Abstract
The gradual replacement of gasoline vehicles with electric vehicles (EVs) and hydrogen fuel cell vehicles (HFCVs) in recent years has provided a growing incentive for the collaborative optimization of power distribution network (PDN), urban transportation network (UTN), and hydrogen distribution network (HDN). However, an appropriate collaborative optimization framework that addresses the prevalent privacy concerns has yet to be developed, and a sufficient pool of system operators that can competently operate all three networks has yet to be obtained. This study proposes a differentiated taxation-subsidy mechanism for UTNs, utilizing congestion tolls and subsidies to guide the independent traffic flow of EVs and HFCVs. An integrated optimization model for this power-hydrogen-transportation network is established by treating these vehicles and the electrolysis equipment as coupling bridges. We then develop a learning-aided decoupling approach to determine the values of the coupling variables acting among the three networks to ensure the economic feasibility of collaborative optimization. This approach effectively decouples the network, allowing it to operate and be optimized independently. The results for a numerical simulation of a coupled system composed of a IEEE 33-node power network, 13-node Nguyen-Dupuis transportation network, and 20-node HDN demonstrate that the proposed learning-aided approach provides nearly equivalent dispatching results as those derived from direct solution of the physical models of the coupled system, while significantly improving the computational efficiency.
Set of child buses for bus
Set of hydrogen demands connected to node
Set of hydrogen refueling stations (HRSs) connected to node m
Set of hydrogen sources connected to node
Set of electrolytic tanks connected to node
Set of charging links powered by bus
Set of electrolytic tanks powered by bus
Set of buses in power distribution network (PDN)
Set of transmission lines in PDN
Set of hydrogen pipelines connected to node
Sets of electric vehicle (EV) and hydrogen fuel cell vehicle (HFCV) paths connecting an origin-destination (O-D) pair rs
Set of hydrogen compressors connected to node
Set of nodes in urban transportation network (UTN)
Set of origin nodes,
Set of destination nodes,
Set of O-D pairs
Set of regular links in UTN
Set of charging links in UTN
Set of hydrogen refueling links in UTN
Charging price at charging station (CS)
Hydrogen refueling price at HRS
Power-to-hydrogen (P2H) conversion coefficient
Ratios of EV and HFCV traffic demands
Lower and upper limits of hydrogen pressure at node
Hydrogen gas density
Lower and upper ratio limits of hydrogen compressor
Monetary cost of travel time
Energy production cost coefficient of distributed generation (DG)
Traffic flow capacity of link
The maximum allowable vehicular flow of charging link
The maximum allowable vehicular flow of hydrogen refueling link
Production cost of hydrogen source
Weymouth constant of hydrogen pipeline
Charging demand of unit traffic flow
Capacity of hydrogen compressor
The maximum production of hydrogen source
Capacity of electrolytic tank
Hydrogen refueling demand of unit traffic flow
Square of current flow capacity of line
Lower and upper limits for active generation of DG at bus i
Lower and upper limits for reactive generation of DG at bus i
Traffic demand between O-D pair
Travel time with free flow on regular link
Travel time with free flow on charging link
Travel time with free flow on hydrogen refueling link
Lower and upper bounds of squared voltage magnitude at bus i
Electricity price of main grid
Voltage magnitude at slack bus
Binary term: if path passes link , ; otherwise,
Binary term: if path passes link , ; otherwise,
Binary term: if path passes link , ; otherwise,
Energy conversion efficiency of hydrogen compressor
Hydrogen pressure at node
Hydrogen pressures of compressor at inlet and outlet
EV travel cost on path between O-D pair rs
HFCV travel cost on path between O-D pair rs
EV flow on path between O-D pair rs
HFCV flow on path between O-D pair rs
Charging service fee on link
Hydrogen refueling service fee on link
Amount of hydrogen demand consumed
Hydrogen consumption of HRSs on link
Predicted hydrogen consumption of HRSs on link
Hydrogen production of hydrogen source
Hydrogen production of electrolytic tank
Predicted hydrogen production of electrolytic tank
Hydrogen flow through compressor
Hydrogen flow through pipeline
Squared current magnitude of line l from buses i to j
Active power demand at bus i
Total charging power demand at bus i
Regular power demand at bus i
Predicted total charging power demand at bus i
Total power demand for hydrogen production at bus i
Predicted total power demand for hydrogen production at bus i
Generated active power at bus i
Active power flow of line l from buses i to j
Reactive power demand at bus i
Generated reactive power at bus i
Reactive power flow of line l from buses i to j
Resistance of line l from buses i to j
Travel time on link
Average time spent by EVs on link
Average time spent by HFCVs on link
Congestion toll of EVs on link
Congestion toll of HFCVs on link
The minimum travel cost of EVs between O-D pair rs
The minimum travel cost of HFCVs between O-D pair rs
Squared voltage magnitude at bus i
Aggregated traffic flow on link
Aggregated EV flow on link
Predicted aggregated EV flow on link
Aggregated HFCV traffic flow on link
