A Stochastic Multi-Agent Optimization Framework for Interdependent Transportation and Power System Analyses
11 A Stochastic Multi-Agent OptimizationFramework for Interdependent Transportationand Power System Analyses
Zhaomiao Guo, Fatima Afifah, Junjian Qi,
Senior Member, IEEE , SinaBaghali,
Student Member, IEEE
Abstract
We study the interdependence between transportation and power systems considering decentralizedrenewable generators and electric vehicles (EVs). We formulate the problem in a stochastic multi-agent optimization framework considering the complex interactions between EV/conventional vehicledrivers, renewable/conventional generators, and independent system operators, with locational electricityand charging prices endogenously determined by markets. We show that the multi-agent optimizationproblems can be reformulated as a single convex optimization problem and prove the existence anduniqueness of the equilibrium. To cope with the curse of dimensionality, we propose ADMM-baseddecomposition algorithm to facilitate parallel computing. Numerical insights are generated using standardtest systems in transportation and power system literature.
Index Terms
Electric vehicle, decomposition, multi-agent optimization, renewable energy, transportation andpower interdependence. N OMENCLATURE
Sets and Indices • Ξ : set of uncertain capacity factors, indexed by ξ Z. Guo ([email protected], corresponding author), F. Afifah, and Sina Baghali are with the Department of Civil, Environmentaland Construction Engineering, and Resilient, Intelligent, and Sustainable Energy Systems(RISES) cluster, University of CentralFlorida, Orlando, FL 32816, USA.J. Qi is with the Department of Electrical and Computer Engineering, Stevens Institute of Technology, Hoboken, NJ 07030USA. a r X i v : . [ m a t h . O C ] J a n • G ( N , A ) / G ( I , E ) : transportation/power graphs, with N , I (indexed by n, i ) being vertexsets and A , E (indexed by a, e ) being edge sets • I S / I C / I T : node sets of renewable/conventional generators/charging stations, indexed by i • I i : node sets connecting to node i , indexed by j • ¯ R / ¯ S ( R / S ): set of origins/destinations of conventional(electric) vehicles, indexed by r, s Parameters and Functions • α : coefficient for augmented Lagrangian function • β /(cid:15) : coefficients/error terms in drivers’ utility function • ν : iteration number • A : node-link incidence matrix of transportation network • B : investment budget for renewable generators • B ij : susceptance of transmission line ( i, j ) • C S,Ii ( · ) /C S,Oi ( · ) : aggregated investment/operation cost functions of renewable generators atlocation i • C Ci ( · ) : aggregated production cost functions of conventional generators at location i • E rs : O–D incidence vector of O–D pair rs with +1 at origin and -1 at destination • e rs : average EV charging demand from r to s • l Ci , u Ci : lower and upper bound of conventional generation at location i • l i : power load (excluding charging load) at node i • P ( ξ ) : probability measure of scenario ξ • Q r : EVs initial quantity at location r • ¯ q rs : conventional vehicle travel demand from r to s • tt a ( · ) : link travel time function of link a • tt rs : equilibrium path travel time from r to s • U rs : deterministic component of utility measures for drivers going from r to s • v a : aggregated traffic flow on link a • z νi : the renewable investment at i at iteration ν Variables • γ : dual variables for non-anticipativity constraints • θ i : phase angle at node i • λ i : charging price at location i • ρ i : wholesale electricity price at location i • τ rsn (cid:48) : dual variable of rs flow conservation at node n (cid:48)• d i : energy purchased by ISO at node i • f ij /v a : transmission/transportation link flow • g Si /g Ci : electricity generation from renewable/conventional generators at node i • p i : charging load at node i • q rs : EV travel demand from node r to node s • u Si : capacity of renewable generator at node i • x rsa : link traffic flow on link a that travels from node r to node s I. I
NTRODUCTION
Transportation and power systems are increasingly interdependent. The majority of moderntransportation management, operations, and control systems are powered by electricity [1]; powersystem supplies, maintenance, and restoration (e.g. crew dispatch and mobile generators) relyon an efficient transportation network [2]. The emergence of transportation electrification (e.g.,electric vehicles (EVs) further strengthens the interdependence between these two systems [3].However, the majority of literature on planning and operation of transportation and powersystems tackles these two systems separately, with outputs of one system served as exogenousinputs for the other [4]. Due to a large-body of literature from both transportation and powerresearch communities, we only highlight representative ones for conciseness. For detailed review,one can refer to [5], [6], [7], [8].In power system literature, EVs with controllable charging and vehicle-to-grid (V2G) ca-pabilities are considered as a part of a smart grid to improve power system efficiency andreliability [9], [10], [11], [12]. To estimate power demand, EVs’ arrival and departure rates aretypically assumed to be known as constants or probability distribution [4]. For example, [13]investigates the provision of reactive power from EVs to reduce the probability of power qualityviolation given EV plug-in and departure time; [14] includes a discussion on EVs participatingin frequency regulation in the power system treating travel demand as given; [3] uses robustoptimization and reformulates transportation equilibrium as charging demand uncertainty set fordistribution network optimization; [15] investigates the potential benefits of dynamic networkreconfiguration to distribution network, with EVs’ spatial-temporal availability and their chargingdemand estimated from transportation network models.
