformers (PSTs) can enhance the capability of power systems to respond to the uncertainties caused by the integration of RES and PEVs on the grid. The OLTCT support adjusting the local voltage and control the transmission power ﬂow by regulating the magnitude of voltages in the limited admissible voltage ranges [17]. In addition, they control the reactive power ﬂow by changing the voltage magnitude [17]. On the other hand, the PSTs can simultaneously control both voltage magnitude and angle [18]. Thus, it oﬀers a useful option for controlling the active power ﬂow under diﬀerent operational circumstances. In [19], the phase shifters redirect power ﬂows in a power grid, to improve security of the power system. Besides, the OLTCT and PSTs can help to manage variable energy ﬂows of transmission lines by increasing grid side ﬂexibility and facilitating the absorption of wind power generation (WPG) [20,16]. For this reason, the OLTCTs and PSTs are being broadly deployed in the recent years to accommodate more renewable energies as well as to minimize the energy loss and most importantly to manage energy ﬂow and voltage proﬁle [20,16]. Hence, including the modeling of these devices in the conventional day-ahead scheduling problem, will open new possibilities to control the active and reactive ﬂows in different optimization problems like OPF [21] and SCUC [22]. Several studies have been done to cover the economic aspects of integrating PEVs into electricity markets; however, they often lacked PFCTs [10]. In [23], PEVs are considered as the system demand in which V2G is not addressed, and the optimal charging sequence is determined according to several criteria. In [7], the modeling of aggregated EV ﬂeets as stationary distributed load and energy storage facilities for high penetration of wind energy has been studied. However, the wind uncertainty has not been addressed in this study. In [24], the inclusion of PEVs in microgrids is considered to reduce the on-peak demand, and improve the economic eﬃciency and increase the environmental sustainability. In this work, a two-stage energy management strategy has been developed for the contribution of PEVs in demand response (DR) programs of commercial building microgrids. Nevertheless, in this reference the PFCTs has not been considered. In [25], the impact of integration of large-scale renewable energy resources into the power grid is presented, but, the impact of PEV has not been addressed. Similarly, in [26,27], the eﬀect of DR in energy management has been reported, nonetheless, the eﬀect of PEV has not been addressed. Furthermore, in [28,29], a framework has been presented for the energy management based on the energy storage, but, the energy storage capability of PEV has not been addressed. With respect to the above mentioned research works in the area of PEVs, this paper is providing a deeper insight into the integration of aggregated PEV ﬂeets into power system operation to beneﬁt from the potential beneﬁts of V2G and mobility in decreasing operation cost and wind power spillage. Also, the eﬀect of the aggregated PEV ﬂeets on the optimal operation of power systems is investigated through several case studies. The V2G, mobility and consumer behaviors and their respective impacts on the optimal operation of power systems are addressed. Most studies in the ﬁeld of PST model have only focused on the DC optimal power ﬂow (DCOPF) to integrate more WPG and enabling the security evaluation [4,30]. However, the use of AC optimal power ﬂow has not been investigated. Also, [4,30] evaluate the impact of PST on the operation cost and wind power spillage based on DC OPF in transmission systems. However, in those research works, the stationary PEV ﬂeet has not been addressed. Besides, to have an OLTCT in the optimization problems, AC optimal power ﬂow (ACOPF) modeling is required to determine bus voltage magnitudes. However, most studies of DCOPF are unsuitable for

Fig. 1. Linkages between the distribution and transmission networks with ISO.

presents the details of linkage between the distribution and transmission networks with ISO which facilitates the bi-directional information ﬂow between individual PEVs, DISCO, transmission system and the ISO. The advanced metering infrastructure (AMI) facilitates the bi-directional communication and control signal ﬂows from PEV customers to DISCO [7]. The PEV customer’s data includes the battery state of charge and capacity, required energy, in addition to the origin and the destination and travel times. The bi-directional mechanism can be applied via connecting a device to each PEV in a ﬂeet which communicates with the DISCO through AMI. The communication hardware would be an extension to the PEV charging station which enables the DISCO and consumers to monitor and control the charging/discharging sequences [8]. The second challenge is insuﬃcient transmission capacity to deliver large amount of wind power from remote areas to the load centers. An obvious approach to improve the transmission capacity and facilitate the wind power integration is to build new transmission lines. Nevertheless, such projects are unappealing since they usually need high investment cost, long building time and stringent environmental approvals [12]. Therefore, it is essential to utilize the existing transmission network as eﬃciently as possible to remove congestion and 227

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several reasons: First, the DCOPF model’s solutions cannot consider AC feasibility, and the system security might be jeopardized when implementing the OLTCT and PSTs [13]. Second, the inaccuracy of the DCOPF may lead to the solutions with the OLTCT and PSTs that underestimate the cost savings or in some cases may result in higher costs [31]. Third, the DCOPF cannot be used to grasp other potential beneﬁts of the OLTCT, such as correcting temporary voltage violations [20]. Also, above-mentioned previous studies have investigated the DCOPF that provides only the active power of the network. The use of only active power for modeling the OLTCTs and PSTs in the optimization problems causes enormous error in the power system studies, especially for implementing OLTCT where the voltage magnitude plays a critical role [17]. The ACOPF is an important part in the power system operation, [32], and plays a key role in modeling the OLTCT and PSTs. Accordingly, it has a great signiﬁcance and motivate the research described in this paper. In general, the AC security constraints in a stochastic day-ahead scheduling problem with the OLTCT and PST devices results in a nonconvex mixed integer nonlinear programming (MINLP) optimization. This non-convexity partly is due to the non-linearity in the active and reactive power ﬂow equations which leads to local optimal solutions for this problem [33]. Accordingly, the MINLP models are generally intractable if directly solved by traditional MINLP solvers, i.e., CONOPT and SBB tools [33]. There are few works in this area that have addressed this computational challenge [33]. In this paper, to overcome the above-mentioned challenges, a linearized AC security constrained generation and PEV scheduling model with linear OLTCT and PST constraints has been developed, which maintains the accuracy within an acceptable level. The global optimality of the solution for the proposed model can be guaranteed by the commercial software and solvers, e.g., CPLEX solver [34]. Finally, generalized Benders decomposition (GBD) technique, as an eﬃcient technique to solve the proposed problem for large-scale systems in an iterative fashion is deployed [33].

