On the Interaction between Autonomous Mobility on Demand Systems and Power Distribution Networks -- An Optimal Power Flow Approach

1 On the Interaction between Autonomous Mobility on Demand Systems and Po wer Distrib ution Networks — An Optimal Po wer Flo w Approach Alv aro Estandia*, Maximilian Schif fer , Federico Rossi, Justin Luke, Emre Kara, Ram Rajagopal, Marco Pa v one Abstract —In future transportation systems, the charging be- havior of electric A utonomous Mobility on Demand (AMoD) ﬂeets, i.e., ﬂeets of electric self-driving cars that service on- demand trip requests, will lik ely challenge power distribution networks (PDNs), causing overloads or voltage drops. In this paper , we show that these challenges can be signiﬁcantly at- tenuated if the PDNs’ operational constraints and exogenous loads (e.g., from homes or businesses) are accounted for when operating an electric AMoD ﬂeet. W e focus on a system-level perspective, assuming full coordination between the AMoD and the PDN operators. From this single entity perspecti ve, we assess potential coordination beneﬁts. Speciﬁcally , we extend pre vious results on an optimization-based modeling approach for electric AMoD systems to jointly control an electric AMoD ﬂeet and a series of PDNs, and analyze the beneﬁt of coordination under load balancing constraints. For a case study of Orange County , CA, we show that the coordination between the electric AMoD ﬂeet and the PDNs eliminates 99% of the overloads and 50% of the voltage drops that the electric AMoD ﬂeet would cause in an uncoordinated setting. Our r esults show that coordinating electric AMoD and PDNs can help maintain the reliability of PDNs under added electric AMoD charging load, thus signiﬁcantly mitigating or deferring the need for PDN capacity upgrades. Index T erms —Electric A utonomous Mobility on Demand, Net- work Flow , Smart Grid, Unbalanced Optimal Po wer Flow . I . I N T R O D UC T I O N A. Estandia is with Marain Inc., Palo Alto, CA 94306, USA. He worked on this paper while he was a visiting student at Stanford Uni versity . Email: alvaro@marain.com . M. Schif fer is with the TUM School of Management, T echnical Univ ersity of Munich, Munich 80333, Germany . Email: schiffer@tum.de . F . Rossi is with the NASA Jet Propulsion Laboratory , California In- stitute of T echnology , Pasadena, CA 91109, USA. He worked on this paper while he was a Ph.D. student at Stanford Uni versity . Email: federico.rossi@jpl.nasa.gov . J. Luke is with the Department of Civil and En vironmental En- gineering, Stanford Univ ersity , Stanford, CA 94035, USA. Email: jthluke@stanford.edu . E. Kara is with eIQ Mobility , Oakland, CA 94612, USA. He worked on this paper while he was with the SLA C National Accelerator Laboratory , Menlo Park, CA 94025, USA. Email: eck@fastmail.com . R. Rajagopal is with the Department of Civil and Environmen- tal Engineering, Stanford University , Stanford, CA 94035, USA. Email: ramr@stanford.edu . M. Pa vone is with the Department of Aeronautics and Astronautics, Stanford University , Stanford, CA 94035, USA. Email: pavone@stanford.edu . *Corresponding author . ©2021 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collectiv e works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. Fig. 1: Integration of an expanded road graph (left) and multiple power distrib ution networks (PDNs) (right). T ypically , a road network spans across multiple PDNs and connects to the PDNs via charging station vertices. Besides charging stations that represent control- lable loads, PDNs contain reference buses (typically substations) highlighted in black and uncontrollable loads from residential and commercial customers. F LEETS of electric self-driving cars servicing on-demand trip requests promise affordable urban mobility with reduced greenhouse gas emissions [1], decreased need for parking [2], and fewer road accidents [3]. Additionally , such systems of fer further beneﬁts stemming from optimized central coordination, e.g., increased vehicle utilization compared to priv ately owned vehicles [2], and increased operational ﬂexi- bility and efﬁcienc y compared to taxi, car-sharing, and ride- hailing services. Furthermore, electric Autonomous Mobility on Demand (AMoD) has the potential to foster the adoption of electric vehicles (EVs) since, in a high-utilization ﬂeet, EVs are more economical than their gasoline-powered counter- parts [1]. Nonetheless, operating an electric AMoD ﬂeet also bears inherent challenges as EVs show range limitations which require time-consuming recharging that adds a sizable load on power distribution networks (PDNs). Studies on priv ate EVs showed that uncoordinated charging may require costly PDN upgrades to secure stabilization, as it can destabilize PDNs due to overloads or under-v oltages [4]–[6]. In contrast, we expect that intelligently coordinating the vehicles’ charging would reduce such neg ati ve impacts, in particular, by reducing or deferring the need for power network upgrades. Controlling AMoD systems entails solving a dispatching problem to assign vehicles to on-demand trip requests. The system’ s performance increases if empty vehicles are proac- tiv ely repositioned (rebalanced) in anticipation of future de- mand [7]. In the past decade, multiple approaches have been presented for the control of AMoD systems with varying de- grees of mathematical complexity . A ﬁrst family of algorithms relies on heuristic rules to dispatch and rebalance a ﬂeet [8], [9]. More sophisticated methods use optimization algorithms 2 to control the AMoD system. Often, network ﬂo w models using ﬂuidic relaxations, i.e., allowing for fractional vehicles and fractionally serviced trip requests are used [7]. Models of this type hav e been extended to consider road capacities and congestion [10]. T o control an electric AMoD system, an operator must keep track of a vehicle’ s state-of-charge (SoC) and recharge a ve- hicle’ s battery accordingly . Again, some heuristic approaches exist [11], [12]. Optimization-based algorithms are so far not amenable to large-scale problems as they rely on mixed-inte ger linear programs (MILPs) with discretized SoCs [13], [14]. At its core, the operation of an electric AMoD system induces a coupling between the power network and the transportation system. Speciﬁcally , the electric AMoD ﬂeet represents a controllable load in time and space. All previously mentioned studies neglect the impact of an electric AMoD system on the power grid, despite the fact that ev en a moderate amount of EVs may signiﬁcantly increase electricity prices [15] and may negati vely inﬂuence the power grid’ s reliability [16], [17]. A few recent studies consider such a coupling implicitly via av ailable capacities [18] or prices [19], but the proposed control algorithms for the electric AMoD ﬂeet do not explicitly account for the ﬂeet impact on the power network. Only Rossi, Iglesias, Alizadeh, et al. [20] consider the ﬂeet impact on the power network explicitly , introducing the Power in the Loop Autonomous Mobility on Demand (P-AMoD) model, a linear model that combines a network ﬂow model for the electric AMoD system and a balanced single-phase DC model of a tr ansmission network. Ho wever , this model does not consider the power distribution network, which is the more appropriate grid stage to analyze mesoscopic EVs ﬂeet operations [21]. Notably , a single-phase DC model is not sufﬁcient to model a PDN as it assumes a constant v oltage magnitude, and neglects reactiv e po wer and link resistances [22]; instead, a three-phase model is necessary [23, Ch. 1]. So far , PDNs were only considered when determining optimal charging schedules for pri v ately owned EVs which ha ve to reach a certain SoC by the end of a gi ven planning horizon [5], [6], [24] as opposed to centrally coordinated ﬂeets. Here, an instance of the Optimal Power Flow (OPF) problem can be solved to balance necessary charging loads with PDN-speciﬁc constraints. In summary , indi vidual aspects of the control problem addressed in this paper, such as the control of an electric AMoD system or considering PDN models to optimally char ge priv ate EVs, hav e been addressed in the literature. Howe ver , to the best of our knowledge, no studies that tightly couple an electric AMoD system and PDN models currently exist. This work addresses this gap. Speciﬁcally , our contribution is threefold. First, we present a benchmark of conv ex three- phase PDN power ﬂow approximations and identify a model compatible with the characteristics of the electric AMoD problem. W e then extend the mesoscopic model in [20] to capture the operations of and interaction between an electric AMoD system and a series of unbalanced PDNs. Second, we embed this model within an optimization problem that assesses achiev able beneﬁts with respect to full cooperation between the two systems. The mesoscopic optimization’ s solution en- ables comprehensi ve analyses to identify bottlenecks in PDNs and inform operator decisions in the day-ahead electricity market. Third, we provide a case study of Orange County , CA where