Nested Optimization Framework Research Articles

Tensor product decomposed multi‐material isogeometric topology optimization with explicit NURBS element stiffness computation

AbstractBased on the explicit stiffness computation formula of NURBS (non‐uniform rational B‐splines) element and tensor product decomposed explicit filter, a novel multi‐material isogeometric topology optimization (MMITO) method is put forward to generate multi‐material structures efficiently. The explicit NURBS element stiffness computation formula is: derived from the Bėzier extraction operator under the combination of scaled and rotational geometric transformations between NURBS elements, and the stiffness matrices of NURBS elements can be computed by the stiffness matrix of a Bėzier element without resorting to the time‐consuming integral procedure. As the presented MMITO is a nested optimization framework, we apply the explicit filtering technique to update the multi‐phase design variables efficiently in combination with the tensor product decomposition method. With the decomposed weight matrices of the filter and the explicit form of the NURBS stiffness matrix, we obtain the equivalent but much simplified preprocessing procedure and sensitivity analysis for the MMITO method. A comprehensive set of multi‐material numerical examples are optimized to verify that, while improving the preprocessing efficiency and reducing the memory overhead, the present MMITO scheme is able to generate semblable two‐dimensional and three‐dimensional multi‐material structures with identical structural performance to the traditional scheme. Therefore, the proposed MMITO method is a promising approach to optimizing multi‐material structures.

An interval-based nested optimization framework for deriving flexibility from smart buildings and electric vehicle fleets in the TSO-DSO coordination

Emerging renewable-based transmission and distribution systems, despite many environmental and economic benefits, due to the intermittent nature of their production resources, compared to traditional systems, need more flexibility capacities, which necessitates the need for more suppliers of flexibility. To deal with these challenges, a nested framework is presented to derive the required flexibility of the transmission system operator (TSO) from distributed energy resources (DERs) and active end-users such as smart buildings (SBs) and electric vehicle (EV) fleets at the distribution level. To this end, a novel mechanism to design the demand response program (DRP) is introduced in which tariffs with time-varying rewards are built based on flexibility requirements. The coordination between TSO and distribution system operator (DSO) is initially modeled as a bi-level non-linear programming (NLP) problem, in which the upper-level is day-ahead (DA) operational planning of DS considering the schedules received from SBs, while the lower-level is DA operational planning of the TS. The bi-level NIL problem is transformed into a single-level linear programming (LP) problem by Krush Kuhn Tucker (KKT) conditions, Big-M method and Strong Duality Theory (SDT), which makes it computationally tractable. Finally, a two-stage interval-based algorithm solves the obtained single-level problem to secure the planning against uncertainties where battery energy storage systems (BESSs) are responsible for dealing with extreme conditions. The simulation results testify that the proposed interval-based nested framework has improved the economic, technical and security aspects of the TSO-DSO coordination since it has reduced the daily costs of the energy and flexibility markets, relieved lines congestion and improved voltage characteristics.

RMAML: Riemannian meta-learning with orthogonality constraints

Meta-learning is the core capability that enables intelligent systems to rapidly generalize their prior experience to learn new tasks. In general, the optimization-based methods formalize the meta-learning as a bi-level optimization problem, that is a nested optimization framework, in which meta-parameters are optimized (or learned) at the outer-level, while the inner-level optimizes the task-specific parameters. In this paper, we introduce RMAML, a meta-learning method that enforces orthogonality constraints to the bi-level optimization problem. We develop a geometry aware framework that generalizes the bi-level optimization problem to the Riemannian (constrained) setting. Using the Riemannian operations such as orthogonal projection, retraction and parallel transport, the bi-level optimization is reformulated so that it respects the Riemannian geometry. Moreover, we observe that a superior stable optimization and an improved generalization ability can be achieved when the parameters and meta-parameters of the method are modeled using a Stiefel Manifold. We empirically show that RMAML can easily reach competitive performances against several state of the art algorithms for few-shot classification and consistently outperforms its Euclidean counterpart, MAML. For example, by using the geometry of the Stiefel manifold to structure the fully-connected layers in a deep neural network, a 7% increase in single-domain few-shot classification accuracy is achieved. For the cross-domain few-shot learning, RMAML outperforms MAML by up to 9% of accuracy. Our ablation study also demonstrates the effectiveness of RMAML over MAML in terms of higher accuracy with a reduced number of tasks and (or) inner-level updates.

