Active Network Management via grid-friendly electromobility control for curtailment minimization

Control Engineering Practice 159 (2025) 106289 Available online 26 February 2025 0967-0661/© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac Active Network Management via grid-friendly electromobility control for curtailment minimization✩ Mladen Čičića,∗, Carlos Vivasb, Carlos Canudas-de-Witc, Francisco R. Rubiob aUniversity of California, Berkeley, USA bDept. de Ingeniería de Sistemas y Automática, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Spain cUniv. Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA-lab, France ARTICLE INFO Keywords: Macroscopic EV Traffic EV charging control Electromobility Active Network Management Curtailment minimization Demand response ABSTRACT We propose an integrated power and transportation system control framework, combining the power grid model with a macroscopic electromobility model including charging stations under V2G operation. In this framework, the electrical vehicles (EVs) act as energy storage, but also as additional virtual power grid links, transporting energy from one point to another. This new holistic approach is used as a basis for optimal control design seeking to provide Active Network Management, in order to minimize curtailment of renewable energy sources and loads at various ports of the network, while accounting for the structural limitation of the grid and other constraints necessary for the optimal operation of the EVs. The proposed control scheme is shown to be able to outperform uncoordinated EV charging in terms of total curtailment in various studied scenarios. Additionally, we study the case when public charging stations are able to incentivize or disincentivise EVs to use them, by dynamically varying their charging price throughout the day, and show that this additional control input can further reduce curtailment in certain scenarios. 1. Introduction As efforts towards decarbonization of all economical sectors become a major priority, Electric Vehicles (EVs) have started to emerge as one of the main components of sustainable transportation systems worldwide. Since EVs are projected to reach around 40% of the total fleet in the EU by 2030 (Conway, Joshi, Leach, García, & Senecal,2021), it is becoming clear that their integration with the city infrastructure (charging stations), and the electrical power supply network (power grid) poses yet unsolved problems that will be critical in the coming years (European Commission & Directorate-General for Energy,2019). It is self-evident that a true decarbonization of the transportation sector can only be achieved if the energy sector is likewise decarbonized through a massive increase of the prevalence of Renewable Energy Sources (RES) in the power mix. These two trends will have a profound impact on the power grid operations in the coming years (Ipakchi & Albuyeh,2009), necessitating major infrastructure improvements, both on the ‘‘physical’’ side, upgrading the transmission and distribution networks, and on the ‘‘cyber’’ side, making the grid smarter (Di Silvestre, Favuzza, Sanseverino, & Zizzo,2018). ✩This work has received support from Knut and Alice Wallenberg Foundation, the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement N 694209) Scale-FreeBack project, and by projects PID2020-115561RB-C32 granted by the Spanish Ministry of Science and Innovation. ∗Corresponding author. E-mail addresses: [email protected] (M. Čičić), [email protected] (C. Vivas), [email protected] (C. Canudas-de-Wit), [email protected] (F.R. Rubio). Traditionally, power system management operated under the fit and forget doctrine, focusing on preemptive investments in network components (e.g., lines, cables, transformers) to prevent congestion and voltage issues without requiring continuous monitoring and control of power flows or voltages. However, the rapid proliferation of renewable and distributed generation implies substantial network reinforcement costs (Cornélusse, Vangulick, Glavic, & Ernst,2015;Wang, Ochoa, & Harrison,2009) because energy flow may reverse, moving from the distribution network to the transmission network, and internal distribution network flows may deviate significantly from historical patterns. Active Network Management (ANM) strategies focus on addressing congestion and voltage issues through short-term decision-making policies (Gemine, Ernst, & Cornélusse,2017), often resorting to curtailing wind or solar generation to maintain system stability in quasi real-time. This field of research is rapidly evolving, with numerous new concepts being proposed, including microgrids and virtual power plants (Mancò, Tesio, Guelpa, & Verda,2023;Palizban, Kauhaniemi, & Guerrero, 2014). Nevertheless, curtailment leads to operational and economic inefficiencies that are particularly wasteful in case of RES generation, https://doi.org/10.1016/j.conengprac.2025.106289 Received 27 February 2024; Received in revised form 13 December 2024; Accepted 10 February 2025 Control Engineering Practice 159 (2025) 106289 2 M. Čičić et al. given their low operational cost and potential to reduce overall power system emissions. Exploring ANM schemes capable of leveraging load flexibility is thus imperative for reducing the reliance on generation curtailment when the supply is higher than the demand. On the other hand, if the demand is higher than the supply, the intermittency of the RES also means that it may not be possible to increase their power generation. In this case, demand response through load modulation is necessary, which may lead to loss of comfort or productivity (Ghazvini et al.,2019), with rolling blackouts as an extreme case. Such action may also compromise modulation capacities in the near future, since flexible loads are often required to consume a specific amount of energy over a certain time period (e.g. heat pumps of EV charging) to keep comfort or usability standards. This makes it important for the ANM to take decisions by planning operations over a sufficiently long time horizon (Gemine, Karangelos, Ernst, & Cornélusse,2013;Gill, Kockar, & Ault,2013;Macedo, Franco, Rider, & Romero,2015), further emphasizing the need for good prediction models. Although the constant increase of the electrification of the transportation systems (electromobility) could be seen as a serious potential strain on the power grid, due to large charging power demands (Arias, Kim, & Bae,2017;Fernandez, San Román, Cossent, Domingo, & Frias, 2010), the massive adoption of EVs will not necessarily hinder the development of future electric power systems. In fact, the potential to use their batteries for energy storage and control their charging to provide flexibility (ability to quickly modify the demand to react to changing conditions) introduces new possibilities to provide ANM (Henry & Ernst,2021) and ancillary services (Le Floch, Kara, & Moura,2016; Wenzel, Negrete-Pincetic, Olivares, MacDonald, & Callaway,2017). From the power system side, a power grid reinforced by these methods is able to integrate a higher portion of intermittent renewable energy sources (RES) (Brouwer, Van Den Broek, Seebregts, & Faaij, 2014;Dixon, Bukhsh, Edmunds, & Bell,2020;Lund & Kempton,2008), making the EVs an important component of power system operations. As the natural interface between the mobility and power networks, the charging stations will play an essential role in the electromobility