Optimizing Partially Defined Black-Box Functions Under Unknown Constraints via Sequential Model Based Optimization: An Application to Pump Scheduling Optimization in Water Distribution Networks

Water stands as the fundamental cornerstone of life, and with the human population growing exponentially, the critical importance of sustainable water management becomes increasingly evident. Water not only serves as the essential working fluid for major chemical processes, including thermal energy generation, but it is also integral to manufacturing and domestic use. Despite its significance, water is a scarce commodity, and the planet continues to lose this precious resource due to theft, pipe bursts, and leakages in water distribution networks (WDNs). The consequences of water loss extend beyond the impact on physical resources, as it is intricately linked to financial challenges. Thus, water loss has the potential to result in substantial financial setbacks for companies, organizations, and government entities, underscoring the interdependence of water management and fiscal sustainability. In response to these challenges, digital technologies have been harnessed to optimize WDNs. A major constraint lies in deciphering the applicable digital technologies due to the multidisciplinary nature of the research area encompassing WDNs and digital technologies; therefore, this study adopts a systematic literature review (SLR) combined with digital tools, to comprehensively extract all aspects of a WDN, ranging from sensors to enterprise resource planning (ERP), including equipment and systems. This literature forms the foundational basis for constructing a framework aimed at creating a content repository for digital WDN sustainability. To further develop the framework, this study reviews international best practices in digital WDN. This is adopted in developing a digital WDN framework based on the literature and classifying a database structure from sensor to ERP for a generic WDN. Furthermore, the study explores tools and systems for optimization as well as updating the theoretical structure with Johannesburg WDN data and provides a comprehensive evaluation for enabling digital WDN in South Africa. The digital WDN framework comprises three distinct components: security, a central management database (CMDB), and best practice architecture. The architecture involves five layers, including the physical layer, communication layer, data storage and analysis layer, benefit layer, and application layer. Utilizing systematic literature review and bibliometric keyword analysis, the study determines the equipment, systems, and their uses, leading to the development of a generic WDN. Validation of the framework using the generic WDN emphasizes the equipment and systems used in each of the five layers. Furthermore, the framework is validated using Johannesburg Water's WDN, leading to the development of a CMDB. The CMDB underscores the possibility of storing physical layer and communication layer configuration items (CIs) and highlights asset management, CI relations, change management, and historian capabilities. Maintaining an accurate and updated CMDB necessitates employing asset discovery solutions, while CMDB relationship mapping plays a crucial role in assessing the ripple effects of CI changes within a complex system. Finally, the measurement of the digital maturity of Johannesburg Water's WDN using the Administração e Gestão de Sistemas de Salubridade (AGS) maturity index reveals a maturity index of 62%. The index emphasizes strong capabilities in GIS integration and data collection through the SCADA system. Despite advancements in planning systems, such as domestic flow metering, there remains an opportunity to enhance asset management capabilities for continuous improvement in WDN operations.

Nowadays, due to increasing population and water shortage and competition for its consumption, especially in the agricultural that is the largest consumer of water, proper and suitable utilization and optimal use of water resources is essential. One of the important parameters in agriculture field is water distribution network. In this research, differential evolution algorithm (DE) was used to optimize Ismail Abad water supply network. This network that is pressurized network and includes 19 pipes and nodes 18. Optimization of the network has been evaluated by developing an optimization model based on DE algorithm in MATLAB and the dynamic connection with EPANET software for network hydraulic calculation. The developing model was run for the scale factor (F), the crossover constant (Cr), initial population (N) and the number of generations (G) and was identified best adeptness for DE algorithm is 0.6, 0.5, 100 and 200 for F and Cr, N and G, respectively. The optimal solution was compared with the classical empirical method and results showed that Implementation cost of the network by DE algorithm 10.66% lower than the classical empirical method.