we study the impact of an electric AMoD system on the PDNs and ev aluate the beneﬁts of coordination. The remainder of this paper is structured as follows: Section II reports the mesoscopic model for an electric AMoD system used in previous work for self-consistency . Section III surve ys existing PDN models and identiﬁes a suitable model for the electric AMoD application. Section IV discusses the interaction between the electric AMoD system and a series of PDNs. Section V details our case study of Orange County , CA, and presents results that characterize the impact of electric AMoD systems on PDNs, highlight- ing the improvement potential stemming from coordination. Section VI concludes this paper with a summary of its main ﬁndings and an outlook on future research. Finally , the Appendix summarizes our notation and nomenclature. I I . M O D E L I N G E L E C T R I C A M O D S Y S T E M S In an AMoD system, a ﬂeet of autonomous vehicles services customer transportation requests, i.e., picks up customers at their origin and brings them to their destination [2]. A ﬂeet operator controls the AMoD ﬂeet by assigning vehicles to customer requests and by routing each vehicle. Besides origin- destination trips of customers, the routing may comprise rebalancing trips in-between two customer trips as spatial and temporal mismatches between origins and destinations of different customer requests arise. In an electric AMoD prob- lem, the ﬂeet operator additionally controls vehicle charging schedules and rebalances vehicles based on anticipated spatial- temporal variations of vehicle SoCs and electricity prices. W e model an electric AMoD system with a network ﬂo w model as originally presented in [20], reported in this section for self-consistency . Sections III and IV then detail our main contribution by integrating this model with PDNs. T o av oid integer variables, the model uses i ) a ﬂuidic vehicle approxi- mation and ii ) a road graph expanded along two dimensions: discrete-time and vehicles’ SoC. a) General r oad network r epresentation: W e model the road network as a graph G R = ( V R , A R ) with a set of vertices v ∈ V R and a set of road segment arcs ( v , w ) ∈ A R . Each arc ( v , w ) ∈ A R is characterized by a distance d v ,w , a trav ersal time t v ,w , and an energy consumption c v ,w . W e consider a set T = { 1 , . . . , T } of discrete equidistant time steps (each of duration ∆ t ∈ R + ), and a set C = { 1 , . . . , C } of equidistant discrete battery charge lev els (each of energy E c ∈ R + ). While some vertices in G R merely represent intersections or access points, others represent charging stations S ⊆ V R that allow recharging of vehicles. Each charging station s ∈ S has a charging rate δ C,s ∈ { 1 , . . . , C } that denotes the amount of SoC that can be recharged in a single time step. Additionally , charging stations hav e a certain number of char ging plugs ¯ S s ∈ N + which limits the number of concurrently charging vehicles. W e model congestion using a threshold model, i.e., we assume that vehicles driv e at the road’ s free-ﬂow speed as long 3 as their number is less than the road’ s capacity ¯ f v ,w ∈ R + , as detailed in [20]. b) Expanded graph repr esentation: W e use an expanded graph to model a vehicle’ s location and SoC over time. The expanded graph G = ( V , A ) is directed and has a vertex set V ⊆ V R × T × C . Each vertex v ∈ V is deﬁned by a tuple ( v v , t v , c v ) that represents a verte x v v of V R at a speciﬁc time t v with a speciﬁc SoC c v . Figure 1 (left) illustrates the concept of SoC expansion; for ease of representation, the time expansion is not shown. The resulting arc set A consists of two subsets A T ∪ A S = A . Arcs ( v , w ) ∈ A T represent trav el in the road network and must meet the following condition A T = { ( v , w ) ∈ A | ( v v , v w ) ∈ A R , t w − t v = t v v ,v w , c v − c w = c v v ,v w } , i.e.: i ) ( v v , v w ) is a road arc, ii ) the time expansion t w − t v equals its trav ersal time t v v ,v w , and iii ) the SoC expansion c w − c v equals its consumption c v v ,v w . Arcs ( v , w ) ∈ A S represent recharging at a charging station and must meet the following condition A S = { ( v , w ) ∈ A | v v = v w = s ∈ S , c w − c v = ( t w − t v ) δ C,s } , i.e.: i ) v v and v w are equal and correspond to a charging station, and ii ) the SoC difference c w − c v equals the amount of energy recharged, that is ( t w − t v ) δ C,s . c) Customer trip requests: In addition to this graph representation, we deﬁne a set of customer trip requests M = { 1 , . . . , M } . Each trip m ∈ M is deﬁned by a quadruple ( v m , w m , t m , λ m ) ∈ V R × V R × T × R + that denotes its origin v m , its destination w m , its departure timestep t m , and the number of customer trip requests (i.e., the number of customers who wish to trav el between v m and w m departing at t m ) λ m . W e assume a deterministic setting in which these requests are known or predicted for all timesteps. T o reduce the number of decision variables, we use precomputed vehicle routes for customer -carrying vehicles, corresponding to shortest-time paths r v → w that do not violate the congestion constraints. As we use a threshold congestion model, we can straightforwardly precompute such feasible shortest-time paths by solving a network ﬂow problem as in [20]. Each shortest- time path has a traveling time t v → w and a charge requirement c v → w . W e denote λ c, dep m as the number of vehicles with charge c departing to serve customer trip request m and λ t,c, arr m as the number of vehicles with charge c arriving at time t after serving customer trip request m . Thus, λ t,c, arr m = ( λ c + c v m → w m , dep m if t m = t − t v m → w m 0 otherwise ∀ t ∈ T , c ∈ C , ∀ m ∈ M . (1) d) Electric AMoD model: W e introduce f 0 ( v , w ) : A → R + to represent the ﬂo w of customer -empty vehicles on arc ( v , w ) , which includes both rebalancing and charging vehicles. Further, N I ( v ) denotes the initial location of the ve- hicles, i.e., the number of vehicles av ailable at vertex v v with charge level c v at t v = 1 and is zero for all other timesteps. Analogously , N F ( v ) denotes the desired ﬁnal location of the vehicles, i.e., the number of vehicles that must be at node v v with charge lev el c v at t v = T . W ith this notation, a multi- commodity ﬂo w representation of the electric AMoD model is given by: X w :( v , w ) ∈A f 0 ( v , w ) + M X m =1 1 v v = v m 1 t v = t m λ c v , dep m + N F ( v ) (2) = X u :( u , v ) ∈A f 0 ( u , v ) + M X m =1 1 v v = w m λ t v ,c v , arr m + N I ( v ) ∀ v ∈ V , C X c =1 λ c, dep m = λ m ∀ m ∈ M , T X t =1 C X c =1 λ t,c, arr m = λ m ∀ m ∈ M (3) Here, 1 x is the indicator function. Equation (2) secures ﬂow conservation for rebalancing and charging vehicles, ensures a suf ﬁcient number of empty v ehicles in each v ertex to cov er originating trip requests, and enforces initial and ﬁnal conditions on the v ehicle locations through N I and N F . Equation (3) distributes the demand for a gi ven trip request m to v ehicles with dif ferent SoC, and accumulates vehicles arriving at different times with different SoC for request m . e) Electric AMoD pr oblem: W e now extend the basic constraints of the electric AMoD model to a full electric AMoD model. Speciﬁcally , we optimize the vehicles’ rebal- ancing routes and charging schedules in order to minimize the cost of operating the electric AMoD system, that is: minimize f 0 , [ λ c, dep m ] c ∈C , [ λ t,c, arr m ] c ∈C ,t ∈T ,N I ,N F V D X ( v , w ) ∈A T d v v ,v w f 0 ( v , w ) + X ( v , w ) ∈A S : v v = v w = s V el ,s [ t v ] δ C,s f 0 ( v , w ) (4a) sub ject to Eqs. (1) to (3) Electric AMoD model X ( v , w ) ∈A T : v v = v ,v w = w,t v = t f 0 ( v , w ) ≤ ¯ f ( v ,w ) ,t ∀ ( v , w ) ∈ A R , t ∈ T (4b) X ( v , w ) ∈A S : v v = v w = s,t v = t f 0 ( v , w ) ≤ ¯ S s ∀ s ∈ S , t ∈ T (4c) g I ( N I ) = 0 g F ( N F ) = 0 (4d) Here, we use the pre viously introduced concept of expanded graph vertices: each vertex v ∈ V is deﬁned by a tuple ( v v , t v , c v ) ∈ V R × T × C . The objectiv e function Eq. (4a) minimizes the operational cost of the electric AMoD system, considering time-in variant operational cost per unit distance (e.g., discounted cost for maintenance, tires, depreciation) V D ∈ R for rebalancing vehicles and time-varying electricity costs V el ,s ∈ R for rechar ging vehicles at a charging station s ∈ S . Figure 1 depicts example arcs that model such rebalanc- ing and charging ﬂo ws ( f 0 ), as well as λ c, dep m and λ t,c, arr m for an example trip m marked with bold arrows. Equations (1) to (3) impose general ﬂow conserv ation while Eq. (4b) applies the threshold congestion model to rebalancing ﬂo ws. As customer- carrying ﬂows are ﬁxed, we do not consider these directly in Eq. (4b). Instead, we use the residual road capacity ¯ f ( v ,w ) ,t 4 which results from subtracting the customer carrying ﬂow on road arc ( v , w ) at time step t from the corresponding road capacity ¯ f v ,w . The pre-routed vehicles may congest a road link. In this case, we set the residual capacity ¯ f ( v ,w ) ,t for that link to zero. Thus, customer -carrying ﬂo ws and residual capac- ity are ﬁxed and constant with respect to the optimization of rebalancing ﬂows. Equation (4c) limits the number of vehicles that can use a charging station concurrently according to the number of charging plugs at each station. W e impose initial and ﬁnal conditions on vehicles with the generic functions g I and g F in Eq. (4d). The brackets in the decision variables denote concatenation. W e will use this con vention in the rest of the paper . The electric AMoD problem Eq. (4) has T C ( |A R | + |S | ) + C M + T C |V R | + C |V R | decision v ariables. The dominant term is C M : there could be at most one customer trip request from every origin to every destination at ev ery time step such that M ≤ |V R | 2 T . It follows that C M ∈ O ( C |V R | 2 T ) . A few comments are in order . First, we consider discrete time steps as well as discrete SoC levels. From a mesoscopic viewpoint, these discretizations bear sufﬁcient accuracy while improving the model’ s computational tractability signiﬁcantly . Second, the network ﬂow model treats vehicles and customers as fractional ﬂows; accordingly , it is not readily suitable for real-time control of electric AMoD ﬂeets. Again, this accuracy loss is acceptable at a mesoscopic lev el and is compatible with our goal of assessing the achie vable performance stemming from the coordination between electric AMoD and PDN operators. Note that our solution can still be used as a reference plan for a lower -lev el microscopic controller [cf. 25]. Third, we limit the vehicle ﬂow on a giv en road link to its capacity and assume vehicles travel at free-ﬂo w speed accordingly . Such a threshold congestion model is in line with the accu- racy requirements of our mesoscopic viewpoint. If necessary , more sophisticated congestion models can easily be integrated into our modeling approach, at the cost of computational tractability . Fourth, our model does not explicitly account for congestion from non-AMoD traf ﬁc. Howe ver , this type of trafﬁc can be considered by subtracting the corresponding ﬂow from the residual road capacity ¯ f ( v ,w ) ,t . Fifth, we assume that future trip requests are known or estimated with a high degree of accurac y . While the dev elopment of tools to estimate AMoD demand is beyond the scope of this paper, remarkably accurate algorithms are av ailable in the literature [e.g., 26]. Sixth, we optimize only rebalancing trips and ﬁx customer trips to their shortest-time-paths. In principle, including the optimization of customer -carrying trips could yield solutions with lower cost; howe ver , our prior work has shown that the inclusion of customer-carrying trips in the optimization problem results in a small decrease in cost at the price of a signiﬁcant increase in computational complexity [20]. Also, note that although the route of customer -carrying trips is ﬁxed, the SoC of customer- carrying vehicles is part of the optimization problem. Finally , the electric AMoD problem Eq. (4) may become infeasible if the number or the distribution of customer trip requests exceeds the customer-carrying capacity of the electric AMoD system. Here, we assume that the problem is always feasible as the ﬂeet operator can reject or postpone trip requests to ensure feasibility . This is in line with common practice in today’ s taxi or ride-hailing platforms. Nonetheless, a mechanism to decide which trips should be rejected or postponed is beyond the scope of this paper . I I I . M O D E L I N G U N BA L A N C E D P O W E R D I S T R I B U T I O N N E T W O R K S This section provides the basics for modeling unbalanced PDNs and presents the identiﬁcation of a compatible con ve x power ﬂow surrogate to model the integration of PDN into an electric AMoD model under a uniﬁed notation framew ork. First, we introduce an unbalanced PDN model in Section III-A. Then, we deﬁne the optimal power ﬂow problem in Sec- tion III-B. Finally , we compare con vex power ﬂow surrogates in Section III-C and justify the selected surrogate. A. Unbalanced power distribution network model In the follo wing, we consider only radial network structures which is the typical conﬁguration for PDNs [23, Ch. 1.1] and base our notation on [22]. A radial PDN is modeled as a directed graph P = ( N , E ) with a tree topology , consisting of a set of buses N = { 0 , . . . , N } and a set of links E ⊂ N 2 . Each PDN has a reference b us which typically denotes a substation that connects the PDN to the transmission network. The set N + = N \ 0 contains all buses other than the reference bus 0. Buses are connected by links (e.g., power lines, transformers, regulators), such that ( n, o ) ∈ E represents a link between n and o for which n lies in the single path between the reference bus 0 and b us o . Note that there is only one such path because, by assumption, P is a tree. W e consider unbalanced PDNs with three phases Φ = { a, b, c } . In line with this, Φ n,o ⊆ Φ is the set of phases in link ( n, o ) ∈ E . Further , the set of phases in bus n ∈ N comprises the phases of all links connected to the bus: Φ n =  ∪ ( m,n ) ∈E Φ m,n  ∪  ∪ ( n,o ) ∈E Φ n,o  ∀ n ∈ N . Each bus n has a time-inv ariant shunt admittance matrix Y n ∈ C | Φ n |×| Φ n | , representing the admittance between the bus and ground. Further , each link ( n, o ) has a time-in v ariant impedance matrix Z n,o ∈ C | Φ n,o |×| Φ n,o | . W e consider a discrete-time model that tracks a series of steady states in the power network and neglects dynamic effects. This is appropriate if the discretization time is substan- tially longer than the time scale for the dynamic effects (i.e., in the order of minutes). W e consider a timespan T = { 1 , . . . , T } with time steps t ∈ T , each having a length ∆ t ∈ R + . Each bus n has a time-dependent complex voltage v φ n [ t ] ∈ C and a complex power injection s φ inj ,n [ t ] ∈ C for each of its phases. Concurrently , each link sho ws a time-dependent current for each of its phases i φ n,o [ t ] ∈ C . For brevity , we use vectors for per-phase quantities: v n = [ v φ n ] φ ∈ Φ n , s inj ,n = [ s φ inj ,n ] φ ∈ Φ n , and i n,o = [ i φ n,o ] φ ∈ Φ n,o . Herein, superscripts represent the projection onto speciﬁc phases. The current on each link obeys Ohm’ s law , that is: i φ n,o [ t ] = Y n,o (( v n [ t ]) Φ n,o − ( v o [ t ]) Φ n,o ) ( n, o ) ∈ E , t ∈ T , 5 with Y n,o = Z − 1 n,o [22]. Each bus is either speciﬁed by its voltage or by its power injection such that the remaining quantity is a dependent variable [27, Ch. 6.4]. W e refer to speciﬁed variables as direct variables and to those that are dependent as indirect variables . The reference bus speciﬁes the reference voltage v φ ref [ t ] ∈ R for the network: v φ 0 [ t ] = v φ ref [ t ] φ ∈ Φ 0 , t ∈ T . (5) Accordingly , the complex voltage v 0 is the direct v ariable and the complex power injection s inj , 0 remains dependent. For all other buses n ∈ N + , the complex power injection s inj ,n is the direct variable, whereas the complex voltage v n remains dependent. These b uses are called PQ buses since the active ( p ) and reactiv e power injection ( q ) are the direct variables. Herein, each PQ-bus has a time-varying uncontrol- lable load with complex po wer s unc ,n [ t ] ∈ C | Φ n | . These loads represent electricity demand from residential and commercial customers. W e consider uncontrollable loads to be exogenous but known in advance within timespan T . Controllable loads ` ∈ L = { 1 , . . . , L } are deﬁned by a tu- ple ( s con ,` [ t ] , n ` ) ∈ C | Φ n ` | × N denoting their complex po wer s con ,` and its corresponding bus n ` . These loads represent dispatchable generators or loads that can be throttled. With this notation, the power injections at PQ-buses are s inj ,n [ t ] = − s unc ,n [ t ] − L X ` =1 1 n = n ` s con ,` [ t ] n ∈ N + , t ∈ T . (6) Note that we model generators as negativ e loads without loss of generality . Further , we consider only wye-connected con- stant power loads which may require performing delta-to-wye con versions for some loads and approximating constant current and constant impedance loads as constant power ones. This simpliﬁcation is common in optimization frameworks [28]. Dependent variables result from the network topology and its controllable and uncontrollable loads. Speciﬁcally , they are related by the power ﬂow equation [29] s inj ,n [ t ] = diag( v n [ t ] v n [ t ] H Y H n ) (7) + X n :( n,o ) ∈E diag( v Φ n,o n [ t ]( v Φ n,o n [ t ] − v Φ n,o o [ t ]) H Y H n,o ) Φ n t ∈ T . Collectiv ely , these equations allow us to model a radial time- in variant unbalanced PDN with time-varying controllable and uncontrollable loads. A few comments are in order . First, we consider a discrete- time model that tracks a series of steady states in the power network. As we are not interested in dynamic ef fects, this model is appropriate, and the level of aggregation is aligned with our mesoscopic transportation model. Second, we con- sider a time-inv ariant PDN which cannot model control ele- ments, e.g., step voltage