ecosystem. Today’s power electronics technology and new DC grid topologies, together with V2G-enabled EVs, allow charging stations to operate as dynamic load/supplies providing ancillary services to the power grid in the form of frequency stabilization and congestion relief (López, Martin, Aguado, & de la Torre,2013). By controlling the charging process of EVs present at charging stations, their operators are able to provide demand response (Yao, Lim, & Tsai,2016), tapping into a new source of revenue in addition to helping the power grid. Additionally, EV charging can be scheduled and coordinated to provide VoltVAr control (Sabillon-Antunez, Melgar-Dominguez, Franco, Lavorato, & Rider,2017), keeping the voltage levels and reactive power flow within acceptable limits. Even when EV charging rates cannot be directly controlled, dynamic pricing control may enable us to shift the charging demand in space (Nie, Wang, & Cheng,2017) and time (Moghaddam, Ahmad, Habibi, & Masoum,2019), helping reduce peak load. Apart from acting as a load/supply while charging/discharging at a charging station, EVs also effectively transport energy as they move between different points in the network (Watanabe et al.,2023). In this sense, the EV traffic flows on the roads could also be seen as additional virtual lines in the power grid, in addition to providing energy storage (Čičić & Canudas-de-Wit,2024). One of the main potential barriers to fully exploit the EVs’ potential is the lack of tools and methods for forecasting EV fleets’ flexibility in both time and space. This entails forecasting when and where EVs move, how their State of Charge (SoC) evolves, and how they interact with the infrastructure. Though some approaches based on historical data do exist (Morlock, Rolle, Bauer, & Sawodny,2019), a modelbased framework is preferable for optimal control purposes. Therefore, combining electromobility models with power grid models will enable the use of EVs’ flexibility potential to minimize curtailment of RES and loads, and improve the use of the existing power transmission Table 1 Table of main notation. Symbols related to the power system are indicated by tilde. Subscript indicators typically determine the spatial coordinate and superscript the temporal coordinate. Symbol Meaning 𝜁Node identifier 𝑘Time step  𝑆Total power  𝛥Curtailed power  𝛤RES generation  𝛬Load 𝑈Charging station power 𝜌Traffic density 𝜀State-of-Charge 𝜂Distribution of EVs by SoC at a charging station 𝑣Traffic speed 𝑐Charging rate 𝜉Vehicle class identifier 𝛽Splitting ratio 𝜋Charging price 𝑟Onand off-ramp flows 𝜇EV flows entering or exiting the charging station  𝑉Node voltage  𝑆Line power 𝑢Relative charging power control  𝛿Relative curtailment control network. Charging of a large population of EVs was modelled and controlled in Le Floch et al. (2016) to follow the load reference and provide balancing services, neglecting the spatial component of EV mobility. In Henry and Ernst (2021), RES curtailment was minimized by means of an optimal control strategy, which was also used to learn a computationally efficient control law based on reinforcement learning. Nevertheless, in that work storage and charging of EVs was assumed to be situated at a single point in the power network, whereas in reality both the EVs and the charging stations are distributed in time (in case of EVs) and space, and connected to different power grid nodes. In Zhou, Zhang, Guo, and Sun (2021), coupled traffic and power grid dynamics were considered, but the traffic flows were only described on graph level. Similarly, Lv, Wei, Sun, Chen, and Zang (2019) tackles congestion of both traffic and power network, coupled through dynamic wireless charging, but without explicitly modelling EV traffic dynamics on the road. It is clear that the coupling between the electromobility and the power system will need to be considered on the operational level, as well as when planning for the future (Xie, Hu, & Wang,2020). In this work we tackle the problem of providing ANM through control of EV charging, aimed at minimizing curtailment of both RES generation and loads, while ensuring safe power grid operation. The main contributions are in proposing the general control framework, applicable to a generic and designing prediction-based optimal control based on a macroscopic combined electromobility and power grid model (Čičić, Vivas, Canudas-de-Wit, Carlos & Rubio,2023) to achieve the control objectives. By adopting a detailed electromobility model, we are able to identify and leverage the complex, long-term influence that current EV charging decisions have on the future state of the system, enabling us to go beyond models relying on random EV arrivals. The proposed framework is applicable to a general case of power and mobility network, and we study its performance on a simple example. The remainder of the paper is structured as follows. First, the control architecture and problem are outlined and described in Section 2. The model is presented in Section 3, combining a multi-class electromobility model, describing the EV traffic and charging, and a power grid model. The electromobility layer is interfaced with the power layer through the charging stations, which act as predictable time-varying energy storage, and whose operation we use to control the overall system. In Section 4we first give the general optimal control formulation, and then discuss some simplifications that can be made in the specific simple road and power network that is considered here. These three Control Engineering Practice 159 (2025) 106289 3 M. Čičić et al. Fig. 1. Layout of the combined electromobility and power grid model used for this study, together with the outline of the control system. The controller provides Active Network Management by taking in the information about the power grid, the electromobility state on the road and at charging stations, and loads and RES generation at the nodes, and outputs charging rates for individual EVs at all charging stations and charging price of the public charging station. Sections, 2–4describe the methodology of this work, and the main notation used therein is presented in Table 1. Finally, in Section 5the control framework is tested extensively in simulations, under different operating scenarios, and the operation of the control laws are analysed and discussed. The paper is concluded with Section 6, where we summarize the results and give directions for future work. 2. Active network management via electromobility control In this work we study how controlled EV charging can be used to provide ANM, using a control system architecture outlined in Fig. 1. The system is comprised of the electromobility layer, consisting of the road network and EV traffic, and the power layer, consisting of the power grid, loads, and generators. The final part of the system are the charging stations, which are of particular importance because they serve as the interface between the electromobility layer and the power grid, and can be used as actuators to improve the situation in the wider power system. We focus on three control objectives: 1. ensuring safe power grid operations by keeping the line powers within their capacities, | 𝑆𝑘|≤  𝑆, and node voltages within their limits,  𝑉≤ 𝑉𝑘≤ 𝑉, 2. minimizing total curtailment | 𝛥𝑘 𝜁|of RES generation  𝛤𝑘 𝜁and loads  𝛬𝑘 𝜁at all nodes 𝜁, 3. and keeping the overall SoC of all EVs within the system 𝜀𝑘 avg close to some reference 