Nonlinear State Constraints Research Articles

Comparative study of kernel- and deep learning-based proxy models for nonlinearly constrained life-cycle production optimization

This study presents a comparative study for the application of deep learning–based and kernel-based proxy models in hydrocarbon production optimization with nonlinear constraints, comparing their performance against high-fidelity simulators (HFS) in terms of computational efficiency and the quality of the results achieved. One of the proxy models employed is referred to as embed to control and observe (E2CO), a deep learning–based model. The other model is a kernel-based proxy model known as least-squares support-vector regression (LS-SVR). Both proxy models can predict production/injection outputs for each well. We use the sequential quadratic programming (SQP) method for nonlinearly constrained production optimization. Our optimization objective revolves around the net present value (NPV), while the nonlinear state constraints involve field liquid production rate (FLPR) and field water production rate (FWPR). NPV, FLPR, and FWPR are estimated using two different types of proxy models. The gradient of the objective function and the Jacobian matrix of constraints are computed analytically for LS-SVR. In contrast, the method of stochastic simplex approximated gradient (StoSAG) is employed for optimization with E2CO and HFS. The reservoir model considered in this study is a two-phase, three-dimensional reservoir characterized by channelized and heterogeneous permeability, sourced from the SPE10 benchmark case. We optimize well controls to maximize NPV in an oil-water waterflooding scenario. All proxy models can achieve optimal NPV results like those obtained through HFS but with significantly lower computational resources. Both proxy models are found to be approximately 15-fold more efficient than using HFS when the number of simulation runs is compared. Forward prediction results show that both proxy models outperform numerical simulator, and the E2CO can predict up to 2400 times faster than HFS whereas speedups gained over HFS go up to 5000-fold with LS-SVR. The training time required for E2CO is at least an order of magnitude longer than that for LS-SVR. The main contributions of the paper are (i) development of a novel methodology that uses an E2CO proxy model within a gradient-based iterative retraining proxy approach for life-cycle production optimization with nonlinear state constraints, (ii) comparison of this methodology in terms of computational cost and accuracy with an LS-SVR proxy model, and (iii) new insights into the accuracy and prediction performances of these machine learning-based proxy models for 3D oil-water systems as well as their efficiency in nonlinearly constrained production optimization for waterflooding applications.

Robust reference governor for input-constrained model predictive control to enforce state constraints at low computational cost

In the setting of discrete-time linear systems with unmeasured set-bounded disturbances, this paper proposes a control scheme that combines a disturbance free input-constrained Model Predictive Control (uMPC) policy and a reference governor (RG) which modifies the reference command to enforce constraints that are not enforced by the uMPC policy. Such a scheme, referred to as the robust RGMPC, can handle (possibly nonlinear) state and input constraints and only requires solving an optimisation problem for MPC with polytopic input constraints, for which fast algorithms exist. Conditions are given which ensure recursive feasibility of the robust RGMPC scheme, finite-time convergence of the modified reference command to the desired reference command and asymptotic convergence of the state to an invariant neighbourhood of the associated set-point. Simulation results for a constrained spacecraft rendezvous manoeuvre demonstrate that the rRGMPC scheme has lower average computational time than several robust state and input-constrained MPC controllers.

Life-cycle production optimization with nonlinear constraints using a least-squares support-vector regression proxy