P flnm = gk (Vn2 − Vn Vm cos δnm) − bk (Vn Vm sin δnm)

(1)

q flnm = −bk (Vn2 − Vn Vm cos δnm) − gk (Vn Vm sin δnm)

(2)

To enhance the computational eﬃciency and the robustness of the proposed solution, the nonlinear Eqs. (1) and (2) are linearized and approximated as follows: P flnm = gk (Vn − Vm − ψnm + 1) − bk δnm

(3)

q flnm = −bk (Vn − Vm − ψnm + 1) − gk δnm

(4)

In the above linear functions, while the typical range of voltage angle diﬀerence across transmission lines is lower than 10°, i.e., |δnm| ⩽ 10° , the function cos (δ) has been approximated by some piecewise linear (PWL) functions, ψ, in the general form as follows,

ψnm = αnm,ℓ (δnm) + βnm,ℓ

(5)

where αnm,ℓ and βnm,ℓ are calculated such that ψmn and cos (δmn) meet each other at break points. Note that, in the implemented PWL approach, there are more than one linearized segment. That is, in the approximation process, it is needed to compromise between eﬃciency and accuracy by changing the number of linear segments. The approximation errors associated with this model can be found in [36]. 2.2. Linearizing power ﬂow including OLTCT and PST The models of the OLTCT, the PST and ﬁxed tap transformer are based on [17]. As shown in Fig. 2, the OLTCT/ PST is connected between buses n and m and adjusts the voltage /phase angle at bus m. Accordingly, the active and reactive power ﬂows are obtained as follows, respectively: P flmn = gk (Vn2 − Tk Vn Vm cos(δmn − φk )) + Tk Vn Vm bk sin(δmn − φk )

(6)

q flmn = −bk (Vn2 − Tk Vn Vm cos(δmn − φk )) + Tk Vn Vm gk sin(δmn − φk )

(7)

The linear approximation of the (6) and (7) are similar to Eqs. (1) and (2). Here, according to the ﬁrst assumption, the tap ratio of the transformers, Tk, is always close to 1.0 per unit (p.u.), while it is similar to the bus voltage magnitudes, i.e., 0.95 ≤ Tk ≤ 1.05.

1.1. Contributions The main contributions of this work are twofold: (i) Developing a linearized AC security constrained generation and PEV scheduling model with linearized OLTCT and PST constraints. (ii) Proposing a linearized AC stochastic security constrained dayahead scheduling (LAC-SCDAS) problem framework to coordinate the OLTCT and PST with stationary PEVs storage, in order to enhance both the PEVs penetration and WPG absorption under wind uncertainty condition.

P flmn = gk (Vn − Vm − Tk − ψmn + 2) − bk (δmn − φk )

(8)

q flmn

(9)

= −bk (Vn − Vm − Tk − ψmn + 2) − gk (δmn − φk )

Assuming that the voltage angle, accounting for the phase shifter angle, ranges between |δmn − φk | ⩽ 10° , similar to (3) and (4) the function cos(δmn − φk ) can be approximated by ψmn as follows:

ψmn = αmn,ℓ (δmn − φk ) + βmn,ℓ

(10)

To have a better explanation about the linearization assumption, please note that it is assumed that the diﬀerence between the voltage angle diﬀerence across transmission lines, i.e., δmn, and the phase shifter angle, i.e., φk, should be ranged as |δmn − φk| ≤ 10°. That is, the range of φk maybe is more than 10 degrees. Noted that, while the range of δmn is not so large, therefore, the total range of δmn – φk should not violate the 10 degrees. However, in the case of applying transmissionswitching action or occurring contingencies for units or lines of the power systems, it is probable to have the large values of the φk which

To the best of the authors' knowledge, no research work in this area has provided the MILP formulation for stochastic day ahead scheduling problem with linearized OLTC and PST models and the stationary PEVs energy storage, in the presence of the wind power uncertainty. 2. Linearization of the non-linear AC power ﬂow The linearized AC power ﬂow model is presented in this section to maintain a linear representation of reactive power as well as bus voltage magnitudes. The main assumptions in linearization of the full AC model proposed in [35] are as follows: (i) the bus voltage magnitude is close to 1.0 per unit (p.u), (ii) the voltage angle diﬀerence across a line, δ, is close to zero (or very small) resulting in sin (δ) ≈ δ and cos (δ) ≈ 1.

Vn

1: Tk e jϕk

Vm g k + jbk

2.1. Power ﬂow linearization procedure Fig. 2. A transmission line with on-load tap changer and phase-shifting transformers.

The active and reactive power ﬂows in transmission line k between buses n and m are written as follows: 228

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function (WDP) and a normal probability distribution function (NPDF) are assigned to the forecast errors of wind generation and PEVs parameters, respectively. Then, using the ARIMA method [39], the WDPs and NPDFs are utilized in scenarios generation. Note that the forecast error value is randomly determined by WDF and NPDF. Then this value is added up to the forecast value of uncertain parameters produced by ARIMA. An eﬃcient method based on the probability distance technique [39] is applied for scenario reduction to have much more computationally eﬃcient optimization framework. Finally, the reduced scenarios are considered as the most possible realization of the system in the proposed optimization problem to obtain the optimal day-ahead scheduling including OLTCT, PST and stationary PEVs energy storage models.

may not satisfy the acceptable linearization range, i.e., |δmn − φk| ≤ 10°. This matter may result in an inaccurate linearization of (8) and (9). Also, if the transformer regulates the voltage /phase at bus n, the active and reactive power ﬂows are, respectively. P flnm = gk (Tk2 Vn2 − Tk Vn Vm cos(δnm − φk )) + bk Tk Vn Vm sin(δnm − φk )

(11) q flnm

=

−bk (Tk2 Vn2

− Tk Vn Vm cos(δnm − φk )) − gk Tk Vn Vm sin(δnm − φk ) (12)

The linear approximation of the (11) and (12) are similar to Eqs. (8) and (9). P flnm = gk (Vn + Tk − Vm − ψnm) − bk (δmn − φk )

(13)

q flnm = −bk (Vn + Tk − Vm − ψnm) − gk (δmn − φk )

(14)

3.3. Mathematical formulation

where the sub-index k identiﬁes the component between buses n and m. Note that, in (13) and (14), Tk is the tap ratio of OLTCT k and φk is the phase angle of the PST k. Also, the Eqs. (8, (9) and (13), (14) are valid for OLTCTs and PSTs subject to following conditions:

Here, a two-stage stochastic programming problem is proposed to deal with the uncertainties. The ﬁrst stage, here-and-now (H&N) decisions, refers to the economic dispatch and commitment of thermal units where the decision variables are determined before the realization of the uncertainties. Also, the second stage, wait-and-see (W&S) decisions, relates to the optimal power ﬂow formulations. The H&N and W&S decision variables are as follows: H&N decisions include

(i) OLTCT: k = (n, m) ∈ ΩLTC , Tk is a variable and φk = 0 . (ii) PST: k = (n, m) ∈ ΩPST , φk is a variable and Tk is a constant. (iii) Fixed tap transformer: k = (n, m) ∈ Ωk , Tk is a constant and φk = 0 . 3. Mathematical stochastic optimization formulation

– active/reactive day-ahead power dispatch of thermal units – the stationary PEVs day-ahead energy storage charging and discharging – commitment statuses of thermal units – the OLTCT and PST setting points according to grid operation condition.

