LPV-Filtered Predictive Control Design for Fault-Tolerant Energy Management ⁎

12th IFAC Symposium on Dynamics and Control of 12th IFAC Symposium on Dynamics and Control of 12th IFAC IFACSystems, Symposium on Dynamics Dynamics and Control Control of of Process including Biosystems 12th Symposium on and Available online at www.sciencedirect.com Process Systems, including Biosystems Process Biosystems 12th IFACSystems, Symposium on Dynamics Control of Florianópolis - SC,including Brazil, April 23-26,and 2019 Process Systems, Biosystems Florianópolis - SC,including Brazil, April 23-26, 2019 Florianópolis SC, Brazil, April 23-26, 2019 Process Systems, including Biosystems Florianópolis - SC, Brazil, April 23-26, 2019 Florianópolis - SC, Brazil, April 23-26, 2019

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IFAC PapersOnLine 52-1 (2019) 166–171

LPV LPV -Filtered -Filtered Predictive Predictive Control Control Design Design LPV -Filtered Predictive Control Design for Fault-Tolerant Energy Management LPV -Filtered Predictive Control Design for Fault-Tolerant Energy Management for Fault-Tolerant Energy Management  for Fault-Tolerant Energy Management Marcelo Menezes Morato ∗∗ Paulo Renato da Costa Mendes ∗∗