When nonlinear constraints such as field liquid or water production rate, injection pressures, etc., as functions of time need to be honored in addition to linear ones, the life-cycle production optimization problem, a component of a closed loop reservoir management, becomes challenging and computationally expensive to perform using a high-fidelity reservoir simulator with the existing gradient-based methods using the adjoint or stochastic approximate gradient methods. Therefore, the objective of this study is to present computationally efficient methods for deterministic production optimization under nonlinear constraints using a kernel-based machine learning method where the cost function is the net present value (NPV). We use the least-squares support-vector regression (LSSVR) to approximate the NPV function. To achieve computational efficiency, we generate a set of output values of the NPV and nonlinear constraint functions, which are field liquid production rate (FLPR) and water production rate (FWPR) in this study, by running the high-fidelity simulator for a broad set of input design variables (well controls) and then using the collection of input/output data to train LSSVR proxy models to replace the high-fidelity simulator to compute NPV and nonlinear state constraint functions during iterations of sequential quadratic programming (SQP). To obtain improved (higher) estimated optimal NPV values, we use the existing so-called iterative sampling refinement (ISR) method to update the LSSVR proxy so that the updated proxy remains predictive toward promising regions of search space during the optimization. Direct and indirect ways of constructing LSSVR-based NPVs as well as different combinations of input data, including nonlinear state constraints and/or the bottomhole pressures (BHPs) and water injection rates, are tested as alternative feature spaces. The results obtained from our proposed LSSVR-based optimization methods are compared with those obtained from our in-house stochastic simplex approximate gradient (StoSAG)-based line-search SQP programming (LS-SQP-StoSAG) algorithm that uses directly a high-fidelity simulator to compute the gradients of the objective function and the nonlinear state functions with StoSAG for the Brugge reservoir model. The results show that nonlinear constrained optimization with the LSSVR ISR with SQP is computationally 3.25 fold more efficient than LS-SQP-StoSAG. In addition, the results show that constructing NPV indirectly using the field liquid and water rates for a waterflooding problem where inputs come from LSSVR proxies of the nonlinear state constraints requires significantly fewer training samples than the method constructing NPV directly from the NPVs computed from a high-fidelity simulator. LSSVR has advantages for its computational efficiency which is the main goal in our research, and robustness against overfitting especially for the cases where we have limited data. However, DNNs and random forest require large training size and require much more computational resources and longer training times. Also, Random Forests and Gradient Boosting Machines can be prone to overfitting and become computationally intensive.

Bi-Objective Optimization of Subsurface CO2 Storage with Nonlinear Constraints Using Sequential Quadratic Programming with Stochastic Gradients

SummaryThis study focuses on carbon capture, utilization, and sequestration (CCUS) via the means of nonlinearly constrained production optimization workflow for a CO2-enhanced oil recovery (EOR) process, in which both the net present value (NPV) and the net present carbon tax credits (NPCTC) are bi-objectively maximized, with the emphasis on the consideration of injection bottomhole pressure (IBHP) constraints on the injectors, in addition to field liquid production rate (FLPR) and field water production rate (FWPR), to ensure the integrity of the formation and to prevent any potential damage during the life cycle injection/production process. The main optimization framework used in this work is a lexicographic method based on the line-search sequential quadratic programming (LS-SQP) coupled with stochastic simplex approximate gradients (StoSAG). We demonstrate the performance of the optimization algorithm and results in a field-scale realistic problem, simulated using a commercial compositional reservoir simulator. Results show that the workflow can solve the single-objective and bi-objective optimization problems computationally efficiently and effectively, especially in handling and honoring nonlinear state constraints imposed onto the problem. Various numerical settings have been experimented with to estimate the Pareto front for the bi-objective optimization problem, showing the trade-off between the two objectives of NPV and NPCTC. We also perform a single-objective optimization on the total life cycle cash flow, which is the aggregated quantity of NPV and NPCTC, and quantify the results to further emphasize the necessity of performing bi-objective production optimization, especially when used in conjunction with commercial flow simulators that lack the capability of computing adjoint-based gradients.

Observer-Based Adaptive Optimized Control for Stochastic Nonlinear Systems With Input and State Constraints.

In this work, an adaptive neural network (NN) optimized output-feedback control problem is studied for a class of stochastic nonlinear systems with unknown nonlinear dynamics, input saturation, and state constraints. A nonlinear state observer is designed to estimate the unmeasured states, and the NNs are used to approximate the unknown nonlinear functions. Under the framework of the backstepping technique, the virtual and actual optimal controllers are developed by employing the actor-critic architecture. Meanwhile, the tan-type Barrier optimal performance index functions are developed to prevent the nonlinear systems from the state constraints, and all the states are confined within the preselected compact sets all the time. It is worth mentioning that the proposed optimized control is clearly simple since the reinforcement learning (RL) algorithm is derived based on the negative gradient of a simple positive function. Furthermore, the proposed optimal control strategy ensures that all the signals in the closed-loop system are bounded. Finally, a practical simulation example is carried out to further illustrate the effectiveness of the proposed optimal control method.

Adaptive finite‐time control of multi‐agent systems with partial state constraints and input saturation via event‐triggered strategy

AbstractThis paper focuses on the finite‐time control problem of multi‐agent systems with input saturation, unknown nonlinear dynamics, external disturbances and partial state constraints via output feedback. Fuzzy logic system and fuzzy state observer are introduced to approximate the uncertain nonlinearities and estimate the unmeasurable states, respectively. The partial state constraints are dealt with by using the barrier Lyapunov function, so that all states of the system do not exceed the preset boundary values. In order to reduce the computational complexity of the virtual controller and save communication resources, a first‐order filter and an event‐triggered mechanism are introduced, respectively. It is proved that the Zeno behavior does not occur via the proposed event‐triggered controller. By stability analysis, the finite‐time convergence of tracking error to a small neighborhood of the origin is proven. The effectiveness of the theoretical results is verified by examples.