In this section, ﬁrstly the uncertainty characterization will be discussed and then the formulation of the stochastic security constrained day ahead scheduling problem will be proposed. 3.1. Assumptions

W&S decisions can be classiﬁed into two sets: For the sake of clarity, the main modeling assumptions are summarized below:

– power dispatch of committed thermal units in the ﬁrst stage – the real-time charging/discharging energy of stationary PEVs storage, operations of OLTCT and PST, power output of thermal units, WES as the second set of variables

(i) Uncertainties associated with WPG, and number of PEVs in a stationary are considered. However, the proposed model is capable of considering load uncertainty and equipment failure as well. (ii) In a stationary PEVs, two groups of PEVs are available which are operated in two diﬀerent manners: ﬁrst, the PEVs operated merely as loads (just charging mode); second, the PEVs operated as energy storage devices absorbing/injecting the energy from/to the grid for peak load shaving, congestion mitigation, and ramping scarcity alleviation purposes. Furthermore, the available energy capacity of the stationary PEVs’ storage can be used to provide spinning/regulation reserve service if necessary.

3.3.1. Objective function The stochastic day-ahead scheduling problem is implemented via the ISO for minimizing the expected operation cost of generating units and stationary PEV ﬂeet to obtain the optimal generation schedule considering the generation, transmission lines and stationary PEV ﬂeet constraints. The ISO optimally dispatches the generation resources available in the system and send signals to the DISCO to control the charging/discharging arrangements of PEV ﬂeets considering the consumer’s energy necessities in the day-ahead electricity market. At the distribution level, individual aggregators would maximize revenues by tracking the PEVs that stay parked for up to 96% time of a day for participating in electricity market and coordinating the hourly dayahead solution at the sub-transmission level by sending the optimal signals to end users to manage charging/discharging of individual PEVs. The proposed two-stage model minimizes the stochastic operation cost including the cost of H&N phase and expected cost of W&S subproblems as follows:

3.2. Uncertainty modeling The optimum coordination of OLTCT, PST and stationary PEV (as energy storage) taps the power systems potential ﬂexibility to mitigate the impact of the uncertainties. In the proposed stochastic security constrained day ahead scheduling problem, two main sources of uncertainties are considered: (i) wind forecast, and (ii) calculation of energy storage amount in stationary PEVs while it depends on uncertain parameters such as number of PEVs in a stationary. The current approaches for predicting possible wind generation scenarios resulting from real data to represent scenario parameters in the day-ahead scheduling along with their associated probabilities have been classiﬁed in [37,38]. However, the potential number of scenarios based on uncertainties could be overwhelming for real-time operation. While the focus of this paper is not scenario generation, however, to deal with the computational burden of scenarios, the following uncertainty characterization procedure is employed. Firstly, the Weibull distribution

min

∑ (CtH & N + ECtW & S )

(15)

t

CtH & N =

∑ (Cg·Pgt + CgtSU ) + ∑ (Cv·ΔPv,t ) g

229

v

(16)

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ECtW & S =

Pgt + Pwt + Pvtc& d − (Dnt + Pvtc ) =

U D U D ∑ πω·⎛⎜∑ Cg·(rgtω + rgtω ) + ∑ Cv·(rvtω + rvtω ) ω

⎝

g

∑ ∀ k (m, n)

flktP

∀ n (18)

v

qgt − Qnt =

⎞ + ρw ∑ WSwtω ⎟ w ⎠

∑

flktq

(19)

∀ k (m, n)

(17)

CtH & N

denotes here and now costs including the generation cost where, of thermal units, which comprises the generation and startup costs of generating units, as well as the operation cost of stationary PEVs energy storage at the normal condition. Likewise, ECtW & S indicates W&S costs resulting from corrections in the thermal units and the stationary PEVs energy storage as corrective actions. Also, the objective function includes the expected cost of corrective actions in scenarios as the consequences of uncertainties. Furthermore, the cost of WES is considered but WES action is not acceptable in normal condition, and the system should use the total forecasted WPG in the ﬁrst stage.

3.3.2. Here and now constraints The constraints of the ﬁrst stage, i.e., H&N decisions, are as follows:

P P ¯loss ∀ k , t flnm , t + flmn, t ⩽ flk

(20)

min max ⎧Vn ⩽ Vn, t ⩽ Vn ⎪ min ⩽ Tk, t ⩽ Tkmax T ⎨ k ⎪ φkmin ⩽ φk, t ⩽ φkmax ⎩

(21)

K gSU ·(ugt − ug, t − 1) ⩽ CgtSU ∀ g , t

(22)

max ugt Pgmin , t ⩽ Pgt ⩽ ugt Pg , t

(23)

max ugt qgmin , t ⩽ qgt ⩽ ugt qg , t

(24)

⎧ En v, t = ηv ·Pc v, t + Pd v, t ⎨ Pvc,&t d = Pc v, t + Pd v, t ⎩

(25)

Input data of (wind scenarios, load, thermal units and stationary PEVs )

Master problem (MIP) (Find optimal hourly scheduling and dispatch of thermal units, wind farms and stationary PEVs)

First Subproblem (H&N condition) (hourly network security check with stationary PEVs, OLTCT and PST)

Violation?

Feasibility Cuts (34)

Second Subproblem (W&S condition) (hourly optimally check with stationary PEVs, OLTCT and PST)

Optimally Cut (65)

Convergence Check≤ε

Find Optimal Solution Fig. 3. Proposed solution strategy based on GBD technique. 230

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− α·P¯vtc ·uc v, t ⩽ Pc vt ⩽ −α· P̲ cvt ·uc v, t

(26)

α· P̲ vtd ·ud v, t ⩽ Pd vt ⩽ α·P¯vtd ·ud v, t

(27)

Ev, t = Ev, t − 1 − En v, t

(28)

|Ev, t − Ev, t − 1| = ΔPv, t

(29)

α·Evtmin ⩽ Ev, t ⩽ α·Evtmax

(30)

uc v, t + ud v, t ⩽ 1

(31)

∑ ∀ k (m, n)

P P ¯loss flnm , tω + flmn, tω ⩽ flk

(46)

Because of the massive number of variables owing to the scenarios, the GBD technique is employed to decompose the original large-scale problem into one master problem and some sub-problems as illustrated in Fig. 3. The initial scheduling of units and stationary PEVs’ energy storage (for the base case) are obtained, in the master problem, based on the available market data. After solving the master problem, the ﬁrst subproblem checks the hourly grid security and ﬁnds the OLTCT and PST setting points. If violations persist, the feasibility cuts are generated and added to the master problem. Otherwise, the master problem solutions are also passed on to the second sub-problem to check the optimality of each scenario. If the convergence check results in an undesirable accuracy level, an optimality cut will be produced and served to the master problem in the next iteration. Finally, the iterative procedure will be stopped when the master solution satisﬁes the convergence check with a reasonable accuracy.

∀n 4.1. Master problem (32)

qgtω + Qnt =

U D ⎧ Pgtω = Pgt + rgtω − rgtω ⎪ U U 0 ⩽ rgtω ⩽ Rg ugt ⎨ D ⎪ 0 ⩽ rgtω ⩽ RgD ugt ⎩

4. Proposed solution methodology

3.3.3. Wait-and-See constraints The constraints of the W&S decision are listed as follows:

q flktω

(45)

Constraints (32)–(46) model the real-time operation of power system through linear AC power ﬂow equations. The constraints (32)–(43) are the real-time counterparts for the (18)–(31). The realtime corrective actions including the real-time dispatch deviation of stationary PEV batteries and generating units from the associated dayahead schedules are enforced by (44)–(46), where rvU, tω/ rvD, tω and rgU, tω/ rgD, tω represents physically acceptable power adjustments of energy storage of stationary PEV batteries and thermal units in ten minutes (i.e., 10/60 of hourly ramping of thermal units) to absorb the uncertain WPGs.

