Scenario-based comprehensive expansion planning model for a coupled transportation and active distribution system

Applied Energy 255 (2019) 113782

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

Scenario-based comprehensive expansion planning model for a coupled transportation and active distribution system

T

Shiwei Xie , Zhijian Hu , Jueying Wang ⁎

⁎

School of Electrical Engineering and Automation, Wuhan University, Wuhan City, Hubei Province, China

HIGHLIGHTS

planning model of coupled transportation and active distribution system is proposed. • AScenario generation technique is extended to incorporate traffic demand. • Scenario-based user equilibrium constraint is derived and embedded into the model. • A three dimensional piecewise method is designed to relax the bivariate expressions. • The importance of considering interdependencies of the coupled system is verified. • ARTICLE INFO

ABSTRACT

Keywords: Active distribution system Transportation network Expansion planning Electric vehicle Scenario generation technique

The widespread utilization of electric vehicles has inspired the emerging trend of coupled transportation and distribution system, which entails the systematic methodologies to model the new planning problems. This paper proposes a scenario-based comprehensive expansion planning model for a coupled transportation and active distribution system. With the aim of minimizing the investment and operation costs, this model determines the best alternatives, locations and sizes for candidate assets, including traffic roads, distribution lines, distribution generators, capacitor banks, static var compensators, voltage regulators, energy storage systems and charging facilities, as well as their operation strategies. First, a generated scenario method is extended to incorporate the uncertainty of the traffic flow demand. Based on multiple scenarios, the steady-state distribution of traffic flow is characterized by the Wardrop user equilibrium principle, and the corresponding equivalent constraints are derived and incorporated into the model. For active distribution system, we formulate the operation constraints for related infrastructures. Considering the interdependency between the two systems, an expansion planning model is proposed, which simultaneously optimizes the investment and operation strategies. Due to the nonlinear nature of the model, we have developed a three-dimensional piecewise linear approximation and applied second order cone relaxation to reformulate the model as a mixed-integer second order conic program; thus, the global optimal solution can be found in a reasonable time frame. Results on a test system reveal that the variations of both systems have some influences on each other, indicating the significance of considering the interdependencies between transportation and active distribution system.

1. Introduction Nowadays, with higher distributed energy source integration, traditional distribution networks are undergoing the transformation from passive unidirectional flow networks to active distribution systems (ADSs) [1], which impose significant challenges to distribution companies in terms of both the economy and technology in engineering. As a system with controllable devices and flexible energies, the concept of the ADS is one of the promoting technologies that can provide the

⁎

optimal solution to accommodate these widespread energy sources and address these challenges [2]. However, the increasing penetration of electric vehicles (EVs) and charging facilities (CFs) during the past few years [3] have led to profound interdependencies between the two systems: the electricity network and the transportation network (TN). In a TN, the temporal and spatial distribution characteristics of traffic flow are influenced by the congestion conditions, thereby affecting the charging demands with uncertainties, as well as the operation and planning strategies in distribution systems [4]. Such a significant

Corresponding authors. E-mail addresses: [email protected] (S. Xie), [email protected] (Z. Hu).

https://doi.org/10.1016/j.apenergy.2019.113782 Received 8 June 2019; Received in revised form 3 August 2019; Accepted 23 August 2019 Available online 28 August 2019 0306-2619/ © 2019 Elsevier Ltd. All rights reserved.

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unit investment cost of CF with type unit cost of replacing/adding distribution lines with type unit cost of electricity purchasing from TS/DGs unit charging price for EV unit energy loss cost unit curtailment cost of power load/DG probabilities of scenario of time block b maximum number of scenarios at each time block duration time of time block b B b rij /x ij resistance/reactance of line ij U¯min/ U¯max minimum/maximum permissible value of bus voltage I¯max maximum permissible value of current line nCB /nSVC / nDG maximum investment number of CB/SVC/ DG VR VR VR ,min / ,max minimum/maximum ratio limit of a type SVR, capacity limit of a type VR q CB / Q SVC , max reactive power capacity of a type CB/SVC P¯ C, max / P¯ D, max maximum charging/discharging power of a type ESS , nom DG P¯pDG / Sp, rated active/apparent power capacity of a type DG , QpDGmax maximum reactive power of a type DG , STS transformer substation capacity 0 P¯max / P¯ max active power capacity of original line/a new type line DG / L maximum curtailment rate of DG/power load p CF power capacity of type CF EV the proportion of EVs demand in all vehicles ea ( ) the average energy requirement of EVs on link a at scenario ta0 free travel time on link a Ca0 original capacity of road link a ca unit capacity of each invested road rs correlation coefficient between traffic link a and path k a, k the unit monetary value of traffic time n0EV number of existing charging facilities on link a P jL, qrs predicted value of power load/traffic demand ubD ( )/ ubT ( ) power load factor/traffic demand factor at scenario ubW ( )/ ub ( ) distributed wind/photovoltaic power generation factor at scenario

Nomenclature

C I , CF C LA/ C RA C TS /CDG CEV CLoss CUL/ C CUR b( ) n

Sets set of links/nodes in the transportation network set of start/end nodes in the transportation network set of paths connecting the origin-destination pair r-s SS set of scenarios at time block b b B set of time blocks B/ L set of all existing and candidate buses/lines in the ADS B/ L set of candidate buses/lines in the ADS CF sets of candidate location for CFs DG type set of the renewable energy generation, DG = {WTG,PVG} L set of lines available to be upgraded KL set of available type for new lines K VR/ K DG /K CB sets of available type for VRs, DGs, CBs, SVCs and / K SVC / K ESS ESSs VR / DG / CB sets of candidate location for VRs, DGs, CBs, SVCs and / SVC / ESS ESSs set of buses whose parent/child is bus j set of all nonnegative integers Z 0+

TA/TN TR/ TS Krs

Variables SS aggregate flow on link a TA at scenario b SS travel time on link a TA at scenario b traffic flow on path k K rs between pair r-s at scenario

x a, ta, f krs,

SS b

unit travel cost and Total traffic cost on path k K rs integer investment variable of road expansion on link a TA L with x ijR, , L binary decision variable for replacing line ij K RL type L with type x ijA, , L binary decision variable for adding line ij AL K ESS K VR/ K ESS x ijVR binary decision variables for VRs/ESSs with , / x ij, CB SVC x ij, / x ij, integer decision variables for CBs/SVCs with K CB / K SVC DG DG and x ij, p, integer decision variables for DGs with type p DG Kp x CF K CF integer decision variables for CFs with j, Pij, / Qij, active/reactive power flow from bus i to j TS PTS active/reactive power injection from main grid j, / Qj, D C P j, / P j, discharging/charging power of ESS CB QjSVC , / Qj, reactive power injection of SVC/CB DG P jDG active/reactive power injection of DGs after power , p, / Qj, p, control P jL, / QjL, active/reactive power load CF ) P jEV charging power of EVs at the CF (bus j, j , ~ ~ Uj, /Iij, The square of bus voltage/line current MAGNITUDE P jL, / P jDG curtailment power of load and DG , p, VR adjustment ratio factor of VR at scenario Um, the voltage for the dummy bus m of VR yjCB operating number of CB at scenario ,

ckrs, z aC

/ c krs,

Acronyms ADS TN T&ADS ADSEP ANM TAP UE EV CF DG WTG PVG CB VR ESS SVC PWL SOCR BPR MINLP MISOCP CDF ED/TD IR

SVC QCB reactive power output of CB/SVC at scenario j, / Q j, C D u j, / u j, charging/Discharging state variable of ESS at scenario

Parameters

C I , VR/ C I , CB /

unit investment cost of VR/ CB/ SVC/ ESS with type C I , SVC /C I , ESS I , DG DG and K pDG Cp, unit investment cost of DG with type p 2

active distribution system transportation network transportation and active distribution system active distribution system expansion planning active network management traffic assignment problem user equilibrium electric vehicle charging facility distribution generator wind turbine generator photovoltaic generator capacity bank voltage regulator energy storage system static var compensator piecewise linear second order cone relaxation Bureau of Public Roads mixed integer nonlinear programming mixed integer second order cone programming cumulative distribution function electric demand/traffic demand Increasing rate