𝜀r ef , at all time instants 𝑘, in order of decreasing priority. The first two control objectives fall within the classical purview of ANM, while the third one yields adequate functioning of the electromobility layer. In case the EV charging demand is uncontrollable and inflexible, the only way ANM may achieve the primary control objective at each time step 𝑘is by curtailing either the generation or the load at each node 𝜁where such action is needed. We define relative curtailment  𝛿𝑘as a control input denoting the portion of RES generation that needs to be curtailed for 0< 𝛿𝑘≤1, or the portion of load that needs to be curtailed for −1 ≤ 𝛿𝑘<0. The actual curtailed power at each port is thus  𝛥𝑘 𝜁=⎧ ⎪ ⎨ ⎪ ⎩ − 𝛿𝑘 𝜁 𝛤𝑘 𝜁, 𝛿𝑘 𝜁≥0,  𝛿𝑘 𝜁 𝛬𝑘 𝜁, 𝛿𝑘 𝜁<0. (1) Since curtailment is wasteful and costly, it is of interest to achieve the primary control objective while curtailing as little total power as possible, as stated by the secondary control objective. If the charging station dynamics can be controlled, we can use them to provide flexibility for ANM and reduce, or even eliminate the need for curtailment of generation and loads. By increasing the charging station power 𝑈𝑘 𝜁, we are able to absorb more power generation at node 𝜁, thus reducing the need for RES curtailment. Conversely, by decreasing 𝑈𝑘 𝜁, we are able to reduce the need for load curtailment. Here we adopt the convention that if 𝑈𝑘 𝜁>0, the charging station is a net power consumer from the perspective of the grid, and is using the grid power to charge the EVs. Otherwise, if 𝑈𝑘 𝜁<0, the charging station is a net power provider to the grid, and is using the energy stored in the batteries of some EVs to provide V2G services. On the upper level of charging station control, the power system operator gets from each charging station 𝜁the range of power that they can consume or generate, denoted by 𝑈𝑘 𝜁and 𝑈𝑘 𝜁, respectively. Based on these limits, the operator can set the normalized power flow to (or from) each charging station 𝜁, denoted 𝑢𝑘 𝜁∈ [−1,1]. The actual charging station powers 𝑈𝑘 𝜁represent the control input to the lower level of control, and are given by 𝑈𝑘 𝜁=⎧ ⎪ ⎨ ⎪ ⎩ 𝑢𝑘 𝜁𝑈𝑘 𝜁, 𝑢𝑘 𝜁≥0, −𝑢𝑘 𝜁𝑈𝑘 𝜁, 𝑢𝑘 𝜁<0, (2) therefore we have 𝑈𝑘 𝜁≤𝑈𝑘 𝜁≤𝑈𝑘 𝜁. On the lower level of control, the direct control input we use in this framework are the charging rates at each charging station 𝜁𝑐𝑘 𝑖, set to potentially different levels for EVs with different SoC. These charging rates are set in order to control the charging station power to achieve the reference charging station power communicated from the upper level of control. We calculate the limits of achievable charging station power 𝑈𝑘 𝜁∈ [𝑈𝑘 𝜁, 𝑈𝑘 𝜁]for the upper level of control based on the limitations on the charging rates and the number and SoC of EVs at each charging station. Given that the range of achievable charging power, and therefore also the amount of flexibility provided to ANM, depends directly on the number of EVs present at a charging station, being able to influence when and where EVs charge can lead to an improvement in overall control performance. In particular, in this work we assume that we are able to control charging prices 𝜋𝑘 𝜁of public charging stations. In doing so, we can incentivize EVs to charge at nodes where there is too much power generation, absorbing the excess energy and reducing RES curtailment, by reducing the charging price of these public charging stations. Conversely, a charging price increase causes EVs to defer charging, or charge elsewhere. Finally, we tackle satisfying the three control objectives using an approach similar to lexicographic optimization (Ehrgott,2005). We resort to curtailing RES generation or load  𝛥𝑘 𝜁only if it is impossible to satisfy the primary control objective by controlling charging station powers 𝑈𝑘 𝜁and charging price 𝜋𝑘 𝜁. If the primary objective is satisfied with minimum curtailment, we control the charging station powers to regulate the overall SoC of the electromobility layer. The reference SoC value is selected such that it keeps the EVs’ batteries operating within an efficient SoC range, while also ensuring that we are able to both increase and decrease charging station power in order to minimize potential future curtailment. The evolution of the system state is described by the Coupled Electromobility and Power Grid Model which is presented in Section 3. The full current state is assumed to be known, and the model is used to calculate the prediction-based control actions. Since we are interested in demonstrating the limits of what can be achieved using such control system, we study the idealized case where the centralized control authority has full control over the charging rates (within some range) of individual EVs according to their SoC, while they are at the charging stations at each node. Furthermore, we assume that the EVs are charged Control Engineering Practice 159 (2025) 106289 4 M. Čičić et al. or discharged without losses, and that all EVs are connected to the charging infrastructure while they are parked. The EVs are free to leave charging stations according to their commuting schedules, and the extent of influence that the controller has on their movement is limited to potentially incentivizing or disincentivising them to enter the public charging station by changing the charging price. In doing so, we hope to be able to attract more low-SoC EVs to the public charging station by setting a low charging price (at which the charging station sells power to the EVs) at times when there is an overproduction of RES, but also when there is a need to transfer energy from these nodes to other nodes. Finally, we assume that all EV users are appropriately compensated for the services that their vehicles provide to the grid, including offsetting potential battery degradation due to V2G discharging, so that they fully comply with the centrally managed charging control. The general setup studied in this work is similar to the one used in Henry and Ernst (2021), abstracting the situation where people commute between home and work using EVs, potentially stopping on the way at a public charging station. As shown in Fig. 1, we consider a power grid with three nodes (Home,Public charging station, and Work, in further text denoted by h,p, and w, respectively) connected both by power lines and road links. The two considered road links connect nodes hand w, with a pair of offand on-ramps at the middle, exiting towards node p. This distinction is adopted in order to gather the spatiotemporal operation of the daily commutes of a large majority of EV users, occasionally making use of public charging infrastructure. We assume that there is intermittent RES power generation and EV charging ports at all three nodes, and that there is some time-varying load at nodes hand w. Note that while for simplicity here we consider the case with one h,w, and pnode each, it is straightforward to extend the framework to the more realistic case when multiple nodes of each type are connected by generic road and power networks. 3. Combined electromobility and power grid model In this section we present the combined electromobility and power grid model which is used to represent and simulate the overall system. We first introduce the new multi-class aggregated electromobility model including the charging stations, then present the grid model, propose a model of profile perturbations, and finally combine the parts into an integrated model. While the combined model is adapted to the specific setup studied in this work, it can readily be adjusted to accommodate other types of road and power networks as well. The model combines the contributions from Čičić, Vivas, et al. (2023) and Čičić, Gasnier, and Canudas-de-Wit (2023), and is included here in full for completeness. 