In the above formulations, the AC power ﬂow equations include the active and reactive power ﬂow in transmission lines (18) and (19), line ﬂow limits (20), voltage magnitude, transformer tap and phase limits (21), the linear startup cost of thermal units (22), active and reactive power generation limits of thermal units (23) and (24). The PEV storage energy constraints are demonstrated in (25)–(31) where (25) refers to the net hourly absorbed/injected energy and the dispatched power of stationary PEV batteries. Here, the round-trip efﬁciency of the stationary PEV batteries is considered. The limitations on stationary PEV batteries charging/discharging imposed by power electronic interfaces and charging stations are shown in (26) and (27). The state of charge in the battery is shown (28). To calculate the operation cost of stationary PEV batteries charge/discharge in (16), the value of ΔPv,t is calculated by (29). It is noted (29) has been substituted by linear formulations. The stationary PEV batteries energy range is addressed in (30) representing the stationary PEV batteries capacity limit. When the stationary PEV batteries is connected to the grid, the PEV batteries will be charged and discharged, or remain in the idle mode as shown by (31). Noted that, α (as a value changes between 0 and 1) in Eqs. (26), (27) and (30) determines the percentage of stationary PEVs that are operating in charging or discharging modes.

c&d c P ⎧ Pgtω + (Pwtω − WCwtω) + Pvtω − (Dnt + Pvtω) = ∑∀ k (m, n) flktω ⎨ 0 ⩽ WCwtω ⩽ Pwtω ⎩

U U ⎧ 0 ⩽ rvtω ⩽ Rvt uc vtω D ⎨ 0 ⩽ rvtω ⩽ RvtD ud vtω ⎩

The master problem is formulated as (47)–(50). The objective function (47) adapts (15), where χ is the expected cost of scenarios:

(33)

min ψ̲ =

∑ (CtH & N ) + χ

(47)

t

(34)

χ ⩾ χ min

min max ⎧Vn ⩽ Vn, t , w ⩽ Vn ⎪ min ⩽ Tk, t , w ⩽ Tkmax T ⎨ k min ⎪ φk ⩽ φk, t , w ⩽ φkmax ⎩

(35)

max ugt Pgmin , t ⩽ Pgtω ⩽ ugt Pg , t

(36)

max ugt qgmin , t ⩽ qgtω ⩽ ugt qg , t

(37)

− α·P¯vc ·uc vt ·NEv, tω ⩽ Pc vtω ⩽ −α· P̲ vc ·uc vtω·NEv, tω

(38)

α· P̲ vd·ud vtω·NEv, tω ⩽ Pd vtω ⩽ α·P¯vd·ud vtω·NEv, tω

(39)

(48)

∑ Pgt + ∑ Pwt + ∑ (Pvt + Pvtc& d) = ∑ Dnt + ∑ Pvtc g

w

v

n

v

(22) − (31)

(49) (50)

Constraint (48) imposes a lower bound to accelerate the convergence. The power balance equation for the base case is given in (49). Constraint (50) enforces all H&N constraints. Other unit constraints including minimum on/oﬀ time, ramping up/down rate limitations are considered here [40]. 4.2. Hourly network evaluation for the Here-and-now constraints

⎧ En vtω = ηv ·Pc vtω + Pd vtω c&d ⎨ Pvtω = Pc vtω + Pd vtω ⎩

(40)

Ev, tω = Ev, t − 1, ω − En v, tω

(41)

α·Evtmin u vt ·NEv, tω ⩽ Evtω ⩽ α·Evtmax u vt ·NEv, tω

(42)

uc v, tω + ud v, tω ⩽ 1

(43)

(En v, tω − En v, t ) = rvU, tω − rvD, tω and rvU, tω, rvD, tω > 0

(44)

The hourly grid security evaluation checks the possible grid violations of the master solution is the ﬁrst subproblem. The objective function of this subproblem is (51).

min R t =

∑ (κ1,n,t + κ2,n,t ) n

(51)

The power mismatch of each bus is presented by (51), where κ1, n, t / κ2, n, t are surplus and deﬁcit variables. The mismatch of power 231

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4.4. Convergence check The upper limit of the objective function in the original problem (63)–(64) at the ith iteration is obtained as follows:

ψ¯ (i) =

H&N

∑ (EĈ tW & S + Ct ̂

) (63)

t

|ψ¯ (i) − ψ̲ (i)| ⩽ ε

(64)

If the value of (64) is not lower than the level of the acceptable accuracy ε , the optimality cut (65) will be utilized for the master problem:

Ψ t(i) +

Fig. 4. The six-bus system with stationary PEVs, OLTCT, PST and wind farm.

∑ (γg(,it) (Pg,t − Pg(̂ ,it)) + ηg(,it) (ug,t − u g(̂ i,)t )) + ∑ ζn(,it) (En v,t − En̂ v(,it)) ⩽ n

g

χ balance should be minimized as shown in (52). Constraint (53) comprises the base case constraints. Decision variables in constraints (54) and (56) are the ﬁxed values calculated by the master problem. Also, in (55) and (57), the arrow denotes the dual variables of the equality constraints to perform the feasibility Benders cuts.

Pg,̂ t + Pw, t + Pvtc& d + (Dn, t + Pvtc + κ1, n, t − κ2, n, t ) =

∑

5. Case studies In this section, a modiﬁed six-bus and the IEEE 118-bus test systems are studied to demonstrate the eﬀectiveness of the proposed coordination scheme of the OLTCT, PST and stationary PEV storage devices from the operation cost viewpoint, while considering diﬀerent penetration levels of WPGs and modeling the associated uncertainties.

flkP, t (52)

∀ k (n, m)

(53)

(18) − (21) and (23) − (24)

5.1. Modiﬁed six-bus system

(i)

Pg, t = Pĝ , t → ςg(,it)

(54)

(i)

Pv, t = Pv̂ , t → νv(,it)

(55)

ug, t = u g(̂ i,)t → ξg(,it)

(56)

The single line diagram of the modiﬁed six-bus system is depicted in Fig. 4. The thermal units from most expensive to cheapest are G3, G1, and G2. The line ﬂow limit for the lines 2–4 and 4–5 is 50 MVA and is 150 MVA for all other lines. The WPG is a function of uncertain wind speed of the wind plant at bus 2. Table 1 displays the forecasted load, WPG and number of PEVs at each hour. Tables 2–4 shows some of the main characteristics of the generating units, transmission line data, OLTCT and PST, and stationary PEV ﬂeet, respectively. In this section, the OLTCT, PST and stationary PEVs storage devices are coordinated through the solution of the stochastic LAC- SCDAS problem for diﬀerent case studies. The proposed stochastic optimization problem is implemented on the modiﬁed six-bus test system, which is enhanced by a set of additional features. The proposed features (i.e.,

When violations occurred in H&N constraints, i.e., R t(i) is higher than a speciﬁed threshold, the Benders feasibility cut (57) will be created and forced by the master problem for the solution of the next iteration as follows: (i)

R ̂t +

(i)

(i)

∑ (ςg(,it) (Pg,t − Pĝ ,t ) + ξg(,it) (ug,t − u g(̂ i,)t )) + ∑ νv(,it) (Pv,t − Pv̂ ,t ) ⩽ 0 v

g

(57) 4.3. Optimality check for each scenario

Table 1 Forecasted load, wind power generation and number of PEVs in 24 h.