This study presents a data-driven approach for optimizing the operating altitude of Airborne Wind Energy (AWE) systems to maximize net energy production. Determining the optimal operating altitude of an AWE system is challenging, as the wind speed constantly varies with both time and altitude. Furthermore, without expensive auxiliary equipment, the wind speed is only measurable at the AWE system's operating altitude. The work presented in this paper shows how tools from machine learning can be blended with real-time control to optimize the AWE system's operating altitude efficiently, without the use of auxiliary wind profiling equipment. Specifically, Bayesian Optimization, which is a data-driven technique for finding the optimum of an unknown and expensive-to-evaluate objective function, is applied to the real-time control of an AWE system. The underlying objective function is modeled by a Gaussian Process (GP); then, Bayesian Optimization utilizes the predictive uncertainty information from the GP to decide the best subsequent operating altitude. In the AWE application, conventional Bayesian Optimization is extended to handle the time-varying nature of the wind shear profile (wind speed vs. time). Using real wind data, our method is validated against three baseline approaches. Our simulation results show that the Bayesian Optimization method is successful in dramatically increasing power production over these baselines.

A major limitation of the widely used evolutionary optimization (EO) methods is their inability to handle constraints directly. To address this issue, penalty function methods have been used in most EO applications involving constrained problems such as those of the water distributions networks (WDNs). However, the disadvantage of penalty function methods is that the parameters involved require great expertise in calibration with numerous time consuming trial runs. In addition, penalty parameters are case sensitive and do not necessarily steer the EO search toward the best solutions in every situation. This paper presents a penalty-free multi-objective evolutionary approach (PFMOEA) for the optimization of water distribution networks. The basis behind this approach is to involve the pressure dependent analysis (PDA) in developing a multi-objective evolutionary search. PDA is capable of simulating both normal and pressure deficient networks and provides a quick and precise means to identify the feasible region of the solution space. The proposed PFMOEA couples the renowned NSGA II with an enhanced seamless version of EPANET 2 which is capable of PDA. The PFMOEA has been applied to several WDN benchmarks. Excellent results in terms of the initial network cost and number of hydraulic simulations required were obtained. Comprehensive discussions along with comparisons of solutions from previous studies are presented herein.

A simulation optimization model, WCANET that combines water cycle optimization algorithm with EPANET hydraulic simulation software to tackle water distribution problem for least cost design is proposed. The developed model is to achieve optimal design outputs within the hydraulic and design constraints of the problem. The model has been put to test against two benchmark water distribution network problems, namely (1) Hanoi water distribution network and (2) Two loop network problem. The study shows promising results for WCANET compared to earlier studies. Analysis of the results suggests WCANET can be considered for planning of water distribution networks which has high economic benefits.

The optimization of a water distribution network (WDN) is a highly nonlinear, multi-modal, and constrained combinatorial problem. Particle swarm optimization (PSO) has been shown to be a fast converging algorithm for WDN optimization. An improved estimation of distribution algorithm (EDA) using historic best positions to construct a sample space is hybridized with PSO both in sequential and in parallel to improve population diversity control and avoid premature convergence. Two water distribution network benchmark examples from the literature are adopted to evaluate the performance of the proposed hybrid algorithms. The experimental results indicate that the proposed algorithms achieved the literature record minimum (6.081 M$) for the small size Hanoi network. For the large size Balerma network, the parallel hybrid achieved a slightly lower minimum (1.921M€) than the current literature reported best minimum (1.923M€). The average number of evaluations needed to achieve the minimum is one order smaller than most existing algorithms. With a fixed, small number of evaluations, the sequential hybrid outperforms the parallel hybrid showing its capability for fast convergence. The fitness and diversity of the populations were tracked for the proposed algorithms. The track record suggests that constructing an EDA sample space with historic best positions can improve diversity control significantly. Parallel hybridization also helps to improve diversity control yet its effect is relatively less significant.

Nowadays, potable water supply has become a problem, involving different aspects such as water quality, pollution, pipes' quality, supply and demand of drinking water, water leaks, and supply cuts, among other aspects that have complicated the efficient use of potable water in urbanized regions. The objective of this chapter is to define potable water distribution network (PWDN) optimization as a complex problem, from the perspective of an NP-Hard problem, and the complexity as a variant of water distribution network optimization. The main result of the proposal is the definition of a problem that identifies the main variables involved in the supply of quality potable water as a global problem where each researcher can solve the main problem as one or propose solutions for different parts. The proposal simulation is oriented to present a brief solution related to the supply and demand of drinking water as a part of the PWDN.