3.1. Electromobility model The dynamics of EV road traffic are described by a discrete-time multi-class simplified Coupled Traffic, Energy, and Charging (CTEC) model, consisting of the macroscopic dynamics of the EVs on the roads and at charging stations, coupled through ramp flows. This model is a cell-based discretization of its PDE counterpart (Čičić & Canudas-de Wit,2022), 𝜕 𝜌(𝑥, 𝑡) 𝜕 𝑡+𝜕(𝑣(𝑥, 𝑡)𝜌(𝑥, 𝑡)) 𝜕 𝑥= 0,(3) 𝜕(𝜌(𝑥, 𝑡)𝜀(𝑥, 𝑡)) 𝜕 𝑡+𝜕(𝜌(𝑥, 𝑡)𝑣(𝑥, 𝑡)𝜀) 𝜕 𝑥=𝜌(𝑥, 𝑡)(𝑣(𝑥, 𝑡)),(4) 𝜕 𝜂(𝜀, 𝑡) 𝜕 𝑡+𝜕(𝑐(𝜀, 𝑡)𝜂(𝜀, 𝑡)) 𝜕 𝜀= 0,(5) where 𝑥is the position on the road, 𝑡the time, 𝜌(𝑥, 𝑡)the traffic density, 𝑣(𝑥, 𝑡)the traffic speed, 𝜀(𝑥, 𝑡)the macroscopic SoC, (𝑣)the battery discharge as a function of EV speed, 𝜂(𝜀, 𝑡)the distribution of EVs at a charging station according to their SoC 𝜀, and 𝑐(𝜀, 𝑡)the charging rate of EVs, potentially different for EVs with different 𝜀. This model is also extended by splitting the traffic flows into different classes (e.g., combustion engine vehicles and EVs) which have the same behaviour while driving on the road, but may have distinct SoC dynamics or behaviour at onand off-ramps. Fig. 2. Splitting ratios of EVs towards the public charging station. As 𝜋𝑘 𝐩increases, 𝜁(𝜋𝑘 𝜁)decreases and fewer EVs enter the public charging station due to higher prices, opting instead to charge at home or at work. 3.1.1. Aggregate traffic density equations Each road link 𝑙is split into 𝑁𝑙 𝑥cells of length 𝐿𝑥,𝐿𝑥≥𝑣f f𝑇, where 𝑣f fis the free flow speed of the traffic, and 𝑇is the discretization time step, appropriately selected to ensure numerical stability. The aggregate macroscopic state of the traffic on each road link 𝑙is given by the traffic density 𝜌𝑘 𝑖in each cell 𝑖at each discrete time instant 𝑘, which is updated according to 𝜌𝑘+1 𝑖=𝜌𝑘 𝑖+𝑇 𝐿𝑥(𝑞𝑘 𝑖−−𝑞𝑘 𝑖+),(6) 𝑞𝑘 𝑖+= min {𝑣f fmin{𝜌𝑘 𝑖, 𝜌cr }, 𝜔(𝜌jam − max{𝜌𝑘 𝑖, 𝜌cr }) −𝑟𝑘 on,𝑖+1 1 −𝛽𝑘 𝑖},(7) 𝑞𝑘 𝑖−=𝑞𝑘 𝑖−1+(1 −𝛽𝑘 𝑖−1) +𝑟𝑘 on,𝑖,(8) Here, 𝑞𝑘 𝑖−denotes the traffic flow entering cell 𝑖at its upstream end, and 𝑞𝑘 𝑖+the traffic flow exiting cell 𝑖at its downstream end. The onramp flow into cell 𝑖is denoted 𝑟𝑘 on,𝑖 and assumed to enter it at its upstream end, and the off-ramp flow from cell 𝑖is assumed to leave it at its downstream end, and denoted 𝑟𝑘 of f,𝑖 =𝛽𝑘 𝑖𝑞𝑘 𝑖+, where 𝛽𝑘 𝑖is the splitting ratio of mainstream traffic flow towards the off-ramp. The traffic is assumed to follow a triangular fundamental diagram with critical density 𝜌cr and jam density 𝜌jam, yielding congestion wave speed 𝜔=𝑣f f 𝜌cr 𝜌jam−𝜌cr . Note that we omit stating to which road link 𝑙we are referring for better readability. 3.1.2. Multi-class traffic density equations The aggregate traffic flow is split into some number of vehicle classes 𝜉∈𝛯, with individual traffic densities denoted 𝜉𝜌𝑘 𝑖. Since here we assume that all vehicle classes have the same traffic behaviour, the evolution of their traffic densities is given by 𝜉𝜌𝑘+1 𝑖=𝜉𝜌𝑘 𝑖+𝑇 𝐿𝑥(𝜉𝑞𝑘 𝑖−−𝜉𝑞𝑘 𝑖+),(9) 𝜉𝑞𝑘 𝑖+= 𝜉𝜌𝑘 𝑖 𝜌𝑘 𝑖 𝑞𝑘 𝑖+,(10) 𝜉𝑞𝑘 𝑖−=𝜉𝑞𝑘 𝑖−1+(1 −𝜉𝛽𝑘 𝑖−1) +𝜉𝑟𝑘 on,𝑖,(11) where the ramp flows defined through 𝜉𝛽𝑘 𝑖−1 and 𝜉𝑟𝑘 on,𝑖 are distinct for each class 𝜉, with 𝜉𝑟𝑘 of f,𝑖 =𝜉𝛽𝑘 𝑖−1 𝜉𝑞𝑘 𝑖+. The per-class quantities relate to the aggregate quantities according to 𝜌𝑘 𝑖=∑ 𝜉∈𝛯 𝜉𝜌𝑘 𝑖, 𝑟𝑘 on,𝑖 =∑ 𝜉∈𝛯 𝜉𝑟𝑘 on,𝑖, 𝛽𝑘 𝑖=∑ 𝜉∈𝛯 𝜉𝜌𝑘 𝑖 𝜌𝑘 𝑖 𝜉𝛽𝑘 𝑖.(12) In the studied setup, once the vehicles reach the end of the road link from hto wor from wto h, all of them exit the road and enter charging stations wand h, respectively, with 𝛽𝑘 𝑁𝑥= 1for both road links. The Control Engineering Practice 159 (2025) 106289 5 M. Čičić et al. splitting ratios towards the public charging stations 𝜉𝛽𝑘 𝑖, where 𝑖=𝑖in 𝐩𝐡 for the link hto w, and 𝑖=𝑖in 𝐩𝐰 for wto h, depend on the SoC of the approaching vehicles 𝜉𝜀𝑘 𝑖and the EV charging price 𝜋𝑘 𝐩, and are defined by logistic function, 𝜉𝛽𝑘 𝑖= 1 −⎛⎜⎜⎝ 1 +𝑒− 𝜉𝜀𝑘 𝑖−𝜁(𝜋𝑘 𝜁) 𝛾𝛽⎞⎟⎟⎠ −1 .(13) One example of such function, parameterized by the charging price 𝜋𝑘 𝜁, is exemplified in Fig. 2. The logistic function models how EVs are more likely to enter a charging station if their SoC is low. Its inflection point is given by 𝜁(𝜋𝑘 𝜁), as a function of the charging price 𝜋𝑘 𝜁, and its slope is calibrated by some parameter 𝛾𝛽. 3.1.3. Multi-class SoC equations The macroscopic SoC of each class is denoted 𝜉𝜀𝑘 𝑖, and its evolution is defined by 𝜉𝜌𝑘+1 𝑖 𝜉𝜀𝑘+1 𝑖=𝜉𝜌𝑘 𝑖(𝜉𝜀𝑘 𝑖+𝜉𝑑𝑘 𝑖𝑇)+𝑇 𝐿𝑥(𝜉𝜙𝑘 𝑖 − −𝜉𝜙𝑘 𝑖+),(14) 𝜉𝜙𝑘 𝑖 − = (𝜉𝑞𝑘 𝑖 − −𝜉𝑟𝑘 on,𝑖)(𝜉𝜀𝑘 𝑖 − 1+𝜉𝑑𝑘 𝑖 − 1𝑇)+𝜉𝑟𝑘 on,𝑖 𝜉𝜀𝑘 on,𝑖,(15) 𝜉𝜙𝑘 𝑖+=𝜉𝑞𝑘 𝑖+(𝜉𝜀𝑘 𝑖+𝜉𝑑𝑘 𝑖𝑇),(16) where the battery discharge rate 𝜉𝑑𝑘 𝑖depends on cell 𝑖traffic speed, 𝜉𝑑𝑘 𝑖=𝜉(𝑣𝑘 𝑖), 𝑣𝑘 𝑖=𝑞𝑘 𝑖+ 𝜌𝑘 𝑖 ,(17) and is defined by discharge functions 𝜉(𝑣), which may be different for each class 𝜉. 3.1.4. Charging stations equations The electromobility model is completed by the charging station dynamics. The SoC space of each charging station 𝜁, is split into 𝑁𝜀 bins of length 𝐿𝜀,𝑁𝜀𝐿𝜀= 1. Each bin 𝑖= 1,…, 𝑁𝜀corresponds to a range of SoC [(𝑖− 1)𝐿𝜀, 𝑖𝐿𝜀], and the state of the charging station is given by the number of EV in each bin 𝜁𝜂𝑘 𝑖. We allow the rate of charging 𝜁𝑐𝑘 𝑖for each charging station 𝜁and for each SoC level 𝑖, to vary in time within some range |||𝜁𝑐𝑘 𝑖|||≤𝐶, with 𝐿𝜀≥𝐶 𝑇required for numeric stability. The charging station state update is thus given by 𝜁𝜂𝑘+1 𝑖=𝜁𝜂𝑘 𝑖+𝑇 𝐿𝜀(𝜁𝑐𝑘 𝑖−1 𝜁𝜂𝑘 𝑖−1 −|||𝜁𝑐𝑘 𝑖|||𝜁𝜂𝑘 𝑖−𝜁𝑐𝑘 𝑖+1𝜂𝑘 𝑖+1)… … +𝑇(𝜁𝜇𝑘 in,𝑖 −𝜁𝜇𝑘 out,𝑖), (18) where 𝜁𝑐𝑘 𝑖= max{0,𝜁𝑐𝑘 𝑖}, and 𝜁𝑐𝑘 𝑖= min{0,𝜁𝑐𝑘 𝑖}, and 𝜁𝜇𝑘 in,𝑖 and 𝜁𝜇𝑘 out,𝑖 are the flows of vehicles of SoC level 𝑖entering and exiting the charging station, respectively. The flows between the road and the charging station 𝜁are 𝑟𝑘 on,𝑖out 𝜁 = 𝑁𝜀 ∑ 𝑖=1 𝜁𝜇𝑘 out,𝑖, 𝑟𝑘 on,𝑖out 𝜁 𝜀𝑘 on,𝑖out 𝜁 = 𝑁𝜀 ∑ 𝑖=1 𝜁𝜇𝑘 out,𝑖(𝑖− 1)𝐿𝜀,(19) 𝑟𝑘 of f,𝑖in 𝜁 = 𝑁𝜀 ∑ 𝑖=1 𝜁𝜇𝑘 in,𝑖, 𝑟𝑘 of f,𝑖in 𝜁 𝜀𝑘 𝑖in 𝜁 = 𝑁𝜀 ∑ 𝑖=1 𝜁𝜇𝑘 in,𝑖(𝑖− 1)𝐿𝜀,(20) where 𝑖in 𝜁and 𝑖out 𝜁are the cells where the offand on-ramp connecting the road to the charging station 𝜁are, respectively, ensuring both the vehicles and the energy are conserved. In this work, we assume that multiple classes of vehicles can enter a single charging station 𝜁, 𝜁𝜇𝑘 in,𝑖 =∑ 𝜉∈𝛯 𝜁 ,𝜉 𝜇𝑘 in,𝑖,(21) and that all vehicles exiting the charging station are of the same class 𝜉out 𝜁, therefore 𝜉𝑟𝑘 on,𝑖out 𝜁 =𝑟𝑘 on,𝑖out 𝜁 with 𝜉=𝜉out 𝜁, and 𝜉𝑟𝑘 on,𝑖out 𝜁 = 0for 𝜉≠𝜉out 𝜁. The flows of each class entering and exiting the charging stations are further determined by 𝜁 ,𝜉 𝜇𝑘 in,𝑖 = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛⎜⎜⎝ 𝑖− 𝜉𝜀𝑘 𝑖in 𝜁 𝐿𝜀⎞⎟⎟⎠ 𝜉𝑟𝑘 of f,𝑖in 𝜁 , 𝑖− 1≤ 𝜉𝜀𝑘 𝑖in 𝜁 𝐿𝜀 < 𝑖, 𝜉∈𝛯in 𝜁, ⎛⎜⎜⎝ 𝜉𝜀𝑘 𝑖in 𝜁 𝐿𝜀 −𝑖+ 2⎞⎟⎟⎠ 𝜉𝑟𝑘 of f,𝑖in 𝜁 , 𝑖− 2≤ 𝜉𝜀𝑘 𝑖in 𝜁 𝐿𝜀 < 𝑖− 1, 𝜉∈𝛯in 𝜁, 0,ot her wise, (22) 𝜁𝜇𝑘 out,𝑖 ∈⎡⎢⎢⎣ 0,⎛⎜⎜⎝ 1 𝑇−|||𝜁𝑐𝑘 𝑖||| 𝐿𝜀⎞⎟⎟⎠ 𝜁𝜂𝑘 𝑖+ 𝜁𝑐𝑘 𝑖 − 1 𝐿𝜀 𝜁𝜂𝑘 𝑖 − 1− 𝜁𝑐𝑘 𝑖+1 𝐿𝜀 𝜁𝜂𝑘 𝑖+1⎤⎥⎥⎦ ,(23) where 𝛯in 𝜁denotes the set of vehicle classes that enter charging station 𝜁, and the exiting flow 𝜁𝜇𝑘 out,𝑖 depends on the particular behavioural logic of each charging station. We assume that each traffic class 𝜉enters at most one charging station 𝜁from each cell 𝑖, but it may enter multiple charging stations in case off-ramps leading to them are in different cells 𝑖in 𝜁1≠𝑖in 𝜁2. In particular, we define the flows of EVs exiting each charging stations