Marcelo Menezes Morato da Costa∗∗Mendes ∗ ∗ ∗ Paulo Marcelo Menezes Morato Paulo∗Renato Renato da Mendes Elias Normey-Rico Carlos Bordons Marcelo Julio Menezes Morato Renato da Costa Costa∗∗ Mendes ∗∗ ∗ Paulo∗ Julio Elias Normey-Rico Carlos Bordons ∗ ∗∗ Marcelo Julio Menezes Paulo∗Renato da Costa∗∗Mendes Elias Normey-Rico Carlos Bordons Julio EliasMorato Normey-Rico ∗ Carlos Bordons ∗∗ Julio Elias Normey-Rico Carlos Bordons ∗ cc˜ a o (( DAS), Universidade ∗ Departamento de Automa¸ a o eee Sistemas Sistemas DAS), Universidade ∗ Departamento de Automa¸ ∗ Departamento de cc˜ ˜ a Sistemas ((Brazil DAS), Universidade Federal de Santa Catarina, Florian´ polis-SC, de Automa¸ Automa¸ ˜ ao o eo Sistemas DAS),(marcelomnzm Universidade at ∗ Departamento Federal de Santa Catarina, Florian´ o polis-SC, Brazil (marcelomnzm at Departamento de Automa¸ c ˜ a o e Sistemas ( DAS), Universidade Federal de Santa Santa Catarina, Florian´ opolis-SC, polis-SC, Brazil (marcelomnzm (marcelomnzm at gmail.com, paulorcm at hotmail.com, julio.normey at ufsc.br). Federal de Catarina, Florian´ o Brazil at gmail.com, paulorcm at hotmail.com, julio.normey at ufsc.br). ∗∗ Federal de Santa Catarina, Florian´ opolis-SC, Brazil a(marcelomnzm at gmail.com, paulorcm at hotmail.com, julio.normey at ufsc.br). de Ingenier´ ıa de Sistemas y Autom´ tica, Universidad gmail.com, paulorcm at hotmail.com, julio.normey at ufsc.br). ∗∗ Departamento de Ingenier´ ıa de Sistemas yy Autom´ a tica, Universidad ∗∗ gmail.com, paulorcm at hotmail.com, julio.normey at ufsc.br). ∗∗ Departamento Departamento Ingenier´ ıa de Sistemas Autom´ a tica, Universidad de Sevilla, Spain (bordons at us.es). Ingenier´ ıa de Sistemas y Autom´ atica, Universidad ∗∗ Departamento de Sevilla, Spain (bordons us.es). Departamento de Ingenier´ ıa de Sistemas y Autom´ de Sevilla, Sevilla, Spain (bordons at at us.es).atica, Universidad Spain (bordons at us.es). de Sevilla, Spain (bordons at us.es). Abstract: This study presents a Filtered Model-based Predictive Control method for the FaultAbstract: This study presents Model-based Predictive Control method for FaultAbstract: This study study presentsofa aaFiltered Filtered Model-based Predictive Control method for the thesources FaultTolerant Energy Management sugarcane microgrid; such plant has several renewable Abstract: This presents a Filtered Model-based Predictive Control method for the Faultsources Tolerant Energy Management of a sugarcane microgrid; such plant has several renewable Abstract: This study presents aa Filtered Model-based Predictive Control method for the FaultTolerant Energy Management of sugarcane microgrid; such plant has several renewable sources (solar, wind and biomass power), being subject to different operational constraints and load Tolerant Energy Management of a sugarcane microgrid; such plant has several renewable sources (solar, wind and biomass power), being subject to different operational constraints and load Tolerant Energy Management of a sugarcane microgrid; such plant has several renewable sources (solar, wind and biomass power), being subject to different operational constraints and load demands. The proposed control policy ensures that these demands are met at every iteration, (solar, wind and biomass power), being subject to different operational constraints and load met demands. The proposed control policy ensures that these demands are at every iteration, (solar, wind and biomass power), being subject to different operational constraints and load demands. The proposed control policy ensures that these demands are met at every iteration, despite the presence of faults, coordinating which energy source to use, maximizing the use the demands. The proposed control policy ensures that thesesource demands are maximizing met at everythe iteration, despite the presence of faults, coordinating which energy to use, use the demands. proposed control policy ensures that these demands are maximizing metisatsynthesized everythe iteration, despite the theThe presence oftofaults, faults, coordinating which energy source tocontroller use, maximizing the usewith the renewables according contract rules. The proposed predictive despite presence of coordinating which energy source to use, use the renewables according to contract rules. The proposed predictive controller is synthesized with despite the according presence of faults, coordinating which energy source tocontroller use, maximizing theVarying usewith the renewables to contract rules. The proposed predictive is synthesized a fault-free model of the plant and coupled with a feedback low-pass Linear Parameter renewables according to contract rules. The proposed predictive controller is synthesized with model of the plant and coupled with aa feedback arenewables fault-free low-pass Linear Parameter Varying according to contract rules. The proposed predictive controller is synthesized with a fault-free model of the plant and coupled with feedback low-pass Linear Parameter Varying )) filter that is according to the level of faults faults detected uponParameter the system. system. Such a(LPV fault-free model of scheduled the plant and coupled with a feedback low-pass Linear Varying level of detected upon the Such filter that is scheduled according to the a(LPV fault-free model ofto thea plant and coupled a feedback low-pass Linear Parameter Varying (LPV )) is filter that scheduled according to the level of faults detected upon the Such system compared standard predictive controller, much improved behavior. (LPV filter that is is scheduled according to with the level of displaying faults detected upon the system. system. Such system is compared to a standard predictive controller, displaying much improved behavior. (LPV filter that is to scheduled according to the level of displaying faults detected the system. Such system) is is compared to a standard standard predictive controller, displaying muchupon improved behavior. system compared a predictive controller, much improved behavior. © 2019, is IFAC (International Federationpredictive of Automatic Control) displaying Hosting by Elsevier Ltd. All rights reserved. system compared to a standard controller, much improved behavior. Keywords: Control; Model Model Predictive Predictive Control; Control; Microgrids. Microgrids. Keywords: Fault Fault Tolerant Tolerant Control; Keywords: Keywords: Fault Fault Tolerant Tolerant Control; Control; Model Model Predictive Predictive Control; Control; Microgrids. Microgrids. Keywords: Fault Tolerant Control; Model Predictive Control; Microgrids. 1. INTRODUCTION mentioned, anyhow, but their scope is much more limited 1. INTRODUCTION mentioned, anyhow, but their scope is much more limited 1. mentioned, anyhow, but their scope is much more limited (focused on specific energy subsystems) than the problem 1. INTRODUCTION INTRODUCTION mentioned, anyhow, but their scope is much more limited (focused on specific energy subsystems) than the problem 1. INTRODUCTION mentioned, anyhow, but their scope is much more limited (focused on specific energy subsystems) than the problem faced by this work: (Maharjan et al., 2010) presents the (focused on specific energy subsystems) than the problem The efficiency of energy generation is of utmost importance faced by this work: (Maharjan et al., 2010) presents the The efficiency of energy generation is of utmost importance (focused on specific energy subsystems) than the problem faced by this work: (Maharjan et al., 2010) presents the The efficiency of energy generation is of utmost importance fault-tolerant control of a battery; in (Odgaard et al., faced by this work: (Maharjan et al., 2010) presents in terms of the search for an eco-friendly development Theterms