regulators. Optimization framew orks commonly neglect these elements (see [24], [29]) as their in- clusion substantially increases complexity while their omission results in a more conserv ati ve optimization. This simpliﬁcation is appropriate for the purposes of a mesoscopic system-lev el analysis. Third, we assume that high-quality estimates of uncontrollable electrical loads are a vailable. While deriving such estimates exceeds our scope, techniques to accurately estimate future power demand exist [e.g., 30]. B. Optimal P ower Flow pr oblem The Optimal Power Flo w (OPF) problem Eq. (8) optimizes a power network’ s state subject to its operational constraints and is often used to support grid-related decisions, e.g., operational or strategic planning, and pricing [31]. Here, we use an OPF problem for operational planning and decide on the controllable loads while optimizing a generic objectiv e function f ( · ) subject to the power ﬂow equation Eq. (7) and additional operational constraints: minimize [[ v n ] n ∈N , s 0 , [ s con ,` ] ` ∈L ] t ∈T f ( · ) (8a) sub ject to Eq. (5) V oltage at reference bus Eq. (6) Powe r injections Eq. (7) Powe r ﬂow equation | v φ n [ t ] | ≥ u φ min ,n φ ∈ Φ n , n ∈ N + , t ∈ T (8b) | v φ n [ t ] | ≤ u φ max ,n φ ∈ Φ n , n ∈ N + , t ∈ T (8c)       X φ ∈ Φ s φ 0 [ t ]       ≤ ˆ s 0 t ∈ T (8d) p φ con , min ,` ≤ p φ con ,` [ t ] ≤ p φ con , max ,` φ ∈ Φ n ` , ` ∈ L , t ∈ T (8e) q φ con , min ,` ≤ q φ con ,` [ t ] ≤ q φ con , max ,` φ ∈ Φ n ` , ` ∈ L , t ∈ T (8f) Equations (5) to (7) denote the general power network model. Equations (8b) and (8c) constrain the v oltage magnitude | v φ n [ t ] | to be within a minimal u φ min ,n ∈ R and a maximal u φ max ,n ∈ R value, according to regulations (e.g., ANSI C84.1). Equa- tion (8d) limits the apparent power injected to the reference bus to be less than ˆ s 0 ∈ R + , typically , to respect the rating of the substation transformer . Equations (8e) and (8f) model the characteristics of controllable loads through lower and upper bounds on acti ve power ( p φ con , min ,` , p φ con , max ,` ∈ R ), and reactive power ( q φ con , min ,` , q φ con , max ,` ∈ R ). The AMoD- OPF joint problem described in Section IV -C will lev erage approximations of the operational constraints in Eq. (8) and include an electricity cost objective term. This OPF problem is non-con ve x because of i) the po wer ﬂow equation Eq. (7) and ii) lower bound constraints on voltage magnitudes Eq. (8b). Ev en the optimization of a balanced single-phase approximation of this problem remains an NP-hard problem [32]. C. Con vex power ﬂow surr ogates W e desire the joint AMoD-OPF problem to be conv ex and ideally linear to preserve computational tractability . Hence, we con ve xify the OPF problem Eq. (8) using a po wer ﬂow surrogate that approximates the po wer ﬂow equation Eq. (7) with a con ve x proxy , making the problem formulation computationally tractable. Using such a power ﬂow surrogate, we lose exact knowledge of the indirect variables. Giv en the high rele v ance of the OPF problem, a v ast literature on power ﬂow surrogates exists [31], [33]. Ho we ver , 6 most of these surrogates, as well as comparati ve studies, consider only balanced single-phase models as typically used in transmission networks [34]. For unbalanced three-phase models, only a few power ﬂow surrogates exist, and, to the best of our knowledge, no survey or benchmark classiﬁes the suitability of these surrog ates for speciﬁc problem structures, such as integration with the electric AMoD problem. T o close this gap, we analyzed and compared three promising surrogates. W e compared a con vex, semi-deﬁnite program (SDP) sur- rogate [22], the branch ﬂow model SDP (BFM-SDP), against two linear surrogates: the branch ﬂo w model LP (BFM-LP) [22], [35] and the linearized power ﬂo w manifold LP (LPFM- LP) [36]. W e used the charger maximization problem, which maximizes the power delivered to a series of charging stations across a distribution network as a benchmark, as it challenges the surrogates by pushing the network’ s operational constraints to its limits. For each surrogate, we ev aluated its accuracy in approximating the indirect variables used in Eqs. (8b) to (8d). Additionally , we analyzed the resulting constraint violations and computational times W e detail the methodology of our comparison in [37] but omit it in this paper due to space limita- tions. In summary , BFM-SDP yielded exact solutions on small instances but performed signiﬁcantly worse than the other two approaches in both solution quality and computational time for large instances. LPFM-LP and BFM-LP sho wed a trade-of f between solution quality and computational time, with a 91% reduction on the mean average error in approximating b us v olt- age magnitudes (LPFM-LP) and 97.3% shorter computational times (BFM-LP), while neither of both violated the substation rating constraint. Based on these results, we use the BFM-LP in this work as it preserves linearity in the joint problem while yielding suf ﬁcient solution quality for our mesoscopic study and relatively short computation times. The BFM-LP assumes ﬁx ed link losses and ﬁxed voltage ratios between phases in a bus [22], [35]: let ˜ i n,o ∈ C | Φ n,o | be the ﬁxed link current in link ( n, o ) ∈ E used to determine the ﬁxed link losses. Let ˜ v n ∈ C | Φ n | be the voltage used to determine the ﬁxed voltage ratios in bus n ∈ N + . Then, the matrix of ﬁxed voltage ratios for link ( n, o ) ∈ E , Γ n,o ∈ C | Φ n,o |×| Φ n,o | , has entries ( Γ n,o [ t ]) ij = (( ˜ v n [ t ]) Φ n,o ) i (( ˜ v n [ t ]) Φ n,o ) j i, j ∈ { 1 , . . . , | Φ n,o |} , ( n, o ) ∈ E , t ∈ T . W e deﬁne the following matrices to ease the notation: W n [ t ] = v n [ t ] v n [ t ] H , Λ n,o [ t ] = diag(( v n [ t ]) Φ n,o i n,o [ t ] H ) , ˜ L n,o [ t ] = ˜ i n,o [ t ] H ˜ i n,o [ t ] ( n, o ) ∈ E , t ∈ T . Assuming ﬁxed link losses and voltage ratios, the power ﬂow equation Eq. (7) admits a linear approximation [29]: X m :( m,n ) ∈E Λ n,o [ t ] − diag ( Z m,n ˜ L n,o [ t ]) − diag ( W n [ t ] Y H n ) + s inj ,n [ t ] = X o :( n,o ) ∈E ( Λ n,o [ t ]) Φ n n ∈ N , t ∈ T , (9) W o [ t ] =( W n [ t ]) Φ n,o − ( Γ n,o [ t ] diag( Λ n,o ) Z H n,o (10) + Z n,o ( Γ n,o [ t ] diag( Λ n,o )) H + Z n,o ˜ L n,o [ t ] Z H n,o ( n, o ) ∈ E , t ∈ T The constraints on voltage magnitudes then read: diag(( W n [ t ]) φ ) ≥ ( u φ min ,n ) 2 φ ∈ Φ n , n ∈ N + , t ∈ T (11) diag(( W n [ t ]) φ ) ≤ ( u φ max ,n ) 2 φ ∈ Φ n , n ∈ N + , t ∈ T (12) Now each non-linear term in Eq. (8) can be replaced with a linear approximation to yield the BFM-LP: i) the po wer ﬂow equation Eq. (7) with the BFM linearization Eqs. (9) and (10) and ii) the voltage magnitude constraints Eqs. (8b) and (8c) with Eqs. (11) and (12). Note that Eq. (8d) constrains a complex scalar to lie within a circle of radius ˆ s 0 in the complex plane. This constraint is non-linear but con vex and can be represented as a second-order cone. T o obtain a linear program (LP), we approximate the circle with a 12-face re gular polygon [38] which covers more than 95% of the circle’ s area. The BFM-LP has T ( P n ∈N | Φ n | 2 + 2 P ( n,o ) ∈E | Φ n,o | + 2 | Φ 0 | + 2 P ` ∈L | Φ n ` | ) decision v ariables. Here, T P n ∈N | Φ n | 2 is the dominant term since, by assumption, P has a tree topology such that |N | = |E | + 1 . Since v oltages are complex-valued (i.e., two components per phase) and Φ n has at most three phases, it follows that | Φ n | 2 ∈ O (1) . Thus, the dominant term grows proportional to the number of buses |N | and the number of time steps T . In line with this, it admits an upper bound O ( T |N | ) . A few comments are in order . First, we use a linear power ﬂow surrogate which entails the approximation of indirect v ari- ables. W e discuss its v alidity and attenuate potential constraint violations in Section V. Second, by using the BFM-LP surro- gate we treat link losses and voltage ratios as ﬁxed parameters. Previous research has shown that BFM-LP achiev es sufﬁcient accuracy e ven under the assumption of zero link losses and perfectly balanced voltage ratios [22]. Our formulation is even more accurate since we use reasonable estimates for the ﬁxed parameters instead of setting them to zero [35]. I V . I N T E R AC T I O N B E T W E E N A N E L E C T R I C A M O D S Y S T E M A N D P OW E R D I S T R I B U T I O N N E T W O R K S In this section, we de velop a model for the joint optimization of an electric AMoD system and a series of PDNs. Speciﬁcally , as an electric AMoD system usually spans across multiple (disconnected) PDNs, we ﬁrst introduce the multi-OPF prob- lem which combines multiple OPF problem instances. Then, we formalize the coupling between the electric AMoD system and the PDNs before we state the joint AMoD-OPF problem. 