Predicted aggregated HFCV traffic flow on link
Reactance of line l from buses i to j
Impedance of line l from buses i to j,
TO effectively reduce the carbon emissions within the transportation sector, the global transition toward renewable energy sources has promoted the increasing use of new energy vehicles [
The increased presence of charging stations (CSs) and EVs has significantly enhanced the coupling between power distribution networks (PDNs) and urban transportation networks (UTNs) [
Existing studies have primarily focused on the coordinated optimization of PDNs and UTNs while considering EVs. For example, [
In terms of the efforts to achieve coordinated optimization of HDNs with UTNs and PDNs when considering HFCVs, [
The learning-aided approaches have also been extensively employed to coordinate the operations of PDN and UTN. For example, [
Although many studies have investigated collaborative optimization of the operations of PDN, HDN, and UTN, the efforts to coordinate the operations of these networks still have two major limitations.
1) As previously discussed, limited attention has been directed toward the coordinated optimization of HDNs while considering HFCVs. An appropriate collaborative optimization framework for power-hydrogen-transportation networks that fully addresses privacy concerns has yet to be developed. In addition, the computational efficiency of an integrated optimization model for PDN, HDN, and UTN is insufficient to satisfy real-world scheduling demands. However, directly employing a data-driven approach to produce the collaborative optimization results of a power-hydrogen-transportation network often violates physical constraints such as voltage constraints.
2) Because PDNs, HDNs, and UTNs are operated by different entities, significant operational challenges are greatly exacerbated by the absence of a single operator that can competently operate all three networks. This challenge is compounded by the implementation of distributed dispatch algorithms, as this condition can produce numerous secondary issues such as significant increases in communication time, computational delays, and convergence problems due to the non-convexity of energy flow models.
This paper addresses these limitations by proposing a learning-aided collaborative optimization framework for an integrated power-hydrogen-transportation network that ensures the economic feasibility of collaborative operations. Accordingly, this paper makes the following contributions.
1) The proposed collaborative optimization framework considers the constraints of the PDN, HDN, and UTN individually. This framework incorporates coupling constraints between ① the PDN and UTN with respect to the charging activities of EV users, ② the HDN and UTN with respect to the refueling activities of HFCV users, and ③ the PDN and HDN with respect to P2H units. We also introduce a differentiated taxation-subsidy mechanism for UTN that guides vehicles toward less congested routes by imposing congestion tolls on regular links and encourages users to consume renewable energy via subsidies at CSs and HRSs. The congestion tolls and subsidies are tailored specifically for EVs and HFCVs. This mechanism enhances the flexibility of UTN management to improve the traffic dispatch efficiency. It also prevents the “collective punishment” phenomenon in vehicle scheduling and ensures that the UTN operates efficiently with the PDN and HDN, thereby reducing the overall operational costs of the three networks.
2) The proposed learning-aided approach employs a model-free deep neural network (DNN) framework to determine the optimal values of the coupling variables acting between the three networks. These optimal values are then applied to decouple the network for independent operation. This greatly simplifies the optimization process, restricts the exchange of private information between the three network operators, and eliminates the need for operators to have specific information about the operations of all three networks. Therefore, the proposed learning-aided approach aligns with the operational independence of each network while simultaneously showcasing the merits of collaborative coordination.
3) The results of numerical computations demonstrate that the proposed learning-aided approach provides dispatching results that are nearly equivalent to those obtained by directly solving the physical models of the coupled system. The proposed learning-aided approach also reduces the required computation time by 96%. We also clarify the potential for PDNs and HDNs integrated with P2H units to accommodate renewable energy generation based on the energy consumption of EVs and HFCVs, respectively.
The remainder of this paper is organized as follows. Section II presents the coupling relationships in a typical power-hydrogen-transportation network, i.e., the combined system with PDN, HDN, and UTN. These relationships are then included with the mathematical formulations of the UTN, HDN, and PDN to establish a collaborative optimization model. Section III presents the DNN architecture of the proposed learning-aided approach and individual optimization models applied to each network following decoupling via the DNN. Section IV presents the results of a numerical simulation of an IEEE 33-node power network, 13-node Nguyen-Dupuis transportation network, and 20-node HDN. Section V concludes this study.