In transportation literature, range anxiety of EV drivers and their charging behaviors havebeen integrated into transportation system modeling and charging infrastructure planning [16],which typically treat power systems as exogenous. For example, [17] studies the coordinatedparking problem of EVs to support V2G services, assuming that EV parking can be centrallycoordinated and the EVs parking demand at each parking facility is given. The literature oncharging infrastructure deployment in the transportation network [18], [19] typically treat loca-tional electricity prices as exogenous variables influencing the facility choice decisions of EVdrivers.Separating transportation and power system analyses is justifiable when EV penetration levelis low and the feedback effects between transportation and power systems are negligible. Withan increasing EV penetration level in the future, the EV travel and charging patterns willlargely affect the spatial and temporal distribution of power demand as well as ancillary servicesavailability; on the other hand, charging costs, power availability, and ancillary service incentivesfrom power system will influence the charging behaviors of EVs and impact transportationmobility. Therefore, alternative modeling strategies are urgently needed to capture the closecouplings and feedback effects between the interdependent transportation and power systems.To address this issue, some recent studies aim to model transportation and power systemssimultaneously using network modeling techniques [4]. Most of them have a bilevel structurewhere a central planner at the upper level optimizes system welfare; traffic equilibrium andoptimal power flow equations are presented at the lower level as constraints. For example, [20]determines the optimal allocation of public charging stations to maximize social welfare con-sidering transportation and power transmission system equilibrium; [21] aims to decide optimalcharging prices to minimize system cost, including power loss in the distribution grid and traveltime in the transportation network, with traffic equilibrium and distribution optimal power flow(OPF) as constraints; [22] minimizes total power and transportation costs by choosing optimalpower and transportation expansion decisions. The problem is linearized and reformulated asan exact mixed integer convex program; [23] proposes a detailed model for transportation andpower systems separately, where a third non-profit entity is responsible for the cooperation ofthe systems by minimizing the total social costs associated with both of them.However, transportation and power system planning are not centrally controlled by a singledecision entity. Therefore, co-optimizing these two systems only provides a lower bound of thecost, which may not be achievable due to the decentralized nature of the decision making in power and transportation systems. In addition, mathematical programming with complementarityconstraints, whose mathematical properties and computation algorithms have been extensivelyinvestigated in the past [24], [25], is still extremely challenging to solve, not to mention forlarge-scale transportation and power networks facing uncertainties. To model a decentralizedtransportation and power interaction and mitigate the computational challenges, [26] uses asimulation-based approach to iteratively solve transportation at least cost vehicle routing problemsand optimal power flow problems, and communicates locational marginal prices and chargingdemand between these two systems. However, the convergence of the proposed algorithm maynot be guaranteed. This work is further extended in [27] to consider power, transportation, andsocial systems as different layers coupled in an internet of thing (IoT) framework. In [28], theauthors assume that each charging station has a renewable energy source installed at the samelocation, and the electricity price is determined based on forecasting the generation of thesesources and the feed-in-tariff. However, [27], [28] ignore the power system operation, e.g., thepower flow constraints.Limited study considers endogenous flow-dependent travel time and charging prices for charg-ing decision making in the context of decentralized multi-agent transportation and power systemmodeling. For example [29], [30], [31] model charging location choice based on travel distanceonly. [32] proposes a more realistic approach for modeling the path selection of EV drivers byallowing en-route charging. However, charging cost is not involved in the decision making. [22],[33] provides a linearized transportation and power distribution system model for the EV chargingstation and transportation system expansion planning. The charging station selection of EVs isbased only on the time of travel, without considering other factors such as the preferability of thecharging station and charging price. [20], [15] consider charging prices as well as flow-dependenttravel time, but charging prices are exogenous.In summary, existing literature suffers from one or more of limitations, including centralizeddecision making, exogenous charging prices, deterministic agent decision making and simplifiedintra- and inter- systems interactions. Each of these limitations hampers the further investigationof interdependent transportation and power systems due to an increasing trend of transportationelectrification, both in terms of planning and operation. First, only if EV behavior is properlymodeled in an analytical framework, we can analyze how to plan and operate transportation in-frastructure systems to influence the EV behavior and system interaction outcomes. Second, someof these characteristics are fundamental characteristics of both systems, such as uncertainty of renewable generation and decentralized decision making structure. Ignoring these characteristicswill only provide a lower bound of system costs, which can cause significant bias for decisionmaking and/or may not be implementable in reality. Third, one of the critical interdependencebetween transportation and power systems is charging demand. Modeling endogenous prices isable to capture a feedback loop of charging demand and charging costs in spatially dependenttransportation and power networks. Modeling endogenous prices also allows for further incentivedesign of charging infrastructure investment and charging costs to optimize both transportationand power systems.Addressing the above mentioned limitations in a holistic modeling framework faces significantchallenges from both modeling and computational perspectives. First, modeling decentralizeddecision making of power and transportation systems with endogenous prices typically leads to ahighly non-convex problem, where equilibrium existence/uniqueness and algorithm convergenceare generally not guaranteed. Second, the curse of dimensionality brought by high dimensionaluncertain parameters, non-convex system interactions, and large-scale transportation and powersystems requires coordination of novel modeling approaches and algorithm design.In this paper, we propose a multi-agent optimization framework to study the interactionsbetween transportation and power systems considering uncertain renewable generation and EVsrouting and charging location choices. More specifically, our main contributions are two-fold.First, we model decentralized key decision-makers with prices and travel time endogenouslydetermined by the model in a unified framework, with equilibrium existence and uniquenessproved. Second, we propose an exact convex reformulation of the non-convex equilibriumproblem based on strong duality that can lead to both scenario and system decomposition andparallel computing.The remainder of this paper is organized as follows. Section II introduces the mathematical for-mulation of each stakeholder’s decision-making problem. Section III presents an algorithm basedon convex reformulation and decomposition. Numerical experiments on small- and medium-scaletest systems overlaying transportation and power networks are discussed in Section IV. Finally,we conclude in Section V with a summary of our results, contributions, and future extensions.II. M
ATHEMATICAL M ODELING
We explicitly model five types of stakeholders: renewable energy investors, conventionalgenerators, ISO, and EV/conventional vehicle drivers, who interact with each other in a unified framework. We assume a perfectly competitive market for the power supply and charging market,i.e. individual decision-makers in both supply and demand sides (e.g., generators, drivers) donot have market power to influence equilibrium prices through unilaterally altering his/herdecisions. For example, power suppliers will decide their investment and/or production quantitiesto maximize their own profits, considering the locational electricity prices in the wholesalemarket. Electricity and charging prices are endogenously determined within the model. Sincewe assume perfectly competitive markets, our models are only applicable when each individualagent does not have significant market power to influence the market price.