This article provides a comprehensive overview of the most approached heuristic optimization techniques of water distribution networks (WDNs) focusing on their optimal design. A single- and multi-objective optimization problem is generally formulated, and the main optimization objectives, decision variables and constraints for the design, rehabilitation and operation of WDNs are synthesized and discussed. Additionally, the main heuristic methods are briefly described and classified. Finally, literature pertaining to the heuristic optimization models of WDNs is overviewed and in conclusion the advantages and disadvantages of the heuristic techniques used for designing WDNs are discussed along with some recommendations for future research directions.

The development of a general‐purpose machine learning algorithm capable of quickly identifying optimal 3D‐printing settings can save manufacturing time and cost, reduce labor intensity, and improve the quality of 3D‐printed objects. Existing methods have limitations which focus on overall performance or one specific aspect of 3D‐printing quality. Here, for addressing the limitations, a multi‐objective Bayesian Optimization (BO) approach which uses a general‐purpose algorithm to optimize the black‐box functions is demonstrated and identifies the optimal input parameters of direct ink writing for 3D‐printing different presurgical organ models with intricate geometry. The BO approach enhances the 3D‐printing efficiency to achieve the best possible printed object quality while simultaneously addressing the inherent trade‐offs from the process of pursuing ideal outcomes relevant to requirements from practitioners. The BO approach also enables us to effectively explore 3D‐printing inputs inclusive of layer height, nozzle travel speed, and dispensing pressure, as well as visualize the trade‐offs between each set of 3D‐printing inputs in terms of the output objectives which consist of time, porosity, and geometry precisions through the Pareto front.

The new power system effectively integrates a large number of distributed renewable energy sources, such as solar photovoltaic, wind energy, small hydropower, and biomass energy. This significantly reduces the reliance on fossil fuels and enhances the sustainability and environmental friendliness of energy supply. Compared to distribution networks in traditional power systems, the new-generation distribution network offers notable advantages in improving energy efficiency, reliability, environmental protection, and system flexibility, but it also faces a series of new challenges. These challenges include potential harmonic issues introduced by the widespread use of power electronic devices (such as inverters for renewable energy generation systems and electric vehicle charging stations) and the voltage fluctuations and flickering caused by the intermittency and uncertainty of renewable energy generation, which may affect the normal operation of electrical equipment. To address these challenges, this study proposes an optimization model aimed at minimizing network losses and voltage deviations, utilizing traditional capacitor adjustments and static var compensators (SVCs) as optimization measures. Furthermore, this study introduces an improved version of the multi-objective particle swarm optimization (MOPSO) algorithm, specifically enhanced to address the unique challenges of reactive power optimization in modern distribution networks. The test results show that this algorithm can effectively generate a large number of Pareto optimal solutions. The application of this algorithm on a 33-node network case study demonstrates its advantages in reactive power optimization. The optimization results highlight the effectiveness and feasibility of the proposed improved algorithm in the application of distribution network reactive power optimization, offering users a uniform and diverse range of reactive compensation solutions.

Optimization of water distribution networks has been of central importance for recent decades. Genetic Algorithms (GA) are the most famous metaheuristics widely used for this purpose with great success. However, the fact that GA basically requires a large number of computations, has led to investigate for faster solvers. In this research, a new approach is proposed in which a simple GA is linked with the Integer-Linear Programming (ILP) method resulting in a hybrid optimization scheme. Using the mathematical method of ILP, the search space is significantly reduced thereby a limited number of evaluations are required to achieve a good solution. The approach is applied to two benchmark pipe-networks in order to show its ability in terms of accuracy and speed. The results are then compared with the previous works. The obtained results indicate that the proposed model is computationally efficient, like classic methods, while is still very promising in finding the global optimum like the nature-inspired metaheuristics.