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interaction introduces an urgent requirement for a systematic methodology to model and analyze the performance of this coupled system based on the promising concept of ADS, which entails interdisciplinary studies. However, the aforementioned have been investigated separately in most of the existing research. In ADS expansion planning (ADSEP) research, active network management (ANM) is recognized as an indispensable part [5], which mainly includes distribution generator (DG) power control, demand management, voltage regulator (VR) adjustment, reactive compensation, and energy storage system (ESS) dispatch. Along this line, a deterministic model for expansion planning with consideration of high penetration rates of DGs was proposed in [6], where the simulation results on ANMs containing the control of VRs and DGs showed the notable advantages of the model. In [7], authors proposed a joint planning model for ADSs to determine the location and size of charging stations, ESSs and DGs, in which the uncertainties of renewable energy was described by scenario sets. In [8], a two-stage optimization method was designed for optimal DG planning. This work utilized the energy storage technology and verified its advantages in compensating the power fluctuations of renewable DGs. The authors in [9] proposed an expansion planning model to facilitate the integration of DGs in an ADS. In the model, the ANM schemes including demand management, network reconfiguration, voltage control and renewable DG control were jointly considered. Ref. [10] proposed a multi-objective model for expansion planning issues, where the coordination of ANM schemes including the controls of DGs and VRs, demand responses, and static var compensators (SVCs) was fully considered. Despite the fact that there are a rich body of studies focused on ADSEP research, the expansion planning models that jointly considers the investment on road, CFs and related infrastructures in ADSs have not been presented yet. On the other hand, the fundamental task in TNs is to solve a wellknown problem, the traffic assignment problem (TAP), and, in turn, to obtain the steady-state of traffic flow. However, with the rapidly growing travel demand, it is no longer enough to rely solely on traffic assignment (or management) to solve the high traffic pressure and congestion problems. Hence, TNs will be regularly improved and expanded in the urban areas [11]. Generally, capacity expansion of the road network can generally interpret as the measure to enhance the maximum number of traffic volume passing the road link in unit time under specific geographical and economic constraints. Although road capacity expansion may face many challenges due to the geography and economy, some certain road sections with expansion conditions are often taken as the expansion objects in this research field [12]. As a classical problem in transportation field, road capacity expansion has been extensively studied in previous research. In [11], a mixed transportation network planning model with its global optimization algorithm was proposed, where the way of increasing road capacity was classified in detail and the traveler’s route choice behavior in a user equilibrium (UE) was incorporated in the model. In [13], a novel network design problem formulation was proposed to determine the optimal new link addition and the corresponding capacities simultaneously. However, these studies have coped with road expansion problems solely, and the interactions between the transportation system and power grid and are not their main concerns. As the number of EVs increases, it has been acknowledged that onroad CFs can introduce notable interactions across distribution system and transportation system. Under this context, [14] and [15] have both corroborated the benefits of the interdependency between two networks and have indicated that ignoring its interaction via on-road CFs may result in an insecure operation. Nevertheless, authors in [14], as well as in [15], have concentrated on operational challenges rather than the planning issues. Recently, a few interdisciplinary studies have been conducted to model the expansion planning of two systems that are coupled by CFs. The deployment planning of charging stations with UE constraints is investigated in [16] and is further improved in [17] by considering the expansion of the road capacity. However, their main

concern has been the traditional distribution system rather than the ADS. More specifically, although the models has successfully incorporated road capacity expansion, a fixed demand and single scenario have also been adopted, which remains to be extended to multiple scenarios to address future uncertainties. In such a context, the theory and methodology for the joint expansion planning of a transportation and active distribution system (T&ADS) are in urgent need of research. To the best of our knowledge, there still remains research gaps from the following aspects: (i) The application of scenario generation techniques combining the uncertainties of sources in both the ADS and TN. There are extensive studies on power system planning using the scenario techniques to model the uncertainties [18]. When the traffic demand is considered, the method remains to be extended to jointly consider the correlated uncertainties of electric demand, traffic demand and renewable-based power generation. (ii) The system-level modeling method for coupled ADS and TN. Most of the existing research focus on modeling either the traditional distribution system or the transportation system while neglect the interactions between the TN and ADS, which are in need of further study. (iii) The comprehensive expansion planning model for coupled ADS and TN. Many current studies on this field only address one, or at most a few, of the items in one system. The joint expansion planning model for distribution lines, DGs, CBs, SVCs, VRs, ESSs, CFs and traffic roads while accounting for the interactions between ADS and TN has rarely been reported so far. Based on the aforementioned discussions, a scenario-based comprehensive expansion planning model for a coupled T&ADS system is proposed in this paper to fill the interdisciplinary research gap. To model the correlated uncertainties, a scenario generation procedure based on [10] is extended to incorporate the uncertainty of traffic flow demand; therefore, the generated scenario set is applied to form the scenario-based stochastic programming framework for both the TN and ADS. In the TN model, the traffic flow pattern is determined by the Wardrop UE principle, which is described in terms of equivalent constraints, and in turn, a scenario-based planning model of a TN that considers road capacity expansion is formulated. In the ADS model, the multiple alternatives, locations and sizes of distribution lines, turbine generators (WTGs), photovoltaic generators (PVGs), CBs, SVCs, VRs, ESSs, and CFs, as well as their corresponding ANM operation strategies are optimized accordingly. On this basis, a scenario-based expansion planning model for a T&ADS system coupled by CFs is established to minimize the total costs while satisfying the technical constraints. For the proposed mixed-integer nonlinear, scenario-based stochastic programming problem, the solution methodology relies on the following applications: (i) a three-dimensional piecewise linear (PWL) approximation method which is developed to solve the bivariate nonlinear function; (ii) a second order cone relaxation (SOCR) approach applied to the non-convex equations; and (iii) certain mathematical techniques, such as a big M method. Finally, the effectiveness of the proposed model is verified using a coupled system composed of an IEEE 33-bus system and a TN system. In the scope of the interdisciplinary research on expansion planning issues, the main contributions of this paper are to fill the research gap of the state-of-the-art in the following points. (i) The scenario generation method incorporating traffic demand. To jointly consider the correlated uncertainties of electric demand, traffic demand and renewable-based power generation, in this paper the scenario generation procedure based on the work in [10] is improved to combine the traffic demand with other sources from distribution system. Compared to the classic scenario generation technique in [19], the proposed scenarios generation 3

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the global optimality is hard to be ensured by the intelligence algorithm. The major contribution to the solution methodology in this paper lies in that certain linear relaxation techniques are designed or applied specially for the proposed non-convex model, which can serve as an effective tool for solving such a kind of planning problem. Since the UE state is determined by solving a TAP, the equivalent complementarity constraints described in [14] are extended to a scenario-based form and then embedded into the model after linearization. To eliminate the non-convexity of bivariate time expression, a novel three-dimensional PWL is developed. Relying on the SOCR that is a mature and effective technology [26], the branch flow equations are transformed into the conic constraints complying with the way in [27]. In contrast to the non-optimized intelligence algorithm, the final optimization problem yields a mixed-integer second order cone program (MISOCP) and thus its global optimal solution can be found by most commercial solvers in tolerable time.

procedure can cover more different levels of scenarios. (ii) The system-level modeling of the ADS and TN while accounting for their interdependency originally. The mathematical formulation of the ADS and TN is more applicable to strategic planning research compared with those in the existing studies. In the TN model, road capacity expansion model is proposed to minimize the total investment and operation costs, while considering the drivers’ routing behavior under multiple scenarios. Compared to the models that use a fixed demand under a single scenario (e.g., [17]), this paper considers multi-scenario traffic demands and employs a scenario-based UE constraint to describe the routing behaviors. This way is more suitable for a planning issue since it incorporates more uncertainties of traffic demand. As for the ADS, conic relaxation-based branch flow equations [20] are adopted to characterize the power balance. The proposed method is more realistic for a low-voltage ADS than using direct-current power equations [21], since it treats voltage magnitude as a variable rather than a constant. To utilize the benefits of the ANM schemes which have been justified in the existing studies, for example [22], the operational modeling for WTGs, PVGs, CBs, SVCs, VRs, and ESSs are fully addressed. (iii) A novel scenario-based comprehensive expansion planning model for a coupled T&ADS system. As discussed above, most literature only deal with the planning problems in one of the systems while disregarding the interdependency between ADS and TN. As one of the first few attempts, research in [17] has well addressed the a single period-based planning model for road, onroad CF, and power grid facilities with consideration of UE constraints. To cover more operating conditions, some prespecified scenarios are applied to model the expansion planning for traditional distribution system in [23]. Although different power levels of EV integration are described via given scenarios in [23], the set of scenarios built on real historical data is obviously more suitable for real engineering applications. Aroused by the promoting concept of ADS, this paper combines the advantages of the interaction framework like [15] and the real data-based scenario generation techniques in [10], and proposes a comprehensive expansion planning model that jointly optimizes the investment and operation costs under multiple scenarios. (iv) The solution methodology designed for the proposed model relying on the mathematical method. An expansion planning model is usually a mixed-integer nonlinear program (MINLP) problem and have proven to be very difficult. Many studies have employed the heuristic algorithms to tackle the MINLP such as the planning model in [24] is solved by a genetic algorithm, and the multi-objective nonlinear problem in [25] is solved by the particle swarm optimization method. However, the finite convergence of