according to their departure demand profile 𝜁𝜇𝑘 t ot , 𝜁𝜇𝑘 out,𝑖 =⎧ ⎪ ⎨ ⎪ ⎩ min {𝜁𝜇𝑘 t ot ,(1 𝑇−|||𝜁𝑐𝑘 𝑖||| 𝐿𝜀)𝜁𝜂𝑘 𝑖}, 𝑖=𝑖𝑘 𝜁, 0, 𝑖≠𝑖𝑘 𝜁. (24) For 𝜁=𝐡and 𝜁=𝐰,𝜁𝜇𝑘 t ot is defined externally through a departure demand profile 𝜁𝜇𝑘 t ot that represent flows of EVs starting their commute from home to work and from work to home. For 𝜁=𝐩, we assume that the EVs stay at the charging station for 1h, so the departure demand will depend on the flow entering the charging station in the past. At each sampling time, we randomly select the SoC of the departing EVs for each charging station 𝑖𝑘 𝜁, with probabilities of each SoC level proportional to the number of vehicles currently at charging station 𝜁. 3.2. Power grid model Here we describe the dynamics of the alternating current (AC) power grid that is employed to model the interaction between a conventional grid and the electromobility layer. To avoid confusion regarding notation conventions, and reduce abuse of notation, we indicate the parameters and variables of the power grid model by tilde. Three-phase AC power systems are the most widely used in electrical engineering today. They consist of three alternating currents, each phase shifted by 120 degrees, providing a continuous and balanced flow of electricity. Phasor description simplifies analysis by representing these currents as rotating vectors in complex plane, aiding in calculations of voltage, current, and power in balanced networks. In this work, the power transmission segment depicted in Fig. 1is assumed to be accurately described as such three-phase balanced network, thus we adopt its equivalent single-phase representation in the following. The distribution grid is usually described as a graph equivalent model where nodes represent grid buses (e.g. generation or load points, substations, etc.) and edges represent transmission lines, power transformers and phase shifters connecting the buses. The power grid is thus modelled as a directed graph  𝐷( , )where  denotes the set of nodes,  ⊆ × is the set of oriented edges, where each link  𝑙∈  connects buses 𝑛 and 𝑚, 𝑙= (𝑛, 𝑚),𝑛, 𝑚 ∈ . For power line modelling, the conventional 𝛱model is employed, as depicted in Fig. 3, where each power line can be defined by five parameters: a series resistance 𝑟 𝑙, a series reactance 𝑥 𝑙, a charging susceptance  𝑏 𝑙, a tap ratio magnitude 𝜏 𝑙and phase shift  𝜃 𝑙. These magnitudes allow to define the link series admittance 𝑦 𝑙, shunt admittance 𝑦sh  𝑙, and the complex tap ratio of transformers,  𝑡 𝑙as 𝑦 𝑙=1 𝑟 𝑙+𝑗 𝑥 𝑙 , 𝑦sh  𝑙=𝑗  𝑏 𝑙 2, 𝑡 𝑙=𝜏 𝑙𝑒𝑗 𝜃 𝑙, 𝑙∈ ,(25) Control Engineering Practice 159 (2025) 106289 6 M. Čičić et al. Fig. 3. 𝛱transmission line model. where 𝑗denotes the imaginary unit. See Zimmerman, Murillo-Sánchez, and Thomas (2011) for a more detailed explanation of the branch convention employed. 3.2.1. Power network equations The complex power flowing from bus 𝑚 to bus 𝑛 is denoted by  𝑆(𝑚, 𝑛), and may be expressed as the product of the complex voltage at bus 𝑚,  𝑉𝑚, and the complex conjugate of the current flowing between buses  𝐼(𝑚, 𝑛). Assuming a generic 𝛱power line model with asymmetric shunt admittances 𝑦sh (𝑚, 𝑛)and 𝑦sh (𝑛, 𝑚), and applying Kirchhoff’s current law, the net current flowing into nodes A and B (see Fig. 3) equals the net power out of them, yielding  𝐼(𝑚, 𝑛)=  𝐼sh (𝑚, 𝑛)  𝑡∗ (𝑚, 𝑛) +𝑦sh (𝑚, 𝑛)  𝑉𝑚 | 𝑡(𝑚, 𝑛)|2,(26)  𝐼(𝑛, 𝑚)= − 𝐼sh (𝑚, 𝑛)+𝑦sh (𝑚, 𝑛) 𝑉𝑛.(27) Applying Ohm’s Law, we have  𝐼sh (𝑚, 𝑛)=𝑦(𝑚, 𝑛)( 𝑉𝑚  𝑡(𝑚, 𝑛)− 𝑉𝑛), and the corresponding complex power flows are thus  𝑆(𝑚, 𝑛)=𝑦∗ (𝑚, 𝑛)(| 𝑉𝑚|2 | 𝑡(𝑚, 𝑛)|2−𝑉𝑚𝑉∗ 𝑛  𝑡(𝑚, 𝑛))+ (𝑦sh (𝑚, 𝑛))∗|𝑉𝑚|2 | 𝑡(𝑚, 𝑛)|2,(28)  𝑆(𝑛, 𝑚)=𝑦∗ (𝑛, 𝑚)(| 𝑉𝑛|2−𝑉𝑛𝑉∗ 𝑚  𝑡∗ (𝑚, 𝑛))+ (𝑦sh (𝑛, 𝑚))∗|𝑉𝑛|2.(29) A convenient formulation of the power flow problem in the setup at hand is the Bus Injection Model (BIM) that parameterizes the problem in terms of node voltages and power injected at nodes. In our problem, at any given bus we can split the injected power into four components: distributed RES generation  𝛤𝜁𝑛 , loads  𝛬𝜁𝑛 , EV charging station power 𝑈𝜁𝑛 , and curtailment of generation or load  𝛥𝜁𝑛 . Here we use the notation 𝜁𝑛 to emphasize the connection between the electromobility node 𝜁𝑛 and power system bus 𝑛. For those buses 𝑛 that have no electromobility nodes associated to them, we take  𝛤𝜁𝑛 = 𝛬𝜁𝑛 =𝑈𝜁𝑛 = 𝛥𝜁𝑛 = 0. Applying Tellegen’s Theorem, we define the net injected power flow at bus 𝑛 as  𝑆𝑛 = 𝛤𝜁𝑛 + 𝛬𝜁𝑛 +𝑈𝜁𝑛 + 𝛥𝜁𝑛 ,(30) which can equivalently be expressed in terms of branch powers  𝑆𝑛 =∑ ( 𝑘, 𝑛)∈    𝑆( 𝑘, 𝑛)+∑ (𝑛, 𝑘)∈    𝑆(𝑛, 𝑘),(31) where the net injected power is expressed as the sum of the incoming (first term) and outgoing (second term) power flows to and from the corresponding adjacent buses of bus 𝑛. From Eqs. (30) and (31) and using Eqs. (26)–(29) the final BIM formulation can be obtained  𝛤𝜁𝑛 + 𝛬𝜁𝑛 +𝑈𝜁𝑛 + 𝛥𝜁𝑛 =⋯ …∑ ( 𝑘, 𝑛)∈  (𝑦∗ (𝑛, 𝑘)(| 𝑉𝑛|2 | 𝑡(𝑛, 𝑘)|2− 𝑉𝑛𝑉∗  𝑘  𝑡(𝑛, 𝑘))+ (𝑦sh (𝑛, 𝑘))∗|𝑉𝑛|2 | 𝑡(𝑛, 𝑘)|2)+⋯ …∑ (𝑛, 𝑘)∈  (𝑦∗ ( 𝑘, 𝑛)(| 𝑉𝑛|2− 𝑉𝑛𝑉∗  𝑘  𝑡∗ ( 𝑘, 𝑛))+ (𝑦sh (𝑛, 𝑘))∗|𝑉𝑛|2). (32) 3.2.2. Power flow solution The standard power flow problem involves solving for the set of voltages  𝑉𝑚 at buses 𝑚 ∈ and complex line power flows  𝑆(𝑚, 𝑛)at branches (𝑚, 𝑛) ∈ for a network corresponding to a specified pattern of power injection at buses at a given time instant 𝑘. Expression (32) yields complex-valued equations that allows to solve for voltages at buses assuming the injected power at the nodes (potentially partially curtailed distributed RES generation and loads, and EV charging station power) are known. Branch power flows can then be determined from expressions (28)–(29). This formulation yields | |complex-valued equations, that can be posed as 2| |quadratic real valued equations with 2| |real unknowns. By convention, a single bus is typically chosen as a reference (slack) bus to serve the roles of both a voltage angle reference and a real power slack. The voltage angle at the reference bus has a known value, but the real power generation at the slack bus is taken as unknown to avoid overspecifying the problem. Unknowns for the problem will thus be active and reactive powers at the slack bus together with complex voltage level at the remaining | |− 1buses. These equations are solved at every time step 𝑘of the problem providing equilibrium powers and voltages at the bus level. The solver of choice employed in this work is based on a standard Newton’s method (Tinney & Hart,1967) using a polar form and a full Jacobian updated at each iteration. Each Newton step involves computing the mismatch, forming the Jacobian based on the sensitivities of these mismatches to changes in and solving for an updated value of the unknowns by factorizing this Jacobian. 