Automated nested co-design framework for structural/control dynamics in flexible space systems

Modern space missions for Earth Observation (EO) purposes often rely on satellites equipped with very large flexible appendages, such as antennas and solar panels, which are demanded to perform agile slew manoeuvres. In most cases, the elasticity of such systems cannot be neglected in the design of the attitude controller, as excessive elastic displacements of the structural elements may compromise their stability and pointing performance. In this scenario, this paper aims at proposing an automated nested optimization framework to simultaneously optimize spacecraft structural and control dynamics, to be applicable to a wide range of flexible spacecraft. The objective of such a co-design architecture is to modify design parameters, at both structural and control levels, to minimize the mass of the spacecraft while maximizing its agility and satisfying imposed requirements. Moreover, as robust multivariable techniques have become more and more applied to ensure satisfactory robust performance margins, this paper's goal is to pose a multi-channel structured H∞ control architecture in the co-design problem. To guarantee the generality of implementation, a structural design tool (MSC Nastran) is interfaced with a coding environment (Matlab/Simulink) to set-up an autonomous exchange of information between structural and control domains. Starting from an initial definition of the spacecraft material, geometry and control requirements (in terms of loop-shaping transfer functions), relevant parameters are extracted from the structural tool and a linearized dynamic model assembled. Then, a controller is synthetized based on the provided requirements, followed by a verification on the nonlinear plant of the satellite. The procedure is repeated until the stop criteria (based on tolerance and max iterations) is satisfied. Finally, the output of the proposed architecture is obtained as an optimized structural model and robust controller tailored for the satellite dynamics.

Risk-averse day-ahead generation scheduling of hydro–wind–photovoltaic complementary systems considering the steady requirement of power delivery

Optimizing day-ahead generation schedules of hydro–wind–photovoltaic (PV) complementary systems (HWPCSs) can help to promote the accommodation of wind and solar energies. However, it is challenging to formulate appropriate generation schedules for the large HWPCS that contains cascade hydropower plants, in particular, a steady requirement of power delivery is considered in the optimization model. To further improve complementary performance of the large HWPCS, we propose a risk-averse day-ahead generation scheduling approach that considers the steady requirement of power delivery. First, a representative scenario set is used to characterize forecast uncertainties of the wind and PV power. Then, a multi-objective optimal generation scheduling model with consideration of the operational risks of electricity curtailment and power shortage is proposed. Finally, a two-layer nested optimization framework is designed to derive the system’s generation schedule. The clean energy base in the upper Yellow River basin, China was selected as a case study. The results show that: (1) forecast uncertainties of wind and PV power are more likely to induce power shortage risk in summer and autumn, but to induce electricity curtailment risk in spring and winter; (2) without using extra constraint handling strategies, the proposed approach could directly yield a stair-shaped power delivery curve, which is good for long-distance power transmission applications; and (3) compared with a traditional method without considering the operational risks, the proposed generation scheduling approach could significantly reduce the comprehensive risk rate by 65% on average, while the cascade hydropower production and peak shaving performance are satisfactory. Therefore, the proposed approach is effective in guiding the day-ahead generation scheduling of the HWPCSs that contain cascade hydropower plants.

Stochastic short-term scheduling of a wind-solar-hydro complementary system considering both the day-ahead market bidding and bilateral contracts decomposition

Complementary operation of hydropower, wind and photovoltaic (PV) power is one of the most cost-efficient solutions for integration of intermittent wind and PV power. In a deregulated market, synergetic optimization of day-ahead (DA) market bids and the decomposition of daily physical bilateral contracts (BCs) can increase profits for a wind-solar-hydro complementary system (WSHCS) by coordinating the spot market and the forward market. However, this short-term scheduling problem is challenging because of complex market mechanisms and uncertainties of multiple energy resources and market prices. A stochastic scheduling model is established to address this issue. First, uncertainties of wind, PV power and DA market prices are characterized by stochastic scenarios simulation with the adoption of the simultaneous backward reduction algorithm. Then, a two-layer nested optimization framework is conducted to optimize both the DA market bidding strategy and the decomposition of daily BCs. A WSHCS in China’s Yalong River Basin is selected as a case study. Results show that: the proposed model can decompose the total energy of the daily BC reasonably by leaving flexible bidding spaces in the DA market according to price variation, thus larger volume of bidding energy can be sold when DA market prices are higher than the contract price. Besides, the optimal decomposition outperforms the average decomposition of daily BCs with larger benefits, less power curtailment, and higher water resources utilization efficiency. These findings verify the effectiveness and applicability of the proposed model for synergetic optimization of both the decomposition of daily BCs and DA market bids for a WSHCS.