A phase field method based on multi-level correction for eigenvalue topology optimization

In topology optimization, finite element methods are usually required to solve state equations repeatedly. Considerable computational costs are required for iterative solvers to nonlinear state constraints, such as eigenvalue problems, especially in 3D with large-scale degrees of freedom. Recently, multi-level correction methods are developed to improve efficiency in solving nonlinear partial differential equations including eigenvalue problems. These methods involve a multi-step correction in a sequence of finite element spaces and can keep accuracy of the eigenpair approximation. The overall computational cost is nearly equivalent to that of solving linear source problems. In this paper, we propose a new phase field method based on multi-level correction for eigenfrequency topology optimization. Both eigenfrequency optimization problems associated with a scalar type elliptic operator and a vectorial type elastic operator are considered. The Allen–Cahn phase field equation discretized by a semi-implicit scheme and finite-elements can have topological and shape changes in arbitrary shaped design regions. The proposed optimization algorithm based on a multi-level correction solver for eigenvalue problems improves efficiency of the traditional phase field method. A variety of numerical examples in 2D and 3D are presented to illustrate the effectiveness and efficiency of the algorithm.

Discrete-Time Differential Dynamic Programming on SO(3) With Pose Constraints

The motion of many robotics systems, such as the rotation of unmanned aerial vehicles, can be modeled by SO(3). However, the difficulties in the parameterization of the SO(3) makes it hard to implement the state-of-the-art collision avoidance algorithm. In this paper, we present a new method for control on SO(3) via the combination of the nonlinear state constraints and geometric control technique. We first define the nonlinear constraints for trajectory optimization on SO(3). Then we solve the trajectory optimization problem by constrained Differential Dynamic Programming (DDP) in a Riemannian geometry framework. The first and second-order expansion of the cost function and dynamics are derived in a coordinate-free manner. The safety condition represented by nonlinear constraints is incorporated into the DDP via an active set method, where we find the active set in the backward path to obtain the optimal control policy. We validated our methods on motion planning of rigid model on SO(3) manifold where pose constraints are imposed. Our methods outperform the baseline methods in terms of convergence speed and numerical robustness to disturbance. The numerical robustness is essential when the system is initialized far from the local optimum.

ALGAMES: a fast augmented Lagrangian solver for constrained dynamic games

Dynamic games are an effective paradigm for dealing with the control of multiple interacting actors. This paper introduces augmented Lagrangian GAME-theoretic solver (ALGAMES), a solver that handles trajectory-optimization problems with multiple actors and general nonlinear state and input constraints. Its novelty resides in satisfying the first-order optimality conditions with a quasi-Newton root-finding algorithm and rigorously enforcing constraints using an augmented Lagrangian method. We evaluate our solver in the context of autonomous driving on scenarios with a strong level of interactions between the vehicles. We assess the robustness of the solver using Monte Carlo simulations. It is able to reliably solve complex problems like ramp merging with three vehicles three times faster than a state-of-the-art DDP-based approach. A model-predictive control (MPC) implementation of the algorithm, running at more than 60 Hz, demonstrates ALGAMES’ ability to mitigate the “frozen robot” problem on complex autonomous driving scenarios like merging onto a crowded highway.

Adaptive distributed tracking control for non-affine multi-agent systems with state constraints and dead-zone input

In this paper, an adaptive distributed control protocol is proposed for non-affine multi-agent system with nonlinear dead-zone input and state constraints under the condition of directed topology. In order to overcome the difficulties caused by non-affine terms in the system, the nonlinear dynamics system is transformed. Then, the neural network technology is introduced to approximate the unknown non-affine terms for the obtained system. State constraints and dead-zone input are common system problems. In order to solve these problems, the barrier Lyapunov function is introduced in this paper. According to the barrier Lyapunov function and backstepping method, an adaptive distributed controller is designed, so that state variables do not violate constraint bounds and the system is not affected by dead-zone input. By Lyapunov stability theory, it is proved that the signals of each follower are cooperative semi-global uniform ultimate boundedness (CSUUB), and the outputs of the followers track the output of the leader. Simulation example is given to demonstrate the effectiveness of the proposed method.