efficiency of energy generation is of utmostdevelopment importance fault-tolerant control of a battery; in (Odgaard et the al., in of the search for an eco-friendly faced bythethis work: (Maharjan et al.,of 2010) presents the fault-tolerant control of aaoperation battery; in (Odgaard et al., The efficiency of energy generation is of utmost importance in terms of the search for an eco-friendly development 2009), fault-tolerant wind turbines is fault-tolerant control of battery; in (Odgaard et al., and a sustainable future. The integration of renewables in terms of the search for an eco-friendly development 2009), the fault-tolerant operation of wind turbines is and a sustainable future. The integration of renewables fault-tolerant control of a battery; in (Odgaard et al., 2009), the fault-tolerant operation of wind turbines is in terms of the search for an eco-friendly development and a sustainable future. The integration of renewables presented; the patent Siri (2002) expose ideas for the fault2009), the fault-tolerant operation of wind turbines is to power systems can be a good alternative to reduce and a sustainable future. The integration of renewables presented; the patent Siri (2002) expose ideas for the faultto power systems can be a good alternative to reduce 2009), the fault-tolerant operation of wind turbines is presented; the patent Siri (2002) expose ideas for the faultand a sustainable future. The integration of renewables to power systems can be a good alternative to reduce tolerant operation of solar systems. presented; the patent Siri (2002) expose ideas for the faultgreenhouse emissions, but a problem to be solved is how to power systems can be a good alternative to reduce tolerant operation of solar systems. greenhouse emissions, but a problem to be solved is how presented; the patent Siri (2002) expose ideas for the faulttolerant operation of solar systems. to power systems can be a good alternative to reduce greenhouse emissions, but a problem to be solved is how tolerant operation of solar systems. to integrate these sources without losing efficiency and greenhouse but awithout problemlosing to be efficiency solved is how to integrateemissions, these sources and Therefore, this work motivated studying the design tolerant operation of is solar systems.by greenhouse but awithout problemlosing to be efficiency solved is how to these and this work is motivated by studying the design dispatchability. to integrate integrateemissions, these sources sources without losing efficiency and Therefore, Therefore, this work is motivated by studying the design dispatchability. of a fault-tolerant EMS for an energy producer based on this work is motivated by studying the design to integrate these sources without losing efficiency and Therefore, dispatchability. of a fault-tolerant EMS for an energy producer based on dispatchability. Therefore, this work is motivated by studying the design of sugarcane a fault-tolerant fault-tolerant EMS plant for an an that energy producer based on a processing has biomass, biogas, of a EMS for energy producer based on The problem of managing energy systems that combine dispatchability. aaof sugarcane processing plant that has biomass, biogas, The problem of managing energy systems that combine a fault-tolerant EMSenergy for anas energy producer based on sugarcane processing plant that has biomass, biogas, The problem of managing energy systems that combine solar and wind power primary energy sources. a sugarcane processing plant that has biomass, biogas, different renewable sources is a significant issue to be invesThe problem of managing energy systems that combine solar and wind power energy as primary energy sources. different renewable sources is a significant issue to be invesa sugarcane processing plant that has biomass, biogas, solar and wind power energy as primary energy sources. The problem of managing energy systems that combine different renewable sources is a significant issue to be invesThis microgrid has been visited in authors’ previous works and wind has power energy asinprimary energy sources. tigated, in order to to sources find the theisoptimal optimal (and most most sustainable differentin renewable a significant issue to be inves- solar This been visited authors’ previous works tigated, order find (and sustainable andetwind power energy primary energy sources. This microgrid microgrid has beenwhere visited in authors’ previous works different renewable sources is a significant issue to bePredicinves- solar tigated, in order to find the optimal (and most sustainable (Morato al., 2018), aaasin realistic simulation model This microgrid has been visited authors’ previous works possible) operation of such plants. For this, Model tigated, in order to find the optimal (and most sustainable (Morato et al., 2018), where realistic simulation model possible) operation of such plants. For this, Model PredicThis microgrid has been visited in authors’ previous works (Morato et al., 2018), where a realistic simulation model tigated, in order to find the optimal (and most sustainable possible) operation of such plants. For this, Model Predicwas developed. Such fault-tolerant EMS must guarantee (Morato et al., 2018), where a realistic simulation model tive Control (MPC ) based schemes have been proven to possible) operation of such plants. Forhave this,been Model Predicwas developed. Such fault-tolerant EMS must guarantee tive Control (MPC ) based schemes proven to (Morato et al., 2018), where a realistic simulation model was developed. Such fault-tolerant EMS must guarantee possible) operation of such plants. For this, Model Predictive Control (MPC based schemes been proven to that this microgrid its safe operation (meeting developed. Suchmaintains fault-tolerant EMS must guarantee be efficient and are ))able able to effectively effectively maxime profits and tiveefficient Controland (MPC based schemes have have beenprofits provenand to was that this microgrid its safe operation (meeting be are to maxime developed. Suchmaintains fault-tolerant EMS mustdespite guarantee 1 and that this microgrid maintains its safe operation (meeting tive Controland (MPC )able based schemes have beenprofits proven to was be efficient are to effectively maxime demands and producing energy as expected), the that this microgrid maintains its safe operation (meeting , as environmental benefits of such renewable microgrids be efficient and are able to effectively maxime profits and 1 demands and producing energy as expected), despite the environmental benefits of such renewable microgrids , as 1 that this microgrid maintains its safe operation (meeting demands and producing energy as expected), despite the be efficient and are able to effectively maxime profits 1 and , as environmental benefits of such renewable microgrids presence of faults in the internal subsystems. demands and producing energy as expected), despite the seen in (Valverde et al., 2012; Garcia-Torres and Bordons, environmental benefits of such renewable microgrids , as presence of faults in the internal subsystems. 1 seen in (Valverde et 2012; and and producing energy as expected), despite the presence of faults in the internal subsystems. environmental benefits of suchGarcia-Torres renewable microgrids , as demands seen inPetrollese, (Valverde et al., al.,Zhang 2012; Garcia-Torres and Bordons, Bordons, presence of faults in the internal subsystems. 2015; 2015; et al., 2015). seen in (Valverde et al., 2012; Garcia-Torres and Bordons, do so, work literature as follows: 2015; et presence ofthis faults incontributes the internaltosubsystems. seen (Valverde2015; et al.,Zhang 2012; Garcia-Torres To 2015;inPetrollese, Petrollese, 2015; Zhang et al., al., 2015). 