energies Article Steam Turbine Rotor Stress Control through Nonlinear Model Predictive Control Stefano Dettori 1, * , Alessandro Maddaloni 1 , Filippo Galli 1 , Valentina Colla 1, * , Federico Bucciarelli 2 , Damaso Checcacci 2 and Annamaria Signorini 2   Citation: Dettori, S.; Maddaloni, A.; Galli,F.; Colla, V.; Bucciarelli, F.; Checcacci, D.; Signorini, A. Steam Turbine Rotor Stress Control through Nonlinear Model Predictive Control. Energies 2021, 14, 3998. https:// doi.org/10.3390/en14133998 Academic Editors: José María Maestre, Carlos Bordons and Juan Manuel Escaño Received: 12 May 2021 Accepted: 28 June 2021 Published: 2 July 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Scuola Superiore Sant’Anna, TeCIP Institute, Information and Communication Technologies for Complex Industrial Systems and Processes (ICT-COISP), Via Moruzzi 1, 56124 Pisa, Italy; a.maddaloni@santannapisa.it (A.M.); filippo.galli@santannapisa.it (F.G.) 2 Nuovo Pignone Tecnologie s.r.l, Baker Hughes, Via Felice Matteucci 2, 50127 Firenze, Italy; federico.bucciarelli@bakerhughes.com (F.B.); damaso.checcacci@bakerhughes.com (D.C.); annamaria.signorini@bakerhughes.com (A.S.) * Correspondence: s.dettori@santannapisa.it (S.D.); valentina.colla@santannapisa.it (V.C.) Abstract: The current flexibility of the energy market requires operating steam turbines that have challenging operation requirements such as variable steam conditions and higher number of startups. This article proposes an advanced control system based on the Nonlinear Model Predictive Control (NMPC) technique, which allows to speed up the start-up of steam turbines and increase the energy produced while maintaining rotor stress as a constraint variable. A soft sensor for the online calculation of rotor stress is presented together with the steam turbine control logic. Then, we present how the computational cost of the controller was contained by reducing the order of the formulation of the optimization problem, adjusting the scheduling of the optimizer routine, and tuning the parameters of the controller itself. The performance of the control system has been compared with respect to the PI Controller architecture fed by the soft sensor results and with standard pre-calculated curves. The control architecture was evaluated in a simulation exploiting actual data from a Concentrated Solar Power Plant. The NMPC technique shows an increase in performance, with respect to the custom PI control application, and encouraging results. Keywords: steam turbine startup; nonlinear model predictive control; rotor stress control 1. Introduction In the current energy market, renewable sources and flexible power plants are increas- ingly exploited in order to meet the energy demand of increasingly connected cities [1], where smart grids are going to become a consolidated reality [2,3]. In this context, the design of new power plants must aim at increasing the level of flexibility, with a resulting multitude of challenges in terms of degradation of the components, restrictions related to the environmental impact, and required level of workforce specialization. For these reasons, the design of Steam Turbines (STs) nowadays must take into account frequent dis- continuous operations. For instance, in the case of Concentrated Solar Power Plants (CSPP), the inlet steam highly depends on the weather conditions [4], while in industrial power generation, the discontinuity is related to the scheduling of the electricity production, opti- mized according to the prices of energy media (natural gas and electricity) that vary on an hourly or daily basis. One of the most important aspects related to discontinuous operation is the need to enforce protection against stress induced by thermal transients, which, due to the continuous changes in the steam conditions, can lead to faster aging of the turbine and, in extreme cases, to component failure due to nucleation and propagation of cracks. While almost every ST component is affected by thermal stress aging, transient thermal stresses in rotors often represent the factor determining the minimum allowable start-up time and the maximum allowable production during start-up. The start-up optimization is carried out Energies 2021, 14, 3998. https://doi.org/10.3390/en14133998 https://www.mdpi.com/journal/energies