Neuro‐adaptive output‐feedback optimized stochastic control for the active suspension systems with state constraints

SummaryThis article investigates an adaptive neural network (NN) output‐feedback optimal control design problem for active suspension systems (ASSs) with stochastic disturbance. The ASSs under consideration contain the characteristics of spring nonlinear dynamics, unmeasured states, and state constraints. The NNs are developed to approximate the unknown nonlinear functions. Meanwhile, observer‐based output feedback control design method is proposed based on the adaptive backstepping technique. Furthermore, the stability of the closed‐loop system is demonstrated by constructing the barrier Lyapunov function, thus ensuring that the full‐state constraints are not exceeded. In particular, the simulation validations are given for the cases of bump, C‐class, and D‐class road displacements inputs. Finally, the simulation results verify the effectiveness of the studied control strategy.

Robust Multiobjective Nonlinear Constrained Optimization with Ensemble Stochastic Gradient Sequential Quadratic Programming-Filter Algorithm

Summary As the crucial step in closed-loop reservoir management, robust life-cycle production optimization is defined as maximizing/minimizing the expected value of a predefined objective (cost) function over geological uncertainties (i.e., uncertainties in the reservoir permeability, porosity, endpoint relative permeability, etc.). However, with robust optimization, there is no control over downside risk defined as the minimum net present value (NPV) among the individual NPVs of the different reservoir models. Yet, field operators generally wish to keep this minimum NPV reasonably large to try to ensure that the reservoir is commercially viable. In addition, the field operator may desire to maximize the NPV of production over a much shorter time period than the life of the reservoir under the limitation of surface facilities (e.g., field liquid and water production rates). Thus, it is important to consider multiobjective robust production optimization with nonlinear constraints and when geological uncertainties are incorporated. The three objectives considered in this paper are; to maximize the average life-cycle NPV, to maximize the average short-term NPV, and to maximize the minimum NPV of the set of realizations. Generally, these objectives are in conflict; for example, the well controls that give a global maximum for robust life-cycle production optimization do not usually correspond to the controls that maximize the short-term average NPV of production. Moreover, handling the nonlinear state constraints (e.g., field liquid production rates and field water production rates for the bottom-hole pressure controlled producers in the robust production optimization) is also a challenge because those nonlinear constraints should be satisfied at each control steps for each geological realization. To provide potential solutions to the multiobjective robust optimization problem with state constraints, we developed a modified lexicographic method with a minimizing-maximum scheme to attempt to obtain a set of Pareto optimal solutions and to satisfy all nonlinear constraints. We apply the sequential quadratic programming filter with modified stochastic gradients to solve a sequence of optimization problems, where each solution is designed to generate a single point on the Pareto front. In the modified lexicographic method, the objective is always considered to be the primary objective, and the other objectives are considered by specifying bounds on them to convert them to state constraints. The temporal damping and truncation schemes are applied to improve the quality of the stochastic gradient on nonlinear constraints, and the minimizing–maximum procedure is applied to enforce constraints on the normal state constraints. The main advantage that the modified lexicographic method has over the standard lexicographic method is that it allows for the generation of potential Pareto optimal points, which are uniformly spaced in the values of the second and/or third objective that one wishes to improve by multiobjective optimization.

Gradient-Enhanced Support Vector Regression for Robust Life-Cycle Production Optimization with Nonlinear-State Constraints

Summary In the context of production optimization, we consider the general problem of finding the well controls that maximize the net present value (NPV) of life-cycle production, where the well controls are either the bottomhole pressure (BHP) or a rate (oil, gas, water, or total liquid) at each well on a set of specified control steps (time intervals), with the limitations on surface facility considered as nonlinear-state constraints [e.g., field-liquid-production rates (FLRs), field-water-production rates (FWRs), and/or field-gas-production rates]. If the reservoir simulation used for reservoir management has sufficient adjoint capability to compute gradients of the objective function and all state constraints, we show that one can develop a significantly more computationally efficient procedure by replacing the adjoint-enhanced reservoir simulator by a proxy model and optimizing the proxy. Our methodology achieves computational efficiency by generating a set of output values of the cost and constraint functions and their associated derivative values by running the reservoir simulator for a broad set of input design variables (well controls) and then using the set of input/output data to train a proxy model to replace the reservoir simulator when computing values of cost and constraint functions and their derivatives during iterations of sequential quadratic programming (SQP). The derivation of the equations for computing the proxy-based model that uses both function and gradient information is similar to that of least-squares support vector regression (LS-SVR). However, this method is referred to as gradient-enhanced support vector regression (GE-SVR) because, unlike LS-SVR, the method uses derivative information, not just function values, to train the proxy. Similar to LS-SVR, improved (higher) estimated optimal NPV values can be obtained by using iterative resampling (IR). With IR, after each proxy-based optimization, one evaluates the cost and constraint functions and their derivatives at the estimated optimal controls using reservoir-simulator output, and then adds this new input/output information to the training set to update the proxy models for predicting NPV and constraints. Using the updated proxies, one applies SQP optimization again. IR continues until the simulator and proxy evaluated at the latest estimate of the optimal well controls give the same value of NPV within a specified percentage tolerance and the constraints evaluated by reservoir simulator at the latest optimal well controls are such that the constraints are satisfied within some small specified tolerance. Our results indicate that proxy-based optimization with iterative resampling might require up to an order of magnitude less computational time than pure reservoir-simulator-based optimization. By comparing the results generated with an LS-SVR proxy with the GE-SVR results, we find that GE-SVR is roughly an order of magnitude more computationally efficient than LS-SVR but also provides a better approximation of a complex cost-function surface so that it is possible to locate multiple optima in cases where LS-SVR fails to identify the multiple optima.