2015). and Bordons, To 2015; Petrollese, 2015; Zhang et al., 2015). To do do so, so, this this work work contributes contributes to to literature literature as as follows: follows: To do so, this work contributes to literature as follows: Real instrumented systems are always susceptible to faults, 2015; Petrollese, 2015; Zhang et al., 2015). (1) sugarcane microgrid is modelled in faulty Real systems are susceptible to To doThe so, studied this work contributes to literature as follows: Real instrumented instrumented systems are always always susceptible to faults, faults, (1) The studied sugarcane microgrid is modelled in faulty which is also very true for energy systems. Possible faults Real instrumented systems are always susceptible to faults, (1) The sugarcane microgrid is in 2) and the global Fault Tolerant which is also very true for energy systems. Possible faults (1) conditions The studied studied(Section sugarcane microgrid is modelled modelled in faulty faulty Real instrumented systems are always susceptible to faults, conditions (Section 2) and the global Fault Tolerant which is also very true for energy systems. Possible faults on these systems may lead the plant not to comply with which is also very true for energy systems. Possible faults (1) The studied sugarcane microgrid is modelled in faulty conditions (Section 2) and anddetailed the global global Fault 3). Tolerant EMS problem is formally (Section on these systems may lead the plant not to comply with conditions (Section 2) the Fault Tolerant which is also very true for energy systems. Possible faults on these systems may lead the plant not to comply with EMS problem is formally detailed (Section 3). its demands, which might result in economic deprivation, on these systems may lead the plant not to comply with conditions (Section 2) and the global Fault Tolerant EMS problem is formally detailed (Section 3). (2) A model-based control policy is proposed its demands, which might in economic deprivation, EMS problem ispredictive formally detailed (Section 3). on these systems may leadresult the lack plant not to comply with its which might result in deprivation, (2) A model-based control policy is proposed environment-related issues and of power availability. its demands, demands, which might result in economic economic deprivation, EMS problem ispredictive formally detailed (Section 3). (2) A model-based predictive control policy is proposed environment-related issues and lack of power availability. for the faultless condition of the plant (Section 4.3), (2) A model-based predictive control policy is proposed its demands, which might result in economic deprivation, environment-related issues and knowledge, lack of of power power availability. for the faultless condition of the plant (Section 4.3), Nonetheless, up to the authors’ very few works environment-related issues and lack availability. (2) A model-based predictive control policy is proposed for the faultless condition of the plant (Section 4.3), which guarantees that demands are met and physical Nonetheless, up to the authors’ knowledge, very few works for theguarantees faultless condition of theare plant (Section 4.3), environment-related issues and lack of power availability. which that demands met and physical Nonetheless, up to the authors’ knowledge, very few works deal with fault-tolerant energy management of microgrids. Nonetheless, up to the authors’ knowledge, very few works for the faultless condition of the plant (Section 4.3), which guarantees guarantees that demands demands are are met met and and physical physical contraints are obeyed. deal with fault-tolerant energy management of microgrids. which that Nonetheless, up to the authors’ knowledge, very few works deal with fault-tolerant energy management of microgrids. contraints are obeyed. That is, with the design of autonomously-adjusting Energy deal with fault-tolerant energy management of microgrids. which guarantees that demands are met and physical contraints are obeyed. (3) Then, the use aa Linear Parameter-Varying (LPV )) That is, with the design of autonomously-adjusting Energy contraints are of obeyed. deal fault-tolerant energy management of microgrids. That is, the design of Energy (3) Then, the use of Parameter-Varying (LPV Management Systems s) that coordinate a given miThatwith is, with with the design(EMS of autonomously-adjusting autonomously-adjusting Energy contraints are is obeyed. (3) Then, the use of aa Linear Linear Parameter-Varying (LPV )) Management Systems (EMS s) that coordinate a given mifeedback filter proposed to be coupled to the MPC (3) Then, the use of Linear Parameter-Varying (LPV That is, with the design of autonomously-adjusting Energy Management Systems (EMS s) that thatSome coordinate ahave given mifilter is proposed to be coupled to the MPC crogrid, despite presence of faults. works to be Management Systems (EMS s) coordinate a given mi(3) feedback Then, the use of a Linear Parameter-Varying (LPV ) feedback filter is proposed to be coupled to the MPC loop (Section 4.5). Such filter is able to adapt itcrogrid, despite presence of faults. Some works have to be feedback filter is proposed to be is coupled to adapt the MPC Management Systems (EMS s) thatSome coordinate given loop (Section 4.5). Such filter able to itcrogrid, presence of works to be crogrid, despite despite presence of faults. faults. Some works ahave have tomibe feedback filter is proposed to be is coupled to adapt the MPC loop (Section 4.5). Such filter is able to adapt it self to both faulty and faultless situations, imposing loop (Section 4.5). Such filter able to itThe authors thank CNPq and de Econom´ y Competcrogrid, despite presence of Ministerio faults. Some works ıahave to be  self to both faulty faultless situations, imposing The authors thank CNPq and Ministerio de Econom´ıa y Competloop 4.5).and Such filter is able to according adapt it self to both faulty and faultless situations, imposing more/less conservatism to the closed-loop thank CNPq and de Econom´  The self to(Section both faulty and faultless situations, imposing itividad de Espa˜ na for financing the projects The authors authors thank CNPq and Ministerio Ministerio de CNPq401126/2014-5, Econom´ıa ıa y y CompetCompetmore/less conservatism to the closed-loop according itividad de Espa˜ n a for financing the projects CNPq401126/2014-5,  self to both faulty and faultless situations, imposing more/less conservatism to the closed-loop according itividad de Espa˜ n a for financing the projects CNPq401126/2014-5, The authors thank CNPq and Ministerio de Econom´ ıa y Competto the level of faults. CNPq303702/2011-7, 305785/2015-0 DPI2016-78338-R. more/less conservatism to the closed-loop according itividad de Espa˜ na forCNPq financing the projectsand CNPq401126/2014-5, to the level of faults. CNPq303702/2011-7, CNPq 305785/2015-0 and DPI2016-78338-R. 1 more/less conservatism to the closed-loop according to the level of faults. CNPq303702/2011-7, CNPq 305785/2015-0 DPI2016-78338-R. itividad de Espa˜ financing the projects CNPq401126/2014-5, A microgrid isnaa for set of generators, loadsand and storage units, as 1 to the level of faults. CNPq303702/2011-7, CNPq 305785/2015-0 and DPI2016-78338-R. A microgrid is a set of generators, loads and storage units, as 1 A microgrid is a set of generators, loads and storage units, as to the level of faults. 1 CNPq303702/2011-7, CNPq 305785/2015-0 DPI2016-78338-R. proposed by Lasseter and Paigi (2004). loadsand A microgrid is a set of generators, and storage units, as proposed by Lasseter and Paigi (2004). proposed by Lasseter and Paigi (2004). 1 proposed by Lasseter andofPaigi (2004). A microgrid is a set generators,