7 A. Multi-OPF problem The multi-OPF problem couples D instances of the OPF problem and results straightforwardly by extending the con- straints for each instance d ∈ D = { 1 , . . . , D } . W e neglect couplings upstream of PDN substations through the transmission network as this paper focuses solely on the interaction between an electric AMoD system and a series of PDNs. Couplings between the electric AMoD system and the power network at the transmission and distribution lev el occur on very dif ferent spatial scales (tens of kilometers vs. hundreds of meters), and result in largely orthogonal effects: speciﬁcally , couplings at the transmission lev el mainly inﬂuence bulk elec- tricity prices [20], whereas couplings at the distribution level inﬂuence bus voltages and power losses. Accordingly , due to the orthogonal nature of the two couplings, we en vision that a nested optimization approach could be used to ﬁrst address transmission-lev el couplings through existing algorithms [e.g., 20], and then optimize distribution-lev el couplings through the tools proposed in this paper . B. Coupling of the electric AMoD system and power distri- bution networks The charging stations, which appear as controllable loads in the PDNs, couple the electric AMoD system to the PDNs (see Fig. 1). F ormally , this coupling is established by two functions, M S , A S and M S , L , deﬁned below . The function M S , A S : S × T → A S maps a charging station s ∈ S for each time step t ∈ T to all arcs in A S that represent charging vehicles at this station: M S , A S ( s, t ) = { ( v , w ) ∈ A S | v v = v w = s, c v < c w , t v ≤ t ≤ t w } . Then, the load at charging station s is gi ven by p s [ t ] = E c δ C,s X ( v , w ) ∈M S , A S ( s,t ) f 0 ( v , w ) s ∈ S , t ∈ T (13) The function M S , L : S → ( L × D ) maps a char ging station s ∈ S to the associated controllable load ` ∈ L and distrib ution network d ∈ D . It follows that charging station s is attached to bus n M S , L ( s ) in PDN d M S , L ( s ) . As we consider three-phase charging stations, we assume equally distributed loads, that is s a con , M S , L ( s ) [ t ] = s b con , M S , L ( s ) [ t ] = s c con , M S , L ( s ) [ t ] = 1 3 p s [ t ] s ∈ S , t ∈ T . (14) Note that we can model in verters that control the load power factor since q φ con , M S , L ( s ) must not necessarily be zero. Although the charging station load is distributed equally among phases, loads in distribution networks are inherently unbalanced, which requires an unbalanced distribution model [23, Ch1.3] Also note that charging stations are commonly modeled to operate at unity power factor (no reactive power consumption) [39]. C. AMoD-OPF problem The joint AMoD-OPF problem results from coupling the electric AMoD problem Eq. (4) with the multi-OPF problem through Eqs. (13) and (14), namely: minimize f 0 , [ λ c, dep m ] c ∈C , [ λ t,c, arr m ] c ∈C ,t ∈T ,N I ,N F , [[ v n ] n ∈N , [ i φ n,o ] ( n,o ) ∈E , s 0 , [ s con ,` ] ` ∈L ] t ∈T , d ∈D V D X ( v , w ) ∈A d v v ,v w f 0 ( v , w ) + X t ∈T ∆ t X d ∈D V el ,d [ t ] X φ ∈ Φ p φ 0 ,d [ t ] (15a) sub ject to Eqs. (1) to (3) and (4b) to (4d) Electric AMoD system [ Eqs. (5), (6), (9) and (10) ] d ∈D and [ Eqs. (8b) to (8f) ] d ∈D PDNs Eqs. (13) and (14) Coupling from charging stations. The objecti ve Eq. (15a) captures operating costs for both the electric AMoD ﬂeet and the PDNs since we consider full cooperation between both operators. Analogously to the isolated electric AMoD problem Eq. (4a), we consider only rebalancing costs for the AMoD ﬂeet given ﬁx ed customer ﬂows. In each distribution netw ork d ∈ D , we account for the electricity cost that results from charging vehicles, uncontrollable loads, and power losses. Note that our joint problem formulation treats both operators as a single entity , assuming complete information and coop- eration. This assumption is in line with our mesoscopic view and scope to estimate the achiev able beneﬁts of coordination and cooperation between the two systems. W e leave the study of game-theoretical aspects to future work where we intend to de velop pricing and coordination mechanisms to align the goals of the electric AMoD operator and the PDN operators, and to lev erage distributed optimization algorithms to compute a solution to the AMoD-OPF problem Eq. (15) in a distributed manner . Further , our joint model assumes that the electric AMoD system is the dominant means of electric transportation, which is in line with our system-level perspectiv e [3]. Howe ver , the model can readily accommodate other EVs by including their trafﬁc ﬂow as residual capacity in Eq. (4b) and their charging as exogeneous loads in Eq. (8). V . C A S E S T U D Y I N O R A N G E C O U N T Y , C A W e ev aluate the impact of an electric AMoD system on the PDNs and the beneﬁt of optimized joint coordination through a case study in Orange County , CA. Our case study considers commuting trips within the cities of Fountain V alley , Irvine, North T ustin, Orange, Santa Ana, T ustin, and V illa Park. In the following, we detail our data (V -A), outline the experimental design (V -B) and, ﬁnally , discuss our results (V -C). A. Model parameters W e focus on an eight-hour commuting cycle from 5 am to 1 pm on July 3, 2015 discretized into six-minute time steps, such that |T | = 80 . As we do not consider future grid storage devices, which would charge/discharge over the span of a day , an eight-hour horizon is suf ﬁcient to model the po wer system. W e chose the time discretization to be close to the traversal 8 time of the shortest road link. As the power system considers hourly prices and excludes transient effects, six-minute time steps are more than sufﬁcient to model PDNs for a mesoscopic analysis. For this period, we model the charging station and transportation networks at a mesoscopic aggregation level that allows a suf ﬁcient le vel of detail to analyze the interaction between an electric AMoD system and the PDNs, and ensures computational tractability . a) T ransportation network data: we derive trip demand from Census Tract Flow data from the 2006-2010 American Community Survey . From these data, we take the estimated commuting ﬂows between the 143 census tracts that are part of our case study . T o align the granularity of aggregated charging station network representations and census tracts, we cluster the 143 census tracts into 20 larger areas using a k-means algorithm. W e neglect commuting ﬂows if they start or end outside the area of our case study or if they start and end within the same cluster since these types of ﬂows cannot be accurately represented in our model. Our planning horizon comprises 122 , 219 trips (32.8% of the total daily trips). The problem of ﬂeet sizing for (electric) AMoD systems [40] is beyond the scope of this paper . For this case study , we heuristically selected a sufﬁcient ﬂeet size, large enough to keep the AMoD-OPF problem Eq. (15) feasible with only a small number of idle vehicles and corresponding to 140% of the peak concurrent number of passenger-carrying trips. W e create an aggregated road network based on Open- StreetMap data with the same granularity as the trip demand data. For this, we select the road network vertices closest to the centroids of the census tract clusters and add arcs between those vertices if a connection exists in the real road network. W e obtain an aggregated road network with 20 vertices and 76 arcs (see Fig. 2), which captures vehicle travel and charging between the separate PDNs of the case study region. Note that computational complexity limits our model to coarse road networks; this is discussed in detail below . F or each aggregated road network vertex, we consider three-phase 50-kilow att DC fast char ging stations with ¯ S s = 40 plugs in total. Accordingly , each verte x has a charging station with a maximum load of Fig. 2: Area considered in the Orange County , CA case study . The aggregated road network is sho wn in orange, representing vertices as dots and arcs as lines. Green dots show the substation locations. Blue lines show the assignment of a charging station to its closest substation. two mega watts ( 0 . 66 mega watts per phase). b) Electric vehicle data: we consider a homogeneous vehicle ﬂeet based on the characteristics of the 2018 Nissan Leaf which has a 40 -kilow att-hour battery and a range of 240 kilometers. Based on fast-charging guidelines, we reduce a vehicle’ s battery capacity and its range to 80 percent of their original values [11], and discretize this effecti ve battery capac- ity into C = 40 lev els, resulting in energy discretizations of 0.8 kWh which remains close to the energy necessary to traverse the lowest energy road link. T o account for the possibility that vehicles might not start the day with fully char ged batteries, we set the SoC at t = 1 to 50% . Furthermore, we require vehicles to recharge the amount of energy used over a planning horizon such that the ﬁnal SoC must be at a minimum 50% again. W e set the vehicle operation cost per unit distance (excluding electricity) to V D = 0 . 3 USD / km [41]. c) P ower distribution networks data: we use a GridLAB- D model of the PL-1 distribution network, a primary feeder operated by the Paciﬁc Gas and Electric Company (PG&E) av ailable for research purposes [42], as a proxy for (sub-)urban distribution networks. The network comprises 322 buses and operates at a nominal voltage of 12 . 