Fig. 1 Coupling relationships in a typical combined system with PDN, HDN, and UTN.
As illustrated in

Fig. 2 Schematic illustrating four types of transportation links applied in UTN.
For the UTN, the mixed UE conditions between EVs and HFCVs can be modeled as follows [
1) Traffic Flow Constraints
(1) |
(2) |
(3) |
(4) |
(5) |
Here, (1) and (2) describe the relationships between traffic demands and traffic flows on paths, and (3)-(5) describe the relationship between traffic flows on links and paths.
2) Travel Time Constraints
For regular links, we correlate the travel time and traffic flow according to the function of the Bureau of Public Roads as:
(6) |
For charging and hydrogen refueling links, the queuing time of EVs and HFCVs at the CSs and HRSs is defined as (7) and (8), respectively, according to the Davidson model.
(7) |
(8) |
where J is typically set to be 0.05.
3) Travel Cost Constraints
The travel costs for EVs and HFCVs are defined as:
(9) |
(10) |
As can be seen, the overall travel costs are the sum of the travel costs on each link and mainly include the time, charging, hydrogen refueling, and response costs to the price regulations of the system operator. We incorporate the response cost from previous studies into the differentiated taxation-subsidy mechanism, guiding users to less-congested road links by imposing separate congestion tolls on EVs and HFCVs. In addition, subsidies are provided at CSs and HRSs to attract EVs and HFCVs to stations with shorter queues or higher renewable energy availability. This pricing mechanism effectively mitigates the traffic congestion and reduces travel costs.
4) UE Constraints
The UE constraints for the traffic flow on path k between origin-destination (O-D) pair rs are based on Wardrop’s first principle.
(11) |
(12) |
This principle posits that a UTN reaches an equilibrium state when all road users are fully aware of network traffic conditions and select the most efficient available routes. Consequently, users cannot further reduce their travel expenditures by modifying their route preferences under UE conditions. Notably, without the incorporation of the differentiated taxation-subsidy mechanism under the UE criterion, EVs and HFCVs can only achieve equilibrium operations in the UTN. In this case, EVs and HFCVs lack sufficient price incentives to alter their routes or choose alternative CSs and HRSs, thereby preventing the UTN from coordinating with the PDN and HDN. By contrast, via the introduction of this novel pricing mechanism, EVs and HFCVs become connecting bridges of the UTN with PDN and HDN.
The UTN constraints include (1)-(12).
The steady-state operational constraints of an HDN are analogous to those of a natural gas network and can be expressed as [
(13) |
(14) |
(15) |
(16) |
(17) |
(18) |
(19) |
As (13) shows, all hydrogen demands at node m of the HDN must be balanced. Constraint (14) corresponds to the supply constraint of the gas source , (15) represents the supply constraint of the electrolytic tank , (16) and (17) represent the operational constraints of the compressor , (18) describes the nonlinear relationship between the hydrogen flow in the pipeline mn and the pressure values at its inlet and outlet, and (19) enforces safety constraints on the pressure values at node m.
The hydrogen consumption of the HRSs on link represents the coupling between the HDN and UTN, which is defined as:
(20) |
Utilizing second-order cone (SOC) relaxation, the nonlinear functional relationship given in (18) can be reformulated as:
(21) |
The standardized SOC representation of (21) is given as [
(22) |
The HDN constraints include (13)-(22).
A PDN is typically described by a conventional DistFlow model. However, this model includes nonlinear terms that are non-convex and challenging to solve. This issue can be addressed effectively by applying the SOC relaxation technique to the nonlinear terms in the DistFlow model, and we can obtain [
(23) |
(24) |
(25) |
(26) |
(27) |
(28) |
Equations (
(29) |
where signifies the coupling between the PDN and UTN, and signifies the coupling between the PDN and HDN. The two demand terms can be expressed as:
(30) |
(31) |
The PDN constraints include (23)-(31).
Under the UE conditions, all routes carrying a positive traffic flow between O-D pair share the same travel costs. Therefore, the overall travel cost for travelers in UTN can be formulated as:
(32) |
The economic cost of PDN encompasses both the generation cost of the DG units and the cost of purchasing electricity from the main grid, which is defined as:
(33) |
The economic cost of HDN is the supply cost of hydrogen sources, which can be formulated as:
(34) |
Finally, the applied economic optimization model for the power-hydrogen-transportation network is derived by minimizing the sum of the defined economic costs as:
(35) |
The proposed collaborative optimization framework ensures the operational independence of the PDN, HDN, and UTN by learning the coupling variables (i.e., EV charging demand, HFCV hydrogen consumption, and power demand for hydrogen production) between them via a learning-aided approach. This approach leverages a DNN to predict the coupling variables, thereby minimizing the need for data sharing and maintaining the confidentiality of operational data in each network. In addition, the model-free architecture of DNN and aggregated anonymized data exchange protect against unauthorized access and potential data breaches. Privacy is further enhanced through strict access control, transparent data usage policies, and regular security audits, all of which contribute to safeguarding sensitive information while allowing for efficient network optimization.
1) Structure and Learning Goals
In general, DNNs are machine learning models that consist of multiple layers, where each layer takes the outputs from the previous layer as its inputs. In standard feedforward neural networks, each node in one layer is connected to all nodes in the next layer. The function that connects these layers is given by:
(36) |
where and are the input and output vectors, and and are the numbers of demands and coupling variables, respectively; is the weight matrix; is the bias vector; and is a nonlinear activation function, and in this paper, a rectified linear unit (ReLU) activation function is applied.