3.3. Profile perturbation model Finally, we model the effects of uncertainty by introducing perturbations to the various profiles. We assume that the future values of departures from 𝐡and 𝐰charging stations 𝜁𝜇𝑘 t ot ,𝜁∈ {𝐡,𝐰}, and RES generation  𝛤𝑘 𝜁and load  𝛬𝑘 𝜁at all nodes, 𝜁∈ {𝐡,𝐩,𝐰}, are only known approximately, with only their current actual values assumed to be known exactly. We denote the expected values of these profiles by  𝑓𝑘, where 𝑓𝑘is 𝜁𝜇𝑘 t ot , 𝛤𝑘 𝜁, or  𝛬𝑘 𝜁, and define their actual values as 𝑓𝑘=𝑎𝑘 𝑓 𝑓𝑘,(33) where 𝑎𝑘 𝑓is the random multiplicative perturbation of 𝑓around its profile  𝑓at each time step 𝑘. Each of the stated profiles is perturbed by its own realization of 𝑎𝑘 𝑓, with an exception for RES generation profiles, which are perturbed by the same 𝑎𝑘  𝛤for all nodes 𝜁. We define 𝑎𝑘 𝑓as a weighted sum of two random walks, one forward in time and the other backwards in time, 𝑎𝑘 𝑓= 1 + 2𝜎max 𝑓 √𝑘end + 1(𝑘end −𝑘 𝑘end − 1𝑊𝑘 𝑓++𝑘− 1 𝑘end − 1𝑊𝑘end+1−𝑘 𝑓−),(34) where 𝑊𝑘 𝑓+and 𝑊𝑘 𝑓−are independent Gaussian random walks with normally distributed steps, 𝑊1 𝑓±∼(0,1), 𝑊𝑘+1 𝑓±−𝑊𝑘 𝑓±∼(0,1),(35) Control Engineering Practice 159 (2025) 106289 7 M. Čičić et al. and 𝜎max 𝑓is a design parameter denoting the maximum standard deviation of 𝑎𝑘 𝑓. This perturbation is defined separately for each day, 𝑘= 1,…, 𝑘end, where 𝑘= 1and 𝑘=𝑘end correspond to the beginning and the end of the day, respectively. This way, the perturbation has the highest variance at the middle of the day, and is zero at its beginning and at its end. As an integral part of prediction-based control, at the current time 𝑘, we are required to predict the future evolution of the full system state until some prediction horizon 𝑘+𝐻. While the future values of the electromobility state on the road and at the charging stations can be predicted well using the electromobility model, the perturbed profiles are predicted using their known current value 𝑓𝑘and the expected profile  𝑓𝑘+ℎ,ℎ= 1,…, 𝐻, 𝑓𝑘+ℎ|𝑘≈ 𝑓𝑘+ℎ  𝑓𝑘𝑓𝑘, ℎ= 1,…, 𝐻 .(36) 3.4. Combined model We denote the encapsulated full state of the mobility model at time 𝑘as 𝑀𝑘, 𝑀𝑘=(𝜉𝜌𝑘 𝑙 ,𝑖|𝑙∈ {𝐡𝐰,𝐰𝐡}, 𝑖= {1,…, 𝑁𝑥}, 𝜉∈ {𝐡∕𝐰,𝐩},… 𝜉𝜀𝑘 𝑙 ,𝑖|𝑙∈ {𝐡𝐰,𝐰𝐡}, 𝑖= 1,…, 𝑁𝑥, 𝜉∈ {𝐡∕𝐰,𝐩},… 𝜁𝜂𝑘 𝑖|𝑖∈ {1,…, 𝑁𝜀}, 𝜁∈ {𝐡,𝐰,𝐩𝐡𝐰,𝐩𝐰𝐡}),(37) capturing the traffic density 𝜉𝜌𝑘 𝑙 ,𝑖 and SoC 𝜉𝜀𝑘 𝑙 ,𝑖 for all vehicle classes 𝜉 and all cells 𝑖on both road links 𝑙, and the numbers of charging vehicles 𝜁𝜂𝑘 𝑖at each SoC level 𝑖at each charging station 𝜁. The electromobility state is updated according to (9),(14), and (18), as defined in Section 3.1, and we write jointly 𝑀𝑘+1 =(𝑀𝑘, 𝑈𝑘, 𝜋𝑘 𝐩),(38) where we denote by 𝑈𝑘the collection of power 𝑈𝑘 𝜁allocated to all physical charging stations, 𝜁∈ {𝐡,𝐰,𝐩}, that indirectly determine the evolution of 𝜁𝜂𝑘 𝑖through (18) by influencing the charging rates, and 𝜋𝑘 𝐩 is the public charging station charging price. Power grid equations at discrete time step 𝑘are given by a static mapping, and can be written compactly as [ 𝑉𝑘  𝑆𝑘]= ( 𝛤𝑘, 𝛬𝑘, 𝑈𝑘, 𝛥𝑘),(39) yielding bus voltages  𝑉𝑘and line powers  𝑆𝑘at time 𝑘, as a function of the EV power injected or retrieved at charging stations 𝑈𝑘, loads  𝛬𝑘, distributed RES generation  𝛤𝑘, and curtailment  𝛥𝑘, for each port 𝜁∈ {𝐡,𝐰,𝐩}. These need to be kept within safe operating limits to ensure proper functioning of the power grid. 4. Control design With the Coupled Electromobility and Power Grid Model introduced, we are now able to formulate control laws that follow the control architecture and achieve the control objectives outlined in Section 2. As discussed, the control can be split into two levels: the upper level, which is tasked with achieving the three stated control objectives by controlling the reference charging station powers and the public charging price, and the lower level, which is tasked with ensuring that the actual power of the charging station follows this reference by controlling the charging rates. We first describe the lower level control, then formulate the general optimal control problem, and finally, discuss the approximations appropriate to the specific studied scenario that are employed to make the problem tractable. 4.1. Charging station power control Given a distribution of EVs at a charging station 𝜁at time 𝑘according to their SoC, 𝜁𝜂𝑘 𝑖, the current total power of each charging station 𝜁is given by 𝑈𝑘 𝜁= 𝑁𝜀 ∑ 𝑖=1 𝜁𝑐𝑘 𝑖 𝜁𝜂𝑘 𝑖𝐵 ,(40) where 𝐵is the average EV battery capacity. Since the lower level of control has no influence over EV arrivals and departures, it may only achieve total power reference tracking by setting the charging rates 𝜁𝑐𝑘 𝑖 for EVs at different SoC levels 𝑖. For each charging station we define the maximum charging rate 𝐶𝜁≥0and maximum discharging rate 𝐶𝜁≤0per EV, yielding 𝐶𝜁≤𝜁𝑐𝑘 𝑖≤𝐶𝜁, respecting the numeric stability requirements |𝐶𝜁|≤𝐶and |𝐶𝜁|≤𝐶. It can be seen that the choice of 𝜁𝑐𝑘 𝑖that makes 𝑈𝑘 𝜁follow some reference is not unique. Instead, we may adopt a charging rate scheme that also achieves some other desirable outcomes. In order to ensure that the EVs have an adequate SoC to continue commuting, as well as to best utilize the power available to the charging station, we employ a hierarchical charging scheme, where vehicles with lower SoC have a higher charging priority. We split the vehicles into two groups according to their SoC: low-SoC EVs, with 0≤𝜀 < 𝜀bnd, and high SoC EVs, with 𝜀bnd < 𝜀≤1. In case not enough power is allocated to charge the low-SoC EVs, or if the upper level of control requests power to be provided to the grid, we utilize the energy stored in batteries of EVs with higher SoC to satisfy the power demands. Given the boundary between these groups of EVs 𝜀bnd, we define 𝑖bnd as the highest 𝑖for which 𝜀𝑖< 𝜀bnd,𝑖∈ {1,…, 𝑁𝜀}. The number of EVs currently in each group is then given by 𝜁𝜂𝑘 lo = 𝑖bnd ∑ 𝑖=1 𝜁𝜂𝑘 𝑖,𝜁𝜂𝑘 hi = 𝑁𝜀 ∑ 𝑖=𝑖bnd+1 𝜁𝜂𝑘 𝑖.(41) We assign a different charging rate to each group of EVs, denoted 𝜁𝑐𝑘 low, and 𝜁𝑐𝑘 high, and given by 𝜁𝑐𝑘 𝑖=⎧ ⎪ ⎨ ⎪ ⎩ 𝜁𝑐𝑘 lo, 𝑖= 1,…, 𝑖bnd, 𝜁𝑐𝑘 hi, 𝑖=𝑖bnd + 1,…, 𝑁𝜀, (42) and the charging station power simplifies to 𝑈𝑘 𝜁=(𝜁𝑐𝑘 lo 𝜁𝜂𝑘 lo +𝜁𝑐𝑘 hi 𝜁𝜂𝑘 hi)𝐵 .(43) While the charging rates of high-SoC EVs may take any value within the allowed range, 𝐶𝜁≤𝜁𝑐𝑘 hi ≤𝐶𝜁, we restrict the charging rates of lowSoC EVs to be nonnegative, 0≤𝜁𝑐𝑘 lo ≤𝐶𝜁, in order to keep their SoC high enough to continue commuting. Given the current provided or demanded power 𝑈𝑘 𝜁, the charging rates of low-SoC and high-SoC EVs are set to 𝜁𝑐𝑘 lo = max {0,min {𝑈𝑘 𝜁−𝜁𝜂𝑘 hi𝐶𝜁 𝜁𝜂𝑘 lo , 𝐶𝜁}},(44) 𝜁𝑐𝑘 hi =⎧ ⎪ ⎨ ⎪ ⎩ 𝐶𝜁,𝜁𝜂𝑘 lo𝐶𝜁> 𝑈𝑘 𝜁−𝜁𝜂𝑘 hi𝐶𝜁, min {𝑈𝑘 𝜁−𝜁𝜂𝑘 lo𝐶𝜁 𝜁𝜂𝑘 hi , 𝐶𝜁},𝜁𝜂𝑘 lo𝐶𝜁≤𝑈𝑘 𝜁−𝜁𝜂𝑘 hi𝐶𝜁, (45) respectively. This charging scheme results in a range of charging station 𝜁power consumption or production [𝑈𝑘 𝜁, 𝑈𝑘 𝜁], 𝑈𝑘 𝜁=(𝜁𝜂𝑘 lo +𝜁𝜂𝑘 hi)𝐶𝜁𝐵 ,(46) 𝑈𝑘 𝜁=𝜁𝜂𝑘 hi𝐶𝜁𝐵 ,(47) which are communicated to the upper level of control. Control Engineering Practice 159 (2025) 106289 8 M. Čičić et al. 4.2. Optimal control formulation We now formulate ANM taking electromobility into account as a receding horizon optimal control problem minimize 𝑢𝜁,𝜋𝐩, 𝛿𝜁 𝐽(𝑘, 𝑢𝜁, 𝜋𝐩, 𝛿𝜁) subject to Mobility and grid dynamics (38)–(39), | 𝑆𝑘+ℎ  𝑙|≤𝑆 𝑙, 𝑙∈ , ℎ= 0,…, 𝐻 ,(48)  𝑉𝑖≤| 𝑉𝑘+ℎ 𝑖|≤ 𝑉𝑖, 𝑖∈ − slack, ℎ= 0,…, 𝐻 ,(49) − 1≤𝑢𝜁≤1, 𝜁∈ {𝐡,𝐩,𝐰}(50) 𝜋≤𝜋𝐩≤𝜋 ,(51) − 1≤ 𝛿𝜁≤1, 𝜁∈ {𝐡,𝐩,𝐰}(52) Here, the decision variables are relative charging powers 𝑢𝜁, public charging prices 𝜋𝐩, and relative curtailments  𝛿𝜁, all given as stacked vectors of current and future control inputs, over an 𝐻-step horizon, 𝑢𝜁=⎡⎢⎢⎢⎢⎢⎣ 𝑢𝑘 𝜁 𝑢𝑘+1 𝜁 ⋮ 𝑢𝑘+𝐻 𝜁 ⎤⎥⎥⎥⎥⎥⎦ , 𝜋𝐩=⎡⎢⎢⎢⎢⎢⎣ 𝜋𝑘 𝐩 𝜋𝑘+1 𝐩 ⋮ 𝜋𝑘+𝐻 𝐩 ⎤⎥⎥⎥⎥⎥⎦ , 𝛿𝜁=⎡⎢⎢⎢⎢⎢⎣ 𝛿𝑘 𝜁 𝛿𝑘+1 𝜁 ⋮ 𝛿𝑘+𝐻 𝜁 ⎤⎥⎥⎥⎥⎥⎦ .(53) Both the relative charging power 𝑢𝑘 𝜁and relative curtailment 𝛿𝑘 𝜁are normalized control actions, allowing to accommodate the constraints on the time-varying mobility reserve powers, loads, and RES generation in a more convenient form. From the side of the power grid, the optimization problem is subject to constraints on the maximum apparent power | 𝑆𝑖𝑗 |(48), and voltage | 𝑉𝑖|deviation at buses (49). It is straightforward to reformulate this problem to a more general setup with multiple 𝐡,𝐩, and 𝐰nodes, connected by a more elaborate road network and power grid. The optimization problem is articulated as constrained minimization of the cost function 𝐽(𝑘, 𝑢𝜁, 𝜋𝐩, 𝛿𝜁) = 𝐻 ∑ ℎ=0(∑ 𝜁∈{𝐡,𝐩,𝐰}(𝑤ℎ 𝛿| 𝛥𝑘+ℎ 𝜁|)+𝑤ℎ 𝜀(𝜀𝑘+ℎ avg −𝜀r ef )2).