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Economic Predictive Control with Periodic Horizon for Water Distribution Networks

Authors

Mirhan Ürkmez, Carsten Skovmose Kallesøe, Jan Dimon Bendtsen, John Leth
Type: Preprint
Publication Date: 2023-01-01
Citations: 0
DOI: https://doi.org/10.48550/arxiv.2301.13598

Abstract

This paper deals with the control of pumps in large-scale water distribution networks with the aim of minimizing economic costs while satisfying operational constraints. Finding a control algorithm in combination with a model that can be applied in real-time is a challenging problem due to the nonlinearities presented by the pipes and the network sizes. We propose a predictive control algorithm with a periodic horizon. The method provides a way for the economic operation of large water networks with a small linear model. Economic Predictive control with a periodic horizon and a terminal state constraint is constructed to keep the state trajectories close to an optimal periodic trajectory. Barrier terms are also included in the cost function to prevent constraint violations. The proposed method is tested on the EPANET implementation of the water network of a medium size Danish town (Randers) and shown to perform as intended under varying conditions.

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Economic Predictive Control with Periodic Horizon for Water Distribution Networks

2023-01-01
Mirhan Ürkmez , Carsten Skovmose Kallesøe , Jan Dimon Bendtsen +1 more
This paper deals with control of pumps in large-scale water distribution networks with the aim of minimizing economic costs while satisfying operational constraints. Finding a control algorithm in combination with … This paper deals with control of pumps in large-scale water distribution networks with the aim of minimizing economic costs while satisfying operational constraints. Finding a control algorithm in combination with a model that can be applied in real-time is a challenging problem due to the nonlinearities presented by the pipes and the network sizes. We propose a predictive control algorithm with a periodic horizon. The method provides a way for the economic operation of large water networks with a low-complexity linear model. Economic Predictive control with a periodic horizon and a terminal state constraint is constructed to keep the state trajectories close to an optimal periodic trajectory. Barrier terms are also included in the cost function to prevent constraint violations. The proposed method is tested on a state-of-the-art EPANET model of the water network of a medium size Danish town (Randers) and shown to perform as intended under varying conditions.
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Minimizing Pumping Energy Cost in Real-Time Operations of Water Distribution Systems Using Economic Model Predictive Control

2021-05-13
Ye Wang , Kevin Too Yok , Wenyan Wu +3 more
Optimizing pump operations is a challenging task for real-time management of water distribution systems (WDS). With suitable pump scheduling, pumping costs can be significantly reduced. In this research, a novel … Optimizing pump operations is a challenging task for real-time management of water distribution systems (WDS). With suitable pump scheduling, pumping costs can be significantly reduced. In this research, a novel economic model predictive control (EMPC) framework for real-time management of WDS is proposed. Optimal pump operations are selected based on predicted system behavior over a receding time horizon with the aim to minimize the total pumping energy cost. Time-varying electricity tariffs are considered while all the required water demands are satisfied. The novelty of this framework is to choose the number of pumps to operate in each pump station as decision variables in order to optimize the total pumping energy costs. By using integer programming, the proposed EMPC is applied to a benchmark case study, the Richmond Pruned network. The simulation with an EPANET hydraulic simulator is implemented. Moreover, a comparison of the results obtained using the proposed EMPC with those obtained using trigger-level control demonstrates significant economic benefits of the proposed EMPC.
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Economic Model Predictive Control of Water Distribution Systems with Accelerated Optimization Algorithm

2024-06-10
Saskia A. Putri , Faegheh Moazeni , Javad Khazaei
Model predictive control (MPC) has emerged as an effective strategy for water distribution systems (WDSs) management. However, it is hampered by the computational burden for large-scale WDSs due to the … Model predictive control (MPC) has emerged as an effective strategy for water distribution systems (WDSs) management. However, it is hampered by the computational burden for large-scale WDSs due to the combinatorial growth of possible control actions that must be evaluated at each time step. Therefore, a fast computation algorithm to implement MPC in WDSs can be obtained using a move-blocking approach that simplifies control decisions while ensuring solution feasibility. This paper introduces a least-restrictive move-blocking that interpolates the blocked control rate of change, aiming at balancing computational efficiency with operational effectiveness. The proposed control strategy is demonstrated on aggregated WDSs, encompassing multiple hydraulic elements. This implementation is incorporated into a multi-objective optimization framework that concurrently optimizes water level security of the storage tanks, smoothness of the control actions, and cost-effective objectives. A fair comparison between the proposed approach with the non-blocking Economic MPC is provided.
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Economic Model Predictive Control for Periodic Operation: A Quadratic Programming Approach