Multiobjective nested optimization framework for simultaneous integration of multiple photovoltaic and battery energy storage systems in distribution networks

The rapid growth of renewables in modern distribution networks results in the spilling of energy due to the limited hosting capacity of these networks, violation of system constraints, reduced network efficiency, and improper utilization of resources. Battery energy storage system (BESS), in spite of its high cost, a shorter life and complex control, offers a flexible solution for the problem. In this paper, a multiobjective nested optimization framework is developed for the simultaneous optimal allocation of multiple solar photovoltaics (SPVs) and BESSs in the distribution networks. The framework involves a two-layered structure; the outer layer provides tentative planning solutions to the inner layer that optimizes the desired objectives of network operations and then returns the functional values back to the outer layer. The purpose of the inner-layer is to satisfy the operational constraints of the networks and ensure the optimal utilization of BESS capacities, suggested by the outer layer, at the time of planning itself. A new BESS operating strategy is proposed for optimum utilization of BESS. The nested multiobjective optimization problem is handled by suggesting a new weighted sum approach in conjunction with a recently developed swarm intelligence-based algorithm, i.e. moth search optimization. Overall, the proposed deterministic model essentially ensures the high penetration of SPVs and the optimal utilization of BESSs to justify their installation. The optimization model is investigated on a benchmark 33-bus test distribution network. The application results highlight enhanced energy efficiency, peak load shaving, high renewable penetration, voltage profile improvement, and mitigation of reverse power flow while effectively absorbing the excess renewable power generation during light load hours.

Combined Plant and Controller Design Using Batch Bayesian Optimization: A Case Study in Airborne Wind Energy Systems

This paper presents a novel data-driven nested optimization framework that addresses the problem of coupling between plant and controller optimization. This optimization strategy is tailored toward instances where a closed-form expression for the system dynamic response is unobtainable and simulations or experiments are necessary. Specifically, Bayesian optimization, which is a data-driven technique for finding the optimum of an unknown and expensive-to-evaluate objective function, is employed to solve a nested optimization problem. The underlying objective function is modeled by a Gaussian process (GP); then, Bayesian optimization utilizes the predictive uncertainty information from the GP to determine the best subsequent control or plant parameters. The proposed framework differs from the majority of codesign literature where there exists a closed-form model of the system dynamics. Furthermore, we utilize the idea of batch Bayesian optimization at the plant optimization level to generate a set of plant designs at each iteration of the overall optimization process, recognizing that there will exist economies of scale in running multiple experiments in each iteration of the plant design process. We validate the proposed framework for Altaeros' buoyant airborne turbine (BAT). We choose the horizontal stabilizer area, longitudinal center of mass relative to center of buoyancy (plant parameters), and the pitch angle set-point (controller parameter) as our decision variables. Our results demonstrate that these plant and control parameters converge to their respective optimal values within only a few iterations.

Integrated design for a CVT: dynamical optimization of actuation and control

Abstract With increasing demands on more energy and fuel efficient vehicles, one can achieve the goal by improving the vehicle powertrain system. A continuously variable transmission (CVT) allows the engine or electric machine to operate on its optimal operation points. The optimal operation points are high efficiency points that lead to reduced energy consumption of the vehicle. However, this type of transmission may still have relatively high actuation losses (depending on the actuation type), which hinders the energy saving benefits. Classically, the plant (e.g., actuation system, variator) of the CVT was separately designed from the control design. In this paper, an integrated optimal CVT variator and actuation control design is presented. The aim of the new design is to minimize the CVT mass (pulley sheaves, belt), tracking error and control effort. To achieve this goal, a nested optimization framework is implemented to obtain an optimal transmission system design over a selected drive cycle. The results show that the optimized CVT design yields non-compromising tracking performance, however, with much smaller variator mass (-46%) and control effort (-62%).