Interior Point Differential Dynamic Programming

This brief introduces a novel differential dynamic programming (DDP) algorithm for solving discrete-time finite-horizon optimal control problems with inequality constraints. Two variants, namely feasible- and infeasible-IPDDP algorithms, are developed using a primal–dual interior-point methodology, and their local quadratic convergence properties are characterized. We show that the stationary points of the algorithms are the perturbed KKT points, and thus can be moved arbitrarily close to a locally optimal solution. Being free from the burden of the active-set methods, it can handle nonlinear state and input inequality constraints without a discernible increase in its computational complexity relative to the unconstrained case. The performance of the proposed algorithms is demonstrated using numerical experiments on three different problems: control-limited inverted pendulum, car-parking, and unicycle motion control and obstacle avoidance.

A constrained extended Kalman filter based on LS-VCE formulated by condition equations with prediction of cross-covariances

A constrained extended Kalman filter (CEKF) based on least-squares variance component estimation (LS-VCE) is generally developed by condition equations since the proper prediction of dispersion matrices is one of the main bottlenecks in the KF algorithms. Here we investigate four problems which have not been simultaneously considered yet. These problems are examination of non-linearty of dynamic model, VCE, general non-linear state constraints and fairly general stochastic model. Although a few contributions proposed some adaptive KF in particular based on Helmert’s VCE method, they developed their filters for special problems with some restrictive conditions such as independence of all variables and/or linearity of the dynamic model. Also some of these filters did not apply VCE methods to all parts of the dynamic model. In this contribution, we try to overcome all of these restrictions. Moreover, LS-VCE method gives some added advantages over other VCE methods. First the new formulation of CEKF is developed by condition equations with prediction of all possible cross-covariances as algorithm 1. Then the LS-VCE method is applied to it after some modifications which results in an adaptive constrained extended Kalman filter (ACEKF) as the second algorithm.

Surrogate-assisted differential evolution for production optimization with nonlinear state constraints

In recent years, evolutionary computation (EC) has gained increasing attention in the field of production optimization due to its powerful global search ability and derivative-free characteristic. However, state constraints are much more challenging to handle in comparison with the explicit constraints. To alleviate this difficulty, this paper presents a framework called surrogate-assisted differential evolution with an effective constraint-handling technique—the feasibility rule with the incorporation of objective function information (SADE-FROFI). The novel constraint-handling technique used in this work achieves an effective balance between constraints and objective function by revising the well-known feasibility rule (FR), which can effectively maintain the diversity of population and help the population jump out of local optimal solution. Moreover, to address the computationally time-consuming issue of numerical simulator, a multi-surrogate strategy is introduced specially for state constraints. Two benchmark functions are tested to verify the effectiveness of new constraint-handling technique (FROFI). The efficacy of the proposed method is validated on two synthetic reservoir models, named three-channel model and PUNQ-S3 model, respectively. The experimental results demonstrated that the proposed method can find better solutions on the basis of satisfying all constraints in comparison with several existing methods.