A. Renewable Energy Investors Modeling
In a perfectly competitive market, since prices are considered exogenous by investors, theinvestment decisions can be calculated by aggregating a large number of investors into a dummyinvestor, whose cost functions are aggregated costs for all investors [18]. The decision-makingproblems of renewable energy investors are formulated as two-stage stochastic programming,shown in (1). max u S , g S ≥ E ξ (cid:88) i ∈I S (cid:104) ρ i, ξ g Si, ξ − C S,Oi ( g Si, ξ ) (cid:105) − (cid:88) i ∈I S C S,Ii ( u Si ) (1a)s.t. g Si ( ξ ) ≤ ξ i u Si , ∀ i ∈ I S , ξ ∈ Ξ (1b) (cid:88) i ∈I S c Si u Si ≤ B (1c)In the first stage, investors decide the renewable energy generation capacity u S considering thefuture uncertain renewable capacity factors ξ at each location in the future, which is determinedby uncertain factors such as renewable radiance intensity, weather, and temperature [34]. Inthe second stage, given a realization of renewable generation uncertainties ξ and the first stagedecision variable u S , renewable generators determine the generation quantities g S . Our second-stage model is at hourly level. We assume investors aim to maximize their long-term expectedprofits [35], which is calculated as total expected net revenue (revenue subtracting operationalcost) in the second stage minus investment cost in the first stage, as shown in (1a). Without lossof generality, we assume C S,Ii ( · ) has a quadratic form, with a positive quadratic coefficient toreflect an increasing marginal cost of land procurement and C S,Oi ( · ) is a linear function [18].Constraint (1b) guarantees the power output of renewable generator does not exceed the capacity u Si times the uncertain renewable intensity parameter ξ i . Constraint (1c) is the budget constraint for the total investment. Note that since we focus on long-term planning, model (1) could beadapted to model different renewable energy resources, such as solar panels and wind turbines,whose total output capacity is uncertain and influenced by natural resource availability. B. Conventional Generators
For each scenario ξ ∈ Ξ , conventional generators solve the following optimization problemto determine their generation quantity g Ci , ∀ i ∈ I C . Notice that the decision variables forconventional generators, ISO, and drivers are all scenario dependent, but we omit the notation ξ for brevity. max g C ≥ (cid:88) i ∈I C (cid:0) ρ i g Ci − C Ci ( g Ci ) (cid:1) (2a)s.t. g Ci ≤ u Ci , ∀ i ∈ I C (2b) g Ci ≥ l Ci , ∀ i ∈ I C (2c)Objective (2a) maximizes the profits of conventional generators at each scenario ξ , which iscalculated as total revenue (cid:80) i ∈I C ρ i g Ci subtracting total production costs (cid:80) i ∈I C C Ci ( g Ci ) . Weassume C Ci ( · ) has a quadratic form, which is consistent with the settings in IEEE test systems .Constraints (2b)–(2c) are the upper and lower bounds for power generation at each conventionalgenerator location i ∈ I C . C. ISO Modeling
While conventional Level 1, Level 2, and DC fast charging infrastructure are connected tolow voltage distribution network, recent proposal on 350kW–1MW ultra fast chargers may needto be connected to transmission or sub-transmission systems [4]. In this study we focus on inter-city travel with ultra-fast charging stations that are directly connected to the sub-transmissionnetwork.ISO monitors, controls, and coordinates the operation of electrical power systems. While ISOhas many specific tasks, we focus on their daily operation to determine the power purchaseand transmission plan to maximize system efficiency, which will implicitly determine locationalmarginal prices. Denote a power system as G P = ( I , E ) . The ISO decision making can bedescribed by model (3). Objective function (3a) minimizes total energy purchasing cost from Source https://matpower.org/docs/ref/matpower5.0/ both renewable and conventional generators, (cid:80) i ∈I S ∪I C ρ i d i , minus total energy revenue fromcharging stations, (cid:80) i ∈I T λ i p i . Notice that when p i < , ISO purchases − p i energy from chargingstation i , and objective function (3a) means minimizing total energy cost for power systems.Constraint (3b)–(3d) are power flow constraints, where (3b) gives line flow patterns under DCpower flow assumptions; (3c) guarantees the power balance at each location i ; and (3d) is thetransmission capacity constraint. Model (3) guarantees that ISO will prefer purchasing powerfrom cheaper generators and supplying power to more demanded charging locations, given gridtopological and physical constraints. min p , d ≥ , θ , f (cid:88) i ∈I S ∪I C ρ i d i + (cid:88) i ∈I T λ i ( − p i ) (3a)s.t. B ij ( θ i − θ j ) = f ij , ∀ ( i, j ) ∈ E (3b) (cid:88) j ∈ I i f ij = d i − p i − l i , ∀ i ∈ I (3c) − u Pij ≤ f ij ≤ u Pij , ∀ ( i, j ) ∈ E (3d) D. Drivers Modeling
EV drivers need to determine their charging locations and travel routes in a transportationgraph, denoted as G ( N , A ) . EVs departing from r ∈ R select a charging station s ∈ S , with theutility function defined in (4). A similar utility function has been adopted in previous literature[15] to describe charging facility location choices, and can be extended to include other relevantfactors based on evolving EV charging behaviors without interfering the fundamental modelingand computational strategies presented in this paper. Utility function (4) reflect the trade-off ofEV drivers between four aspects: locational attractiveness β ,s , travel time − β tt rs , charging cost − β e rs λ s , and an error term (cid:15) . For example, an EV driver may not choose a charging facilityon the shortest path or with the lowest charging cost. Instead, he/she will balance locationalpreference, travel time, and charging costs. Notice that we use average charging demand foreach rs pair. This modeling framework can be naturally extended to incorporate the variationof charging demands by creating dummy rs pairs. For home charging or workplace charging,where travelers have fixed destinations, the corresponding locational attractiveness (i.e. β ,s ) willdominate the other utility components. U rs = β ,s − β tt rs − β e rs λ s + (cid:15) (4) Different assumptions on the probability distribution of (cid:15) result in different