The sub-problem for scenario ω and time period t is formulated as (58)–(62):

Ψ t(i) = min

∑ (ECtW & S ) t

(58)

(32) − (37) and (44) − (46)

(59)

(i)

) Pg, t = Pĝ , t → γg(,itω ,

γg(,it) =

∑ πω·γg(,itω) ω

) En v, t = En̂ v(,it) → ζ v(,itω ,

ζ v(,it) =

∑ πω·ζ v(,itω) s

) ug, t = u g(̂ i,)t → ηg(,itω ,

ηg(,it) =

∑ πω·ηg(,itω) ω

(65)

(60) (61) (62)

The optimality sub-problem (58)–(62) checks the optimality conditions of master solution in scenario ω and time period t. The objective function (58) represents the total operation costs in W&S condition. Constraint (59) is related to the second-stage constraints. Constraints (60)–(62) ﬁx the values of the complicating variables to the obtained values from the master problem. The complicating variables are ﬁxed by constraints (60)–(62), whose dual variables provide the sensitivities to be used in building optimality Benders’ cuts for the master problem. 232

Hour

Load forecasted (p.u)

WPG (p.u)

Number of PEVs

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.50 0.52 0.53 0.50 0.51 0.53 0.58 0.64 0.70 0.85 0.89 0.92 0.95 0.95 0.97 1.00 1.00 0.96 0.96 0.93 0.93 0.89 0.89 0.90

0.94 0.95 0.84 0.83 0.73 0.66 0.71 0.87 0.91 0.84 0.88 0.82 0.85 0.78 0.43 0.37 0.27 0.15 0.11 0.18 0.11 0.13 0.30 0.18

2973 2980 2879 2818 2823 2108 1909 975 1052 1034 1172 969 1184 1166 968 1597 2120 3055 3443 3309 3948 3989 3541 3554

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Table 2 Data of generating units and transmission line for the 6-bus test system. Generating units data Units

Energy bid price ($/MWh)

Start up/Shut down cost ($)

Pmax (MW)

Pmin (MW)

Qmax (MVAr)

Qmin (MVAr)

Min up (h)

Min down (h)

Ramp up/down rate (MW/h)

G1 G2 G3

23 20 35

100/0 100/0 100/0

220 200 50

100 10 10

200 70 50

−80 −40 −40

4 3 1

4 2 1

55 50 20

Transmission line data Line no.

From bus

To bus

X (p.u.)

R (p.u.)

Max. line ﬂow (MW)

2 3 4 5 6

1 2 2 3 4

4 3 4 6 5

0.258 0.037 0.197 0.018 0.037

0.003 0.022 0.007 0.005 0.002

150 150 150 150 50

stationary of PEVs is applied on the modiﬁed six-bus system. Also, here, the stationary of PEVs are operating only as load (only charging mode) at bus 2. The charging mode behavior has been given in Table 1. The stochastic operation cost and unit commitment (UC) results have been given in Tables 5 and 6. The UC with the linear AC network security constraints is used to ﬁnd the results shown in Table 6. As can be seen in Table 6, the cheaper unit G2 is ON at all hours while unit G1 is used at peak hours to satisfy the remaining load and minimize the cost. Noted that in this test system, since the wind farm installed in bus 2, with more WPG, the power ﬂow of some transmission branches are increased (i.e., lines 1–4 and 4–5), which is caused ﬂow violations on these lines. In order to mitigate the mentioned lines violation, two primary options are usually utilized: additional commitment of unit G3 and/or WPS. In this case, both options are used. For example, the transmission network encounters ﬂow violations on lines 1–4 and 4–5, this fact causes that expensive unit G3 to be turned on at peak hours to help mitigating these violations. Line 4–5 is congested at peak hours 11–24, which leads to a fewer commitment of unit G1 and a higher operating cost. This matter has been shown in Table 6. On other hand, since only using more commitment of unit G3 is supposed to be an expensive decision here, accordingly, using more WPS option would be the cost-eﬀective choice. This matter has been shown in Fig. 5. It can be seen from the data in Fig. 5 that excepted wind power spillage (EWES) is 55.35 MWh. The other issue which impose higher operating costs is low voltage magnitudes in power system. In other words, here, the lower voltage magnitude in operation causes to increase congestion in system. As shown in Fig. 6, for the Case 1, the voltage magnitude at almost buses are less than 0.98 per unit, which imposes a new critical issue in system operation and leads to the higher cost. Therefore, to overcome this issue, it is needed to have more commitment of new (expensive) unit, i.e., unit G3 for this case to increase reactive power generation to support system voltage magnitude. Looking at Table 5, it is apparent that for Case 1, the total stochastic operating cost is $100550.69.

Table 3 Data of tap-changing transformer and phase shifter. Transformers

From bus

To bus

X (p.u.)

Max tap/angle

Min tap/angle

OLTCT PST

5 1

6 2

0.037 0.018

1.05 10°

0.95 −10°

Table 4 Characteristics of stationary PEV ﬂeet. Number of PEV

Min cap. (MWh)

Max cap (MWh)

Min charge/ discharge (MWh)

Max charge/ discharge (MWh)

Price ($/MW)

ηv (%)

4000α

10α

100α

10α /10α

10α /10α

8.21

85

conventional thermal units, the OLTCT, PST, wind farm, stationary PEVs operating in charging mode only or charging/discharging modes, which are always connected to a speciﬁc bus) are embedded in the test system, as shown in Fig. 4. In this test system, one stationary for PEVs is considered which consists of 4000 vehicles that can be connected to bus 2. Table 1 shows the ratio of number of available PEVs in a stationary at each hour to total vehicles, i.e., 4000 EVs. The available energy, max/min capacity and charge/discharge power of individual vehicles are aggregated in PEV ﬂeet characteristics representing max/min capacities, and max/min charging/discharging capabilities. Table 4 shows PEV ﬂeet characteristics. Wherein α , in Table 4, is the percentage of PEVs that are operating as an energy storage device, in the stationary, also, the α value changes between 0 and 1. Here, 2000 scenarios are generated and scenario reduction techniques are used to obtain 15 scenarios [38]. Five diﬀerent cases are simulated and compared to analyze the proposed methodology as follows:

Case 2. In this case, the eﬀects of the share of the stationary PEVs (in per unit) as an ES system on the Case 1 results are discussed. Accordingly, Cases 2.1 and 2.2 are simulated as follows:

Case 1: The proposed stochastic LAC- SCDAS problem with stationary PEV (operating only in charging mode or as a load) and without OLTCT and PST is investigated. This case has been considered as a reference case. Case 2: This case study allocates 20% and 50% of the total available PEVs in the stationary to participate in the system operation, i.e., they are operating in energy storage mode (charge or discharge modes). Case 3: Assessing the impact of including the OLTCT on Case 1. Case 4: Assessing the impact of including the PST on Case 1. Case 5: Assessing the impact of co-operation of the stationary PEVs storage, OLTCT and PST in Case 1. Case 1. In this case, the proposed stochastic LAC-SCDAS problem with

Table 5 comparison of cost results in six cases. Stochastic operating cost ($) Case Case Case Case Case Case

233

1 2.1 2.2 3 4 5

100550.69 98129.02 90534.65 99345.49 98287.12 93999.52

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Table 6 Hourly commitment status of units for six cases. Hours (24 h) Case 1

G1 G2 G3

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 1

0 1 1

0 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

Case 2.1

G1 G2 G3

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

Case 2.2

G1 G2 G3

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

1 1 0

1 1 0

1 1 0

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 0

1 1 0

Case 3

G1 G2 G3

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

1 1 0

1 1 0

1 1 0

1 1 0

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

Case 4

G1 G2 G3

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

1 1 0

1 1 0

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

Case 5

G1 G2 G3

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

0 1 0

1 1 0

1 1 0

1 1 0

1 1 0

1 1 0

1 1 0

1 1 0

1 1 0

1 1 0

1 1 0

1 1 0

1 1 0

1 1 0

55.35

60

53.23

50.91

49.04

48.31

Case 4

Case 5

EWES [MW]

50 40

which causes to reduce operation cost as compared to Case 1. Besides, by considering the 20% of the total available PEVs in a stationary to be operated as an ES system (by controlling their charging/discharging behavior) in the scheduling problem, the system could overcome uncertainty of wind farm at bus 2. As shown in Fig. 7, the stationary PEVs are charged at low load hours and high wind energy hours (i.e., hours 1–15) and provides the stored energy to be injected back to the grid when wind energy is low and load is high (i.e., 16–24 h). Accordingly, in this case, both resources (unit G2 and stationary PEVs) with together could alleviate uncertainty of wind farm at bus 2. Furthermore, as can be seen in Fig. 5, the EWES, as compared to Case 1, has been reduced in this case. By comparison of the obtained results for Cases 1 and 2.1, it can be seen that the wind energy absorption in uncertain condition has been increased by employing the stationary PEVs as an ES system. Overall, these results indicate that storage capability of stationary PEVs can decrease operation cost as well as WPS.

31.96

30 20 10 0

Case 1

Case 2.1

Case 2.2

Case 3

Fig. 5. Expected wind power spillage in six cases.

Case 1

Case 2.1

Case 3

Case 4

Case 5

Voltage [p.u]

1.04 1.02 1

Case 2.2. In this case, the share of PEVs as an ES system in power grid are increased from 20% (in Case 2.1) to 50%. Table 6 shows the hourly commitment of thermal units in Cases 2.1 and 2.2, respectively. From the Table 6 we can see that by increasing the share of the PEVs as an ES system will result in a similar UC results in Case 2.1, but the commitment of expensive unit G3 is decreased. It can be seen from the data in Table 6 that for Case 2.1, unit G3 are committed 10 h, but in this case, this unit is committed 5 h. It is obvious in Table 5 that the stochastic operating cost has been reduced compared to Cases 1 and 2.1. In Case 1, the WES may occur either due to transmission congestion or thermal ramping limitations. However, further analysis showed that the coordination between thermal unit output (i.e., unit G2) and the stationary PEVs charging/discharging behavior could decrease the EWES.

0.98 0.96 0.94 0.92 0.9 1

2

3

4

5

6

Bus Fig. 6. Average of voltage magnitude at peak hours, hours 12–24.

POWER EXCHENG [MW]

Case 2.1. This case is to analyze the impact of the share of availability of the PEVs in the stationary which operates in both charging and discharging modes as an ES device, on the operation results. Accordingly, this case takes the 20% of available PEVs at the stationary operating as an ES device which can charge or discharge. In the previous case, system congestion (i.e., caused by lines 1–4 and 4–5) is the main obstacle to mitigate the wind energy uncertainty at bus 2. In this case, if there was no participation of the stationary PEVs as an ES system in the scheduling, the system would somehow overcome wind uncertainty issue by commitment of thermal unit G2 and expensive unit G3, thus, as can be seen in Table 5, the unit G3 is ON at hours 11–24 to provide spinning reserve and compensate wind energy uncertainty. Here, it can be seen from the data in Table 5 that the application of stationary PEVs as an ES system will result in a similar UC results, but the commitment of expensive unit G3 is decreased by three hours,

15

Case 2.1

Case 2.2

10 5 0 -5

1

3

5

7

9

11

13

15

17

19

21

23

-10

TIME [H] Fig. 7. Total charge and discharge proﬁle at bus 2 for Case 2 and 2.1. 234

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A. Nikoobakht et al.

is apparent that the PST application in this case mitigates the congestion in lines 1–4 and 5–6, and increase the commitment of G1 by 13 h as expected because this device can better control power ﬂows. The results of the UC, stochastic operating cost and EWES values for Cases 1 and 3 can be compared in Tables 5, 6 and Fig. 5. The results in these tables and ﬁgure imply that the higher utilization of the economic unit G1 leads to a lower stochastic operating cost and EWES values.

As compared to Case 2.1 and as shown in Fig. 7, the stationary PEVs will more store wind energy at oﬀ-peak hours or when there is an excess WPG in the system (e.g., hours 1–15), and more generates at hours when the WPG is low or the demand is high (e.g., hours 16–24). A low storage capacity (in Case 2.1) is less eﬀective than high capacity (in Case 2.2) in reducing the volatility of hourly WPG because the charging and discharging of storage may be limited at (oﬀ) peak hours and some other hours, i.e., at hours 1–15 (as shown in Fig. 8). Accordingly, the results, as shown in Fig. 5, indicate that the EWES is decreased with increasing the share of the PEVs as the energy storage system. Besides, by comparing Cases 1, 2.1 and 2.2 in Table 5 and Fig. 5, it is inferred that the stochastic operating cost and EWES in Case 2.2 has been more reduced by operating 50% of total available PEVs as the energy storage system.