2025-01-14
J.A. Borja-Conde , Juan Moreno Nadales , Filiberto Fele +1 more
Periodic dynamical systems, distinguished by their repetitive behavior over time, are prevalent across various engineering disciplines. In numerous applications, particularly within industrial contexts, the implementation of model predictive control (MPC) … Periodic dynamical systems, distinguished by their repetitive behavior over time, are prevalent across various engineering disciplines. In numerous applications, particularly within industrial contexts, the implementation of model predictive control (MPC) schemes tailored to optimize specific economic criteria was shown to offer substantial advantages. However, the real-time implementation of these schemes is often infeasible due to limited computational resources. To tackle this problem, we propose a resource-efficient economic model predictive control scheme for periodic systems, leveraging existing single-layer MPC techniques. Our method relies on a single quadratic optimization problem, which ensures high computational efficiency for real-time control in dynamic settings. We prove feasibility, stability and convergence to optimum of the proposed approach, and validate the effectiveness through numerical experiments.
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Receding Horizon Control for Drinking Water Networks: The Case for Geometric Programming

2019-01-01
Shen Wang , Ahmad F. Taha , Nikolaos Gatsis +1 more
Optimal, network-driven control of Water Distribution Networks (WDN) is very difficult: valve and pump models form non-trivial, combinatorial logic; hydraulic models are nonconvex; water demand patterns are uncertain; and WDN … Optimal, network-driven control of Water Distribution Networks (WDN) is very difficult: valve and pump models form non-trivial, combinatorial logic; hydraulic models are nonconvex; water demand patterns are uncertain; and WDN are naturally large-scale. Prior research on control of WDN addressed major research challenges, yet either (i) adopted simplified hydraulic models, WDN topologies, and rudimentary valve/pump modeling or (ii) used mixed-integer, nonconvex optimization to solve WDN control problems. The objective of this paper is to develop tractable computational algorithms to manage WDN operation, while considering arbitrary topology, flow direction, an abundance of valve types, control objectives, hydraulic models, and operational constraints---all while only using convex, continuous optimization. Specifically, we propose new Geometric Programming (GP)-based Model Predictive Control (MPC) algorithms, designed to solve the water flow equations and obtain WDN controls, i.e., pump/valve schedules alongside heads and flows. The proposed approach amounts to solving a series of convex optimization problems that graciously scale to large networks. The proposed approach is tested using a 126-node network with many valves and pumps and shown to outperform traditional, rule-based control. The developed GP-based MPC algorithms, as well as the numerical test results are all included on Github.
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Receding Horizon Control for Drinking Water Networks: The Case for Geometric Programming

2020-01-07
Shen Wang , Ahmad F. Taha , Nikolaos Gatsis +1 more
Optimal, network-driven control of water distribution networks (WDNs) is very difficult: valve and pump models form nontrivial, combinatorial logic; hydraulic models are nonconvex; water demand patterns are uncertain; and WDNs … Optimal, network-driven control of water distribution networks (WDNs) is very difficult: valve and pump models form nontrivial, combinatorial logic; hydraulic models are nonconvex; water demand patterns are uncertain; and WDNs are naturally of large scale. Prior research on control of WDN addressed major research challenges, yet either i) adopted simplified hydraulic models, WDN topologies, and rudimentary valve/pump modeling or ii) used mixed-integer, nonconvex optimization to solve WDN control problems. The objective of this article is to develop tractable computational algorithms to manage WDN operation, while considering arbitrary topology, flow direction, an abundance of valve types, control objectives, hydraulic models, and operational constraints-all while only using convex, continuous optimization. Specifically, we propose new geometric programming (GP)-based model predictive control (MPC) algorithms, designed to solve the water flow equations and obtain WDN controls, i.e., pump/valve schedules alongside heads and flows. The proposed approach amounts to solving a series of convex optimization problems that graciously scale to large networks. The proposed approach is tested using a 126-node network with many valves and pumps and is shown to outperform traditional, rule-based control. The developed GP-based MPC algorithms, as well as the numerical test results, are all included on Github.
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Mixed-Integer Path-Stable Optimisation, with Applications in Model-Predictive Control of Water Systems

