Continuous-time Optimization Problem Research Articles

Continuous-time optimization of integrated networks of electricity and district heating under wind power uncertainty

The integrated operation of the electricity and district heating systems (EDHS) attracted lots of attention in recent years due to considerable impacts on the power system’s flexibility. The time intervals and mathematical methods used in the optimization procedure are essential, especially when flexible operation in the presence of intermittent renewable resources is an objective because of the sub-hourly dynamics. Due to the intrinsic deficiencies of the traditional discrete-time hourly models in handling the sub-hourly variation of the load and renewable generation, in this paper, a new continuous-time optimization model is proposed to model the look-ahead operation of EDHS. The proposed continuous-time model is approximated by the linear spline-based trajectories and represented by the cubic splines of Bernstein function space to capture EDHS’s sub-hourly load and wind generation fluctuations. The EDHS of Barry Island is employed to investigate the proposed model and obtain results compared with the discrete-time procedure. Also, to measure the impact of uncertainties on both the continuous-time and discrete-time models, the information gap decision theory (IGDT) is utilized. The examination results illustrate that the proposed continuous-time model brings a saving of 0.91% in the costs when compared with the discrete-time model on a small test system. In addition, the results of the IGDT technique show more opportunities by wind increasing and fewer threats by wind reduction using the proposed continuous-time optimization problem compared to the discrete-time model.

Solving a class of nonlinear delay Fredholm integro-differential equations with convergence analysis

The main idea proposed in this article is an efficient shifted Legendre pseudospectral method for solving a class of nonlinear delay Fredholm integro-differential equations. In this method, first we transform the problem into an equivalent continuous-time optimization problem and then utilize a shifted pseudospectral method to discrete the problem. By this method, we obtained a nonlinear programming problem. Having solved the last problem, we can obtain an approximate solution for the original delay Fredholm integro-differential equation. Here, the convergence of the method is presented under some mild conditions. Illustrative examples are included to demonstrate the efficiency and applicability of the presented technique.

Distributed Continuous-Time Algorithms for Time-Varying Constrained Convex Optimization

This paper is devoted to the distributed continuous-time optimization problems with time-varying objective functions and time-varying constraints. Different from most studied distributed optimization problems with time-invariant objective functions and constraints, the optimal solutions in this paper are time varying and form a trajectory. First, for the case where there exist only time-varying nonlinear inequality constraints, we present a distributed control algorithm that consists of a sliding-mode consensus part and a Hessian-based optimization part coupled with the log-barrier penalty functions. The algorithm can guarantee the asymptotical tracking of the optimal solution with a zero tracking error. Second, we extend the previous result to the case where there exist not only time-varying nonlinear inequality constraints but also linear equality constraints. An extended algorithm is presented, where quadratic penalty functions are introduced to account for the equality constraints and an adaptive control gain is designed to remove the restriction on knowing the upper bounds on certain information. The asymptotical convergence of the extended algorithm to the vicinity of the optimal solution is studied under suitable assumptions. The effectiveness of the proposed algorithms is illustrated in simulation. In addition, one proposed algorithm is applied to a multi-robot multi-target navigation problem with experimental demonstration on a multi-crazyflie platform to validate the theoretical results.

Predictive maintenance scheduling for multiple power equipment based on data-driven fault prediction

In view of the maintenance of multiple power equipment operating in abnormal conditions in large-scale power plants, based on the prediction of fault state, a predictive maintenance scheduling method is proposed to schedule the maintenance activities. Firstly, based on the actual operating data, combined with the influencing factors of fault state deterioration by Pareto analysis, a time-variant function is improved to predict the deterioration state of potential fault in future maintenance interval. Then, maintenance priority is divided based on the fault state, considering the constraints of maintenance resources and equipment downtime, with the objective of minimizing the total maintenance cost, a scheduling model is built for the maintenance of multiple equipment. Finally, aiming at the continuity of maintenance time, a two-stage algorithm is proposed, which divides the maintenance time-window to transform the complex continuous time optimization problem into a combinatorial optimization problem of time periods, and then develops the optimal maintenance scheme. Taking the maintenance of multiple power transformers as an example, combined with the data resources provided by Yunnan power grid of China, the effectiveness of the improved prediction function of fault state is proved. In addition, by comparing with the traditional maintenance strategy based on the principle of first-fault-first-repair, the superiority of the proposed maintenance scheduling method is verified in reducing cost and improving system stability.