proposed by Lasseter and Paigi (2004). loads and storage units, as proposed Lasseter and(International Paigi (2004).Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. 2405-8963by © 2019, IFAC Copyright © 2019 IFAC 166 Copyright ©under 2019 IFAC 166Control. Peer review responsibility of International Federation of Automatic Copyright © 166 Copyright © 2019 2019 IFAC IFAC 166 10.1016/j.ifacol.2019.06.055 Copyright © 2019 IFAC 166

2019 IFAC DYCOPS Florianópolis - SC, Brazil, April 23-26, 2019 Marcelo Menezes Morato et al. / IFAC PapersOnLine 52-1 (2019) 166–171

(4) High-fidelity simulation results are presented to assess the performance of the proposed fault-tolerante EMS policy (Section 5), which is compared to other standard techniques. Synthetically, this work proposes the use of a filtered predictive controller, as a Fault Tolerant EMS. Overall good results are obtained with such scheme; conclusions are drawn in Section 6. 2. CONSIDERED MICROGRID Firstly, the studied microgrids has to be detailed more minutely: it is based on a sugarcane power plant and has been modelled in (Morato et al., 2018). This plant is composed of the following subsystems: two boilers, with different efficiencies; two steam turbines, with different efficiencies; a combined heat and power system (CHP ); a water chiller; a hot water tank; photovoltaic panels; water heating solar (WHS ) panels; a wind turbine; two pressure reduction valves; one heat exchanger; stocks of bagasse, straw and compressed biogas and a battery bank. The use of these intermediate stocks allows the system to accumulate energy when the renewable generation is high and use it when the renewables are not sufficient. The inputs to this microgrid, from an energy management point-of-view, are given by uj (set-points to the internal subsystems), and the measured outputs yj stand for the available transmission lines (electric network, gas pipes, steam lines, etc). This plant’s operation is coordinated by a hierarchical EMS : i) a supervisory controller determines hourly set-points for each subsystem, such that the plant demands (these are explained in the sequence) are met and energy generation is maximized, while ii) the lower-level loops guarantee reference tracking within each hour. 2.1 Microgrid Demands and Operational Constraints The studied microgrid demands are:

167

2.2 Faulty Plant Model This studied power plant is subject to multiple faults. These faults can represent different situations, such as: the accumulation of debris and residues on the boilers; a bearing temperature reduction on the turbines; a pressure valve malfunction; oil leakages on pressure valves; power-house management issues; bottleneck clogging on water flow and even emergency stops from manual operators. These possible faults have been minutely detailed in (Morato et al., 2019), based on historical data and empirical observations. To model these faults, this work opts for multiplicative loss of effectiveness factors αi upon each subsystem j. This kind of representation has been firstly presented in (Hern´andez-Alc´antara et al., 2016), and introduces a solid framework for the modelling of actuator faults. Roughly speaking, it is assumed that the EMS sets set-points uj for the subsystems that are not properly tracked and, thus, the actual output from these subsystems are given, individually, by αj .uj . Then, the (faulty) control-oriented microgrid model is: uf (k)

   x(k + 1) = Ax(k) + B1 w(k) + B2 Λ(k)u(k) ,

z(k) = C1 x(k) + D11 w(k) + D12 Λ(k)u(k) ,

y(k) = C2 x(k) + D21 w(k) + D22 Λ(k)u(k) , where: Faults upon subsys. j

ufj (k) =

   αj (k)

The operational contraints of this plant are given with full details in (Morato et al., 2018). In fact, these are related to the physical properties of each subsystem and are mathematically expressed as saturation limits and slew-rate constraints upon the states x and control policies uj . As an example: the higher-efficiency turbine unit must operate within 5 to 20 MW, which converts into 5 ≤ u1 (k) ≤ 20, and increase / decrease power, at most, by 5 MW per hour, which gives −5 ≤ u1 (k) − u1 (k − 1) ≤ 5. 167

.uj (k) ,

  α1 (k) 0 . . . u1 (k) 0 0   u2 (k)   0 α2 (k) . . .  .  . uf (k) =  .. ..  ... . . 0   ..  0 0 . . . α13 (k) u13 (k)       

Λ(k)

(1) due to the sugar and ethanol production process, to produce steam in different pressures (used to boost shredders, spray pumps and other equipments), as well as cold water (internal refrigeration needs); (2) to be autonomous, the plant must sustain itself, which converts into, in average, 8000 kW of electric power to be made available; (3) to sell excedent electric energy to a local Distribution Network Operator (DNO). This is regulated by a contract that defines the goal of 753 M W h of sold energy per day; this generation can be continuous, maintaining a constant value, or time-varying, with different values at each hour of the day.