2020-01-01
Jorn Baayen , Jakub Mareček
Many systems exhibit a mixture of continuous and discrete dynamics. We consider a family of mixed-integer non-convex non-linear optimisation problems obtained in discretisations of optimal control of such systems. For … Many systems exhibit a mixture of continuous and discrete dynamics. We consider a family of mixed-integer non-convex non-linear optimisation problems obtained in discretisations of optimal control of such systems. For this family, a branch-and-bound algorithm solves the discretised problem to global optimality. As an example, we consider water systems, where variations in flow and variations in water levels are continuous, while decisions related to fixed-speed pumps and whether gates that may be opened and closed are discrete. We show that the related optimal-control problems come from the family we introduce -- and implement deterministic solvers with global convergence guarantees.
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Nested Algorithms for Optimal Reservoir Operation and Their Embedding in a Decision Support Platform

2020-04-30
Blagoj Delipetrev
Reservoir operation is a multi-objective optimization problem, and is traditionally solved with dynamic programming (DP) and stochastic dynamic programming (SDP) algorithms. The thesis presents novel algorithms for optimal reservoir operation, … Reservoir operation is a multi-objective optimization problem, and is traditionally solved with dynamic programming (DP) and stochastic dynamic programming (SDP) algorithms. The thesis presents novel algorithms for optimal reservoir operation, named nested DP (nDP), nested SDP (nSDP), nested reinforcement learning (nRL) and their multi-objective (MO) variants, correspondingly MOnDP, MOnSDP and MOnRL. The idea is to include a nested optimization algorithm into each state transition, which reduces the initial problem dimension and alleviates the curse of dimensionality. These algorithms can solve multi-objective optimization problems, without significantly increasing the algorithm complexity or the computational expenses. It can additionally handle dense and irregular variable discretization. All algorithms are coded in Java and were tested on the case study of the Knezevo reservoir in the Republic of Macedonia. Nested optimization algorithms are embedded in a cloud application platform for water resources modeling and optimization. The platform is available 24/7, accessible from everywhere, scalable, distributed, interoperable, and it creates a real-time multiuser collaboration platform.This thesis contributes with new and more powerful algorithms for an optimal reservoir operation and cloud application platform. All source codes are available for public use and can be used by researchers and practitioners to further advance the mentioned areas.
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Learning-based hierarchical control of water reservoir systems

2020-01-01
Pauline Kergus , Simone Formentin , Matteo Giuliani +1 more
The optimal control of a water reservoir systems represents a challenging problem, due to uncertain hydrologic inputs and the need to adapt to changing environment and varying control objectives. In … The optimal control of a water reservoir systems represents a challenging problem, due to uncertain hydrologic inputs and the need to adapt to changing environment and varying control objectives. In this work, we propose a real-time learning-based control strategy based on a hierarchical predictive control architecture. Two control loops are implemented: the inner loop is aimed to make the overall dynamics similar to an assigned linear through data-driven control design, then the outer economic model-predictive controller compensates for model mismatches, enforces suitable constraints, and boosts the tracking performance. The effectiveness of the proposed approach as compared to traditional dynamic programming strategies is illustrated on an accurate simulator of the Hoa Binh reservoir in Vietnam. Results show that the proposed approach performs better than the one based on stochastic dynamic programming.
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Management strategies for run-of-river hydropower plants -an optimal switching approach