discrete choicemodels. In this study, we adopt a multinomial logit model, in which (cid:15) follows an extreme valuedistribution. The outputs of discrete choice models have two interpretations. First, discrete choicemodels calculate the probability of a vehicle choosing from different destinations. Second, theresults describe the traffic distribution to different destinations at an aggregated level.The utility function (4) partially depends on travel time tt rs , which is determined by thedestination and route choices of all the drivers (including EVs and conventional vehicles). Weassume conventional vehicles have known destination choices, with origin-destination flow ¯ q rs .The destination choice of EVs q rs and path travel time tt rs are coupled. On one hand, selection ofcharging location s will increase the travel demands on certain paths from r to s and influence thetravel time of the transportation network; on the other hand, path travel time will affect destinationchoices as travel time is a factor in the utility function (4). To capture these couplings, we adoptcombined distribution and assignment (CDA) model [36] to model their destination choices androute choices, as shown in (5). min x , q ≥ (cid:88) a ∈A (cid:90) v a tt a ( u )d u + 1 β (cid:88) r ∈R (cid:88) s ∈S q rs (ln q rs − β e rs λ s − β ,s ) (5a)s.t. v = (cid:88) r ∈R ,s ∈S x rs + (cid:88) r ∈ ¯ R ,s ∈ ¯ S ¯ x rs (5b) ( τ rs ) A x rs = q rs E rs , ∀ r ∈ R , s ∈ S (5c) A ¯ x rs = ¯ q rs E rs , ∀ r ∈ ¯ R , s ∈ ¯ S (5d) (cid:88) s ∈S q rs = Q r , ∀ r ∈ R (5e)The Bureau of Public Roads (BPR) function [37] is used here to determine the time of travel tt a ( · ) . (cid:80) a ∈A (cid:82) v a tt a ( u )d u in objective function (5a) is the summation of the area under all the linktravel cost functions tt a ( · ) , which is the total travel time (cid:80) a ∈A v a tt a ( v a ) subtracts externalitiescaused by route choices. The second part consists of the entropy of traffic distribution q rs (ln q rs − and utility terms in (4). Objective function (5a) does not have a physical interpretation [36], andit is a potential function constructed to guarantee the optimal solutions of (5) are consistent with the first Wardrop principal (a.k.a. user equilibrium ) [38] and the multinomial logit destinationchoice assumption. These conditions can be guaranteed by sufficient and necessary Karush-Kuhn-Tucker conditions of model (5) regarding x rs and q . For detail proofs, one can refer to [36].Constraint (5b) calculates link flows by summing link flows of EVs and conventional vehiclesover all origin and destination pairs. Constraints (5c)–(5d) are the vehicle flow conservation ateach node for EV travel demand q rs and conventional vehicle travel demand ¯ q rs , respectively.Constraint (5e) guarantees the summation of EV traffic flow distribution to each s equals tothe total EV travel demand from r , Q r . The equilibrial travel time for each OD pair rs can becalculated as tt rs . = τ rsr − τ rss , where τ rsi is the dual variable for constraint (5c).Notice that an implicit assumption of model (5) is that EV drivers will charge at theirdestinations. In [39], an extension of combined distribution and assignment (CDA) model isintroduced, denoted as generalized combined distribution and assignment model (GCDA), wherevehicle can charge at their origins, destinations, or en-route. In addition, the GCDA model alsoconsider both one-way and round-trip travel. The reason we do not include that model is toavoid over complication of the presented modeling framework and to keep a clear focus ondeveloping a multi-agent power transportation interaction model considering renewable energyand effective computational strategies. Replacing CDA model with GCDA model is a relativelystraightforward process and does not make a major difference to the modeling framework, andthe proposed computational strategies in Section III still apply. E. Market Clearing Conditions
The power purchased and supplied by ISO at each location i , d i and p i , need to be balancedwith locational power generation and charging demand, respectively, in a stable market. Other-wise, some of the supply or demand cannot be fulfilled and the market-clearing prices will beadapted accordingly. Since we aim to provide a steady-state modeling framework for long-termplanning problems, real-time agent deviation from the equilibrium will not be considered and thehourly market clearing conditions can be stated as (6), where (6a) guarantees that total energypurchased by ISO is equal to total energy generated at each location; equation (6b) enforces thebalance between charging supply and charging demand of EVs. i ( s ) denotes the node index in The journey times in all routes used are equal and less than those that would be experienced by a single vehicle on anyunused routes. the power graph that charging location s connects to. Equation (6b) does not include regularpower demand, denoted by l i , which has been considered in the ISO power balancing constraint(3c).Locational prices of electricity ρ i and charging λ s can be interpreted as the dual variables forthe market clearing conditions, respectively. Notice that we will have market clearing condition(6) for each scenario ξ , which could result in different prices for peak and off-peak hours. ( ρ i ) d i = g Si + g Ci , ∀ i ∈ I S ∪ I C (6a) ( λ s ) (cid:88) r ∈R e rs q rs = p i ( s ) , ∀ s ∈ S (6b)III. C OMPUTATIONAL A PPROACH
The system formulations for each stakeholder need to be solved simultaneously, as the decisionprocesses of one agent depend on the decisions of the others. For example, locational prices aredetermined by the collective actions of all agents, as described in the market clearing conditionsin (6).Since each stakeholder’s optimization problem is convex with mild assumptions on the costfunctions of travel, investment, and generation, one can reformulate each optimization problemas sufficient and necessary complementarity problems (CPs), and solve all the CPs together[40].However, solving CPs directly is challenging because of the non-convexity and high dimen-sionality. The main computational contribution of this paper is to propose an exact convexreformulation for this problem, as more formally stated in Theorem 1.