Active Distribution System Active power

Rective power

WTG

SVC

Electric grid

EES

The remainder of this paper is organized as follows. In Section 2, the structure of the coupled system is introduced. In Section 3, the scenario generation approach is introduced. The mathematical models of the TN and ADS will be established in Sections 4 and 5 respectively, and its corresponding solution methodology is described in Section 6. Section 7 carries out the case study simulation and validation of the proposed method. Finally, the conclusions are drawn in Section 8. 2. The structure for a coupled transportation and active distribution system As further explanation, Fig. 1 provides a framework of the coupled T &ADS. In the ADS part, a variety of devices are contained such as the EES, VR, SVC, CB, WTG and PVG. In order to facilitate the efficient use of energy and ensure the supply-demand balance of electricity, they can execute control commands following the dispatch decision in real time, which is called ANMs. Thus, active and reactive power through the network can be flexibly controlled in an ADS. In the transportation system, each driver (of EV or other vehicles) is allowed to be informed about real-time traffic condition and to select his own route. Specifically, the route of EV can affect the spatial and temporal distribution of charging demand through the on-road CFs, shown in Fig. 1. In this framework, the connection of the two systems is represented by a coupling unit, where the electricity can be consumed by plug-in EVs. 3. Stochastic scenario generation method Electrical demands and renewable energy-based sources are affected by many weather factors and have a certain correlation [28]. The

Coupling Unit: CF System

Electrified Transportation System

CF

EV

VR PVG

Electric vehicles Other vehicles

CB

Fig. 1. A schematic diagram of coupled active distribution and transportation system. 4

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authors in [19] have noted that the aforementioned are not statistically independent of each other, and thus have proposed a scenario generation method that incorporates the correlations among the demand, wind speed and solar irradiation. In fact, the traffic flow is also greatly affected by the environment and weather [29]. The work in [30] has verified through historical data that the precipitation, temperature, visibility and other elements have different effects on the traffic flow, thus indicating that there is a certain relation between the traffic flow and weather. However, correlated uncertainties characterizing the demand, renewable power generation and traffic flow have not been jointly modeled yet. Based on the approach in [19], and improved in [10], this paper extends the scenario generation method to include the uncertainty of traffic flow, and accounts for the correlation between the data and the other three sources. The main steps of the method incorporating the traffic demand is presented as follows:

wind and photovoltaic power are normalized as per-unit factors firstly by dividing each of them by the corresponding maximum value. Step 2: Sort the hourly electric demand data in descending order while keeping the correlation between the different hourly data of wind power, photovoltaic power, and traffic demand. Thus, the load duration curve, as well as the other three types of curves which are arranged according to the load duration curve, can be obtained. Note that the full chronological information may lost through such a sorting method. Step 3: Define time blocks. In order to accurately capture the effect of peak load, which imposes a great impact on investment decisions, the first time block should be narrower than the others. An example with a three-time-block division is shown in Fig. 2(a). For each block, the hourly factors for wind power, photovoltaic power, and traffic demand are sorted by electric demand factor in the same descending order, shown in Fig. 2(b)–(d). In this way, the time sequence for each set of data is changed, and therefore the abscissas in

Step 1: Historical hourly data of system electric and traffic demand,

Traff ic demand (pu)

Phot ovoltaic power (pu)

Wind power (pu)

Elect ri c demand (pu)

Time block 1

Time block 2

Time block 3

1

0.5

0

Time (h)

8760

Time (h)

8760

Time (h)

8760

Time (h)

8760

(a)

1

0.5

0

(b)

1

0.5

0

(c)

1

0.5

0

(d)

Fig. 2. Time block divisions of electric demand, wind power, photovoltaic power and traffic demand. (a) Electric demand data in descending order and the division of three time blocks with duration hours of 988, 4622, 3174. (b) Wind power curve sorted by electric demand in each time block. (c) Photovoltaic power curve sorted by electric demand in each time block. (d) Traffic demand curve sorted by electric demand in each time block. 5

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Fig. 2 are composed of 8760 unordered individual hours (refer to [19]). Step 4: The cumulative distribution functions (CDFs) of the factors in each time block is used for representing uncertainty. For each time block, the CDF for each ordered curve can be built accordingly. Fig. 3 depicts the CDFs corresponding to the curves given in Fig. 2. Step 5: Each obtained CDF is divided into a pre-specified number of segments with the corresponding probabilities. Then, several equal intervals are obtained by dividing each segment equally. Therefore, each interval can be characterized by the average value of factors within this interval, shown in the Fig. 4. Step 6: The system scenarios are formed by combining the average factor values of four types of data randomly in each time block. Each scenario comprises the four selected average factors, and thus the final scenario sets bSS of time block b is formulated as follows. SS b

= {ubD ( ), ubW ( ), ub ( ), ubT ( )}

= 1,

n

;

b

Cumulative probability

1

0.8

0.6

0.4

Segment 2 0.2

0 0.5

0.6

0.7

0.8

0.9

1

Electric demand (pu)

Fig. 4. Cumulative distribution function curves of electric demand in time block 1 (divided into two segments with 8 selected points each).

(1)

B

Selected point

Segment 1

The probability of scenario of time block b, b ( ), is calculated as the product of the corresponding four probability values in each segment. In addition, the nodal demand in each scenario can be obtained as the product of the predicted value P jL and power load factor ubL ( ) . Similarly, the traffic demand is calculated by the product of the predicted value qrs and traffic demand factor ubT ( ) .

origin-destination pair r-s expressed as Krs , with r TR and s TS . For SS scenario , b , the total traffic flow from start point r to end node s is expressed as qrs , and the traffic flow on road link a is expressed as x a, . Using the widely used Bureau of Public Roads (BPR) function, the scenario-based travel time can be rewritten as follows:

4. Mathematical modeling of traffic network

ta, (xa, ) = ta0 [1 + 0.15·(xa, / Ca ) 4],

The constructed TN can be represented by a connected graph GT = {TN TA} , in which TN / TA represents a node/link set of the road. Thus, any link a in the network can be represented by a TA . Let TR and TS represent the start node set and the end node set of the traffic demand flow (TR, TS TN ). Therefore, the set of paths that connects the

where ta0 represents the free travel time on link a and Ca is the maximum capacity of link a. Previous work has shown that the expansion of road capacity can alleviate the pressure created by the growth of the traffic demand and reduce the degree of road congestion. To explore the mutual effect of

Cum ulative probability

Cum ulative probability

Cum ulative probability

C um ulative probability

Time block 1

Time block 2 1

1

0.5

0.5

0.5

0

0

0

1

1

0.5

0.5

0

0 0.5 Wind power (pu)

1

0 0.5 1 Electric demand (pu)

0

(a)

b

B

(2)

0 0.5 1 Electric demand (pu)

0.5

0 0.5 Wind power (pu)

1

(b)

0

1

0.5

0.5

0.5

0

0

0

0 0.5 1 Photovoltaic power (pu)

(c)

0.5

0.5

0.5

0

0

0

(d)

0.5 1 Wind power (pu)

0 0.5 1 Photovoltaic power (pu)

1

0 0.5 1 Traffic demand (pu)

0

1

1

0 0.5 1 Traffic demand (pu)

SS b ,

1

1

0 0.5 1 Photovoltaic power (pu)

TA,

Time block 3

1

0 0.5 1 Electric demand (pu)

a

1

0 0.5 1 Traffic demand (pu)

Fig. 3. Cumulative distribution function curves of four sources in three pre-determined time blocks. (a) Electric demand factor curves. (b) Wind power factor curves. (c) Photovoltaic power factor curves. (d) Traffic demand factor curves. 6

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both system, the capacity expansion is also considered in this paper. In our model, the expansion investment integer variable z aC is introduced, and the capacity of link a is expressed as the sum of the original capacity and new expansion capacity, formulated as follows:

Ca = Ca0 + ca· z aC ,

a

where and CRoad are the annual investment coefficient and unit capacity cost of road investment, respectively. 5. Active distribution system planning model

(3)

TA

5.1. Branch flow model in the active distribution system

In this scenario, the traffic flow must not exceed the limited capacity after road expansion according to the following:

x a,

Ca,

a

TA,

As the penetration of distributed energy resources continues to increase, the traditional distribution system may face the challenge of bidirectional power flow. On this basis, a radial ADS can be represented by a directional graph [ B , L], where distribution line ij denotes the B positive direction of power flow from bus i to bus j, with i, j L. and, ij The branch flow equations proposed in [35] are applied to describe the power flow in our model. By integrating the DGs, CBs, SVCs, ESSs, and VRs in the ADS, the scenario-based Dist-Flow equation can be rewritten as follows:

(4)

SS b

The path traffic flow is introduced and it satisfies the following constraint:

f krs,

0,

r,

s,

k,

(5)

According to the TN topology, the relationship between the link traffic flow and the path traffic flow can be obtained as follows:

x a, = r

s

k

f krs, ·

rs a, k ,

a

SS b

TA,

(6)

where ars, k is the correlation coefficient between the link and the path, and if road link a is located on the kth path connecting the r-s, then rs rs a, k = 1; otherwise, a, k = 0 . Furthermore, in all scenarios, path flow should meet the traffic demand, as follows: k

f krs, = qrs ·ubT ( ) ,

k

SS b ,

Krs ,

b

B

a

t a, ·

rs a, k )·x a,

,

,

k,

r,

(ckrs,

µ rs )=0 rs

u ) 0

,

k,

r,

s,

a

CRoadz aC +

s.t. {(2), (3), (4), (5), (6), (7), (8), (9)}

k

L,

SS b ,

DG ,

p

b

B

(11)