2020-01-01
Niklas L. P. Lundström , Marcus Olofsson , Thomas Önskog
The mathematical theory for optimal switching is by now relatively well developed, but the number of concrete applications of this theoretical framework remains few. In this paper, we bridge parts … The mathematical theory for optimal switching is by now relatively well developed, but the number of concrete applications of this theoretical framework remains few. In this paper, we bridge parts of this gap by applying optimal switching theory to a set of production planning problems related to hydropower plants. In particular, we study two different cases involving small run-of-river hydropower plants and show how optimal switching can be used to create fully automatic production schemes in these cases, with non-zero cost of switching between different states of production. Along the way of deriving these schemes, we also create a model for the random flow of water based on stochastic differential equations and fit this model to historical data. This stochastic flow model, which should be of independent interest, mimics the long term seasonal behaviour of the flow while still allowing for stochastic fluctuations and can incorporate a given forecast to damp the impact of such fluctuations in near time. We benchmark the performance of our model using actual flow data from a small river in Sweden and find that our production scheme lies close to the optimal, within 2 % and 5 %, respectively, in a long term investigation of the two plants considered.
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Management strategies for run-of-river hydropower plants -an optimal switching approach

2020-09-22
Niklas L. P. Lundström , Marcus Olofsson , Thomas Önskog
The mathematical theory for optimal switching is by now relatively well developed, but the number of concrete applications of this theoretical framework remains few. In this paper, we bridge parts … The mathematical theory for optimal switching is by now relatively well developed, but the number of concrete applications of this theoretical framework remains few. In this paper, we bridge parts of this gap by applying optimal switching theory to a set of production planning problems related to hydropower plants. In particular, we study two different cases involving small run-of-river hydropower plants and show how optimal switching can be used to create fully automatic production schemes in these cases, with non-zero cost of switching between different states of production. Along the way of deriving these schemes, we also create a model for the random flow of water based on stochastic differential equations and fit this model to historical data. This stochastic flow model, which should be of independent interest, mimics the long term seasonal behaviour of the flow while still allowing for stochastic fluctuations and can incorporate a given forecast to damp the impact of such fluctuations in near time. We benchmark the performance of our model using actual flow data from a small river in Sweden and find that our production scheme lies close to the optimal, within 2 % and 5 %, respectively, in a long term investigation of the two plants considered.
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Optimal Control for Scheduling and Pricing Intra-day Natural Gas Transport on Pipeline Networks

2019-01-01
Anatoly Zlotnik , Kaarthik Sundar , Aleksandr Rudkevich +2 more
We formulate an economic optimal control problem for transport of natural gas over a large-scale transmission pipeline network under transient flow conditions. The objective is to maximize economic welfare for … We formulate an economic optimal control problem for transport of natural gas over a large-scale transmission pipeline network under transient flow conditions. The objective is to maximize economic welfare for users of the pipeline system, who provide time-dependent price and quantity bids to purchase or supply gas at metered locations on a system with time-varying injections, withdrawals, and control actions of compressors and regulators. Our formulation ensures that pipeline hydraulic limitations, compressor station constraints, operational factors, and pre-existing contracts for gas transport are satisfied. A pipeline is modeled as a metric graph with gas dynamics partial differential equations on edges and coupling conditions at the nodes. These dynamic constraints are reduced using lumped elements to a sparse nonlinear differential algebraic equation system. A highly efficient temporal discretization scheme for time-periodic formulations is introduced, which we extend to develop a rolling-horizon model-predictive control scheme. We apply the computational methodology to a pipeline system test network case study. In addition to the physical flow and compressor control solution, the optimization yields dual functions that we interpret as the time-dependent economic values of gas at each location in the network.
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Optimal Control for Scheduling and Pricing Intra-day Natural Gas Transport on Pipeline Networks

2019-12-01
Anatoly Zlotnik , Kaarthik Sundar , Aleksandr Rudkevich +2 more
We formulate an economic optimal control problem for transport of natural gas over a large-scale transmission pipeline network under transient flow conditions. The objective is to maximize economic welfare for … We formulate an economic optimal control problem for transport of natural gas over a large-scale transmission pipeline network under transient flow conditions. The objective is to maximize economic welfare for users of the pipeline system, who provide time-dependent price and quantity bids to purchase or supply gas at metered locations on a system with time-varying injections, withdrawals, and control actions of compressors and regulators. Our formulation ensures that pipeline hydraulic limitations, compressor station constraints, operational factors, and pre-existing contracts for gas transport are satisfied. A pipeline is modeled as a metric graph with gas dynamics partial differential equations on edges and coupling conditions at the nodes. These dynamic constraints are reduced using lumped elements to a sparse nonlinear differential algebraic equation system. A highly efficient temporal discretization scheme for time-periodic formulations is introduced, which we extend to develop a rolling-horizon model-predictive control scheme. We apply the computational methodology to a pipeline system test network case study. In addition to the physical flow and compressor control solution, the optimization yields dual functions that we interpret as the time-dependent economic values of gas at each location in the network.
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Model Predictive Control for Incorporating Transport of Water and Transport over Water in the Dry Season