Fig. 3 Architecture of DNN.
Formally, the resulting predictor learns the mapping . The input of the learning task is a dataset , and represent the observations of the demands and coupling variables, respectively, and . The output is a function , which ideally represents the outcome of the following optimization problem:
(37) |
where the Huber loss function is expressed as (38), which offers improved robustness over the commonly-used loss functions by effectively reducing the sensitivity to outlier [
(38) |
where the parameter is set to be a default value of 1.
In this paper, we use Bayesian search to tune the hyperparameters of DNN, which navigates vast hyperparameter spaces systematically and efficiently by leveraging probabilistic models to pinpoint configurations that maximize the model performance [
As previously discussed, the optimal values of the coupling variables between the three networks are predicted by the DNN, and these predicted values are applied to decouple these networks. Accordingly, we first replace in (13) with the predicted value and replace and in (29) with the predicted values and , respectively. From (31), we can deduce as:
(39) |
Thus, (13) can be transformed into (40) with predicted values:
(40) |
The newly obtained HDN constraints include (15)-(21), (39), and (40). The decoupled HDN model can be described succinctly as:
(41) |
Similarly, (29) is transformed into:
(42) |
The newly obtained PDN constraints include (23)-(28), (42). The decoupled PDN model can be expressed succinctly as:
(43) |
To decouple the UTN, the power demand and hydrogen consumption values predicted by the DNN for the PDN and HDN essentially represent the predicted distributions of the EVs and HFCVs at the CSs and HRSs, respectively. Based on (20) and (30), the flow distributions of EVs and HFCVs can be expressed as:
(44) |
(45) |
In addition, (7) and (8) can be transformed into:
(46) |
(47) |
The newly obtained UTN constraints include (1)-(3), (6), (9)-(12), and (44)-(47). The decoupled UTN model can be described succinctly as:
(48) |
In the real world, the application of proposed learning-aided approach may encounter regulatory and policy challenges. It is imperative to harmonize regulations across various departments to ensure cohesive and effective policy implementation. In addition, the stringent adherence to data privacy regulations is essential to safeguard the sensitive information of individuals and organizations.
Furthermore, structural reforms are needed. Market structures must be reconfigured to incentivize participation by all relevant stakeholders. Concurrently, the integration of supporting technologies is critical, where the establishment of requisite infrastructure and systems is required to facilitate the implementation of the proposed learning-aided approach.
The role of the predictor can be fulfilled by specialized agencies or departments within existing utility companies that already perform data analysis and forecasting tasks. These entities are well-equipped with the necessary expertise and infrastructure to extend their services to the proposed collaborative optimization framework.

