Data-driven Bi-level Optimization of Thermal Power Systems with embedded Artificial Neural Networks

TL;DR

This work tackles the computational challenges of hierarchical optimization in thermal power systems by introducing a data-driven ANN-KKT framework. Upper- and lower-level objectives are approximated with artificial neural networks and the lower-level KKT conditions are embedded to produce a single-level MPEC, with Fischer-Burmeister reformulations enhancing numerical stability. The framework is validated on benchmark bi-level problems and applied to real-world cases: a 660 MW coal plant and a 395 MW gas turbine system, achieving near-bilevel-optimal solutions in sub-second times and enabling robust operating envelopes via Mahalanobis-distance constraints. The results demonstrate scalable, fast, and robust data-driven optimization for industrial power systems, with potential to support Industry 5.0 through domain-compliant AI-assisted operation. Key contributions include (1) the ANN-KKT reformulation for bi-level problems with surrogate objectives, (2) a FB-based complementarity handling approach, (3) demonstration on large-scale power plants with robust optimization capabilities, and (4) analysis showing practical computation times suitable for real-time or near-real-time operation control.

Abstract

Industrial thermal power systems have coupled performance variables with hierarchical order of importance, making their simultaneous optimization computationally challenging or infeasible. This barrier limits the integrated and computationally scaleable operation optimization of industrial thermal power systems. To address this issue for large-scale engineering systems, we present a fully machine learning-powered bi-level optimization framework for data-driven optimization of industrial thermal power systems. The objective functions of upper and lower levels are approximated by artificial neural network (ANN) models and the lower-level problem is analytically embedded through Karush-Kuhn-Tucker (KKT) optimality conditions. The reformulated single level optimization framework integrating ANN models and KKT constraints (ANN-KKT) is validated on benchmark problems and on real-world power generation operation of 660 MW coal power plant and 395 MW gas turbine system. The results reveal a comparable solutions obtained from the proposed ANN-KKT framework to the bi-level solutions of the benchmark problems. Marginal computational time requirement (0.22 to 0.88 s) to compute optimal solutions yields 583 MW (coal) and 402 MW (gas turbine) of power output at optimal turbine heat rate of 7337 kJ/kWh and 7542 kJ/kWh, respectively. In addition, the method expands to delineate a feasible and robust operating envelope that accounts for uncertainty in operating variables while maximizing thermal efficiency in various scenarios. These results demonstrate that ANN-KKT offers a scalable and computationally efficient route for hierarchical, data-driven optimization of industrial thermal power systems, achieving energy-efficient operations of large-scale engineering systems and contributing to industry 5.0.

Figures (12)

• Figure 3.1: Visualizations of benchmark problems optimization landscape for problems of nature (a) C & C, (b) C & NC, and (c) NC & NC. The background gradient represents the upper-level objective function value, while the dashed lines denote the lower-level objective. Solid black lines define the feasible solutions. (d) CPU time requirement for solving the benchmark problems - red: Bi-level-KKT and blue: ANN-KKT. The approximation error of ANN-KKT framework is also mentioned.
• Figure 3.2: 3D visualization of the turbine heat rate response surface as a function of coal flow rate and power output. The surface exhibits significant non-convexity with multiple local optima, characterized by irregular ridges and valleys across the operating domain. The complex topography exhibits the nonlinear interdependencies between the variables of thermal power plant. Note that this visualization represents the raw data surface rather than a fitted model profile and it serves a purely visual purpose to illustrate the data's complexity and is not an accurate depiction of the isolated relationship between these variables.
• Figure 3.3: Optimal operating points from bi-level optimization across Mahalanobis distance tolerances. (a) IPOPT convergence profiles for objective function and primal infeasibility constraint for bi-level optimization across tolerance levels. Primal infeasibility trajectories on logarithmic scale, with feasible solutions achieving constraint satisfaction below $10^{-6}$ and infeasible solutions stagnating above $10^0$ (red dashed reference line). (b) The surface shows turbine heat rate as a function of coal flow rate and power output. Star markers indicate optimal solutions for tolerance levels from 81% to 95%, with yellow stars representing feasible solutions (lower THR) and purple stars indicating infeasible solutions at constraint boundaries (higher THR). Black contour lines show historical operational density. (c) CPU time consumed to compute solutions for different values of $\tau$ for Mahalanobis constraint.
• Figure 3.4: Parallel coordinates plot visualizing the multidimensional operating space of a 395 MW gas turbine system. The visualization highlights complex non-linear mapping network between the variables and non-convex function space of performance variables (Power and THR) built with the operating variables.
• Figure 3.5: Optimal operating points from bi-level optimization of gas turbine system across Mahalanobis distance tolerances. (a) IPOPT convergence profiles for objective function and primal infeasibility constraint for bi-level optimization across tolerance levels. Primal infeasibility trajectories on logarithmic scale, with feasible solutions achieving constraint satisfaction below $10^{-6}$ and infeasible solutions stagnating above $10^0$ (red dashed reference line). (b) The surface shows turbine heat rate as a function of gas fuel flow rate and power output. Star markers indicate solutions computed for $\tau$ levels from 75% to 95%, with yellow stars representing feasible solutions and purple stars correspond to infeasible solutions. Black contour lines show historical operational density. (c) CPU time consumed to compute solutions corresponding to different values of $\tau$ for Mahalanobis constraint.