(54) The first part of the cost function corresponds to total curtailment, and is minimized for 𝛿𝑘+ℎ= 0, in case this can be achieved without violating constraints (48)–(51). The second part corresponds to deviation of the average overall SoC 𝜀𝑘 avg from its reference value 𝜀r ef . These two parts are weighed by 𝑤ℎ 𝛿and 𝑤ℎ 𝜀, including an exponential forgetting factor, and we have 𝑤ℎ 𝛿≫ 𝑤ℎ 𝜀, i.e., reducing curtailment is prioritized over regulating the average SoC, as outlined in Section 2. While it is able to capture a wide variety of scenarios, this optimization problem is very difficult to solve in its exact form, due to the strongly nonlinear nature and high order of the underlying model, as well as the large number of decision variables in case the control horizon is taken to be appropriately long. Therefore, in order to make the problem tractable, enabling us to implement the proposed ANM control, we need to adopt some simplifications and approximations, as well as to decompose the problem into subproblems, all of which will be discussed in the remainder of this section. 4.3. Approximate formulation First, in order to ease the numeric burden, we solve the optimization problem at a time scale different than that of the electromobility model. We take the optimization sampling period  𝑇as a multiple of electromobility sampling periods 𝑇, and denote the optimization time step  𝑘. The resulting relative charging power control 𝑢 𝑘 𝜁is then applied to the electromobility layer over a number of time steps, 𝑢𝑘 𝜁=𝑢⌊𝑘𝑇  𝑇⌋+1 𝜁,(55) where ⌊⋅⌋denotes rounding down. The charging price 𝜋𝐩is updated at a lower rate still, with time step  𝑇𝜋taken as a multiple of  𝑇, in order to further reduce the computational burden, but also to improve realism by allowing the EV drivers more time to react to the change. We first study minimizing curtailment at the current time step  𝑘, which can only be affected by the current charging control 𝑢 𝑘 𝜁. In our case, due to the simple architecture of the grid, it is enough to only consider the power lines directly connected to 𝐡,𝐩, and 𝐰. For each port, constraint (48) simplifies to | 𝛤 𝑘 𝜁+ 𝛬 𝑘 𝜁+𝑈𝛿|≤𝑆𝜁,(56) where 𝑈𝛿is some purely real, positive or negative, charging power. In this case, it is easy to find the range of 𝑈𝛿for which there is no curtailment, 𝑈𝛿≤𝑈𝛿≤𝑈𝛿, assuming |Im{  𝛤 𝑘 𝜁+ 𝛬 𝑘 𝜁}|< 𝑆𝜁. Therefore, if [𝑈𝛿, 𝑈𝛿] ∩ [𝑈 𝑘 𝜁, 𝑈  𝑘 𝜁]≠∅, where 𝑈 𝑘 𝜁and 𝑈  𝑘 𝜁are the current minimum and maximum charging power at 𝜁, respectively, it is possible to achieve zero curtailment if 𝑢 𝑘 𝜁is restricted to max{𝑈 𝑘 𝛿, 𝑈 𝑘 𝜁} 𝑈 𝑘 𝜁 =𝑢  𝑘 𝜁≤𝑢 𝑘 𝜁≤𝑢  𝑘 𝜁= min{𝑈  𝑘 𝛿, 𝑈  𝑘 𝜁} 𝑈  𝑘 𝜁 ,(57) possibly tightening constraint (50). Otherwise, if 𝑈 𝑘 𝛿> 𝑈  𝑘 𝜁, we replace constraint (50) by equality constraint 𝑢 𝑘 𝜁= 1, RES generation will need to be curtailed, since there are not enough EVs that can absorb it, and  𝛿 𝑘 𝜁can be found by solving | 𝛿 𝑘 𝜁 𝛤 𝑘 𝜁+ 𝛬 𝑘 𝜁+𝑈  𝑘 𝜁|≤𝑆𝜁,0< 𝛿 𝑘 𝜁≤1.(58) Conversely, if 𝑈  𝑘 𝛿< 𝑈 𝑘 𝜁, we replace (50) by 𝑢 𝑘 𝜁= −1, load will need to be curtailed, since there are not enough EVs that can return power to the grid, and  𝛿 𝑘 𝜁can be found by solving | 𝛤 𝑘 𝜁− 𝛿 𝑘 𝜁 𝛬 𝑘 𝜁+𝑈 𝑘 𝜁|≤𝑆𝜁,−1 ≤ 𝛿 𝑘 𝜁<0.(59) We may use the current relative charging powers 𝑢 𝑘 𝜁at those ports where there is no curtailment at the current step to regulate the average SoC at the next time step 𝜀 𝑘+1 avg to the reference value 𝜀r ef , ignoring the SoC at the subsequent steps by setting 𝑤ℎ 𝜀= 0for ℎ >1. We formulate a simple myopic proportional controller that achieves this, 𝑢 𝑘 𝜁= max {𝑢  𝑘 𝜁,min {𝑢  𝑘 𝜁,−𝐾𝑝(𝜀  𝑘 𝑇 𝑇 avg −𝜀r ef )}},(60) with limits 𝑢  𝑘 𝜁and 𝑢  𝑘 𝜁given by (57). This control scheme will be used in simulations as a simple benchmark for comparison with the approximate optimal control laws. Note that this controller essentially ignores the dynamics of the electromobility layer, relying solely on the battery storage of the EVs that are currently present at each charging station to attempt to reduce current curtailment. It can therefore be seen as a simplified proxy for the more conventional ANM approaches relying solely on stationary energy storage, albeit without taking advantage of load and generation predictions. Since curtailment at any port is highly unlikely to change sign during the prediction horizon (from RES curtailment to load demand response, or vice versa), we adopt a two-step-long control horizon, effectively assuming that the relative charging rates take some constant value 𝑢> 𝑘 𝜁after the current time step  𝑘, 𝑢 𝑘+ℎ 𝜁=𝑢> 𝑘 𝜁, ℎ= 1,…, 𝐻 .(61) Instead of finding the exact  𝛿 𝑘+ℎ 𝜁for ℎ= 1,…, 𝐻, we calculate the predicted future curtailment approximately, using Boltzmann operator smooth maximum,  𝛥 𝑘+ℎ 𝜁=||| 𝛤 𝑘+ℎ 𝜁+ 𝛬 𝑘+ℎ 𝜁+𝑈 𝑘+ℎ 𝜁||| 1 +𝑒−𝛾𝛥||| 𝛤 𝑘+ℎ 𝜁+ 𝛬 𝑘+ℎ 𝜁+𝑈 𝑘+ℎ 𝜁||| ,(62) Control Engineering Practice 159 (2025) 106289 9 M. Čičić et al. with the future charging power taken as 𝑈 𝑘+ℎ 𝜁= max {𝑢> 𝑘 𝜁,0}𝑈( 𝑘+ℎ−1)  𝑇 𝑇+1 𝜁+ max {−𝑢> 𝑘 𝜁,0}𝑈( 𝑘+ℎ−1)  𝑇 𝑇+1 𝜁,(63) for ℎ= 1,…, 𝐻. Although the current relative charging power does affect future 𝑈𝑘 𝜁 and 𝑈𝑘 𝜁, and will thus affect  𝛥 𝑘+ℎ 𝜁,ℎ= 1,…, 𝐻, this influence is in practice low enough that it can be neglected when finding 𝑢> 𝑘 𝜁that minimize approximate future curtailment. It is enough to consider 𝑢> 𝑘 𝜁 taking is extreme values or zero, [𝑢> 𝑘 𝐡𝑢> 𝑘 𝐩𝑢> 𝑘 𝐰] ∈ {−1,0,1}3, and select the future relative charging powers that minimize curtailment. Next, we update the charging price 𝜋𝑘 𝐩at a lower rate than the relative charging powers, with time step  𝑇𝜋taken as a multiple of  𝑇, in order to reduce the computational burden, but also to improve realism by allowing the EV drivers more time to react to the change. Since the effects of changing the price only manifest in the future, it is enough to evaluate 𝐽𝜋( 𝑘, 𝜋𝐩;𝑢> 𝑘 𝜁) =  𝑇𝜋  𝑇 ∑ ℎ=1 ∑ 𝜁∈{𝐡,𝐩,𝐰} 𝑤ℎ 𝛿| 𝛥 𝑘+ℎ 𝜁|,(64) for two cases of future relative charging powers, 𝑢> 𝑘 𝐩= 1, 𝑢> 𝑘 𝐡=𝑢> 𝑘 𝐰= ±1.(65) This simplification can be made because in the studied case there is only a single charging station with only generation and no load at its node. The charging price is updated to the minimum value for which the curtailment is zero for at least one of the cases of 𝑢> 𝑘 𝜁, or if this is not achievable, it is set to its minimum value 𝜋. Finally, the approximated optimization problem is minimize 𝑢 𝑘 𝜁  𝐽(𝑘, 𝑢 𝑘 𝜁;𝑢> 𝑘 𝜁, 𝜋 𝑘 𝐩)(66) subject to Mobility and grid dynamics (38)–(39), 𝑢  𝑘 𝜁≤𝑢 𝑘 𝜁≤𝑢  𝑘 𝜁,(67) with the control input constraints resulting from minimizing curtailment of the current step (57), and the cost function  𝐽(𝑘, 𝑢 𝑘 𝜁;𝑢> 𝑘 𝜁, 𝜋 𝑘 𝐩) =𝑤𝜀(𝜀  𝑘 𝑇 𝑇 avg −𝜀r ef )2+ 𝐻 ∑ ℎ=1 ∑ 𝜁∈{𝐡,𝐩,𝐰}(𝑤ℎ 𝛿| 𝛥 𝑘+ℎ 𝜁|).