Theorem 1 (Convex Reformulation): If tt a ( · ) , C S,Ii ( · ) , C S,Oi ( · ) , and C Ci ( · ) are convex, theequilibrium states of agents’ interactions in perfectly competitive market, i.e., (1), (2), (3), (5),and market clearing conditions (6) are equivalent to solving a convex optimization, as formulatedin (7). min ( u,g,p,d,x,q ) ≥ (cid:88) i ∈I S C S,Ii ( u Si ) + E ξ (cid:20) (cid:88) i ∈I S C S,Oi ( g Si, ξ )+ (cid:88) i ∈I C C Ci ( g Ci, ξ ) + β β (cid:88) a ∈A (cid:90) v a, ξ tt a ( u )d u + 1 β (cid:88) r ∈R (cid:88) s ∈S q rs, ξ (ln q rs, ξ − − β ,s ) (cid:21) (7a)s.t. (1 b ) − (1 c ) , (2 b ) − (2 c ) , (3 b ) − (3 d ) , (5 b ) − (5 e ) , (6 a ) − (6 b ) Proof:
See Appendix A.
Remark 1 : The intuition behind reformulation (7) is the reverse procedures of Lagrangianrelaxation, where we move the penalty terms (e.g., ρ i d i , ρ i g Si , and ρ i g Ci ) from the objectivefunctions back to constraints (e.g. (6a–6b)). This convex reformulation allows us to apply alter-nating direction method of multipliers (ADMM) , which leads to decomposition with guaranteedconvergence properties[41], in contrast to heuristic diagonal methods. Furthemore, the existenceand uniqueness of systems equilibrium is stated in Corollary 2. Corollary 2 (Existence and Uniquness of Systems Equilibrium): If tt a ( · ) , C S,Ii ( · ) , C S,Oi ( · ) , and C Ci ( · ) are strictly convex functions, the system equilibrium exists and is unique. Proof:
See Appendix A.To develop a decomposition-based algorithm, we first relax the investment decision u Si to bescenario dependent and introduce non-anticipativity constraints , as shown in (8). Then, problem(7) can be reformulated as (9). ( γ i ( ξ )) u Si, ξ = z i , ∀ i ∈ I S , ξ ∈ Ξ (8)min ( u,g,p,d,x,q ) ≥ E ξ (cid:20) (cid:88) i ∈I S ( C S,Ii ( u Si, ξ ) + C S,Oi ( g Si, ξ ))+ (cid:88) i ∈I C C Ci ( g Ci, ξ ) + β β (cid:88) a ∈A (cid:90) v a, ξ tt a ( u )d u + 1 β (cid:88) r ∈R (cid:88) s ∈S q rs, ξ (ln q rs, ξ − − β ,s ) (cid:21) (9a)s.t. (1 b ) − (1 c ) , (2 b ) − (2 c ) , (3 b ) − (3 d ) , (5 b ) − (5 e ) , (6 a ) − (6 b ) , (8) Notice that (9) can be decomposed by scenarios and by systems if we relax non-anticipativityconstraints (8) and charging market clearing conditions (6b), respectively. We propose a solutionalgorithm based on ADMM, as summarized in Algorithm 1. Since Algorithm 1 is an applicationof the general ADMM approach on a convex optimization problem, convergence is theoreticallyguaranteed [41]. Decisions made before uncertainties being revealed should be not be measurable by a specific scenario ξ . We use an augmented Lagrangian approach to relax constraints (6b) and (8), with dual variables λ and γ , respectively. The decision variables of model (9) can be divided into two groups, ( u , g , p , d ) and ( x , q , z ) , and be updated iteratively. When fixing ( x , q , z ) and add augmentedLagrangian terms (cid:80) i ∈I S [ γ νi, ξ ( u Si, ξ − z νi ) + α || u Si, ξ − z νi || ] and (cid:80) i ∈I T [ λ νi, ξ ( − p i, ξ + (cid:80) r ∈R q νri, ξ ) + α || − p i, ξ + (cid:80) r ∈R q νri, ξ || ] in model (9), we will have model (10) to update ( u , g , p , d ) ; likewise,when ( u , g , p , d ) are fixed, we will have model (11) to update ( x , q ) , and the z updates canbe derived analytically, see equation (12), using the fact that (cid:80) ξ ∈ Ξ γ νi, ξ = 0 [42]. Notice thatthe updates of both ( u , g , p , d ) and ( x , q ) can be decomposed by scenarios, as described inStep 1 and Step 2 in Algorithm 1. Step 3 updates the dual variables λ and γ , with step sizeequal to augmented Lagrangian parameter α , so that the dual feasibility with respect to Step 2is guaranteed in each iteration (see [41]). Decomposing transportation and power systems alsoallows for taking advantage of sophisticated algorithms in respective domains, which is not thefocus of this paper and will be left for future research. min ( u,g,p,d ) ≥ (cid:88) i ∈I S (cid:20) C S,Ii ( u Si, ξ ) + C S,Oi ( g Si, ξ ) (cid:21) + (cid:88) i ∈I C C Ci ( g Ci, ξ )+ (cid:88) i ∈I S (cid:20) γ νi, ξ ( u Si, ξ − z νi ) + α || u Si, ξ − z νi || (cid:21) + (cid:88) i ∈I T (cid:20) λ νi, ξ ( − p i, ξ + (cid:88) r ∈R q νri, ξ ) + α || − p i, ξ + (cid:88) r ∈R q νri, ξ || (cid:21) (10a)s.t. (1 b ) − (1 c ) , (2 b ) − (2 c ) , (3 b ) − (3 d ) , (6 a ) min ( x,q ) ≥ β β (cid:88) a ∈A (cid:90) v a, ξ tt a ( u )d u + 1 β (cid:88) r ∈R (cid:88) s ∈S q rs, ξ (ln q rs, ξ − − β s )+ (cid:88) i ∈I T (cid:20) λ νi, ξ ( − p ν +1 i, ξ + (cid:88) r ∈R q ri, ξ ) + α || − p ν +1 i, ξ + (cid:88) r ∈R q ri, ξ || (cid:21) (11a)s.t. (5 b ) − (5 e ) IV. N
UMERICAL S IMULATION
A. Three Node Test System
To generate insights on the interdependence between transportation and power systems, westart with a three-node test system, as shown in Fig. 1. The solid and dashed lines in Fig. 1 aretransportation and power links, respectively. A traffic flow of 50 departing from node 1 has twopossible charging destinations, nodes 2 and 3. Each node has 50 units of existing power load in Algorithm 1:
ADMM-Based Decomposition Algorithm