Fig. 4 Topology of coupled system.
We establish a DNN consisting of three hidden layers. To train and test the DNN, we extract the hourly load ratio variation data recorded for the
We evaluate the performance of the proposed differentiated pricing mechanism in scheduling a heterogeneous fleet comprising EVs and HFCVs along with its advantages in reducing user travel costs. We first investigate the congestion tolls imposed on various regular links under differentiated and non-differentiated pricing mechanisms in the economically optimal conditions of tri-network coordinated operation. As shown in

Fig. 5 Congestion tolls under different pricing mechanisms.
Furthermore, we investigate the traffic flow distribution in the UTN under different pricing mechanisms.

Fig. 6 Traffic flow distribution under different pricing mechanisms. (a) Non-differentiated pricing mechanism. (b) Differentiated pricing mechanism.
1) Bayesian Hyperparameter Tuning
The hyperparameters of the established DNN subject to Bayesian hyperparameter tuning comprise the number of neurons in each of the three hidden layers as well as the learning rate and batch size. The tuning framework includes neuron counts in the hidden layers 1-3 in the integer ranges of [64, 256], [256, 1024], and [64, 256], respectively. The learning rate is evaluated over a logarithmic range of [0.0001, 0.1], and the batch size is evaluated with three commonly employed values of 32, 64, and 128.
Hyperparameter | Neurons of hidden layer 1 | Neurons of hidden layer 2 | Neurons of hidden layer 3 | Learning rate | Batch size | value |
---|---|---|---|---|---|---|
Initial | 150 | 600 | 200 | 0.05000 | 32 | 0.74 |
After tuning | 140 | 566 | 252 | 0.00546 | 64 | 0.91 |

Fig. 7 Comparison of loss values obtained before and after tuning. (a) Before tuning. (b) After tuning.
2) Comparison with Solutions Obtained by Directly Solving Coupled Physical Models
We first compare the computational time between directly solving the coupled physical models of the PDN, HDN, and UTN using the sample data in the test dataset and the proposed learning-aided approach, as shown in
Approach | PDN time (s) | HDN time (s) | UTN time (s) | Prediction time (s) | Total time (s) |
---|---|---|---|---|---|
Directly solving | 212.11 | ||||
Proposed | 0.25 | 0.17 | 2.32 | 4.35 | 7.09 |
Compared with PDN, HDN exhibits more complex characteristics, which leads to greater difficulties in calculating the hydrogen consumption of HRSs and the power demand for hydrogen production, and brings greater differences in optimization costs. In addition, the proposed learning-aided approach relies on DNNs to predict coupled variables. If not properly accounted for, any inaccuracies in these predictions can lead to suboptimal HDN operations and increased costs. These findings demonstrate that the proposed learning-aided approach retains the economic advantages of collaborative optimization, even though the scheduling of each network is solved independently. This approach also significantly enhances the computational efficiency. Although the proposed learning-aided approach exhibits strong performance under normal operating conditions, its adaptability must be addressed under constraints such as power line overloads and voltage violations. To address these abnormal conditions, the proposed learning-aided approach requires regular retraining of the DNN using a more extensive dataset that includes cases that violate constraints. In addition, it is necessary to employ centralized optimization methods to coordinate solutions when dealing with these issues. This ensures that the proposed collaborative optimization framework remains effective with operational abnormalities.
3) Operation Results from Model-driven Centralized Approach and Learning-aided Decomposed Approach
We compare the dispatch solutions by directly solving the coupled physical models of the PDN, HDN, and UTN (model-driven approach) and the proposed learning-aided approach.