6 kilovolts. W e set the uncontrollable loads to the model’ s time-varying loads. W e take the location of substations from the utility’ s data [43] and attach a model of the PL-1 distribution network to each substation. W e set the electricity price at each substa- tion to the corresponding locational marginal price [44] and conservati vely assume a base load utilization of 75 percent at the substation transformer . T ypically , distribution networks are operated at 50 to 75 percent of their load capacity so that loads can be transferred from one distribution network to another if needed [45]. Accordingly , we set the substation transformer rating ˆ s 0 to 1 / 0 . 75 times the v alue of the peak base load (i.e., without charging stations), yielding ˆ s 0 = 10 . 42 MV A . In addition, we set the lower voltage magnitude limit to 0 . 96 per-unit and the upper limit to 1 . 04 per -unit, which is 0 . 01 per-unit tighter than required by ANSI C84.1 to allow for the voltage drop in the secondaries of the network. W e connect each charging station to the distribution network whose substation is nearest. Since no data on the coordinates of the distribution network buses exist, we randomly attach the charging station to one of the PDN buses. Thus, the PDN is the same for each substation, except for the varying number and location of charging stations. In total, we consider 14 distribution networks, each with one or two charging stations. W e set the price of electricity at each charging station to be equal to the electricity price at the respective substation, such that V el ,s [ t ] = V el ,d M S , L ( s ) [ t ] holds. Since we focus on the total beneﬁt from a system perspective and treat both operators as a single entity , only the spatial variation of electricity prices that are closely linked to the substation prices af fects our solution. Some comments on the distrib ution network modeling are in order . First, we used the same network model and load values for each distribution network, considering loads from a single summer day . As PDNs are treated as critical infrastructure and load data is usually conﬁdential to protect customers, more accurate data is not publicly av ailable for research purposes [46]. Ho wever , our model can be rerun with more accurate 9 data at any time. Second, we set the electricity price at each substation to the corresponding locational marginal price. Locational marginal prices result from the po wer consumption at the transmission grid level. As our focus is on the interaction of the electric AMoD ﬂeet with the distrib ution grids and the power used for recharging represents only a negligible fraction at the transmission grid le vel, neglecting the impact of this consumption on the marginal prices only minimally affects the accuracy of our results. Third, we assume the electricity price for charging at a certain station to be equal to the electricity price at the respectiv e substation. Neglecting the possible difference in electricity prices among nodes in a single distribution network is consistent with our mesoscopic transportation model. The resulting AMoD-OPF problem has 6,224,240 deci- sion variables, 1,463,600 from the electric AMoD part and 4,760,640 from the multi-OPF part. Since the multi-OPF part comprises D PDNs, the number of v ariables in it admits the upper bound O ( T D |N | ) . Thus, the number of decision variables in the whole AMoD-OPF problem admits the fol- lowing upper bound: O ( T ( C |V R | 2 + D |N | )) . Recall that the complexity of solving the LP with an interior point method is polynomial in the number of v ariables with an exponent lower than 3.5 (depending on the implementation) [47]. Nominally , the size of the electric AMoD part of the problem increases quadratically with the number of road v ertices. Howe ver , if more vertices are added for the same area, the road segment arcs will become shorter , and T and C should be increased to capture the reduced travel duration and energy consumption in the shorter road segment arcs. Thus, in practice, the electric AMoD part of the problem gro ws more than quadratically with the number of road vertices. This limits our formulation to coarse road networks. In future work, we will explore methods that improve the scalability of the AMoD-OPF problem, extending its applicability to ﬁner networks. B. Experimental design T o quantify the impact of an electric AMoD system on the PDNs and the beneﬁt of optimized joint coordination, our e xperiments consider tw o cases. First, we analyze the impact of an electric AMoD system on the PDNs without coordination, i.e., the uncoordinated case. This study sho ws how electric AMoD systems can neg ativ ely affect PDNs. Then, we focus on the coordinated case in which the electric AMoD system and the distribution networks are jointly optimized. Comparing the results of both cases allow us to quantify the potential of optimized coordination between these systems. In both cases we generate results as follows: a) Computing contr ollable loads: we determine the load at each charging station that results from the operation of the electric AMoD system. Depending on the studied case, we solve either Eq. (4) (uncoordinated) or Eq. (15) (coordinated). b) Solving the power ﬂow equation: to assess the quality of a solution from step (a), we solv e the e xact po wer ﬂow equation Eq. (7) to deriv e the true values of the indirect variables (i.e., complex power injection at the reference b us and complex voltage in all other buses). c) Evaluating constraint violations: in step (a), we deter- mine controllable loads without an exact model of the PDNs as it is either neglected (uncoordinated case) or approximated (coordinated case). Hence, it is often the case that solutions do violate some of the constraints. T o quantify these violations, we ev aluate integral constraint violations as we consider a time-variant model. Speciﬁcally , regulations require voltage magnitudes to be kept within a giv en percentage of a nominal value (e.g., ANSI C84.1). Hence, we analyze the integral absolute voltage magnitude constraint violation u viol , int = ∆ t X t ∈T X d ∈D X n ∈N + d X φ ∈ Φ n,d | u φ viol ,n,d [ t ] | where u φ viol ,n,d [ t ] = min( u φ n,d [ t ] − u φ min ,n,d , 0) + max( u φ n,d [ t ] − u φ max ,n,d , 0) is the voltage magnitude constraint violation at phase φ ∈ Φ n in b us n ∈ N . Note that u φ viol ,n,d is neg ati ve when the voltage magnitude is lo wer than u φ min ,n,d , positive when it is larger than u φ max ,n,d , and zero when it is in-between. Additionally , substations typically connect distribution networks to the higher-v oltage transmission network, requiring a transformer to lower the v oltage. T o av oid ov erloading this transformer , the power draw must be less than the transformer rating. Hence, we analyze the integral substation transformer rating violation ˆ s 0 , viol , int = X t ∈T ∆ t X d ∈D ˆ s 0 ,d, viol [ t ] where ˆ s 0 ,d, viol [ t ] = max( | X φ ∈ Φ 0 ,d s φ 0 ,d [ t ] | − ˆ s 0 ,d , 0) is the substation transformer rating violation for d ∈ D . d) Evaluating ener gy consumption and cost: we analyze the energy consumption of the electric AMoD system and its cost. The total energy consumption E total , which includes the ener gy consumed by exogenous loads and the electric AMoD system, results from summing the energy dra w of all substations. The total energy consumption in the base case E total , base results analogously without considering an electric AMoD system. Consequently , the difference of E total and E total , base represents the additional energy consumption caused by the electric AMoD system: E AMoD = E total − E total , base = X t ∈T ∆ t X d ∈D X φ ∈ Φ 0 ,d ( p φ 0 ,d [ t ] − p φ base ,d [ t ]) . Here, p φ base ,d ∈ R is the power drawn in phase φ ∈ Φ 0 from substation d ∈ D in the base case. Due to losses in the distribution networks, not all of E AMoD relates to charging stations. The energy deli vered to the charging stations is giv en by E charge , AMoD = X t ∈T ∆ t X d ∈D X ` ∈L d X φ ∈ Φ n ` ,d p φ con ,`,d [ t ] . 10 The difference between E AMoD and E charge , AMoD represents the link losses caused by the electric AMoD system: E loss , AMoD = E AMoD − E charge , AMoD . Analogously , the cost of these losses is given by V el , loss , AMoD = V el , AMoD − V el , charge , AMoD where V el , AMoD is given by V el , AMoD = X t ∈T ∆ t X d ∈D V el ,d [ t ] X φ ∈ Φ 0 ,d ( p φ 0 ,d [ t ] − p φ base ,d [ t ]) and V el , charge , AMoD is the cost of E charge , AMoD : V el , charge , AMoD = X t ∈T ∆ t X d ∈D V el ,d [ t ] X ` ∈L d X φ ∈ Φ n ` ,d p φ con ,`,d [ t ] . Our implementation builds on top of the authors’ AMoD T oolkit 1 which relies on Y ALMIP [48] to formulate and solve electric AMoD problems. Additionally , we built a general codebase for unbalanced OPF problems, the Unbalanced OPF T oolkit 2 . T o support future research in this ﬁeld, we released both the AMoD T oolkit and the Unbalanced OPF T oolkit under an open-source license. C. Results and discussion Follo wing our experimental design, we ev aluate constraint violations (Fig. 3), as well as energy consumption and costs. T able I summarizes the key results. Figure 3a sho ws a histogram with all voltage magnitude constraint violations u φ viol ,n,d ; each event represents the con- straint being violated in one phase during one of the six-minute time steps. The base case shows no violations and, hence, is not plotted. In contrast, violations appear in both cases that include the electric AMoD system. ANSI C84.1, the power quality standard for v oltage ranges used across the United States, advises that service voltage violations must be limited in extent, frequency , and duration. Optimized coordination between the electric AMoD system and the PDNs helps to decrease voltage constraint violations signiﬁcantly . The number of voltage constraint violations is reduced by 3 . 