(1)

(2)

(3)

u(k)

Remark 1. The diagonal matrix Λ(k) stands for the collection of faults (on each subsystem) αj (k). If αj (k) = 1, subsystem j is completely healthy, whereas if αj (k) = 0, a complete failure/breakdown has occurred in subsystem j. The system state vector x is defined as the collection of the normalized percentages of the microgrid storage units, i.e. x(k) = col{xnx (k)} for nx = 1 , . . . , 5. Respectively, these units are: the battery bank, bagasse stock, straw stock, biogas stock and the hot water tank. In Eq. (1), the input vector u stands for zero-order-held set-points for the internal subsystems. This set-points are determined by the supervisory EMS every Ts = 1 h and tracked by the lower-level controllers within this period (if no fault occurs). The complete control vector is the collection of the thirteen set-points to each subsystems j, i.e. u(k) = col{uj (k)}. Respectively, these set-points uj are related to the following subsystems: higher-efficiency turbine (“A”), lower-efficiency turbine (“B”), power house (that regulates the total energy power generation), steamboiler, high-to-middle pressure reduction valve, middlepressure steam escape valve, low-pressure steam escape

2019 IFAC DYCOPS 168 Marcelo Menezes Morato et al. / IFAC PapersOnLine 52-1 (2019) 166–171 Florianópolis - SC, Brazil, April 23-26, 2019

valve, CHP unit, water chiller, heat exchanger, battery bank energy flow, middle-to-low pressure reduction valve, hot water escape valve. In the used model, two different outputs are considered: i) the available measurements, which in fact coincide with the states, i.e. y(k) = x(k); ii) the controlled outputs z(k) = col{znz (k)} (for nz = 1 , . . . 5) are those related to the microgrid demands, detailed in Section 2.1, respectively: flow of low pressure steam ( Mhg ), flow of middle pressure steam ( Mhg ), flow of chilled water 3 ( mh ), power available for the internal needs (kW ), power to be sold to the external network (kW ) according to contract; note that some of these outputs are not directly measurable and, therefore, not fed back to the EMS ; anyhow, system (1) is controllable and observable. The disturbances to the system w are defined as the collection of renewable sources available to the microgrid, i.e. w(k) = col{wnw (k)} for nw = 1 , . . . , 5 and respectively as: the wind speed ( km h ) present in the canebrakes, the W solar irradiance ( m 2 ) upon the (photovoltaic and water heating) solar panels; and the bagasse, straw and biogas incomes ( Mhg ) from the canebrakes to the processing units.

(2) Design a model-based predictive controller that reads yf (k), taking Gn (z) as model, such that a cost function J is minimized considering the system constraints and demands (see Section 2.1). This step is coherent with (Morato et al., 2017) and a-priori yields good results in faultless situations. (3) Then, find a scheduling parameter 0 ≤ σ(k) ≤ 1, a bounded scalar that represents the level of faults on the system, computed according to Gf (z), which allows one to directly quantify how much have the faults deteriorated the microgrid. (4) Finally, find a low-pass LPV filter F that filters the measured outputs y(k), giving yf (k). This filter must vary according to σ(k), being scheduled by this parameter. For faultless situations, F is an identity matrix. This is the main inovation of this work, as it F is able to deal with faults in a very simple manner.

3. FAULT-TOLERANT ENERGY MANAGEMENT Fig. 1. Outline of Problem and Proposed Solution Assumption 1. Faults upon the microgrid, represented by Λ(k), are adequately estimated by an online Fault Detection and Diagnosis FDD scheme. This means that the fault-tolerant controller that is designed herein has access to Λ(k) at every iteration. This FDD scheme can be based on parity-space approaches, as in (Odendaal and Jones, 2014), or even on LPV methods, as the one recently proposed by authors (Morato et al., 2019). Assumption 2. The external disturbances w(k) are, basically, renewable sources and, thereby, derived from meteorological data (solar irradiance and wind on the canebrakes). As extensively discussed in (Vergara-Dietrich et al., 2017), such curves can be estimated reasonably well within 24 h horizons. Therefore, it is assumed that the EMS has access to some estimation of these curves, w(k) ˆ for the next 24 h. Problem 1. Given the accurate knowledge of Λ(k), find a hourly-discrete controller (EMS ) that determines the set-points u(k) to the microgrid subsystems, so that the controlled outputs z(k) abide by the demands (presented in Section 2.1), despite the presence (and magnitude) of faults on each subsystem αj (k) and the disturbances w(k). 4. LPV -FILTERED PREDICTIVE CONTROL The following Procedure presents the proposed solution to Problem 1, which is depicted via block-diagrams in Fig. 1: Procedure 1. A Fault-Tolerant EMS for the studied problem can be designed by performing the four following steps: (1) Firstly, find a cascaded model of the system, for which the faults are decoupled from the main dynamics. This is G(z) = Gn (z).Gf (z), where G(z) stands for the complete faulty model, Gn (z) for a nominal fault-free model and Gf (z) for the fault model. This allows one to clearly separate the microgrid internal dynamics from the fault events. 168

Remark 2. Proof that this Procedure guarantees the expected stability and performance goals is immediate from linear superposition and linear differential inclusion. The use of the filter to overcome faults is likewise the one in (Normey-Rico et al., 2015) used to compensate dead-time. Yet, stability must be a priori guaranteed at both extreme situations, at σ = 0 (100 % failure) and σ = 1 (faultless). 4.1 Why a Feedback Filter? Nowadays, the state-of-practice of most sugarcane processing industries 2 consists of control policies derived from high-level automata or PLC s, see (Morato et al., 2019). Thus, the use of a feedback filter is much more suitable than controller reconfiguration approaches, once it can be directly added to such PLC s; moreover, the filter option is less costly, as it has smaller implementation complexity and can be done with a single additional microcontroller. 4.2 Step (1): Fault Decoupling Considering the (faulty, complete) microgrid model (G(z)) described by the state-space representation in Eq. (1), the fault decoupling procedure is quite straightforward: the fault model Gf (z) is the fault distribution matrix Λ(k), whereas the nominal plant model Gn (z) is given by Eq. (1), while taking uf (k) as inputs. To illustrate this point, Fig. 2 shows the decoupled model: the transfer from u to uf corresponds to Gf (z), while the one from uf to y stands for Gn (z); finally the transfer from u to y stands for the complete model G, which is by definition given by Gn (z).Gf (z) (cascaded models), and thereby abides to the fault decoupling prodecure. 2

Specially in Brazil, the world’s largest sugarcane producer.