Real-time Bidding for Time Constrained Impression Contracts in First and Second Price Auctions - Theory and Algorithms

We study the optimal bids and allocations in a real-time auction for heterogeneous items subject to the requirement that specified collections of items of given types be acquired within given time constraints. The problem is cast as a continuous time optimization problem that can, under certain weak assumptions, be reduced to a convex optimization problem. Focusing on the standard first and second price auctions, we first show, using convex duality, that the optimal (infinite dimensional) bidding policy can be represented by a single finite vector of so-called ''pseudo-bids''. Using this result we are able to show that the optimal solution in the second price case turns out to be a very simple piecewise constant function of time. This contrasts with the first price case that is more complicated. Despite the fact that the optimal solution for the first price auction is genuinely dynamic, we show that there remains a close connection between the two cases and that, empirically, there is almost no difference between optimal behavior in either setting. This suggests that it is adequate to bid in a first price auction as if it were in fact second price. Finally, we detail methods for implementing our bidding policies in practice with further numerical simulations illustrating the performance.

Optimal investment strategies for pension funds with regulation-conform dynamic pension payment management in the absence of guarantees

In this article we consider the post-retirement phase optimization problem for a specific pension product in Germany that comes without guarantees. The continuous-time optimization problem is defined consisting of two specialties: first, we have a product-specific pension adjustment mechanism based on a certain capital coverage ratio which stipulates compulsory pension adjustments if the pension fund is underfunded or significantly overfunded. Second, due to the retiree’s fear of and aversion against pension reductions, we introduce a total wealth distribution to an investment portfolio and a buffer portfolio to lower the probability of future potential pension shortenings. The target functional in the optimization, that is to be maximized, is the client’s expected accumulated utility from the stochastic future pension cash flows. The optimization outcome is the optimal investment strategy in the proposed model. Due to the inherent complexity of the continuous-time framework, the discrete-time version of the optimization problem is considered and solved via the Bellman principle. In addition, for computational reasons, a policy function iteration algorithm is introduced to find a stationary solution to the problem in a computationally efficient and elegant fashion. A numerical case study on optimization and simulation completes the work with highlighting the benefits of the proposed model.

Trajectory Design via Convex Optimization for Six-Degree-of-Freedom Asteroid Powered Landing

This paper presents a convex programming algorithm based on modified Rodrigues parameters for a six-degree-of-freedom asteroid powered landing trajectory design problem. The trajectory design problem is formulated as a nonconvex fuel-optimal control problem with nonlinear coupled translational and rotational dynamics, nonconvex state and control constraints. In order to tackle the fuel-optimal control problem in the convexification framework, the original nonconvex continuous-time optimization problem is converted into a convex discrete-time optimization subproblem by linearizing and discretizing the dynamics and constraints. The effect of the coupling of translational and rotational dynamics on the convexification of nonconvex control constraints is discussed. The successive convexification method of solving a sequence of constrained convex subproblem is used to generate the optimal trajectory. The validity of the proposed algorithm for generating fuel-optimal trajectories and the effect of asteroid gravity on generated trajectories are examined through simulations of landing on different irregular asteroids. The Monte Carlo simulation is performed to examine the calculation performance of the proposed algorithm comparing to the Radau pseudospectral method algorithm and the proposed algorithm with the rotational motion parameterized by quaternions.

Distributed Time-Varying Optimization With State-Dependent Gains: Algorithms and Experiments

This brief addresses distributed continuous-time optimization problems with time-varying objective functions. The goal is for multiple agents to cooperatively minimize the sum of local time-varying objective functions with only local interaction and information while explicitly taking into account distributed adaptive gain design. Here, the optimal point is time varying and creates an optimal trajectory. First, for the unconstrained case, a distributed nonsmooth algorithm coupled with a state-dependent gain is proposed. It is shown that the interaction gain for each agent can be computed according to the variation of the Hessian and gradient information of the convex local objective functions so that the algorithm can solve the time-varying optimization problem without imposing a bound on any information about the local objective functions. Second, for the case where there exist common time-varying linear equality constraints, an extended algorithm is presented, where local Lagrangian functions are introduced to address the equality constraints. The asymptotic convergence of both algorithms to the optimal solution is proved. Numerical simulations are presented to illustrate the theoretical results. In addition, the one proposed algorithm is experimentally implemented and validated on a multi-Crazyflie platform.