B

(12)

~ SVC Iij, rij ) + QTS + QCB j, j, + Q j ,

(j ) B,

j

P Cj,

SS b ,

DG ,

p

b

~ 2(Pij, rij + Qij, xij ) + (rij2 + x ij2) Iij, ,

~ Ui,

ij

2(Pij, rij + Qij,

b

(13)

B

~ xij ) + (rij2 + x ij2) Iij, + M (1 SS b ,

B,

i, j

SS b ,

B/ B,

i, j

b

K LA

x ijLA , ),

B (14)

~ Uj,

~ Ui, L,

ij

2(Pij, rij + Qij,

~ xij ) + (rij2 + x ij2) Iij, SS b ,

B,

i, j

b

M (1

K LA

x ijLA , ),

B (15)

~ ~ I i j, Ui, = Pij2, + Qij2, , (U¯min )2

~ Iij,

~ Uj,

(I¯max

(U¯max )2 ,

SS b ,

L,

ij j

b

SS b ,

B,

SS b ,

(16)

B b

B

(17)

(18) ~ ~ It should be noted that, to avoid the quadratic terms, Uj, and Iij, are used to represent the square of bus voltage and line current (i.e., ~ ~ Uij, = Uij2, and Iij, = Iij2, ). Eqs. (11) and (12) describe the power balance equations, where the left-hand sides of both are the power injected into bus j and the right-hand side is the power withdrawn from bus j. Eq. (13) presents the forward voltage drop equation. Eq. (16) defines the apparent power at the head bus i of each line ij. Eqs. (17) and (18) define the bounds on the magnitudes of nodal voltage and line current. It is noteworthy that, on account of an expansion planning problem, B and L are used to represent the set of candidate buses and lines, reL , (i.e., xijLA spectively. If a feeder is invested in line ij , = 1), the constraints (14) and (15) would be imposed to be equivalent to (13). Otherwise, a large constant M could make the voltage drop equation invalid.

0

(9)

( )

QjL, ,

L/ L,

~ Uj,

Clearly, constraint (9) is consistent with the Wardrop UE condition in the literature [34] that does not consider multiple scenarios. If u rs . If f krs, f krs, = 0 , then ckrs, 0 , ckrs, = µ rs . The latter means that in any scenario, when TAP model (A1) reaches its optimal solution, the dual multiplier µ rs of OD pair r-s is smaller than or equal to the cost of any path ckrs, , . Thus, µ rs is the travel cost in the equilibrium state. Based on the discussion above, to consider the traffic flow balance under operation conditions, we can add the scenario-based equilibrium equivalence constraint (9) to the model so that the optimal solution satisfies the Wardrop UE principle. Therefore, the proposed TN expansion planning model that considers multiple scenarios can be summarized as follows: T T min finv + foper = ·

i

ij

(8)

B,

j (Qij,

(j )

~ ~ Uj, = Ui,

where is the monetary value; ckrs, is the travel time of path k, and is also called unit travel cost in many studies (e.g., [31]). In traffic theory, it is usually assumed that the user selects the route with the shortest travel time based on the destination and that the transit time of the road link is the result of the travel flow. Therefore, the transit times and travel choices interact and restrict each other and finally achieve the traffic flow balance state, which is the well-known Wardrop UE principle [32]. At this equilibrium state, the system is stable, and the traffic costs incurred on all paths used by the traveler are the same and not greater than the travel expenses on the unused routes. In other words, no traveler can reduce their travel expenses by changing routes. For determining equilibrium flow patterns, TAP model has been investigated in many studies. The previous works like [33] have confirmed that its optimal solution satisfies the Wardrop UE condition. To use this rule in the proposed model with multiple scenarios, this paper derives its equivalent scenario-based constraints. The detailed derivation process is shown in Appendix A. On this basis, the following equivalent constraint of scenario-based UE can be obtained:

f krs, (ckrs,

(j )

P jL, ,

+ QjDG , p,

(7)

s

i

Qjk, = k

In this paper, the travel expense (c krs, ) of path k is as follows:

c krs, = ckrs, ·xa, =(

(j )

+ P jDG , p,

~ D Iij, rij ) + PTS j, + P j,

(Pij,

Pjk, = k

)2 ,

ij

L,

b

B

5.2. Extended active distribution system expansion planning model The expansion planning model of ADS is developed to attain an optimal expansion scheme over the planning horizon with minimized investment and operation costs and technical constraints. The model identifies the optimal types, locations and sizes of the distribution lines, WTGs, PVGs, CBs, SVCs, VRs, ESSs, and CFs, combined with their corresponding operation strategies in a coordinated manner. Considering the EV charging demand, the objective and related

c krs, (10) 7

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constraints of the proposed ADS expansion planning model are given in the following.

A VR is modeled by a distribution line in series with an ideal transformer with an adjustable ratio VR . If feeder ij is equipped with a VR, the voltage of bus j can be adjusted within a continuous range [37]. Voltage Um, for fictitious bus m can be calculated by Eq. (16) relying on head bus voltage Ui, , and the relational expression between Um, and Uj, is formulated as Eq. (31a). The ratio limit and capacity limit of a type VR are restricted by Eqs. (31b) and (31c), respectively. For simplicity, the tap positions of the VR are defined as continuous variables. As noted in [38], this assumption is reasonable for a planning issue, and has also been adopted by [37].

(1) Objective function D D min f D = finv + foper = C I + (C E + C L + C U )

CI =

L

ij

+

K LA

j

DG

KpDG p

j

SVC

KSVC

+

C LA x ijLA , + DG

L

ij

C I , SVC x jSVC + ,

CB

j

ESS

j

C LR x ijLR , +

K LR

DG CpI,, DG SpDG , x j, , p +

(19)

K ESS

K CB

VR

ij

KVR

C I , VR x ijVR ,

C I , CB x CB j,

VR ,min

(20) n

CE =

b(

)

b

b B

(

j

n

CL =

b(

)

bC

Loss

b B n

CU =

b(

)

b

b B

TS

(

( B

j

C TSPTS j, +

DG p

j

DG

CDGP jDG , p,

)

~ Iij, rij

)

~ TS Ij, z + TS

j

KTS

L

ij

CUL P jL, +

WTG p

j

DG

C CUR P jDG , p,

K LR

K LA

KVR

ij

L,

K LR

xijLA ,

1,

x ijLA ,

{0, 1},

ij

L,

K LA

x ijVR ,

K ESS

K CB

KSVC

p

DG

x jESS , x CB j, x jSVC ,

KpDG

1, 1,

x ijVR , x jESS ,

nCB , nSVC , x jDG , ,p

{0, 1},

x CB j,

{0, 1}, Z 0 +,

x jSVC , nDG ,

CB ,

j j

Z 0 +,

VR ,

ij

K VR

x CB j, ,

K CB

,

yjCB ,

Z 0 +,

CB ,

j

,

j

CB

(32a)

K CB

(32b)

(5) Energy storage system (ESS) In this paper, a generic storage model [39] is applied and mathematically represented as follows. (i) Charge/discharge state limit

uCj, + ujD,

1,

ESS

j

(33a)

(ii) Charge/discharge power limit

0

P Cj,

uCj,

K ESS

¯ C , max , x jESS , P

j

ESS ,

0

P jD,

ujD,

x ESS P¯ D, max , K ESS j,

j

ESS ,

B,

j

(33b)

(iii) ESS transition function

(

C

SS b

P Cj, )

(

P jD, D

)=0,

b

ESS ,

k

K ESS

(33c)

where and are two binary variables to describe the operating state of the ESS. Constraint (33a) ensures that each ESS cannot charge and discharge simultaneously. In Eq. (33b), the charge or discharge power of an ESS unit is bounded between the lower and upper values. x ESS = 1) and one of Only if the ESS is equipped according to ( K ESS ij, C D the state variables equals 1 (uj, = 1 or uj, = 1) at the same time will the charge/discharge power be nonzero. In (33c), the ESS transition function is formulated to be achieved at each time block following the way introduced in [39]. To model long-term planning problem by the proposed scenario technique, historical data has to be arranged in scenarios where chronological information between individual hours is missing [40]. As also indicated in [40], the time block division for data may reduce the loss of information about the sequence, but the time sequence of data is still failed to be retained. Due to the adopted scenario-based framework that may lose some chronological information,

uCj,

(28) (29)

p

,

where is the number of CB units in operation at bus j in scenario . Constraint (32a) indicates that the maximum CB operation number is not higher than the CB installation number. Eq. (32b) represents the total reactive power output of the CBs in each scenario.

(27)

K pDG,

)

x ijVR , ,

yjCB ,

(26)

K SVC DG ,

KVR

(

CB CB QCB yj , , j, = q

(25)

K CB

j

2 SVR , + M 1

yjCB ,

0

(24)

K ESS

SVC ,

(31b)

VR ,max ,

Based on [38], the CB reactive power output is defined by discrete variables. For further consideration of the investment variables x CB j, , the CB operation constraints can be extended as follows:

)

K VR

ESS ,

j

Z 0 +, x jDG , ,p

VR ,

ij

K VR

(ij)

(4) Capacity banks (CBs)

(22)

Investment decisions in new assets are modeled by binary or integrality variables, which are shown as follows.