Fig. 8 Comparison of scheduling results for PDN and HDN. (a) Power generation. (b) Hydrogen production.

Fig. 9 Comparison of EV and HFCV distributions obtained at CSs and HRSs. (a) EV distribution. (b) HFCV distribution.
The results in Figs.
4) Effects of Renewable DG Output Levels on Collaborative Optimization Results
The renewable DG output levels directly influence the hydrogen output of the P2H units and the distribution of EVs and HFCVs at the CSs and HRSs, respectively. Therefore, we evaluate the dispatch solutions and P2H flow obtained using the proposed learning-aided approach under different renewable DG output levels of 0.5, 1, and 2 MW, as shown in

Fig. 10 Comparison of EV and HFCV distributions and P2H flow obtained under different renewable DG output levels.
As the renewable DG output level increases, the full consumption of renewable energy sources is supported by dispatching an increasing proportion of electricity and hydrogen from DG3 and DG4 to CS3, CS4, HRS3, and HRS4. In addition, the P2H flow steadily increases with the increasing renewable DG output levels. This indicates that the P2H technology plays a significant role in ensuring full consumption of renewable energy sources.
5) Effects of Hydrogen Loads on HFCV Distributions
We also evaluate the dispatch solutions obtained using the proposed learning-aided approach with hydrogen load factors of 0.9, 1.0, 1.1, and 1.2, as shown in

Fig. 11 Comparison of HFCV distributions at different HRSs under various hydrogen load factors.
These results can be attributed to the inherent losses in hydrogen transport in the HDN, which increases as the HDN becomes more heavily loaded. Therefore, the dispatch solutions obtained with increasing hydrogen load factors may mitigate losses to some extent by increasing the proportion of hydrogen directly dispatched from hydrogen suppliers W1 and W2 located at nodes 1 and 8 of the HDN, respectively.
This study develops a coordinated optimization model for a power-hydrogen-transportation network and employs a differentiated taxation-subsidy mechanism to guide EVs and HFCVs more effectively, thereby facilitating the coordinated operation of the UTN with the PDN and UTN while reducing user travel costs by approximately 1.3%. This study also addresses significant limitations in current efforts to coordinate the operations of coupled PDN, HDN, and UTN by proposing a learning-aided approach. Applying the predicted values capitalizes on the economic benefits of joint optimization and enables the networks to be decoupled, thereby enabling them to operate and be optimized independently. This greatly simplifies the optimization process, restricting the exchange of private information between the three network operators and eliminating the requirement of operators to have specific information regarding the operations of all three networks. Given the nonlinear complexity of the physical system models, Bayesian hyperparameter tuning is applied to determine the optimal hyperparameters of DNN, which improves its prediction performance. The results of numerical simulations of a coupled system composed of an IEEE 33-node power network, 13-node Nguyen-Dupuis transportation network, and 20-node HDN demonstrates that the dispatching results yielded by the proposed learning-aided approach differ from those by the model-driven approach by less than 1% while improving the computational efficiency by more than 96%.
This study is important for the application of learning machine approaches in improving the computational efficiency of traditional large-scale optimization models of energy systems. In addition, the information privacy of multi-energy networks is preserved by learning the coupling power information, which contributes to a sufficiently coordinated solution while maintaining the existing operational independence of each energy system. Notably, the data-driven approaches have limitations in terms of data quality, availability, and the possibility of model overfitting. To address these limitations, we incorporate strategies such as rigorous data preprocessing to ensure quality and regularization techniques within the DNN to prevent overfitting.
This study focuses on static PDN, HDN, and UTN models. In the future work, we hope to explore the dynamic versions of these networks while considering the potential integration of physical or graph neural networks. For a dynamic version, we suggest incorporating a temporal discretization approach into the model to allow it to capture the dynamic evolution of network states. In addition, we suggest implementing a rolling-horizon optimization strategy coupled with event-triggered updates to ensure the computational efficiency and to redirect the focus on critical decision-making moments.
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