2015-01-01
Xin Tian , Rudy R. Negenborn , P. J. van Overloop +2 more

Learning-based hierarchical control of water reservoir systems

2022-01-13
Pauline Kergus , Simone Formentin , Matteo Giuliani +1 more
The optimal control of a water reservoir system represents a challenging problem, due to uncertain hydrologic inputs and the need to adapt to changing environment and varying control objectives. In … The optimal control of a water reservoir system represents a challenging problem, due to uncertain hydrologic inputs and the need to adapt to changing environment and varying control objectives. In this work, we propose a real-time learning-based control strategy based on a hierarchical predictive control architecture. Two control loops are implemented: the inner loop is aimed to make the overall dynamics similar to an assigned linear model through data-driven control design, then the outer economic model-predictive controller compensates for model mismatches, enforces suitable constraints, and boosts the tracking performance. The effectiveness of the proposed approach is illustrated on an accurate simulator of the Hoa Binh reservoir in Vietnam. Results show that the proposed approach outperforms stochastic dynamic programming.
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Scenario Based Cost Optimization of Water Distribution Networks Powered by Grid-Connected Photovoltaic Systems

2023-01-01
Mirhan Ürkmez , Carsten Skovmose Kallesøe , Jan Dimon Bendtsen +1 more
The paper presents a predictive control method for the water distribution networks (WDNs) powered by photovoltaics (PVs) and the electrical grid. This builds on the controller introduced in a previous … The paper presents a predictive control method for the water distribution networks (WDNs) powered by photovoltaics (PVs) and the electrical grid. This builds on the controller introduced in a previous study and is designed to reduce the economic costs associated with operating the WDN. To account for the uncertainty of the system, the problem is solved in a scenario optimization framework, where multiple scenarios are sampled from the uncertain variables related to PV power production. To accomplish this, a day-ahead PV power prediction method with a stochastic model is employed. The method is tested on a high-fidelity model of a WDN of a Danish town and the results demonstrate a substantial reduction in electrical costs through the integration of PVs, with PVs supplying $66.95\%$ of the required energy. The study also compares the effectiveness of the stochastic optimization method with a deterministic optimization approach.
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Scenario Based Cost Optimization of Water Distribution Networks Powered by Grid-Connected Photovoltaic Systems

2023-08-16
Mirhan Ürkmez , Carsten Skovmose Kallesøe , Jan Dimon Bendtsen +1 more
The paper presents a predictive control method for the water distribution networks (WDNs) powered by photovoltaics (PVs) and the electrical grid. This builds on the controller introduced in a previous … The paper presents a predictive control method for the water distribution networks (WDNs) powered by photovoltaics (PVs) and the electrical grid. This builds on the controller introduced in a previous study and is designed to reduce the economic costs associated with operating the WDN. To account for the uncertainty of the system, the problem is solved in a scenario optimization framework, where multiple scenarios are sampled from the uncertain variables related to PV power production. To accomplish this, a day-ahead PV power prediction method with a stochastic model is employed. The method is tested on a high-fidelity model of a WDN of a Danish town and the results demonstrate a substantial reduction in electrical costs through the integration of PVs, with PVs supplying 66.95% of the required energy. The study also compares the effectiveness of the stochastic optimization method with a deterministic optimization approach.
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Learning Optimal Control of Water Distribution Networks Through Sequential Model-Based Optimization

2020-01-01
Antonio Candelieri , Bruno G. Galuzzi , Ilaria Giordani +1 more
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Real-time control of water reservoir operations: a learning-based hierarchical approach.

2020-01-01
Pauline Kergus , Simone Formentin , Matteo Giuliani +1 more
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Periodic optimal control of nonlinear constrained systems using economic model predictive control

2020-06-29
Johannes Köhler , Matthias A. Müller , Frank Allgöwer
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