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4.5 Step (4): LPV Filter Synthesis

Fig. 2. Fault decoupling of the studied microgrid 4.3 Step (2): Predictive Controller Synthesis In order for the energy plant to abide to the energy production rules, supplying the DNO with 753 M W h per day, the following stage cost is established: J=

N c −1 i=0

+

Np −1

 i=0

||D(k + i)||2QP + ||u(k + i)||2Qu

(4)

||ˆ x(k + i) − x ˆref (k + i)||2Qx ,

 Econ − Esum (k + i) , (5) Ts where Esum represents the electric energy that has already been produced; Econ stands for the energy sales contract value; ESell (k) is the total energy sold to the DNO, given by the fifth controlled output z5 (k) and computed according to the fault-free model Gn (z); x ˆref is a state reference; Np = 12 h represents the prediction horizon, while Nc = 5 h represents the control horizon. Note that (Econ − Esum ) represents how much energy the microgrid still has to produce until the end of the day, due to contract with the DNO. 

D(k + i) = ESell (k + i) −

The MPC synthesis, then, resides in minimizing J every Ts = 1 h., considering Gn (z), such that the following constraints are obeyed (equivalent to those in Section 2.1): xj ≤ xj (k + i + 1) ≤ xj

uj ≤ uj (k + i) ≤ ui

∂uj ≤ uj (k + i) − uj (k + i − 1) ≤ ∂ui

(6)

The LPV filter is, essentially, the core of this work. Such filter, considering the chosen scheduling signal σ, must be low-pass to decelerate the EMS control policy u in the case of faults (i.e. σ closer to 0), ensuring more conservatism to the closed-loop. To achieve such goal, two conditions have to be obeyed for its synthesis: i) it must be an identity block when σ = 1 (maintain solely the MPC ); and ii) it should be a low-pass filter with time constant τ when σ = 0 (imposing conservatism). These conditions are expressed mathematically below, with abuse of notation: 1 (i) F (·)|σ=1 = 1 , (ii) F (·)|σ=0 = z−1 . (11) τ Ts + 1 Remark 3. Condition (ii) is, simply, a first-order Eulerdiscretized filter, with sampling period of Ts . Therein, similarly to (Normey-Rico and Camacho, 2009), τ is a design parameter that changes the filter’s bandwidth and, thus, enhances robustness. In the literature (Santos et al., 2016), it has been noted that the stability conditions of schemes such as the one in Fig. 1 depend on the used feedback filter: a larger τ imposes more conservatism, while slows the controlled output’s response (which is expected due to the trade-off between robustness and performance). This feedback filter should become more low-pass in stronger faulty situations (smaller σ) and, less when small faults occur (σ closer to 1). Moreover, it must filter the measured outputs y(k), giving the filtered outputs yf (k). For such goal, the following LPV filter arises:     τ + σTs − Ts Ts − σTs xf (k + 1) = xf (k) + y(k) , τ τ yf (k) = (1 − σ)xf (k) + σy(k) , (12)

where xf are the filter’s internal states. To illustrate its behaviour, Fig. 3 shows the filter’s frequency response and how it varies, as expected, according to σ.

(7) (8)

zn (k + i) = Demands(k + i) for n = 1 , . . . , 4 (9) for i = 0, . . . , Np − 1, where Qu , QP and Qx are adequate positive definite weighting matrices 3 . 4.4 Step (3): Scheduling Parameter Now, the scheduling parameter σ(k), that measures the level of faults on the microgrid is presented: Tr{Λ(k)} , (10) size{Λ(k)} which is a known signal bounded within the set [0 , 1]. Notice how this scheduling signal varies as expected: for completely faultless situations, Λ(k) = diag{113 } and, thus, σ(k) = 1; for a complete failure on all subsystems, Λ(k) = diag{013 } and, thus, σ(k) = 0. One can directly see the level of faults on the microgrid through σ. σ(k) =

3

This cost function J is similar to the one used in (Morato et al., 2017); values used for Nc ,Np , Ts and weighting variables and other details are given therein. Remark that the variables in J are normalized by the weighting matrices, so to achieve adequate results.

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Fig. 3. Filter: Bode Diagram 4.6 Implementation of the proposed Fault-Tolerant EMS The implementation of the complete proposed scheme is the coupling of the MPC with the LPV feedback filter. Note that MPC policies make use of an open-loop plant model to make predictions of the future behaviour of the plant, according to future control action (to be

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computed), measured outputs and disturbance predictions w(k). ˆ The case in the proposed scheme is the same, but the model taken by the MPC is the fault-free one, Gn (z). Furthermore, to make future predictions (at k + i, from k), the measurements y(k) must be available but, in the proposed scheme, the filtered outputs yf (k) are taken instead. Therefore, the filter acts to decelerate (or accelerate) the action of the predictive controller, according to the level of faults. This converts, in practice, into a re-coordination of the plant by the MPC that senses that the outputs are not behaving as expected. Notice that the filter itself is a simple add-on to a closed MPC block. This whole implementation of the proposed scheme is illustrated by Figure 4.