85 percent in the coordinated case, from 46 , 910 to 45 , 106 . Notably , coordination reduces the number of serious violation ev ents (i.e., those exceeding 0 . 005 p.u. which are the most concerning, see Fig. 3a) by 74 . 85 percent, from 21 , 734 to 5 , 467 . All in all, there is a 50 . 28 percent reduction in in- tegral absolute v oltage magnitude constraint violation, from 24 . 04 per-unit hour to 11 . 95 per -unit hour . Consequently , coordination between the two systems helps to achiev e better compliance with re gulations that require the voltage magnitude to be kept close to its nominal value. Figure 3b shows a histogram with all substation transformer rating violations ˆ s 0 ,d, viol ; each event represents the constraint being violated in one substation transformer during one of the six-minute time steps. Optimized coordination nearly eliminates substation capacity constraint violations, reducing 1 https://github .com/StanfordASL/AMoD- toolkit 2 https://github .com/StanfordASL/unbalanced- opf- toolkit their count by 94 . 05 percent from 168 to 10 . The number of substations that experience a transformer rating violation is reduced from six to two. All in all, there is a 99 . 71 percent reduction in integral substation transformer rating violation, from 7 . 89 meg a volt-ampere hour to 0 . 02 mega volt-ampere hour . T ransformers represent a signiﬁcant inv estment by utilities. For example, installing a transformer with a rating similar to the one used in this case study ( ˆ s 0 = 10 . 42 MV A ) has a cost in the order of 1.7 million USD [49]. Gi ven transformers’ substantial cost, increasing their useful life by reducing trans- former capacity threshold violations (as done by coordination) can lead to signiﬁcant monetary savings for utilities. W e lea ve the precise quantiﬁcation of these savings for future research. Figure 4 shows the load at one representativ e substation along with the applicable transformer rating. The load is sho wn for the three cases: base, uncoordinated and coordinated. The base case represents the substation load arising from the uncontrollable loads. The other two cases show higher loads due to the recharging v ehicles. In the uncoordinated case, there is a signiﬁcant transformer rating violation between 8 am and 11 am. Coordination helps to resolve the violation, as charging loads that exceed the capacity constraint are shifted to later time steps. Figure 5 sho ws the number of charging vehicles and the electricity price over time. The coordinated case sho ws steady charging activity after 11 am. In contrast, charging activities decrease signiﬁcantly after 11 am in the uncoordinated case. The char ging acti vity mirrors the substation load in Fig. 4 which is higher for the coordinated case in later time steps. The increased char ging activity later in the day and the ensuing load leads to increased electricity expenditure as the electricity price is higher later in the day . T able I sho ws the impact of coordinating an electric AMoD ﬂeet with PDNs. The total operational costs of the electric AMoD system during the studied 8-hour time span increase slightly by 3 . 13 percent ( 3 , 329 . 61 USD). Rebalancing costs show an increase of 3 . 28 percent ( 3 , 206 . 47 USD) as vehicles charge at more distant charging stations due to an increase in rebalancing detours. The shift of charging activity to later in the day due to coordination causes electricity costs to increase by 1 . 42 percent ( 123 . 15 USD). The small increase in operational costs reﬂects the price paid for reducing system constraint violations, which improves voltage proﬁles and prolongs transformer life. The energy deliv ered to the charging stations (see T able I) increases by 1 . 82 mega watt-hour ( 0 . 68 percent) in the coordi- nated case because of increased rebalancing detours. Howe ver , the energy attributable to the electric AMoD system consumed at the substations increases only by 1 . 24 megaw att-hour ( 0 . 44 percent). The difference of 0 . 58 mega watt-hour is due to energy losses being reduced by 5 . 24 percent. Reduced energy losses reﬂect more ef ﬁcient power distribution: a greater share of the energy leaving the substations reaches the charging stations in the coordinated case ( 96 . 29 percent compared with 96 . 07 percent). The optimization was performed on an A WS r4.xlarge in- stance (4 vCPU at 2 . 3 GHz , 30 . 5 GB RAM). The AMoD-OPF 11 (a) V oltage magnitude (b) Substation transformer rat- ing Fig. 3: Histograms of voltage magnitude ( u φ viol ,n,d ) and substation transformer rating ( ˆ s 0 ,d, viol ) violations. For clarity , we do not show the cases where the violation is zero. All v oltage magnitude violations are negati ve because the upper limit u φ max ,n,d is ne ver exceeded. The vertical line indicates the threshold for serious v oltage magnitude violation events (i.e., those exceeding 0 . 005 p.u.) For both quantities, constraint violations are signiﬁcantly lower in the coordinated case. Unit Uncoord Coord Change V oltage violation p.u. h 24 . 04 11 . 95 − 50 . 28 % Capacity violation MV Ah 7 . 89 0 . 02 − 99 . 71 % Electricity cost, USD 8 . 35 k 8 . 49 k 1 . 67 % charging Electricity cost, USD 0 . 35 k 0 . 33 k − 4 . 59 % losses Electricity cost, USD 8 . 69 k 8 . 82 k 1 . 42 % AMoD Rebalancing cost USD 97 . 79 k 101 . 00 k 3 . 28 % T otal cost, AMoD USD 106 . 49 k 109 . 82 k 3 . 13 % Energy , charging MWh 268 . 82 270 . 63 0 . 68 % Energy , losses MWh 11 . 01 10 . 43 − 5 . 24 % Energy , AMoD MWh 279 . 82 281 . 06 0 . 44 % T ABLE I: Impact of coordinating an electric AMoD ﬂeet with PDNs. Coordination signiﬁcantly reduces constraint violations at the cost of slightly higher operational costs. problem Eq. (15) was solved in 554 iterations over 8 . 1 hours, whereas solving the electric AMoD problem Eq. (4) took 51 iterations over 0 . 7 hours using Gurobi Optimizer . Thus, the presented solution approach is currently not suitable for real- time operations–the design of an operational v ersion of this framew ork is left for future research. One potential avenue for reducing the computation time would be improving the scaling of the AMoD-OPF problem Eq. (15) to reduce the number of iterations required by the solver . Despite the computation times, the mesoscopic analyses presented herein can be used to identify bottlenecks in PDNs that point at necessary grid extension in vestments. Additionally , a grid operator can use this approach to compute the amount of spinning reserves needed to hedge on the day-ahead market to secure a reliable operation of its PDNs. V I . C O N C L U S I O N W e presented the AMoD-OPF problem, which integrates an electric AMoD problem with a multi-OPF problem. In this context, we discussed power ﬂow surrogates to ob- tain a computationally tractable con vex problem formulation. The resulting AMoD-OPF problem allo ws one to assess the achiev able beneﬁt of coordinating an electric AMoD system and a series of PDNs. W ith this methodological framework, Fig. 4: Load at one representative substation and the corresponding transformer rating. The base case shows the load from uncontrollable loads. The uncoordinated and coordinated cases sho w increased load due to charging vehicles. The substation transformer rating is exceeded in the uncoordinated case. In the coordinated case, charging vehicles later during the day resolves the violation. Fig. 5: Number of charging vehicles and electricity price over time. Coordination shifts the charging load to later time steps when the electricity price is higher . The resulting cost increase is the price paid for reducing system constraint violations, which improv es voltage proﬁles and prolongs transformer life. we in vestigated the impact of an electric AMoD system on the PDNs. Herein, we especially focused on the beneﬁts of coordination between the two systems and discussed results for a case study in Orange County , CA. W e showed that in an un- coordinated system, the electric AMoD ﬂeet neg ati vely affects the distribution networks: the charging behavior of the electric AMoD v ehicles caused overloads at substation transformers and violated (lower) v oltage magnitude limits. Furthermore, we showed that a coordinated system helps to balance the load in the PDNs in time and space. Speciﬁcally , link losses were slightly reduced, substation overloads were nearly eliminated, and voltage violations were halved. Nonetheless, these reduc- tions in constraint violations increased the cost of operating the electric AMoD system by 3 . 