{0, 1},

,

VR

j

(31c)

(2) Investment decision constraints

x ijLR ,

(31a)

,

(21)

As has been done in [36], the objective of Eq. (19) centers on minimizing the overall annual investment and operation costs composed of the investment, production, energy loss, and curtailment costs. In (20), the investment costs are characterized by the sum of terms related to: (i) the addition and upgrade of lines, and (ii) the installation of new infrastructures, including DGs, CBs, SVCs, ESSs, and VRs. Three terms are formulated as scenario-based deterministic equivalents: (i) the production costs related to substations, WTGs and PVGs in (21); (ii) the energy losses costs in (22); and (iii) the penalty costs related to curtailment of loads and DGs in (23).

1,

VR

·Um, ,

Pij2, + Qij2,

(23)

x ijLR ,

VR

Uj, =

C I , ESS x jESS ,

DG

(30) As per (24)–(30), the maximum of investment is limited for each system component. It is assumed that (1) only one possible investment of candidate alternatives for VRs, ESSs and lines is allowed, as in equations (24)–(27); (2) more than one investment for SVCs, CBs and DGs is permitted at each potential location, as in equations (28)–(30). (3) Voltage regulators (VRs) constraints 8

ujD,

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x LR = 1) is P¯ max , otherwise the limit power equals the original K LR ij, 0 one P¯max .

the ESS capacity limit is replaced by the charge-discharge balance Eq. (33c) following the way adopted in [39] and [40]. (6) Distributed generations (DGs)

5.3. Coupled transportation and active distribution system expansion planning model

In an ADS, smart inverters enable the DGs to provide not only active power, but also controllable reactive power in a specific range via their inverter interfaces [41], as shown in Fig. 5. To supply/absorb the required reactive power, the capacities of the DG inverters may have to be slightly oversized. Thus, the application of active power curtailment to the DGs and reactive power control are included in our scenario-based model, which is extended as follows: DG,nom 2 2 (SpDG ) + (QpDGmax )2 , , ) = (Pp, ,

* DG, PRE P jDG , p, = P j, p,

P jDG , p,

QjDG , p,

P jDG , p, ,

DG · P DG * , j, p,

,

p,

,

j

s.t. {Cons - ADS\}

In this comprehensive model, the objective is the sum of the investment and operation costs leveraged by a discount factor , where f T and f D are defined in Eqs. (10) and (19), respectively. The ADS constraints are expressed as Cons - ADS = \{ (20) (38)\} , and the constraints on the TN are given by Cons - TN = {(2) (9)} . Since a CF is considered as the bridge between two systems, the C C related costs of investment ( finv ) and operation ( f oper ) are also included in (39), which is characterized as follows:

(34b)

* 1 min{QpDGmax , P jDG , , p, ·tan(cos (pflim ))}

,

p,

,

DG

j

(34c) ,nom ubW ( )· P jDG ,p

if p = WTG,

, PRE P jDG = , p,

KpDG

,nom ub ( )· P jDG ,p

if p = PVG,

KpDG

x pDG , ,

x pDG , ,

, ,

j j

DG DG

C f inv =

(34d)

K SVC

x jSVC , ,

,

j

SVC

P jL,

LP L j,

=

,

P jL· ubL (

, )

,

b,

,

j

(35)

B

(36)

(9) Main grid power supply TS 2 2 (PTS j, ) + (Qj, )

(S TS ) 2 ,

,

TS

j

n

C I , CF x CF j,

( )

b B

b j

(37)

(10) Distribution lines

P¯ max ·

K

LA

x ijLA ,

P¯ max ·

Pi j,

K

LA

x ijLA , ,

,

ij

CF

CEV P jEV ,

(40)

a C (j )

P

L

DG S p,k

(38a)

¯ max

Pij,

{P

Pij,

P¯ max

0 x LR + P¯max K LR ij, 0 x LR + P¯max K LR ij,

(1 (1

)} , )

x LR K LR ij, x LR K LR ij,

,

ij

EV e ·x a a,

(41)

where C (j ) is the set of links coupled with bus j; EV is the proportion of EVs in all vehicles; and ea is the average energy requirement for a EV in all scenarios, which can assumed to be a constant (e.g., [42]). Following the way described in [15], and well applied in [14], this assumption is versatile and rational for modeling the impact on a power grid from the transportation side. Especially for a system-level study of a long-term expansion planning issue, rather than a dispatch problem that focuses on dynamic modeling, it provides a convenient and versatile way to model the interdependency between the TN and ADS. Thus, this general method is adopted and extended to a scenario-based form by constraint (41) in this paper. Thus, the coupling constraints can be formulated as follows:

B

j P jL,

K CF

D L P jL, = ubL ( )·P jL + P jEV , (x a, ) = ub ( )· P j +

(8) Load shedding management

P jL,

CF

where x CF is a binary variable for the type CF and P jEV is the , j, equivalent charging power of the EV. Without losing generality, we assume that the energy demand on each CF follows an increasing function of traffic flow x a, . Based on [14], it is also reasonable to assume that each electric bus serves one load P jL, combined with the EV charging requirement depending on the traffic pattern. In this regard, the nodal power load is expressed by the following interface equation:

(7) Static var compensators (SVCs)

Q SVC , max

j

C f ope =+

In our model, two forms of energy source (wind power and photovoltaic power) are considered for DGs. To represent the difference, notation p , p {WTG, PVG} is used as an index for generator types. The maximum apparent power of a type DG is given in Eq. (34a). The active power curtailments for the DG units is formulated in Eq. (34b), , PRE where P jDG /P jDG , p, is the DG output power before/after curtailment, , p, and curtailed power P jDG is limited by a ratio, namely DG . The , p, constraints (34c) indicate that the injected DG reactive power can be adjusted in a specific range. Finally, Eq. (34d) shows that the DG power at each scenario is calculated by generation factors of WTG or PVG (i.e., ubW ( ) or ub ( ) ), as well as the installed number x pDG , that is used to represent the existence of a type p, DG.

QjSVC ,

{Cons - Couple\} (39)

* 1 min{QpDGmax , P jDG , , p, ·tan(cos (pflim ))}

QjDG , p,

{Cons - TN\}

DG

DG

j

T D C T D C min f T + f D + f C = ·(finv + finv + finv ) + (foper + foper + f oper )

(34a)

p,

p,

By combining the model (10) in Section 4, with the model proposed in this section, the coupled transportation and active distribution system expansion planning is summarized as follows:

DG,nom Pp,k DG,PRE Pj,p,ω

ΔPj,DG p,ω

L

(38b)

Pj,DG* p,ω

-Q

The power flow limits for lines of type to be added and replaced are detailed in Eq. (38). In Eq. (38a), the active power would be imposed to be zero when candidate line ij is not selected to be installed. Eq. (38b) implies that the limit power through an upgraded line (if

Q

pf lim

-Q

DGmax p,k

-Q

DGmax* j, p,ω

Q

DGmax* j, p,ω

Q

DGmax p,k

Fig. 5. DG active and reactive power control model. 9

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P jEV , Cons - Couple =

p0CF · naCF + p CF

K CF

x CF j, ,

P jL, = ubL ( )· P jL + P jEV , (x a, ), P jEV , (x a, ) =

a C (j )

,

EV e · x a a,

,

, b, ,

M

CF

j

m, n = a, m=0 n=0 m, n 0, am, , n a,

CF

j

CF

j

x a, = z aC =

6. Solution methodology

ta, =

The main bottlenecks for solving the mixed integer nonlinear programming (MINLP) model (39) can be summarized as follows: i) the bivariate travel time function of Eq. (2); (ii) the bilinear complementarity condition in Eq. (9); (iii) some nonlinear terms in Eqs. (16), (31), (33) and (37). The equivalent linear expressions will be derived as follows.

x a, +

ca·z aC, m

,

m,

n,

m, n a, )

m=0 n=0 M N

m = 0 n= 0 M N

m, n a, x a , , n m, n C a, z a, m , m, n m, n a, ta,

m=0 n=0

M

m,

n,

a,

(44)

+ xaS, ,

a,

a + taS, ,

a,

(45)

taS,

N m, n m, n a, )·

xaS, ,

xaS,

[0, N / N ],

m=0 n=0

a,

(46)

where m, n is the slope between two discrete dots (xa, , n , tam, , n) and (xa, , n + 1, tam, , n + 1). This PWL approximation is proved to be acceptable with appropriate number of segments [43]. Since (46) contains a new bilinear term: xaS, · am, , n , we introduce a variable am, , n and hold the relationship am, , n = xaS, · am, , n , m , n, a, . Then, (46) can be replaced by using a big constant M, as follows:

0

a,

0

xaS, m, n a,

(43) By introducing an auxiliary weight variable ( of the location of optimal solution that satisfies

N

taS, =(

4 ,n

{0, 1},

where / is the slack variable of the traffic flow/time. Due to the integer nature of z aC , the feasible regions are narrowed to the blue dotted lines in X-Y plane and corresponding time values are denoted by curves with different colors. Thus, the exact solution for z aC can be obtained without introducing the slack variable. In Fig. 6, each function segment between two dots is replaced by a line. For each fixed z aC, m , the relational expression between xaS, and taS, can be given as:

Since the number of new roads to be invested is considered as variable z aC , the bivariate nonlinear function ta, (xa, , z aC ) is very difficult to linearize, and has rarely been studied thus far. For solving the function, we develop a three-dimensional PWL approximation method. Fig. 6 shows the shape of time ta, (xa, , z aC ) and the related feasible regions. It is partitioned into M × N disjoint parts by equidistant breakpoints x a, , n , n = 0, 1, , N and z aC, m ,m = 0, 1, , M . Then, the discrete function value (black dots in Fig. 6) corresponding to each x a, , n and z aC, m is denoted by:

Ca0

M

xaS,

6.1. Linearizing the travel time function

z aC, m) = ta0 1 + 0.15

a,

we can present any point using a convex combination of the extreme points in the rectangle regions with additional increment slack variables, formulated as follows:

where the first constraint imposes that the charging demand cannot be in excess of the capability severed by the CFs. The second and third constraints are the interface equations which have been defined in equation (41).

,n,

1,

(xa, , n , z aC, m , tam, , n)

(42)

tam, , n (xa,

N

M

as an indication

m, n a, M · am, , n ,

M ·(1 m,

m, n a, ),

n,

m, a,

a, (47)

N

taS, = m=0 n=0

m, n m, n · a, ,

a,

Fig. 6. Three dimensional PWL approximation approach for the bivariate travel time function. 10

n,

(48)

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6.2. Linearizing the complementarity condition By introducing a binary variable vkrs, replaced as:

0

f krs,

M (1

0

ckrs,

u rs

vkrs, ),

r,

s,

k,

M ·vkrs, ,

r,

s,

k,

the ESS charging/discharging limit of (33b) into convex constraints based on the big M method as follows:

{0, 1} , constraint (9) can be

M (1

0

(49)

u rs In constraint (57), a large M enforces that when vkrs, = 1, ckrs, 0 , which is and f krs, = 0 , otherwise, when vkrs, = 0 , ckrs, = u rs and f krs, equivalent to the original expression (9). 0 (

PTr QTr j, j,

(

T 2

SVR, + M 1

T 2

S Tr ,

,

j

KVR

)

x ijVR , ,

Tr

,

ij

VR ,

(

~ |Uj,

~ U

,

j

~ Uj, |

2 (U¯ max

(ij)

VR

m,

(

P jD,

P

D x jESS , ) + u j, D j, ,

C j, ,

K ESS

D j, ,

(54)

M (1

D x jESS , ) + uj,

M ·x jESS ,

P¯ D, max

K ESS

D j, ,

(55)

C j, ,

or

D j, ,

C j, ,

will be 0.

or

D j, ,

will have a value of 1.

6.4. The final mixed-integer second-order cone program formulation On the basis of the previous discussion, the constraints of the model described by (39) can be redefined as follows:

K VR

① Cons - ADS* = {(20 )

(51)

, (34) (36),(38),(50) (55)} ② Cons - TN* = {(2) (8), (43) (49)}

(52)

KVR

C x jESS , ) + u j,

M · x jESS ,

(30), (31b), (32), (33a), (33c)

Thus, the resulting MISOCP model can be given as follows: T D C T D C min f T + f D + f C = ·(finv + finv + finv ) + (fope + fope + f ope )

s.t. {Cons - ADS*\}

{Cons - TN*\}

VR 2~ , max ) U m, 2 U¯ min )

M (1

It can be observed that only when the investment variable of ESS D ) and the state variable uCj, or ujD, equals 1 at the same K ESS j, ,

Otherwise,

Now, Eqs. (31a) and (33b) still remain nonconvex. According to [38], Eq. (31a) including the bilinear term VR· Um, , can be reformulated by the following linear constraints: VR 2~ , min ) U m,

¯ C, max

C j, ,

time, the auxiliary variable

In (50), SOCR is performed by replacing “=” with “ ” and has proved to be exact enough in [45]. We can observe that constraints (31c) and (37) have a similar form to (16), and hence can also be rewritten in the conic form as follows:

Qij,

C j, ,

P Cj,

M· x jESS ,

Since the branch flow equation adopted in this paper is similar to [44], the SOCR method introduced in [44] is also applied to reformulate the nonlinear expression (16), shown as follows: ~ ~ ~ ~ SS L, 2Pij, 2Qij, Iij, Uij, 2T Iij, + Ui, , ij (50) b

Pij,

M· x jESS ,

M (1

6.3. Linearizing other expressions

C x jESS , ) + u j,

x ijVR ,

{Cons - Couple\}

(56)

7. Case study

(53)

7.1. Basic parameter setting

~ where Um, is the square of Um, . (53) ensures that the voltage on sec~ ondary side (Uj, ) can be adjusted in a certain range. Finally, two auxiliary variables jC, , and jD, , are introduced to

This section presents the numerical results of a coupled system that involves a ring expressway TN and a radial ADS. The topology and related information (candidate locations, coupled buses) are shown in

substitute the bilinear terms uCj, x jESS and ujD, x jESS , , , thereby transforming

Fig. 7. Schematic of a coupled transportation and active distribution system (including 4 newly added load buses, 8 candidate distribution lines and some potential candidate units). 11

Applied Energy 255 (2019) 113782

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Fig. 7. The related TN parameters can be obtained from [17], and the ADS is based on a well-known IEEE 33-bus system whose data can be referred to [46]. By modification, four load buses (220 kW and 110 kVar) and eight lines are added to the original ADS. Details of the candidate lines are given in Table 1, and the alternatives for all elements are listed in Table 2. It is noticeable that the cost of upgrading lines is higher than that of adding new lines, since upgrading lines involves removing and installing steps. The cost parameter of distribution lines can be obtained from [10]. Additionally, it is assumed that the base power value is 10 MVA. The active power flow limit for all original 0 = 5 MW . The magnitude bounds of the bus voltage and line lines is P¯max current are U¯min = 0.94 , U¯max = 1.06 (in p.u. with a base voltage value of 12.66 kV) and I¯max = 2 A . At the slack bus, the supply is STS = 50 MVA and the reference voltage is 1.0 p.u. The capacity of each new road is ca = 15 (p.u). The existing capacity of CF is p0CF = 0.1 MW and there are two available types of new CF ( pkCF ) whose capacities are 0.5 MW ($12050) and 0.8 MW ($14000). The capital discount factor is = 0.1628. The cost of building on the additional roads is $10,000 and the monetary value of the unit travel time is $10/h. The cost coefficients of the energy supply by the substation and DG, as well as that of power loss are all chosen as $500/MWh. The penalty costs of the DG power and load power curtailment are set to $300/MWh. The DG power factor is pflim = 0.8, and the curtailment rates ( DG / L ) are set to 20%. The data of the EVs is given as EV = 20\% , ea = 200kW . The parameters of PWL are m = 4 and n = 15. All existing roads and distribution lines are considered for expansion or upgrade, and all load buses are schedulable except the slack bus. The CFs are allowed to be built at the roadside and connected to one electric bus such as [14]. The proposed model is coded in MATLAB and solved by calling CPLEX.

invest more roads and CFs. Meanwhile, more lines can also be upgraded when electric demand remains unchanged. The main reason for this is higher charging demand from EV, which leads to more active power through the distribution lines installed with CFs. Therefore, not only do the traffic costs increase, but the operation costs in ADS are also higher. (3) for Case 4, increasing both electric and traffic demands leads to more investments in both TN and ADS, and much higher investment and operation costs. It is also interesting to note that although Case 2 is 40% more electric demand than Case 1, only two more DGs (13 in all) are installed. However, when traffic demand is increased by 40% as well (i.e., Case 4), six more DGs (17 in all) are built. Furthermore, in terms of other network elements in ADS, Case 4 results in two more CBs, one more SVC and a VR with larger capacity (type: II) compared with Case 2. Meanwhile, the curtailment cost related to ANMs in Case 4 is higher than that in Case 2. This result apparently shows that the variation in traffic flow in TN has a major impact on the investment and operation of the ADS’s infrastructures, indicating the necessity to consider the interdependency of ADS and TN in system expansion planning problems. To verify the benefits of road expansion for TN under multiple scenarios, the results of maximum travel time of Case 1, before and after expansion planning, is given in Fig. 9 and Tables 4–5. As can be seen from Fig. 9(a) and (b), the maximum travel time after capacity expansion (the red line) is not more than that before expansion (the blue line) on each road link or under each scenario. It can be observed from Table 4, there has a significant reduction in travel time after expansion, especially for road link 8 and 14 (reduced by 31.475% and 29.899%), which implies that the planning strategy is effective for relieving traffic pressure on some key roads. Moreover, Table 5 shows the results of time reduction under the critical scenarios, which are highlighted in Fig. 9(b), indicating that the capacity expansion plays a vital role on handling the high level of traffic demand. It can be observed that the high-traffic demand factors in the scenarios 12, 14, 28 and 46 can reach 0.726, 0.872, 0.854 and 0.873, respectively. Clearly, these unfavorable situations become the major contributor to traffic congestion for system, and thus roads are invested mainly for them. A few more observations can be made about the interactions in the process of operation. Table 6 provides three other cases with partial consideration of ANMs. It can be observed that without the participation of some active devices from the grid side, the operation costs will increase not merely in the ADS, but also in the TN. This indicates that the coordination of ANMs has a certain impact on traffic operation, since the corresponding operation strategies can help on-road CFs better adapt to the variation of EV charging demand. Moreover, it can be noted that the ESS dispatch plays a more important role in reducing operational costs due to its benefits pertaining to peak load shaving. Fig. 10 exhibits the change of voltage distribution with traffic demand. From Fig. 10(a) and (b), it can be seen that some buses installed with CFs have a lower level of voltage (e.g. buses 11, 12, 16, 17, 22, 23, 32, 33). Although the coordinated operation of the ANMs keeps the