Result: u ν , g ν , p ν , d ν , x ν , q ν , z ν , λ ν , γ ν initialization: u , g , p , d , x , q , z , λ , γ , α = 1 , ε = 0 . , ν = 1 , gap = ∞ ;define: y = ( u ν ξ , g ν ξ , p ν ξ , d ν ξ ) , y = ( x ν ξ , q ν ξ ) , K = { y | y ≥ , (1 b ) − (1 c ) , (2 b ) − (2 c ) , (3 b ) − (3 d ) , (6 a ) } , K = { y | y ≥ , (5 b ) − (5 e ) } while gap ≥ ε do Step 1: ∀ ξ ∈ Ξ , u ν ξ , g ν ξ , p ν ξ , d ν ξ ∈ arg min y ∈K (10 a ) ;Step 2: ∀ ξ ∈ Ξ , x ν ξ , q ν ξ ∈ arg min y ∈K (11 a ) z ν = (cid:88) ξ ∈ Ξ P ( ξ ) u ν ξ (12)Step 3: ∀ i ∈ I S , γ νi, ξ = γ ν − i, ξ + α ( u S,νi, ξ − z νi ) (13) ∀ i ∈ I T , λ νi, ξ = λ ν − i, ξ + α (cid:18) − p νi, ξ + (cid:88) r ∈R q νri, ξ (cid:19) (14)Step 4: gap = max ξ ∈ Ξ ,i ∈I S | u S,νi, ξ − z νi | /z νi gap = max ξ ∈ Ξ ,i ∈I T | − p νi, ξ + (cid:88) r ∈R q νri, ξ | /p νi, ξ gap = max { gap , gap } , let : ν := ν + 1 end addition to the charging load. All the other link parameters are shown in Fig. 1. Fig. 2 shows thedistribution of uncertain renewable capacity factors for 10 scenarios. We implement our modelsand decomposition algorithms on Pyomo 5.6.6 [43] and solve the sub-problems using Cplex12.8 and IPOPT 3.12.13, with 0.1% optimality gap. All the numerical experiments presented inthis paper were run on a 2.3 GHz 8-Core Intel Core i9 with 16 GB of RAM memory, underMac OS X operating system, with parallel computing enabled.We compare three cases. Case 1 is the base case, as shown in Fig. 1. In case 2, the trans-portation capacity for link 1-3 reduces from 10 to 5. In case 3, the transmission capacityfor link 1-3 reduces from 200 to 50. The reduction of transportation or transmission capacitymay have multiple conflicting impacts on the system, including influence of equilibrium prices, Fig. 1. Three-node test system. Fig. 2. Variation in ξ . renewable energy investment, system costs, and redistribution of traffic or power flow. Due tothe interconnection and interdependence between transportation and power systems, these effectscannot be properly quantified without a modeling framework that can include transportation andpower system decision making as endogenous variables and capture the feedback effects betweenthese two systems. The results presented in this section aim to demonstrate the capability ofour models on capturing the systems interdependence and quantifying the impacts of capacityreduction on system interaction outcomes.
1) Effects on Equilibrium Prices:
From Fig. 3a, with sufficient transmission link capacity(cases 1 and 2), the energy prices at node 2 and 3 are identical, otherwise there will be incentivesfor the generators and ISO to supply more energy to the node with higher prices. Notice thatthe reduction of transportation link capacity does not impact the energy prices when there aresufficient transmission capacity and flexibility. When we reduce transmission capacity for link1-3, prices at both nodes 2 and 3 increase. The reason is that the transmission capacity of thewhole network will not be better off with a reduction of transmission capacity on any links.With a limitation on overall transmission capacity, power may have to be generated at a moreexpensive locations, which leads to higher marginal prices. But prices on node 3 increase toa larger extend than node 2, because node 3 is directly impacted by the transmission capacitylimitation, which makes node 3 more challenging to receive energy. (a) Energy Prices (b)
Investment (c)
Sytem Costs
Fig. 3. Impacts of link capacity on energy prices and system costs.
2) Effects on Renewable Investment:
Since nodes 2 and 3 have the same cost parameters,the locational investment amount of renewable generators is determined by the equilibrium pricesof energy ρ and the distribution of capacity factors ξ , see model (1). For example, a locationwith higher equilibrium price and a higher capacity factor will be strictly preferred. But in thecase when these two factors are conflicting with each other, the investment amount will dependon whose influence is dominant. The investment results are shown in Fig. 3b. In cases 1 and 2,all investments are made in node 2. This is because node 2 has slightly higher capacity factorson average (see Fig. 2), and locational energy prices are the same at nodes 2 and 3 in cases 1, 2(see Fig. 3a). But in case 3, the energy prices at node 3 is higher than the prices at node 2. Theinfluence of energy prices on investment outperforms the influence of renewable capacity factors,which leads to more investment in node 3 in case 3. Comparing between cases 1 and 3, we cansee that a reduction of transmission capacity for link 1-3 will increase the energy price at node3 more than node 2, which will lead to a relocation of 46 units of investment from node 2 tonode 3. This observation indicates that renewable energy investment could offset some negativeimpacts of high energy costs and energy scarcity due to limited transmission capacity. Notice thatwe are only able to numerically quantify the influence of energy prices on renewable investmentbecause prices are endogenously determined by the models and calculating the marginal impactsof prices on renewable investment based on model (1) may be misleading.