A new numerical method to solve pantograph delay differential equations with convergence analysis

The main aim presented in this article is to provide an efficient transferred Legendre pseudospectral method for solving pantograph delay differential equations. At the first step, we transform the problem into a continuous-time optimization problem and then utilize a transferred Legendre pseudospectral method to discretize the problem. By solving this discrete problem, we can attain the pointwise and continuous estimated solutions for the major pantograph delay differential equation. The convergence of method has been considered. Also, numerical experiments are described to show the performance and precision of the presented technique. Moreover, the obtained results are compared with those from other techniques.

A New Convergent Pseudospectral Method for Delay Differential Equations

Delay differential equations have a wide range of applications in science and engineering. When these equations are nonlinear, we cannot usually obtain an exact solution. Hence, we must utilize an efficient numerical method with high convergence rate and low error to approximate the solution. In this paper, we propose a new shifted pseudospectral method to solve nonlinear delay differential equations. First, we convert the problem into an equivalent continuous-time optimization problem and then use a pseudospectral method to discretize the problem. By solving the obtained discrete-time problem, we achieve an approximate solution for the main delay differential equation. Here, we analyze the convergence of the method and solve some practical delay differential equations to show the efficiency of the method.

Distributed Continuous-Time Optimization of Second-Order Multiagent Systems With Nonconvex Input Constraints

This article discusses the distributed continuous-time optimization problem (DCTOP) of second-order multiagent systems (SOMASs). It is assumed that the inputs are required to be in some nonconvex sets, the team objective function (TOF) is a combination of general differentiable convex functions, and each agent can only obtain the information of one local objective function. Based on the neighbors’ information, a new distributed continuous-time optimization algorithm (DCTOA) is first proposed for each agent, where its gradient gains are nonuniform. By introducing a scaling factor and a model transformation, the corresponding system is changed into a time-varying nonlinear system which does not contain constraint operator in form. Then, it is proven that all agents’ states could reach an agreement and the TOF could be minimized by constructing some new Lyapunov functions. Finally, the effectiveness of the algorithm is shown by simulation results.

Optimal Investment Strategies for Pension Funds with Regulation-Conform Dynamic Pension Payment Management in the Absence of Guarantees

In this article we consider the post-retirement phase optimization problem for a specific pension product in Germany that comes without guarantees. The continuous-time optimization problem is defined consisting of two specialties: first, we have a product-specific pension adjustment mechanism based on a certain capital coverage ratio which stipulates compulsory pension adjustments if the pension fund is underfunded or significantly overfunded, and second, due to the retiree's fear of and aversion against pension reductions, we introduce a total wealth distribution to an investment portfolio and a buffer portfolio to lower the probability of future potential pension shortenings. Due to the inherent complexity of the continuous-time framework, the discrete-time version of the optimization problem is considered and solved via the Bellman principle. In addition, for computational reasons, a policy function iteration algorithm is introduced to find a stationary solution to the problem in a computationally efficient and elegant fashion. A numerical case study on optimization and simulation completes the work with highlighting the benefits of the proposed model.

Handling power losses in a DC microgrid through constrained optimization

Abstract This paper extends a hierarchical control approach for power balancing in a meshed DC microgrid while minimizing the power losses in the central transmission network. The control strategy is divided into three layers in hierarchical framework: i) the high level solves a continuous-time optimization problem which minimizes the DC-bus power loss and the electricity cost from the external grid power purchase through the combined use of differential flatness with B-splines parametrization; ii) the middle level employs a tracking Model Predictive Control (MPC) method which mitigates the discrepancies among the optimal and the actual profiles; iii) the low level controller handles the switching activity of the converters. The proposed approach is validated in simulation for a specific meshed DC microgrid system.

A Full Rank Condition for Continuous-Time Optimization Problems with Equality and Inequality Constraints

First and second order necessary optimality conditions of Karush-Kuhn-Tucker type are established for continuous-time optimization problems with equality and inequality constraints. A full rank type regularity condition along with an uniform implicit function theorem are used in order to achieve such necessary conditions.

AN OPTIMAL CONSUMPTION AND INVESTMENT PROBLEM WITH QUADRATIC UTILITY AND SUBSISTENCE CONSUMPTION CONSTRAINTS: A DYNAMIC PROGRAMMING APPROACH

In this paper, we analyze the optimal consumption and investment problem of an agent who has a quadratic-type utility function and faces a subsistence consumption constraint. We use the dynamic programming method to solve the optimization problem in continuous-time. We further provide the sufficient conditions for the optimization problem to be well-defined.