7.2. Numerical results For comparison, the scenario generation method proposed in [19] and in this paper are both performed based on the historical data of wind power [47], photovoltaic power [48], electric and traffic flow demand [49]. Three time blocks are set with durations of 988 h, 4622 h, and 3174 h, and two segments in each time block is determined by probabilities of 0.6 and 0.4. Thus, a total of 48 scenarios can be obtained, shown in Fig. 8. It can be observed that there are only 6 levels for each source in the Fig. 8(a), while total 48 different levels for each source can be found in Fig. 8(b). Since the corresponding CDFs are represented by two levels in the conventional way, combining the levels as the representative scenarios will reuse each of them 8 times and lead to similar scenarios. This is also noted in [10]. Compared with it, each CDF is equally divided into 16 intervals (see Step 5 in Section 2) in this paper, and thus the obtained scenario set based on the combination of different values can cover total 48 levels. Based on the scenario set obtained by the proposed method (shown in Fig. 8(b)), a sensitivity analysis was performed on the electric demand (ED) and traffic demand (TD) by setting different increasing rates (IR), and the results are given in Table 3. Table 3 lists the results of different demands for cases 1 to 4. It can be seen that different increments cause the variation of investments and related operation costs. Taking the Case 1 as a benchmark,

Table 1 Impedance parameters for candidate distribution lines.

(1) for Case 2, by increasing electric demand only, more lines are upgraded and more DGs, SVCs and CBs are invested in, thus increasing the investment costs by 12.036%. As for operation costs, the production, energy losses and curtailment costs in ADS have increased by 53.251%, 49.257%, and 69.581% respectively. Since the charging ability of CFs depends on power grid supply capacity that is affected by electric demand level, traffic distribution will change in some extent. Therefore, traffic costs have increased 2.979% when power load grows. (2) for Case 3, by increasing traffic demand by 40%, it is possible to 12

No.

From bus

To bus

Resistance (r)

Reactance (x)

33 34 35 36 37 38 39 40

7 8 21 22 31 32 18 33

34 34 35 35 36 36 37 37

0.0250 0.0187 0.0156 0.0281 0.0094 0.0187 0.0125 0.0218

0.0117 0.0140 0.0117 0.0211 0.0070 0.0140 0.0094 0.0164

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seen from Fig. 10(d), there are more the blue areas (values under 0.96 pu) compared to that in Fig. 10(b), indicating that traffic demand has a major impact on voltage management which cannot be neglected. To investigate the impact of scenario sets on investment decisions and further improve the result’s confidence, four scenario sets of different sizes are generated by setting different numbers of time block (i.e., 2, 3, 4 and 5). Thus, the sizes of the scenario set reaches 32, 48, 64, and 80, respectively. Fig. 11 compares the results of computation time and investment cost from solving the proposed planning model (66) with the four scenario sets. It can be seen that the investment cost tends to be stable when the time block number reaches 3, while the computational time continues to increase with the sizes of scenario sets. For a planning problem, there is generally a compromise between accuracy and computational efficiency. Based on the results, 3 time blocks can be considered as the best compromise since the accuracy of the solution is no longer significantly affected by the size of scenario sets. Furthermore, the computation time for solving the above cases considering three time blocks is provided in Table 7. According to the results, the consuming time of the model implementation is 2191.884 s on average, which indicates the practical applicability of the proposed T&ADS expansion planning approach. Although it will increase with the actual system scale, this will not become a limitation for the real applications since the time scale of the expansion planning problems addressed is long [17]. In order to verify the exactness of the three-dimensional PWL approach, the relative average error index is defined as follows:

Table 2 Available distribution lines, DGs, CBs, SVCs, ESSs and VRs. Existing lines to be upgraded

Candidate lines to be added

Power limit (MW)

Cost ($/km)

Power limit (MW)

Cost ($/km)

I II

1.4 6.0

12,840 19,020

1.4 6

8030 12,520

I II

Wind turbine generator (WTG) Capacity (MWA) Cost ($) 0.8 11,500 1.8 22,400

Photovoltaic generator (PVG) Capacity (MWA) Cost ($) 1.2 5900 2.0 13,200

I II

Capacity bank (CB) Capacity (MVar) 0.02 0.08

Static Var compensator (SVC) Capacity (MVar) Cost ($) 0.5 2500 0.8 4500

I II

Energy storage system (ESS) Efficiency Power limit (MW) 98% 0.2 98% 0.5

Cost ($) 6000 14,000

I II

Voltage regulator (VR) Capacity (MVA) Voltage ranges (pu) 5.0 0.95–1.05 8.0 0.94–1.06

Cost ($) 10,000 14,000

Cost ($/unit) 140 370

voltages under the desired range, the voltage of these buses is reduced by the plug-in EV charging demand during the scenarios with high traffic demand (e.g. ω = 8–16, 24–31, 40–48). In contrast, when the traffic demand increases from 40% to 80%, the voltages have larger fluctuations over all scenarios, shown in Fig. 10(c). More specifically, as

AE(

)=

1 na

ta, a

ta0 [1 + (xa, / Ca) 4 ] 0 ta [1 + (xa, / Ca ) 4]

(57)

(a)

(b) Fig. 8. The results of multiple scenarios in three time blocks (with the duration hours of 988, 4622 and 3174 respectively). (a) The resulting scenarios by the classic scenario generation method in [19]. (b) The resulting scenarios by the proposed scenario generation method in this paper. 13

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Table 3 Planning results from the increased electric and traffic demand. IR (ED,TD) Roads Link (#number)

Case 1 40%, 40%

Case 2 80%, 40%

Case 3 40%, 80%

Case 4 80%, 80%

2(1),6(1),7(1),8(1), 9(2),10(1),14(3), 16(1),18(1),20(1) Sum: 13 I: 2(1),5(1),29(1), 12(1),16(1),17(1), 11(1),26(3); II:22(1),6(1),23(1), 32(2),33(2),17(1) Sum: 18

4(1),6(1),7(1),10(3), 8(1),9(2),13(1),14(3), 15(1),16(1),18(1) Sum: 16 I: 22(2),5(1),32(1) 29(1),12(3),17(3), 16(1),11(1); II:2(1),6(1),23(1), 26(1),32(2),33 (2) Sum: 21

4(2),7(1),8(1),9(1), 10(3),12(3),14(3), 15(1),16(1),18(1) Sum: 14 I:22(2),5(1),32(1),33(3), 29(3),12(2),17(2); II:2(1),6(1),23(3),26(2), 32(1),16 (1),11(1) Sum: 21

I: 8,12,16,18,22, 23,26,27,29,30 II: 2–7,9–11,13,19–21,25,28,31,32 Sum: 27

I: 23 II: 2–13,15,16,18, 21,22,25–32 Sum: 26

I: 34 II: 35,37,38 I: 12(4),22(3), 36(5); II: — Sum: 12

Upgraded lines (Type: lines)

I: 8–10, 22,23,26–29; II: 2–4,11,12, 15,17–21,25,30–32 Sum: 24

Built lines

I: 33 II: 36,37,39 I: 36(1); II: 12(3), 22(3),36(4) Sum: 11

6(1),7(2),8(1),9(1), 10(1),11(1),14(1) 17(1), 20(1) Sum: 13 I: 22(2),6(2),5(1) 32(2),33(3),17(1), 29(2); II:2(1),23(2),26(1), 32(1),11(1),17(1) Sum: 20 I: 6,8,12,13,22,23, 26,28,30 II: 2–5, 7,9–11,18, 21,25, 27,29,31,32 Sum: 26 I: 34 II: 36,38,39 I: —; II: 12(4), 22(4),36(5) Sum: 13