The paper presents a novel approach to applying Bayesian Optimization (BO) in predicting an unknown constraint boundary, also representing the discontinuity of an unknown function, for a feasibility check on the design space, thereby representing a classification tool to discern between a feasible and infeasible region. Bayesian optimization is a low-cost black-box global optimization tool in the Sequential Design Methods where one learns and updates knowledge from prior evaluated designs, and proceeds to the selection of new designs for future evaluation. However, BO is best suited to problems with the assumption of a continuous objective function and does not guarantee true convergence when having a discontinuous design space. This is because of the insufficient knowledge of the BO about the nature of the discontinuity of the unknown true function. In this paper, we have proposed to predict the location of the discontinuity using a BO algorithm on an artificially projected continuous design space from the original discontinuous design space. The proposed approach has been implemented in a thin tube design with the risk of creep-fatigue failure under constant loading of temperature and pressure. The stated risk depends on the location of the designs in terms of safe and unsafe regions, where the discontinuities lie at the transition between those regions; therefore, the discontinuity has also been treated as an unknown creep-fatigue failure constraint. The proposed BO algorithm has been trained to maximize sampling toward the unknown transition region, to act as a high accuracy classifier between safe and unsafe designs with minimal training cost. The converged solution has been validated for different design parameters with classification error rate and function evaluations at an average of <1% and ∼150, respectively. Finally, the performance of our proposed approach in terms of training cost and classification accuracy of thin tube design is shown to be better than the existing machine learning (ML) algorithms such as Support Vector Machine (SVM), Random Forest (RF), and Boosting.

Bayesian Optimization is proposed for data-efficient learning of optimal control strategies aimed at minimizing the energy related costs for operating pumps in a water distribution network. The control strategies are defined as pressure-based rules, whose parameters are the decision variables of the optimization problem. A probabilistic description is used to model the optimization problem from parameters to energy cost. The probabilistic model is learned from data obtained by testing a set of parameters via software hydraulic simulation. Bayesian Optimization selects the next values of the parameters to evaluate in a principled way, proving to be able to find globally optimal control strategies, within relatively few trials (i.e., software simulation runs). The proposed Bayesian Optimization framework deals with a quite general formalization of the control problem, including constraints, also black box. Relevant results on a real-life water distribution network are reported, also in comparison with Pure Random Search and Genetic Algorithms.

Water distribution systems are an integral part of the economic infrastructure of modern-day societies. However, previous research on the design optimization of water distribution systems generally involved few decision variables and consequently small solution spaces; piecemeal-solution methods based on pre-processing and search space reduction; and/or combinations of techniques working in concert. The present investigation was motivated by the desire to address the above-mentioned issues including those associated with the lack of high-performance computing (HPC) expertise and limited access in developing countries. More specifically, the article’s aims are, firstly, to solve a practical water distribution network design optimization problem and, secondly, to develop and demonstrate a generic multi-objective genetic algorithm capable of achieving optimal and near-optimal solutions on complex real-world design optimization problems reliably and quickly. A multi-objective genetic algorithm was developed that applies sustained and extensive exploration of the active constraint boundaries. The computational efficiency was demonstrated by the small fraction of 10-245 function evaluations relative to the size of the solution space. Highly competitive solutions were achieved consistently, including a new best solution. The water utility’s detailed distribution network model in EPANET 2 was used for the hydraulic simulations. Therefore, with some additional improvements, the optimization algorithm developed could assist practitioners in day-to-day planning and design.

Water distribution networks require huge investment for construction. Involved people, especially researchers, are always seeking to find a way for decreasing costs and achieving an efficient design. One of the main factors of the network design is the selection of proper diameters based on costs and deficit of flow pressure and velocity in the network. The reduction in construction costs is accomplished by minimizing the diameter of network pipes which leads to the pressure drop in the network. Supplying proper pressure in nodes is one of the important design principles, and low pressure will not provide a complete water supply at the consumption site. Therefore, in this research, the problem of optimization in several sample networks was defined with the objectives of cost minimization and minimization of pressure deficit in the whole network. The EPANET software was used for hydraulic analysis of sample networks, and the multi-objective optimization process was performed by coding NSGA-II and MOPSO algorithms in the MATLAB software environment and linking them to EPANET. The cost function was initially defined only by considering the relationship between cost and diameter and the length of pipes, and in the next definition, the cost resulted by violation of the allowable pressure range was added to this function In both cases, the schedule for achieving the optimal answer was executed. The results showed that these algorithms have a high ability to find optimal solutions and are able to optimize the network in terms of cost and pressure by finding the appropriate pipe diameter. The time for reaching convergence was reduced by considering the cost of violation of the allowable pressure limits significantly and the optimal answer is obtained in a small number of repetitions. In NSGA-II and MOPSO algorithms in two-looped network with 20 and 30 iterations and run time of 0.66 and 0.8 s, respectively, and in Lansey network with 150 and 250 iterations and run time of 5.7 and 9.5 s, the optimal solutions were obtained.