Table 1. Effect of τ on performance τ 1 2 3 4 5 10 15

J( %) 0.03 0.2 0.4 0.6 0.77 3.19 5.23

FT No Yes Yes Yes Yes Yes Yes

DC Yes Yes Yes Yes No No No

Ms 1.7484 1.7031 1.5486 1.4824 1.4456 1.3475 1.3268

Conservative No No OK Best Yes Yes Yes

5.2 Scenario 2: Fault-Tolerant Energy Management For the second scenario (main results), τ is fixed at 4 h and a day with average sun and wind is considered, refer to (Vergara-Dietrich et al., 2017). The faults now occur only in the main energy generation subsystem (Turbine “A”, at t = 10 h, Turbine “B”, at t = 7 h and CHP, t = 10 h), and represent, for instance, effectiveness losses due to a reduction of the bearing temperature, see Figure 5. Note that the faults that occur on the CHP and Turbine “A” are concomitant, to require further FT control action.

Fig. 4. Implementation Outline 5. RESULTS AND ANALYSIS This Section presents some simulation results in order to illustrate the effectiveness of Fault-Tolerant Energy Management System, proposed in Procedure 1. All the strategies presented herein were synthesized in Matlab. 5.1 Scenario 1: Filter Tuning

Fig. 5. Scenario 2: Simulated Faults

It has to be shown how the filter parameter τ was chosen, so that good results in terms of fault-tolerance can be achieved. For this test, multiple faults were randomly simulated on all energy subsystems, Λ(k) ∈ [013 , 113 ], considering a daily scenario with no renewable generation (cloudy and windless). Table 1 synthesizes the obtained results, showing the influence of τ (given in hours) upon the average values obtained for J (related to economic performance) and module margin (robustness) and whether the system presents fault-tolerance (FT, i.e. if the strategy acts to compensate the effect of the faults, when these are sensed by the FDD scheme, by re-distributing the energy generation goal between the subsystems), the sufficient compliance to the internal demands (DC ) and if it is too conservative (qualitatively) 4 . The overall best results were obtained with τ = 4 h, where good a compromise is achieved: performances are not deteriorated too much (small variation from the Jnom ), the system is able to overcome the effect of faults, demand compliance is maintained and a sufficiently good robustness margin is obtained.

The achieved control results are given by Figures 6 and 7, which stand, respectively, for the performance of the (faulty) energy generation subsystems and global energy production throughout the day. Therein, FTC stands for the proposed filtered predictive controller and, for comparison goals, SMPC stands for the “Standard ” MPC from (Morato et al., 2017), that solves the same minimization problem J but has no feedback filter F (z, σ) and, thus, does not consider the effect of the faults. Note that, if no faults occur (Λ(k) = I13 , F as an unitary gain) would lead the operation of both FTC and SMPC to be the same.

4 J: Values given in percentage, in respect to the nominal case, Jnom = 1.877e12 , with no feedback filter; Ms is computed with the unconstrained MPC, as the inverse of the minimal distance in the Nyquist diagram to the critical point (−1, 0j), with σ = 0, as defined in (Sideris and Pena, 1988); As remarked in many robust control textbooks, good values for this margin are Ms < 2.

Fig. 6. Scenario 2: Energy Generation Subsystems

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Evidently, as it can be seen in Figures 6 and 7, the FTC approach is able to overcome the effect of faults much faster and with ease (changing set-points as soon as faults are detected), being able to produce the contract

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Fig. 7. Scenario 2: Energy Production energy goal Econ , at the end of the day, even though some energy systems fail. The FTC acts to re-distribute this goal between the subsystems: it dictates Turbine “B” to produce more energy, while reducing the generation by the CHP. Note that from instant t = 7 h (first fault), the FTC approach manipulates the set-point in order to ensure that the end-of-the-day energy goal is achieved. On the other hand, SMPC takes much more time to perceive and compensate the fault effects (faults are overlapped when uF T C = uSM P C , around 5 h after the first fault, treated by the SMPC as disturbance rejection). This can be complicated, as the system does not comply with the energy needs, which may imply in profit loss. Figure 7 makes the advantages of the use of the feedback filter very clear: with it, in a faulty situation, the microgrid is able to produce almost 100 MWh of extra energy. Note that the SMPC takes too long to try to compensate the faults that occur, once it only senses these faults after their effect appear on the measured outputs y. Indeed, this happens after the dynamics of the plant itself translates the fault to y. The FTC approach, much on the contrary, is adjusted instantaneously by the FDD scheme via its filter and, thereby, is able to adequately re-coordinate the microgrid and guarantee demand compliance. 6. CONCLUSION This work presented the issue of managing a sugarcane microgrid that is subject to faults on its subsystems. Considering that these faults are accurately detected, an LPV Filtered Predictive Controller is designed to re-coordinate the plant in faulty conditions, while always attending to the demands and producing the defined amount of energy. The results enlighten the interest of the proposed LPV Filter plus MPC paradigm to the development of FaultTolerant Energy Management Systems. Results show that the proposed scheme can accurately re-adjust the control law so that effects of faults are mitigated. For future works, analysis shall presented for the case when both renewable disturbances and faults are badly estimated. REFERENCES Garcia-Torres, F. and Bordons, C. (2015). Optimal economical schedule of hydrogen-based microgrids with hybrid storage using model predictive control. Industrial Electronics, IEEE Transactions on, 62(8), 5195–5207. Hern´ andez-Alc´ antara, D., Tud´ on-Mart´ınez, J.C., Am´ezquita-Brooks, L., Vivas-L´ opez, C.A., and Morales-Men´endez, R. (2016). Modeling, diagnosis and estimation of actuator faults in vehicle suspensions. Control Engineering Practice, 49, 173–186. 171

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