I: 33 II: 7 I: 23(6) II: 34(6) Sum: 12 I:16(1),29(1) II:— Sum: 2

I: 7 II: 33 I: 23(17) II: 34(3) Sum: 20 I:16(2),29(2) II:— Sum: 4

I: 33 II: 7 I: 34(6) II: 23(6) Sum: 12 I:16(1),29(1) II:— Sum: 2

I: — II: 34,36,38,40 I: 6(5),12(1); II: 12(4),22(2),36(5) Sum: 17 I: 7 II: 33 I: 23(19); II: 34(3) Sum: 22 I:16(3),29(2) II:— Sum: 5

Type: I 62424.551 2371.858 1070.442 3440.947 4414.469

Type: I 70965.874 3634.894 1597.713 5835.198 4545.954

Type: I 71253.654 3573.678 1232.362 4561.317 5713.558

Type: II 77408.675 6115.176 1588.524 7857.435 5911.702

CFs

Type: bus (number)

DGs

Type: bus (number) ESSs Type: bus CBs Type: bus (number) SVCs Type: bus (number)

VR CI ($) CE ($) CL ($) CU ($)

T ($) foper

Fig. 12 depicts the average error scatter in different scenarios. It is clear that the average errors are very small and kept in 1%. For a planning issue with long time scale, the approximation method in this paper is efficient in the practical application. As a new type of distribution network in a real application, the ultimate goal of T&ADS planning is to make decisions on building DGs Travel time before expansion

and other active devices to meet the growth and uncertainty of power and traffic demand with consideration of operational aspects. Based on the simulation results, the proposed model and solution in this paper can effectively deal with the planning issues under the current trend of independency for ADS and TN.

Travel time after expansion

#1~20 Link number 14

#6

#8

#7

50 s

#5

#4

#9

19 20 21 22

#3

25 s #10

#2

#11

#20

#16

9

8

7

6

5

4 3 2

#17

1

48 47 29 30 31

#18 #15

11 10

27

28

#19

#14

12

25 26

#13

13

23 24

#1

#12

18

17

16 15

1~48 Scenario

32

33

42 41 34 35 40 36 37 38 39

45 44 43

46

(b)

(a)

Fig. 9. The comparison results of the travel time. (a) The maximum travel time of each link (#1–20) among all scenarios. (b) The maximum travel time under each scenario (ω = 1–48) among all links. 14

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Time reduction

Corresponding scenario (ω)

# # # # # # #

18.065% 31.475% 17.629% 10.872% 29.899% 9.568% 16.507%

28 12 14 14 14 14 46

6 8 9 10 14 16 18

Investment cost ($)

Link number.

Time cost (s)

Table 4 Travel time reductions on some key links.

Table 5 Travel time reductions under certain scenarios. Scenario (ω)

Time reduction

Traffic demand factor: ubT ( )

1 12 14 26 28 31 41 46

29.899% 31.475% 21.051% 18.222% 27.432% 10.754% 9.368% 29.282%

0.380 0.726 0.872 0.716 0.854 0.652 0.664 0.873

T ($) foper

4 0

2

3

5

4

5

(a)

105

8

4

Time block number

6 4 2

3

Time block number

(b)

Fig. 11. The results from solving the proposed model with different scenario sets of 2, 3, 4 and 5 time blocks. (a) The comparison of time costs with different numbers of time block. (b) The comparison of investment costs with different numbers of time block.

by the climate and is not statistically independent and, for the first time, we have combined the historical data of traffic flow with electric demand, wind and photovoltaic power to generate scenario sets. Based on multiple scenarios, comprehensive mathematical formulations for coordinated planning are established as a mixed-integer nonlinear program model. To solve the model, we provide the related mathematical transformation method and develop a three-dimensional piecewise linear approximation method to reformulate the model into a mixedinteger second order conic program. The simulation results reveal that it is of great importance for expansion planning problems to consider the interdependencies between the transportation and active distribution system. The conclusions based on the cases studies can be summarized into the following aspects:

Table 6 Comparison of operation costs between different cases with full and partial considerations of the ANM.

CE ($) CL ($) CU ($)

103

8

Case 1

Case 5: Without DG control

Case 6: Without ESS dispatch

Case 7: Without SVC and CB

2371.858 1070.442 3440.947 4414.469

7727.181 1737.318 2790.547 4637.587

9982.317 3863.963 4824.439 4895.396

3372.097 1351.412 3372.097 4725.343

8. Conclusion The simultaneous expansion planning model for the coupled transportation and active distribution system presented in this interdisciplinary research is one of the first attempts to study planning issues considering the interdependencies between the active distribution system and transportation network. We find that traffic flow is affected

Voltage (pu)

Bus Number

(1) The comparison of planning results shows that the variation of traffic demand can affect the investment and operation strategies of active distribution system simultaneously. Similarly, a higher level

Scenario

(a) Voltage (pu)

Bus Number

(b)

Scenario

(c)

(d)

Fig. 10. Voltage magnitude in ADS over 48 scenarios. (a) Voltage distribution of Case 1 (T = 40%). (b) Voltage contour map of Case 1. (c) Voltage distribution of Case 3 (T = 80%). (d) Voltage contour map of Case 3. 15

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Table 7 CPU computation time of solving different cases by using CPLEX (in seconds). Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Case 7

Mean value

2403.679

2386.332

2279.427

2686.449

1773.299

1940.453

1873.555

2191.884

Fig. 12. Average error scatter of travel time Cases 1, 2, 3 and 4 under different scenarios.

of electric demand also incurs a relatively higher traffic operation cost. (2) The planning strategies on traffic roads from solving the scenariobased model exhibit a considerable reduction of travel time, especially for some key roads during the high-traffic demand period. Based on the results, some unfavorable scenarios with high-level traffic demand can be considered as the major contributor to traffic congestion, indicating the great impact of scenario sets on road investment decisions. (3) The coordinated control of active network management has a certain impact on traffic flow distribution. The case with full consideration of active network management has a lower traffic cost compared to the case with partial consideration. In addition, the different levels of traffic demand also influence the voltage

distribution under each scenario, and will in turn change the optimal operation strategies of the active distribution system. Future research based on this work can be highlighted in developing more sophisticated models that combine more advanced technologies in the active distribution system and more uncertain pattern in the transportation network, as well as the more efficient and accurate algorithms for solving non-convex optimization problems. Acknowledgement This work was supported by National Natural Science Foundation of China under grant 51977156.

Appendix A. The derivation of the equivalent constraint of scenario-based UE state The scenario-based TAP model can be formulated by an integral form of time as follows:

{

x a,

min FTAP = ·

a TA

ta, (xa, ) dxa, | s.t. (5)(6)(7)

0

}

(A1)

Thus, the Lagrange function of (A1) can be defined as follows:

L (x , µ , ,

) = FTAP + +

a,

a

r, s

µ rs ·[qrs ·ubT ( )

·(xa, r, s k

f krs, ·

rs a, k ),

k

f krs, ] + ,

r, s k

b,

r,

rs k,

·f krs,

s

(A2)

where µ , a, and are dual variables of corresponding constraints (7), (6), (5); µ , and are corresponding matrices. The corresponding first-order condition includes constraints (3), (4), (5), (6), as well as the following equations: rs

rs k,

L / f krs, = 0,

k,

r,

s,

(A3)

rs k,

·f krs, = 0,

k,

r,

s,

(A4)

rs k,

rs

0, µ ,

a,

: free,

k,

r,

(A5)

s,

where (A1) means that the partial derivatives of (A2) respect to the path flow f krs, are equal to 0. Deriving the first item of (A2) and combining with (6) can be simplified as follows:

x a, FTAP = { f krs, f krs, x a,

x a, a

0

ta, (xa, ) dxa, } =

rs a, k

ta , (x a, )

(A6)

a

Deriving the second item, we find the following:

f krs,

µ rs ·(qrs ·ubT ( ) r, s

k

f krs, ) =

µ rs rs a, k

(A7)

After deriving the third item, we find the following: 16

Applied Energy 255 (2019) 113782

S. Xie, et al. rs k,

f krs,

r, s

· f krs, =

k

rs k,

(A8)

After deriving the fourth item, we find the following: f krs,

=

a,

a

=

a

r, s k

fkrs,

f krs,

x a,

(

f krs, ·

(x a,

a

x a,

a,

rs a, k · a,

a,

fkrs, rs a, k )

·

rs a, k )

a r, s k

a,

·f krs, ·

rs a, k

=0

(A9)

Then, (A3) can be written as follows:

L / f krs, =

a

· ta·

rs a, k

µ rs +

rs k,

= 0,

(A10)

It is easy to find that the first item in Eq. (A10) is equivalent to the unit travel cost

ckrs,

µ rs =

rs k,

,

k,

Finally, based on

f krs,

(ckrs, rs (ck,

rs k,

r,

ckrs,

, and then we have: (A11)

s,

0 and (A5), the following equivalent constraint can be obtained:

rs

µ )=0 u rs ) 0

,

k,

r,

s,

(A12)

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