A Sequential-Quadratic-Programming-Filter Algorithm with a Modified Stochastic Gradient for Robust Life-Cycle Optimization Problems with Nonlinear State Constraints

SummarySolving a large-scale optimization problem with nonlinear state constraints is challenging when adjoint gradients are not available for computing the derivatives needed in the basic optimization algorithm used. Here, we present a methodology for the solution of an optimization problem with nonlinear and linear constraints, where the true gradients that cannot be computed analytically are approximated by ensemble-based stochastic gradients using an improved stochastic simplex approximate gradient (StoSAG). Our discussion is focused on the application of our procedure to waterflooding optimization where the optimization variables are the well controls and the cost function is the life-cycle net present value (NPV) of production. The optimization algorithm used for solving the constrained-optimization problem is sequential quadratic programming (SQP) with constraints enforced using the filter method. We introduce modifications to StoSAG that improve its fidelity [i.e., the improvements give a more accurate approximation to the true gradient (assumed here to equal the gradient computed with the adjoint method) than the approximation obtained using the original StoSAG algorithm]. The modifications to StoSAG vastly improve the performance of the optimization algorithm; in fact, we show that if the basic StoSAG is applied without the improvements, then the SQP might yield a highly suboptimal result for optimization problems with nonlinear state constraints. For robust optimization, each constraint should be satisfied for every reservoir model, which is highly computationally intensive. However, the computationally viable alternative of letting the reservoir simulation enforce the nonlinear state constraints using its internal heuristics yields significantly inferior results. Thus, we develop an alternative procedure for handling nonlinear state constraints, which avoids explicit enforcement of nonlinear constraints for each reservoir model yet yields results where any constraint violation for any model is extremely small.

Exploiting nonlinear invariants and path constraints to achieve tighter reachable set enclosures using differential inequalities

This article presents a new method for computing sharp bounds on the solutions of nonlinear dynamic systems subject to uncertain initial conditions, parameters, and time-varying inputs. Such bounds are widely used in algorithms for uncertainty propagation, robust state estimation, system verification, global dynamic optimization, and more. Recently, it has been shown that bounds computed via differential inequalities can often be made much less conservative by exploiting state constraints that are known to hold for all trajectories of interest (e.g., path constraints that describe feasible trajectories in the context of dynamic optimization, or constraints that explicitly describe invariant sets containing all system trajectories). However, effective bounding algorithms of this type are currently only available for problems with linear constraints. Moreover, the theoretical results underlying these algorithms do not apply to constraints that depend on time-varying inputs and rely on assumptions that prove to be very restrictive for nonlinear constraints. This article contributes a new differential inequalities theorem that permits the use of a very general class of nonlinear state constraints. Moreover, a new algorithm is presented for efficiently exploiting nonlinear constraints to achieve tighter bounds. The proposed approach is shown to produce very sharp bounds for two challenging case studies.

Model Predictive Coverage Control

Cooperative robotic problems often require coordination in space in order to complete a given task, important examples include search and rescue, operations in hazardous environments, and autonomous taxi deployment. Events can be quickly detected by partitioning the working environment and assigning one robot to each partition. However, a crucial factor that limits the effectiveness and usage of coverage algorithms is related to the ability of taking decisions in the presence of constraints. In this paper, we propose a coverage control algorithm that is capable of handling nonlinear dynamics, and state and input constraints. The proposed algorithm is based on a nonlinear tracking model predictive controller and is proven to converge to a centroidal Voronoi configuration. We also introduce a procedure to design the terminal ingredients of the model predictive controller. The effectiveness of the algorithm is then highlighted with a numerical simulation.

A dynamic user equilibrium model for multi-region macroscopic fundamental diagram systems with time-varying delays

Macroscopic fundamental diagram (MFD) has been widely used for aggregate modeling of urban traffic network dynamics to tackle the dimensionality problem of microscopic approaches. This paper contributes to the state-of-the-art by proposing a dynamic user equilibrium (DUE) model that enables simultaneous route choice and departure time choice under the MFD framework for various applications such as park-and-ride, vehicle dispatching and relocation. To better capture the traffic flow propagation and to adapt to the fast time-varying demand, the state-dependent travel time function is integrated into the MFD dynamics as an endogenous time-varying delay. The multi-region MFD dynamics with saturated state and inflow constraints is then used as the network loading model to formulate the DUE model through the lens of the differential variational inequality. Necessary conditions for the DUE are analytically derived using the Pontryagin minimum principle. Difficulties raised in handling the dynamic state-dependent nonlinear travel time functions, state and inflow constraints are addressed without model linearization nor enforcing constant delay assumption as conventionally done in the literature. The additional cost induced by inflow capacity and accumulation constraints can capture the hypercongestion represented by the downward sloping part of the MFD without actually activating traffic congestion. Numerical examples solved by using time-discretization solution algorithm illustrate the DUE characteristics and the corresponding dynamic external costs induced by constraints.