Case 5. In the previous Cases 2–4, the eﬀect of the stationary of PEVs as the storage device, OLTCT and PST were studied separately. Case 5 illustrates the eﬀect of their simultaneous operation. The concurrent eﬀects of these options will bring more energy storage capability, voltage and power ﬂow control and can lead to comparatively cheaper solution. Indeed, the shortcomings of the previous cases can be improved by their simultaneous cooperation. For instance, here, the shortcoming of Case 3 is resolved by employing the PST device as shown in Tables 5, 6 and Figs. 5, 6. In the Fig. 6, the top plot shows the average voltage levels at peak hours for each bus in all cases. It is evident from Fig. 6 and Table 6 that the OLTCT device improves the average voltage, at peak hours, in the most of buses which results in decreasing the commitment of the expensive unit G3 by zero hour. On the other hand, based on the results of Table 6, it is apparent that the PST device improves the transmission congestion control by independent control of voltage angle at buses 1 and 2 and increases the commitment of economic unit G1. The OLTCT and PST devices by resolving voltage issue and transmission congestion cause to increase the eﬃciency of stationary PEVs as a storage device. As it can be observed in Fig. 8, the charging and discharging capability of the stationary PEVs storage device in bus 2 at most 24 h are increased by relieving transmission congestion. For example, as shown in Fig. 8, the stationary of PEVs has more capability to be charged and discharged in Case 5 with respect to Case 2. Also, it can be seen from Table 5 that the OLTCT and PST devices by resolving voltage issue and relieving transmission congestion in the power grid could eﬃciently transfer the stored energy in the stationary PEVs into the power grid and reduce the stochastic operating cost value, i.e., 93999.52 $, as well as mitigating uncertainty of WPG. Similarly, from Fig. 6 we can see that the inclusion of the stationary PEVs storage device with the OLTCT and PST devices, has eﬀectively reduced the EWES to 48.31 MW. Finally, from Fig. 5 and Table 5, it is apparent that except Case 2.2, in the case of considering concurrent operation of the OLTCT, PST devices and the stationary PEVs storage system, the system operator would have the maximum ﬂexibility and the lowest stochastic operating cost and EWES values consequently.

Case 3. In this case, the OLTCT is connected in the line 5–6. Simulations results for this case is presented in Tables 5, 6 and Figs. 5, 6. As can be seen from the Table 5, for the stochastic operating cost between 99345.49 $ and 100550.69 $, no power adjustment is needed. Also, for these values of the cost, the binding constraints are mainly voltage limits (in particular, at bus 6). It is apparent from this table that stochastic operating cost for this case is reduced by 99345.49 $. In the absence of the OLTCT, to satisfy voltage constraints, the reactive power generated by the thermal units would satisfy voltage constraints partially with additional operation cost. For instance, in Fig. 6, we can observe this consideration for the Case 1, the average voltage magnitudes at the most buses for peak hours are lower than 0.98 p.u., for this reason, with the lower average voltage magnitude in buses 4 and 5, the system congestion at peak hours may not be completely removed. This is improved by additional reactive power injection at these buses by the expensive unit G3 which is turned on at most peak hours (as shown in Table 6), which causes to increase the operating cost value. On other hand, it is observed in Fig. 6 that the OLTCT is able to signiﬁcantly improve the average voltage magnitude at the most buses and overcome this issue by adjusting transformer taps. Accordingly, it can be seen from the results of Table 6 that the commitment of expensive unit G3 is decreased by four hours caused to slightly reduce the operating cost compared to Case 1. Similarly, in this case, the optimal utilization of the OLTCT is eﬀective to decrease the EWES value. Finally, the results of this case and previous Cases 1 and 2 can be compared in Table 5 and Fig. 5. The important consideration about the results in this table and ﬁgure are that increasing the share of the stationary of PEVs as an energy storage device is better option than the OLTCT option to decrease the stochastic operating cost and EWES values. Case 4. In this case, in comparison with Case 1, a PST is installed in line 1–2. In Case 1, the security constraints, including voltage magnitude and power ﬂow limits of lines (in particular, on lines 1–4 and 5–6) result in no reduction of the stochastic operating cost more than 1.2%. It can be seen from the data in Table 5 and Fig. 5 in Case 3, by employing the OLTCT device to control the voltage levels, the stochastic operating cost and EWES values can be reduced. Nevertheless, the eﬀects of this device on power ﬂow through transmission lines are negligible, as this device controls only voltage levels. However, the voltage control cannot improve the cost reduction beyond 1.2% and it is associated with the constraints that limits the power ﬂow through lines (in particular, through lines 1–4 and 5–6). As shown in Table 5 and Fig. 5, the PST is most eﬀective device for the stochastic operating cost and EWES reduction in comparison with the OLTCT, as this device can control the power ﬂow of transmission lines. From the Table 6 we can see that in Case 1, due to the congestion in lines 1–4 and 5–6, the economic unit G1 can be utilized only for 12 h. On other hand, the shortage of the wind generation is compensated by the expensive thermal unit G3 instead of G1 and G2. Also, as can be seen from the Table 6 in Case 1, due to the congestion in lines 1–4 and 5–6, the economic unit G1 can be utilized only for 10 h. From Table 6 it

5.2. The modiﬁed IEEE 118-bus system

POWER EXCHANGE [MW]

The modiﬁed IEEE 118-bus system is used to illustrate the eﬀectiveness of the proposed stochastic LAC-SCDAS problem for larger systems. The system consists of 54 thermal units, 5 wind farms, 186 branches and 91 loads. The hourly active and reactive loads as well as 4

Case 2.1

3

Case 5

2 1 0 -1

1

3

5

7

9

11

13

15

17

19

21

23

-2 -3

TIME [H]

Fig. 8. Total charge and discharge proﬁle at bus 2 for Case 2 and 5. 235

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(linear) OLTCT and PST models and the stationary PEVs storage system simultaneously in both of the OPF models. Considering the full AC OPF results as the reference, the maximum calculation errors in 24-hour are given by:

the characteristics of units and lines are taken from [41]. Based on load ﬂow analysis, the heavily loaded lines with low capacity in the modiﬁed IEEE 118–bus system are lines [11 (bus 5–bus 11), 55 (bus 39–bus 40), 70 (bus 49–bus 50), 136 (bus 85–bus 89) and 168 (bus 104–bus 105)] where the OLTCT devices are located. Similarly, for lines 5–8, 37–38, 59–63, 61–64, 65–66, 68–69 and 80–81, the PST devices are located. Additionally, ﬁve 200 MW wind farms are connected to buses 3, 9, 14, 18, and 22. There are three WPGs at buses 36, 77 and 96. The proﬁle of wind farm generation, which is located at bus 36, follows the same pattern as that of the previous test system which is scaled by factor of 4. Similarly, the WPG proﬁle of the other two wind farms is scaled by factor of 2. The maximum capacity of these wind farms is 800 MW. The system peak load is 3733 MW. The installed WPG is 21.43% of the system peak load. The total energy of WPGs is 12,197.63 MWh and the total energy of the demanded load is 68,933.51 MWh. Hence, the penetration level of the available WPG is 17.69% in this system. The cost of WES is set at 100 $/MWh. Three stationary PEVs storage devices are attached at buses with wind power generation that track the same pattern as of the former test system which is scaled by factor 2. The proposed stochastic LAC-SCDAS model with the linear OLTCT and PST models and the stationary PEVs storage system is examined for this test system as done in previous case. Using the Monte Carlo Sampling method, 2000 scenarios are generated and scenario reduction techniques are used to obtain 15 scenarios as done in the previous case. Six cases as described in the previous subsection were studied again for this test system. The proposed stochastic LAC-SCDAS optimization for large-scale test system is an NPhard problem with the linear OLTCT and PST models that dramatically has increased the computation burden. Therefore, the GBD technique is applied to solve the proposed problem for this test system. The solution time needed to solve the proposed problem is less than 32 min, which is reasonable for this test system. Also, the results obtained for these cases are consistent with those of the previous system. At ﬁrst, similar to the pervious test system, the results obtained from Case 1 is considered as the reference case, for this test system. In Case 2.1, the application of 20% stationary PEVs as the ES system will result in pervious test system results, which is caused to reduce the stochastic operating cost and EWES in comparison with Case 1. But, similar to the previous test system, the stochastic operating cost and EWES have the lowest values for the Case 2.2. As shown in Table 7, in Case 3, the OLTCT is able to resolve voltage issue at most load buses and overcome this issue by adjusting transformer tap, which is caused to slightly reduce the stochastic operating cost with respect to Case 1. Table 7 shows the cost and EWES values, for diﬀerent cases from 3 to 5 as well. The results for the 118-bus system are consistent with the results of the modiﬁed six-bus system. For instance, in Case 1, the voltage magnitude issues are partially overcame by increasing the commitment of expensive thermal units, which increases the operational cost. In Case 3, the transmission congestion plays major role for the stochastic operating cost reduction beyond 2.4%. The results in Case 4 indicate the PST device plays an important role in the cost reduction by managing the congestion in the transmission lines which has lower limits similar to the previous 6-bus system. Finally, as shown in Table 7, in Case 5, the concurrent utilization of the OLTCT and PST devices and the stationary PEVs storage system, except in Case 2.2, have the highest wind energy utilization at uncertain condition and the lowest cost for the large-scale system. In order to investigate the accuracy of the proposed model with OLTC and PST, in this section, the proposed LACOPF and full ACOPF [33] have been compared in the IEEE 118-bus system. The simulations are performed to obtain the active power ﬂow (APF) in 186 transmission line and the active power generation (APG) for 54 generating unit using the proposed linear AC OPF model and the full AC OPF model in [33]. Furthermore, we have considered the

∼ σk, t = |flkP, t − f lkP, t |,

σkmax = max{σk,1, σk,2, …σk,24}

(66)

∼ σg, t = |Pg, t − Pg, t| σgmax = max{σg,1, σg,2, …, σg,24}

(67)

Eq. (66) is the calculation error of the APF in line k at 24 h which is obtained from the proposed linear ACOPF model, i.e., flktP , and the full ∼ ACOPF model [33], i.e., f lktP . Then, the maximum calculation error of the APF for each transmission line at 24 h is obtained by σkmax = max{σk,1, σk,20, …σk,24} . Also, the maximum calculation error of the APG for each unit in 24 h is obtained by (67) similar to the procedure by (66). As can be seen in Figs. 9 and 10, the maximum value of the error calculated for σkmax and σgmax , in our proposed linear ACOPF model, are 0.0228p.u and is 0.01225p.u, respectively. As shown in Table 7, the diﬀerence between the EOC (EWES) of the proposed linear ACOPF and the full ACOPF models are negligible. But, the elapsed time to solve the proposed problem with linear ACOPF model and the full ACOPF model are less than 32 min and more than 250 min, respectively. These results indicate that the maximum value of the error can be negligible, but, the proposed linear ACOPF model is much faster than the full ACOPF model. It should be noted that providing solution results with negligible error and faster computational time is more important than achieving global optimality with high computational time. These results show the eﬃciency of the proposed linear ACOPF model for the large-scale systems as well. 6. Conclusion This paper has argued that the coordination of the WPG and PEV can help ISOs to decrease the grid operation costs and wind power spillage. However, two main challenges faced by ISOs to provide this coordination are: (i) wind uncertainty (ii) transmission lines congestion. The ﬁrst challenge related to the uncertainty of WPG can be managed properly by charging/discharging capability of the stationary PEV ﬂeets. The simulation results in this paper indicate that the optimal scheduling of PEV ﬂeets as the stationary storage services can reduce the operation costs, and enable superior daily wind energy consumption proﬁles with minimal hourly wind power spillage. The optimal V2G implementation is an especially promising method for ensuring that the wind power supply would match the hourly demand as well as smooth out the uncertainty of wind farms. The other challenge is related to the actual injections of WPGs into a transmission grid can be considerably diﬀerent than their forecasted values, leading to signiﬁcant transmission congestion which requires to be alleviated by the ISOs to maintain the power system reliability. The OLTCT and PST devices as ﬂexible PFCs can play an important role in managing congestion which does not incur any signiﬁcant operational cost to the ISOs. The results conﬁrm that the cost savings and maximum wind power integration achieved via a combination of OLTCT and PST devices can be substantially higher compared to the exclusive employing either of the technologies. Also, in this paper, in order to coordinate the OLTCT and PST devices, a stochastic linear AC security Table 7 Comparison of results for Cases 1–5 for IEEE 118-bus system. OPF model

Results

Case 1

Case 2.1

Case 2.2

Case 3

Case 4

Case 5

LACOPF

EOC (M$) EWES (MW) EOC (M$) EWES (MW)

1.241 743.5

1.123 613.2

1.028 332.7

1.143 697.2

1.132 512.4

1.103 414.5

1.244 751.2

1.128 621.4

1.033 339.5

1.148 705.1

1.138 519.3

1.106 418.2

ACOPF [33]

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0.02 0.015 0.01 0.005 0

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115 121 127 133 139 145 151 157 163 169 175 181

APF calculation error [p.u]

0.025

Line lable Fig. 9. APF calculation error for stochastic SCDAS model with the OLTCT and PST modeled by LACOPF; the maximum calculation error in 24-hour period.

APG calculation error [p.u.]

0.014 0.012 0.01 0.008 0.006 0.004 0.002 0

1

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Generating Unit Lable Fig. 10. APG calculation error for stochastic SCDAS model with the OLTCT and PST modeled by LACOPF; the maximum calculation error in 24-hour period.

constrained day-ahead scheduling model under WPG and PEV uncertainties has been proposed. To overcome the shortcomings of the DC power ﬂow model widely used in the stochastic OPF formulations, a tractable linear AC power ﬂow model has been developed. Numerical studies have demonstrated that the proposed linear AC OPF formulation have improved the accuracy as well as the computational tractability. With the aim of decreasing the computational burden for large-scale systems, the proposed model has been formulated based on Benders decomposition. An extension of this study could assess the short-term eﬀects of the OLTCT, PST devices and PEV ﬂeets as stationary storage systems on power system ﬂexibility. Also, further research should be undertaken to explore how to assess the ﬂexibility needs of the power system. Hence, in the next research we will present a novel uniﬁed ﬂexibility formulation for a stochastic day-ahead scheduling model based on power system ramping capacity.

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