(68) Therefore, in this formulation we only directly optimize the current relative charging powers 𝑢 𝑘 𝜁, with the future relative charging powers 𝑢> 𝑘 𝜁and the public charging price 𝜋 𝑘 𝐩that parametrize the problem, affecting the model dynamics (38)–(39) and the cost function. Crucially, since 𝑢> 𝑘 𝜁,𝜁∈ {𝐡,𝐰,𝐩}, is set to reduce future curtailment, in case there is no curtailment at some port 𝜁,𝑢 𝑘 𝜁can be optimized to further reduce possible future curtailment at other ports  𝜁. This is only possible due to the electromobility model capturing the flows of EVs from one port to another, carrying their SoC with them. In certain simple cases, the optimal control is straightforward to derive. For example, in case there is no current curtailment at any port at time  𝑘, and the model predicts future RES generation will be necessary at 𝐩at time  𝑘𝛿, the optimal solution needs to cause the future 𝑈  𝑘𝛿 𝐩, and consequently, also 𝜁𝜂𝑘𝛿 lo +𝜁𝜂𝑘𝛿 hi , to increase. In order for this to be achieved, the optimal current charging rates will be 𝑢 𝑘 𝐡=𝑢 𝑘 𝐰= −1, leading to a lower SoC of EVs entering the road from 𝐡and 𝐰,𝜉𝜀𝑘 on,𝑖,𝑖∈ {𝑖out 𝐡, 𝑖out 𝐰}, which in turn will cause 𝜉𝛽𝑘 𝑖in 𝐩 to increase, and more vehicles to enter 𝐩. Conversely, in case the model predicts future load curtailment will be necessary at 𝐰 at time  𝑘𝛿, the optimal solution needs to increase the future 𝑈  𝑘𝛿 𝜁, and consequently, also 𝜁𝜂𝑘𝛿 hi , leading to 𝑢𝜁 𝐡= 1being optimal. Such effects are shown in simulations. In summary, the procedure for finding the current approximate optimal control is as follows: Table 2 Simulation parameters and their values. Symbol Meaning Value 𝐿t ot al Total road length 25 km 𝑁EV Total number of EVs 4400 𝑁𝑥Number of road cells 14 𝐿𝑥Road cell length 1.79 km 𝑇Electromobility time step 60 s 𝑣f fFree flow speed 100 km/h 𝜌cr Critical density 30 veh/km 𝜌jam Jam density 360 veh/km 𝐿𝜀SoC cell length 0.1 𝑁𝜀Number of SoC cells 11 0Battery discharge constant term −1.40 ⋅10−1 1/h 1Battery discharge linear multiplier −1.85 ⋅10−3 1/km 2Battery discharge quadratic multiplier −1.02 ⋅10−6 h/km2 𝐵EV battery capacity 60 kWh 𝐶 𝐡∕𝐰Maximum charging rate at 𝐡and 𝐰0.05 1/h 𝐶 𝐩Maximum charging rate at 𝐩2.5 1/h 𝛾𝛽Splitting ratio parameter 10 0Price sensitivity constant term 1.2 1Price sensitivity linear multiplier −0.2 𝜎max 𝑓Maximum profile perturbation SD 0.2 𝑆Power line 𝐡,𝐩, and 𝐰capacities 30 MVA  𝑇Control time step 0.5 h 𝐻Control horizon 8 𝜀bnd Boundary SoC 0.3 𝜀r ef Reference SoC 0.5 1. if ⌊𝑘 𝑇  𝑇𝜋⌋= 0, update the public charging price 𝜋 𝑘 𝐩, 2. find [𝑢> 𝑘 𝐡𝑢> 𝑘 𝐩𝑢> 𝑘 𝐰] ∈ {−1,0,1}3that results in minimal curtailment, and 3. find 𝑢 𝑘 𝜁by solving (66). As discussed earlier, if at some time  𝑘curtailment cannot be avoided at some node 𝜁, constraints (67) degenerate to an equality constraint for that 𝑢 𝑘 𝜁, and only the remaining relative charging powers are optimized, thus reducing the computational load. 5. Simulation results and discussion We complete the study of the proposed ANM system using electromobility control by putting it to the test in simulations. After introducing the general simulation setup, we consider three groups of simulations. In the first group, we execute a single simulation run of two different scenarios where we assume the generation, load, and mobility profiles are known. The aim of these simulations is to illustrate in detail how electromobility control can be used for ANM in the ideal, full-information case. The scenarios are one work week long each, one set in summer and one in winter, with correspondingly different RES generation and load profiles. The second group of simulations consists of 20 simulation runs of asummer and winter work day with perturbed generation, load, and mobility profiles, with the aim towards demonstrating robustness to uncertainty with more statistical significance. The third group of simulations study a larger network structure, with the Home and Work ports split into multiple nodes, demonstrating how the approach can be scaled up. Note that in all cases the vehicles decide whether or not they enter the public charging station depending on their SoC, according to (13), which is only directly influenced by charging price control. 5.1. Simulation setup The overall structure of the simulated system is introduced in Section 2, and a list of all relevant simulation parameters and their values is given in Table 2. The simulated road network consists of two links of length 𝐿t ot al, connecting 𝐡and 𝐰in both directions. The state of the EV traffic evolves according to the electromobility model presented in Control Engineering Practice 159 (2025) 106289 16 M. Čičić et al. allowing us to predict the times when significant control action will be needed, and to adjust the charging control accordingly to preempt this. We compare three cases of control: simple myopic control that only reacts to the current state, and two cases of prediction-based receding horizon control, with or without public charging station price control. The proposed control laws are tested in simulations on summer and winter scenarios, with or without perturbations to the various power and mobility profiles, as well as using a more complex power network where the Home and Work ports are split into multiple sub-ports. The results show that we are able to significantly reduce curtailment through EV charging control, even if a simple myopic controller is used. Both prediction-based controllers are able to further reduce curtailment, emphasizing the need for good prediction models. The improvement is especially pronounced in case we are able to control the public charging price, indicating that providing incentives to EV drivers can play an important role in implementing grid-friendly electromobility. Although in this work we mainly focus on a fairly simplistic system structure, with three electromobility nodes, the overall control framework, including the models and control approaches, lends itself to extensions to a more general case. These extensions include larger and more complex road and electrical networks, more realistic behavioural and technological models of EVs and their drivers, and more intricate control objectives, e.g., taking into account balancing regulation market dynamics. While a simple model of EV drivers’ sensitivity to charging price is included in the model, and the charging price is used as the most straightforward additional control input, the actual incentive mechanisms that could be applied in reality are still unclear. There remains much work to be done on modelling the impact that these incentives have on real-world EV drivers in the nowadays largely hypothetical scenario of high EV market penetration rate. While in this work we use a detailed macroscopic electromobility model, in order to capture the complex interactions between the EVs and the power grid, it may be possible to achieve similar results using a simplified model. One potential approach is to represent the coupling between pairs of charging stations as EV-implemented virtual power lines. In this case, the parameters and limitations of such EV virtual power lines can be identified by studying the full system, represented by the detailed model. The acquired abstract simplified representation could then be used for control, significantly improving the scalability of the system. Such analysis remains as future work. There are several practical obstacles to implementing approaches similar to the ones discussed in this work the real world. Very few places currently have a large portion of EVs, severely limiting the impact that any EV charging control scheme could have. The charging infrastructure is still not sufficiently developed, and is largely incapable of accepting V2G power flows. The incentive structures that would fairly compensate EV owners for potential inconvenience and additional battery wear are not yet in place, and the specifics of the business case are unclear. The first steps towards real-world testing and validation of the approach should likely focus on small-scale isolated systems, where it might be easier to control all the involved components, such as in islanded microgrids. 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