3) Effects on System Costs:
System costs include travel costs and energy costs. Travel costsdepend on traffic distribution and the capacity of transportation infrastructure. Fig. 3c shows thedistribution of travel costs and energy costs. Both cases 2 and 3 have higher travel time thancase 1, but for different reasons. For case 2, the increased total travel time is because of a reduction on the transportation capacity on link 1-3, which leads to more congestion for thewhole transportation network. The reason for higher travel time in case 3 than case 1 is becausenode 2 has a cheaper energy price compared to node 3 in case 3. More traffic will choose totravel through link 1-2. Since the total travel demand is fixed (i.e. 50), an imbalanced trafficpattern will cause more congestion. Energy costs are the total costs of conventional energy andrenewable energy production. Since renewable energy is cheaper and the total energy demand isfixed, cases with more renewable energy utilized will have lower total energy costs. Cases 1 and2 do not have transmission congestion, so all the renewable energy can be utilized. The totalenergy in cases 1 and 2 will be identical and less than case 3, where transmission constraintsprevent the effective usage of renewable energy. These observations are consistent with commonbeliefs, which provides evidence that the modeling framework is effective to describe the maininteraction between transportation and power systems.
4) Effects on Flow Distribution:
Comparing between Fig. 4a and Fig. 4b, transportationcongestion on link 1-3 shifts 8.1 units of travel demand from link 1-3 to link 1-2. The increasingof charging demand on node 2 leads to redistribution of power flow on each transmission lineto preserve power flow conservation and power physics laws. Similarly, comparing between Fig.4a and Fig. 4c, constraining the transmission capacity of link 1-3 results an increase of thepower flow on links 1-2 and 2-3. In addition, the reduced transmission capacity on link 1-3will increase the energy prices at node 3 more than the prices at node 2 (see Section IV-A1),which will discourage EVs from traveling to node 3. Notice that few vehicles are chargingat node 3 in case 3 (see Fig. 4c), and the traffic flow from node 1 to node 2 splits into twopaths, − and − − , to avoid traffic congestion. Again, these results also illustrate theinterdependence between transportation and power systems. Without properly considering thecomplicated interaction and feedback effects, the analyses results may be biased.
5) Sensitivity Analysis of Load and EV Penetration:
We have investigated the sensitivityof existing power load and EV penetration on the system. The results for both case 1 and 2showed similar patterns in energy price and link traffic flow except that the traffic flow splitsunevenly between the links in case 2. Therefore, we have presented the results only for case 2and 3 here to maintain conciseness.Firstly, we change the existing load demand from 20 to 50 in the increments of 5. In case 2,the energy price increases linearly in all three nodes (see Fig. 5a). Since we have the same pricefor all the nodes and all the scenarios in each increment of load change, both nodes 2 and 3 are i (a) Case 1 (b)
Case 2 (c)
Case 3
Fig. 4. Impacts of link capacity on link flow. equally desirable for charging in terms of charging cost. However, the road capacity of link 1-3is less than that of link 1-2, and the traffic flow splits unevenly between these two links (seeFig. 6a).We reach more intriguing results for case 3; the energy prices start increasing for the threenodes until the load reaches 50 (Fig. 5b). After that, the problem becomes infeasible, whichis sensible because node 3 can not receive its required energy from node 1 and node 2 due totransmission capacity limitation (3d) and phase angle constraints (3b). Furthermore, node 3 hasa higher charging price as the locational load increases than node 2. Therefore, the traffic flowto node 2 is typically more than the flow to node 3. While most of the drivers use link 1-2 toreach node 2, some drivers choose links 1-3 and 3-2 to avoid congestion on link 1-2 (see Fig.6b).We also change the initial EV traffic flow from 20 to 50 with increments of 5 with the loaddemand to be 50 in all nodes. As the EV penetration increases, the traffic flow on both links1-2 and 1-3 also increases. With limited road capacity on link 1-3 in case 2, however, we seemore traffic on link 1-2 compared to link 1-3 (see Fig. 5c). On the other hand, the energy pricesincrease linearly for all nodes, because the EV penetration acts as load demand on power system(see Fig. 6c). For case 3, the limitation on transmission line 1-3 changed the linear increase inenergy prices (Fig. 5d), and similar to the case 3 for the load change, the optimization problembecomes infeasible when EV penetration exceeds 50. The difference on the locational energyprices has also caused different route selection of EVs, changing the dynamic of traffic flow onthe links (see Fig. 6d). (a) Case 2 (b)
Case 3 (c)
Case 2 (d)
Case 3
Fig. 5. Impacts of existing load demand and EV penetration on Energy Price.
The results found for sensitivity analysis of both EV penetration and existing load demandshow the necessity to model the interdependence of transportation and power systems, wherelimitations on one system may cause drastic impacts on the other.
B. Sioux Falls Road Network and IEEE 39-bus Test System
In addition to insights generated by small test cases above, this section aims to demonstratethe convergence properties of our algorithms on larger networks. We use Sioux Falls roadnetwork (see Fig. 8) and IEEE 39-bus test system (see Fig. 7), two test systems widely used intransportation and power system literature, respectively. The correspondence between the nodeindexes in transportation and power systems is shown in Table I. To avoid infeasibility due toincreased power demand from charging. we scale down the travel demand and road capacity to Data: https://github.com/bstabler/TransportationNetworks/ Data: https://matpower.org/docs/ref/matpower5.0/case39.html (a) Case 2 (b)
Case 3 (c)
Case 2 (d)
Case 3
Fig. 6. Impacts of existing load demand and EV penetration on link traffic flow. be 1% of the original values. Sioux Falls network is at city level, while IEEE 39-bus systemis a regional-level transmission network. To better reflect the inter-city travel, we scale up thefree flow link travel time in Sioux Falls network to 10 times of its original value. A similarapproach was adopted by [20], [18]. The computational results presented here are for illustrationpurposes only. Future research may need to collect transportation and power network data fromthe same geographical area to gain practical insights. The candidate EVs charging locations andPV investment locations are marked in green in Fig. 8. The charging load accounts for 18.5%of the total load. The uncertain renewable generation factors ξ are randomly sampled fromuniform distribution [0 . , . . The optimality gap is set to be 1%. The convergence patterns andcomputing time are shown in Fig. 9. The algorithm we developed converges reliability within100 iterations for up to 100 scenarios. The computing time is almost linearly increasing (from1.1 minutes to 301.0 minutes) with the scenario number for our algorithm, but with a higherincreasing rate when solving the whole problem using IPOPT. IPOPT cannot solve for 10 or Fig. 7. IEEE 39-bust test systems.