13 percent caused by vehicles driving to charge in less congested but more distant stations and charging when electricity prices are higher . This indicates that distribution networks can support more electric vehicles before upgrades are needed if the vehicles are charged in coordination with exogenous loads in the PDNs. Due to our system-optimal objective, these ﬁndings remain an assessment of the ov erall beneﬁt of coordination between an electric AMoD ﬂeet and PDNs. Our ﬁndings open the ﬁeld for multiple directions of future research. First, our AMoD-OPF problem is mesoscopic and assumes perfect knowledge of future loads and trip requests. T o design a real-time algorithm, the integration of forecasts to capture the stochastic nature of the problem is an inter - esting avenue for further research. Second, we modeled the operators of the AMoD ﬂeet and the PDNs as a single entity , implying full cooperation. In future work, one should address 12 the interplay between these two stakeholders, with the goal of designing incentive mechanisms, and in vestigate market dynamics, e.g., the price of stability and the price of anarchy . Third, our case study provides preliminary results about the beneﬁt of coordinating electric AMoD ﬂeets with PDNs. T o provide decision support to practitioners, additional case studies that capture dif ferent PDNs, dif ferent road network characteristics, varying instance sizes, and distributed renew- able energy generation are required. Fourth, our case study did not consider the EVs potential to feed power back into the PDN. Hence, extending our modeling approach for vehicle- to-grid options, ev aluating regulation and operating reserv e potentials, remains a promising avenue for future research. A C K N O W L E D G M E N T S The authors thank Sa verio Bolognani, David Chassin and Rafﬁ Sevlian for sharing their insights on power systems. A. Estandia was supported by the Zeno Karl Schindler Founda- tion with a Master’ s Thesis Grant. This research was supported by the National Science Foundation under the CAREER and CPS programs, the T oyota Research Institute (TRI), and the Stanford Bits & W atts EV50 Project. This article solely reﬂects the opinions and conclusions of its authors and not NSF , TRI, any other T oyota entity , or Stanford. A P P E N D I X A N O M E N C L A T U R E AMoD system A set of expanded graph arcs A R set of road arcs A S set of expanded graph arcs representing a recharging process A T set of expanded graph arcs representing a physical time- dependent movement in the road network C set of discrete battery charge levels c v SoC associated to expanded verte x v ∈ V c v,w energy consumption for traversing road arc ( v , w ) ∈ A R d v,w distance of road arc ( v , w ) ∈ A R E c amount of energy in a charge le vel f 0 network ﬂow for rebalancing vehicles ¯ f v,w maximum capacity of road arc ( v , w ) ∈ A R ¯ f ( v,w ) ,t residual road capacity M set of customer trip requests M S , A S maps a charging station s for each time step t to all arcs in A S that represent charging vehicles at this station M S , L maps a charging station s to the associated controllable load ` and distribution network d S set of chargers in the road network ¯ S s number of charging plugs in charging station s ∈ S T set of time steps t m departure timestep of trip request m ∈ M t v time step associated to expanded vertex v ∈ V t v,w time to traverse road arc ( v , w ) ∈ A R V set of expanded graph vertices V D vehicle operation cost per unit distance (excluding electricity) V el ,d price of electricity at the substation of network d ∈ D V el ,s price of electricity in charging station s ∈ S v m origin of trip request m ∈ M V R set of road vertices v v road vertex associated to expanded vertex v ∈ V w m destination of trip request m ∈ M δ C,s charging rate of charger s ∈ S ∆ t length of a time step λ m customer rate of trip request m ∈ M λ c, dep m number of vehicles with charge c departing to serve customer trip request m λ t,c, arr m number of vehicles with charge c arri ving at time t after serving customer trip request m OPF pr oblem D set of distribution networks E set of links i φ n,o complex current through link ( n, o ) ∈ E L set of controllable loads N set of buses n ` reference bus of controllable load ` ∈ L s con ,` complex po wer of controllable load ` ∈ L s φ inj ,n complex po wer injection at phase φ ∈ Φ n in bus n ∈ N s unc ,n complex po wer of uncontrollable load in bus n ∈ N + v φ n complex voltage at phase φ ∈ Φ n in bus n ∈ N Y n shunt admittance matrix of bus n ∈ N Z n,o impedance matrix of link ( n, o ) ∈ E Φ set of phases R E F E R E N C E S [1] G. 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Gondzio, “Interior point methods 25 years later, ” Eur opean Journal of Operational Resear ch , v ol. 218, no. 3, 2012. [48] J. L ¨ ofberg, “Y ALMIP : A toolbox for modeling and optimiza- tion in MA TLAB, ” in IEEE Int. Symp. on Computer Aided Contr ol Systems Design , 2004. [49] M. R. Sarker, D. J. Olsen, and M. A. Ortega-V azquez, “Co- optimization of distribution transformer aging and energy arbitrage using electric vehicles, ” IEEE T rans. on Smart Grid , vol. 8, no. 6, 2017. Alvaro Estandia is a Software Engineer at Marain Inc. He earned an MSc in Robotics in 2018 and a BSc in Mechanical Engineering in 2015, both from ETH Zurich. He dev elops software to simulate and algorithms to control ﬂeets of electrical au- tonomous vehicles providing mobility-on-demand in urban environments. More broadly , he is interested in the applications of optimization for improving the performance of transportation systems and the power network. 14 Maximilian Schiffer is an Assistant Professor of Operations and Supply Chain Management at T ech- nical Univ ersity of Munich. He recei ved a Ph.D. degree in Operations Research from R WTH Aachen Univ ersity in 2017. His main research interests are in operations research, machine learning, and in- telligent systems, with an emphasis on transporta- tion and logistics topics, especially electric vehicles and autonomous systems. He is a recipient of the INFORMS TSL Dissertation Prize and the GOR Doctoral Dissertation Prize. Federico Rossi is a Robotics T echnologist at the Jet Propulsion Laboratory , California Institute of T echnology . He earned a Ph.D. in Aeronautics and Astronautics from Stanford University in 2018, a M.Sc. in Space Engineering from Politecnico di Milano, and the Diploma from the Alta Scuola Politecnica in 2013. His research focuses on optimal control and distributed decision-making in multi- agent robotic systems, with applications to planetary exploration and coordination of ﬂeets of self-driving vehicles for autonomous mobility-on-demand. Justin Luke is a Ph.D. Candidate at Stanford Univ ersity in the Autonomous Systems Laboratory and Sustainable Systems Laboratory . He earned a B.S. in Energy Engineering from the University of California, Berkeley in 2018. His research focuses on optimization methods for integration of elec- tric autonomous mobility-on-demand ﬂeets into the electricity grid, particularly in scenarios with high- penetration of renewable generation. Emre Kara receiv ed the Ph.D. degree from the Carnegie Mellon Uni versity focusing on infrastruc- ture systems, machine learning, and data science. He is currently leading the engineering and data science efforts with eIQ Mobility . His research inter- ests include data-driven methods to integrate HV A C, electric vehicles, and battery storage systems into the electricity grid as ﬂexibility assets. Ram Rajagopal is an Associate Professor of Electri- cal Engineering as well as Civil and En vironmental Engineering at Stanford Univ ersity , where he directs the Sustainable Systems Lab, focused on large-scale monitoring, data analytics, and stochastic control for infrastructure networks, in particular , power net- works. He received the Ph.D. degree in Electrical Engineering and Computer Sciences from the Uni- versity of California, Berkele y . His current research interests are in the integration of renewables, smart distribution systems, and demand-side data analytics. Marco Pa vone is an Associate Professor of Aero- nautics and Astronautics at Stanford University , where he is the Director of the Autonomous Systems Laboratory . He received a Ph.D. degree in Aero- nautics and Astronautics from MIT in 2010. His main research interests are in the development of methodologies for the analysis, design, and control of autonomous systems, with an emphasis on self- driving cars, autonomous aerospace vehicles, and future mobility systems. He is currently an Associate Editor for the IEEE Control Systems Magazine.

On the Interaction between Autonomous Mobility on Demand Systems and Power Distribution Networks -- An Optimal Power Flow Approach

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