Robust hydroelectric unit commitment considering integration of large-scale photovoltaic power: A case study in China

Hybrid generation of large-scale photovoltaic (PV) power together with hydropower offers a promising option to promote the integration of PV power, because hydro units can complement variable PV generation rapidly at relatively low cost. However, the strong variations in PV generation create uncertainties for the operation of the hydro units. To improve guidelines for large-scale hydro–PV plant operation, a stochastic hydro unit commitment model considering the uncertainty in forecasting PV power is presented. This model seeks robust solutions (i.e., hydro unit status) to minimize the hydro plant’s water consumption when external load demands are imposed onto the hybrid system. A two-layer nested optimization framework is proposed to solve the model in a hierarchical structure. In the outer layer, a cuckoo search algorithm, combined with a novel encoding strategy, optimizes the number of online units that can meet the load demand under all PV generation processes. In the inner layer, load dispatch schemes for the given number of online units are determined by dynamic programming. China’s Longyangxia hydro–PV plant was selected as a case study. Operational results were compared for three scenarios: actual operation, deterministic operation without consideration of PV forecast errors, and stochastic operation with consideration of PV forecast errors. The results indicate that: (1) the encoding strategy can handle the minimum online and offline time constraints, and can decrease optimization dimensionality; (2) the two-layer nested optimizer can make robust and effective decisions within an acceptable period when using the pre-stored optimization results provided by dynamic programming; and (3) water consumption in deterministic and stochastic operation scenarios decreased by 1.5% and 1.0%, respectively, compared to that of actual operation. These findings verify the applicability and effectiveness of the proposed methods, and also highlight the importance of decreasing uncertainty in the PV power forecast.

A surrogate model based nested optimization framework for inverse problem considering interval uncertainty

The inverse problem is a kind of engineering problem that estimates the input through the given output. In this paper, the given output described as interval uncertainty parameter is concerned, and the interval is formed by the interval midpoint and the interval radius. The two-step framework that estimates the midpoint and the radius separately is used. A novel nested optimization framework is proposed to estimate the input interval radius with more inputs than outputs. The nested framework has two loops: (i) the inner loop quantifies the lower and upper bounds of the output with given interval radiuses of inputs from the outer loop by two optimizations, and the results will be feedback to the outer loop as constraint values of outer loop; (ii) the outer loop maximizes the input interval radiuses to reduce the cost while meeting the constraints transformed from the given interval. The nested framework induces a high number of forward model computations in the loops and may lead to an unacceptable computational burden for most engineering applications. Therefore, the surrogate model is suggested. The Radial Basis Functions (RBF) surrogate model is used to relieve the computational burden. The effectiveness and the accuracy of the framework are verified through a mathematical example, a cantilever tube example and an airfoil example.

Nested Plant/Controller Co-Design Using G-Optimal Design and Extremum Seeking: Theoretical Framework and Application to an Airborne Wind Energy System

This paper presents a unique nested optimization framework for the co-design of a physical system (plant) and controller, which leverages optimal Design of Experiments (DoE) techniques for the plant optimization and extremum seeking for the control system optimization. At each iteration of the optimization, candidate plant parameters are generated by using G-optimal DoE. Unlike gradient-based approaches, the use of optimal DoE enables efficient global exploration of a plant design space that can contain multiple local optima. For each candidate plant design, the corresponding controller optimization is performed in real time, using extremum seeking. This enables the real-time adjustment of controller parameters during the course of simulations or experiments, thereby expediting the overall optimization process. The co-design process is carried out iteratively, where sub-optimal plant designs are rejected based on a response surface characterization and hypothesis testing. The co-design framework was validated in simulation for a Buoyant Airborne Turbine (BAT). Here, the optimized plant parameters were a reference area scale factor (scales the horizontal and vertical stabilizer areas uniformly) and center of mass location, whereas the optimized control parameter was the pitch angle setpoint. After four complete iterations, the flight performance index improved and the feasible plant design space (i.e., the locus of plant design parameters that could possibly be optimal, based on hypothesis testing) shrunk by 99%.