A Transportation Network for Illustration Purpose
Sioux Falls Test Network
Edited based on graph by Hai Yang and Meng Qiang Hong Kong University of Science and Technology
Fig. 8. Sioux falls transportation network.TABLE IN
ODE C ORRESPONDENCE BETWEEN S YSTEMS
System Node IndexTransportation 1 2 4 5 10 11 13 14 15 19 20 21Power 1 4 6 11 13 16 19 2 23 25 27 32 more scenarios in 24 hours. We note that our algorithm has the potential for more scenariosdue to the solution strategies of scenario decomposition and parallel computing. But one mayneed to increase the number of CPU cores to full materialized the benefits brought by parallelcomputing. In addition, since our main computation strategies decompose the transportation andpower systems, the algorithm could be further optimized by using more advanced algorithmsfor transportation network assignment and solving power flow equations. These are beyond thescope of this paper and will be left for the future. O p t i m a li t y G a p s Energy Balance GapNon-anticipativity Gap C o m p u t i n g T i m e ( m i nu t e s ) ADMM-basedIPOPT
Fig. 9. Convergence patterns and computing time.
V. C
ONCLUSION
We study the interdependence between transportation and power systems considering decen-tralized renewable generation and electric vehicles. We propose a stochastic multi-agent optimiza-tion framework to formulate the complex interactions between stakeholders in transportation andpower systems, including EV/conventional vehicle drivers, renewable/conventional generators,and independent system operators, with locational electricity and charging prices endogenouslydetermined by markets. We develop efficient computational algorithms to cope with the curseof dimensionalities based on exact convex reformulation and decomposition-based approaches.We prove the existence and uniqueness of the equilibrium. Numerical experiments show thatour algorithms outperform the existing commercial solvers, and provide insights on the closecoupling between transportation and power systems.This work can be extended in several directions. From a modeling perspective, we adoptedclassic transportation and power flow models. Investigating other alternatives, such as trafficequilibrium models with intermediate stops, may generate more accurate descriptions of trafficflow. Second, our current model could be extended to consider other stochastic factors in additionto renewable generation, such as load, travel, and charging demand uncertainties. But moresophisticated models and high dimensional stochasticity may be more computational demandingto solve. Third, we assume a perfectly competitive market for renewable energy investment. It is valuable to investigate different market settings in order to gain a better understandingabout the system interactions. From a computation perspective, adopting more advanced so-lution algorithms for transportation and power systems will further speed up the solution forsubproblems. In addition, effectively grouping uncertain scenarios will accelerate the convergencerate. From an application perspective, this model can be served as a foundation to capture theinterdependence between transportation and power systems, upon which further system planning,operation, restoration, and incentive design strategies can be modeled and investigated to improveefficiency, reliability and resilience of both systems.A
PPENDIX A Proof: (Theorem 1) The Lagrangian of (7) can be written as (15) after relaxing equilibriumconstraints (6). L = (cid:88) i ∈I S C S,Ii ( u Si ) + E ξ (cid:20) (cid:88) i ∈I S C S,Oi ( g Si, ξ )) + (cid:88) i ∈I C C Ci ( g Ci, ξ )+ β β (cid:88) a ∈A (cid:90) v a, ξ tt a ( u )d u + 1 β (cid:88) r ∈R (cid:88) s ∈S q rs, ξ (ln q rs, ξ − − β ,s )+ (cid:88) i ∈I S ∪I C ˜ ρ i, ξ ( d i, ξ − g Si, ξ − g Ci, ξ ) + (cid:88) s ∈S ˜ λ s, ξ (cid:16) (cid:88) r ∈R e rs q rs, ξ − p i ( s ) , ξ (cid:17)(cid:21) = (cid:88) i ∈I S C S,Ii ( u Si ) + E ξ (cid:20) (cid:88) i ∈I S C S,Oi ( g Si, ξ ) − ˜ ρ i, ξ g Si, ξ + (cid:88) i ∈I C C Ci ( g Ci, ξ ) − ˜ ρ i, ξ g Ci, ξ + β β (cid:88) a ∈A (cid:90) v a, ξ tt a ( u )d u + 1 β (cid:88) r ∈R (cid:88) s ∈S q rs, ξ (cid:16) ln q rs, ξ − β ˜ λ s, ξ e rs − β ,s (cid:17) + (cid:88) i ∈I S ∪I C ˜ ρ i, ξ d i, ξ + (cid:88) s ∈S ˜ λ s, ξ ( − p i ( s ) , ξ ) (cid:21) (15)Define feasible set X . = { u, g, p, d, x, q ≥ | (1 b ) − (1 c ) , (2 b ) − (2 c ) , (3 b ) − (3 d ) , (5 b ) − (5 e ) } .Models (1), (2), (3), (5), and (7) are all convex optimization and satisfy linearity constraintqualifications. So strong duality holds. (7) is equivalent to (16) due to strong duality. min ( u,g,p,d,x,q ) ∈X max ˜ λ, ˜ ρ L = max ˜ λ, ˜ ρ min ( u,g,p,d,x,q ) ∈X L (16) min ( u,g,p,d,x,q ) ∈X L is equivalent with (1, 2, 3, 5) for any given ˜ λ, ˜ ρ . In addition, (6) holds forthe optimal ˜ λ, ˜ ρ . Proof: (Corollary 2) If tt a ( · ) , C S,Ii ( · ) , C S,Oi ( · ) , and C Ci ( · ) are strictly convex functions,model 7 is strict convex optimization problem, which has a unique optimal solution. FollowingTheorem 1, the system equilibrium therefore exists and is unique.R EFERENCES [1] S. B. Miles, N. Jagielo, and H. Gallagher, “Hurricane isaac power outage impacts and restoration,”
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