Distributed Continuous-Time Optimization With Scalable Adaptive Event-Based Mechanisms

This paper investigates the distributed continuous-time optimization problem, which consists of a group of agents with variant local cost functions. An adaptive consensus-based algorithm with event triggering communications is introduced, which can drive the participating agents to minimize the global cost function and exclude the Zeno behavior. Compared to the existing results, the proposed event-based algorithm is independent of the parameters of the cost functions, using only the relative information of neighboring agents, and hence is fully distributed. Furthermore, the constraints of the convexity of the cost functions are relaxed.

Cost-Aware Renewable Energy Management With Application in Cellular Networks

In this paper, we propose a cost minimization strategy for facilities with shared access to a renewable energy farm. We assume space-time varying electricity prices, and model the facilities as non-deferrable time-varying loads. A practical application of such an arrangement can be found in cellular networks, where operators can deploy a centralized energy farm to (partially) power their base stations in a given geographical area. Unlike existing works, we take into account the non-linear behavior of the battery at the energy farm, and obtain closed-form results by solving the corresponding optimization problem in continuous time . We also derive a closed-form expression which estimates the maximum cost saving in terms of the total energy harvested across the entire planning horizon. The results presented in this paper can be used to devise and benchmark low-complexity online strategies, and to understand the relationship between the maximum cost saving, the optimal consumption rate of renewable energy, and system parameters such as battery efficiency and nominal output power.

A numerical method for solving shortest path problems

Chebyshev pseudo-spectral method is one of the most efficient methods for solving continuous-time optimization problems. In this paper, we utilize this method to solve the general form of shortest path problem. Here, the main problem is converted into a nonlinear programming problem and by solving of which, we obtain an approximate shortest path. The feasibility of the nonlinear programming problem and the convergence of the method are given. Finally, some numerical examples are considered to show the efficiency of the presented method over the other methods.

Distributed Continuous-Time Optimization: Nonuniform Gradient Gains, Finite-Time Convergence, and Convex Constraint Set

In this paper, a distributed optimization problem with general differentiable convex objective functions is studied for continuous-time multi-agent systems with single-integrator dynamics. The objective is for multiple agents to cooperatively optimize a team objective function formed by a sum of local objective functions with only local interaction and information while explicitly taking into account nonuniform gradient gains, finite-time convergence, and a common convex constraint set. First, a distributed nonsmooth algorithm is introduced for a special class of convex objective functions that have a quadratic-like form. It is shown that all agents reach a consensus in finite time while minimizing the team objective function asymptotically. Second, a distributed algorithm is presented for general differentiable convex objective functions, in which the interaction gains of each agent can be self-adjusted based on local states. A corresponding condition is then given to guarantee that all agents reach a consensus in finite time while minimizing the team objective function asymptotically. Third, a distributed optimization algorithm with state-dependent gradient gains is given for general differentiable convex objective functions. It is shown that the distributed continuous-time optimization problem can be solved even though the gradient gains are not identical. Fourth, a distributed tracking algorithm combined with a distributed estimation algorithm is given for general differentiable convex objective functions. It is shown that all agents reach a consensus while minimizing the team objective function in finite time. Fifth, as an extension of the previous results, a distributed constrained optimization algorithm with nonuniform gradient gains and a distributed constrained finite-time optimization algorithm are given. It is shown that both algorithms can be used to solve a distributed continuous-time optimization problem with a common convex constraint set. Numerical examples are included to illustrate the obtained theoretical results.

Stochastic Gradient Descent in Continuous Time

Stochastic gradient descent in continuous time (SGDCT) provides a computationally efficient method for the statistical learning of continuous-time models, which are widely used in science, engineering, and finance. The SGDCT algorithm follows a (noisy) descent direction along a continuous stream of data. SGDCT performs an online parameter update in continuous time, with the parameter updates θt satisfying a stochastic differential equation. We prove that limt→∞ ∇g(θt) = 0 where g is a natural objective function for the estimation of the continuous-time dynamics. The convergence proof leverages ergodicity by using an appropriate Poisson equation to help describe the evolution of the parameters for large times. SGDCT can also be used to solve continuous-time optimization problems, such as American options. For certain continuous-time problems, SGDCT has some promising advantages compared to a traditional stochastic gradient descent algorithm. As an example application, SGDCT is combined with a deep neural network to price high-dimensional American options (up to 100 dimensions).