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Decentralized Multiparametric Model Predictive Control for Domestic Combined Heat and Power Systems Nikolaos A. Diangelakis,†,‡ Styliani Avraamidou,†,‡ and Efstratios N. Pistikopoulos*,‡,§ †

Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, SW7 2AZ London, U.K. Artie McFerrin Department of Chemical Engineering and §Texas A&M Energy Institute, Texas A&M University, College Station, Texas 77843, United States

‡

S Supporting Information *

ABSTRACT: In an effort to provide affordable and reliable power and heat to the domestic sector, the use of cogeneration methods has been rising in the past decade. We address the issue of optimal operation of a domestic cogeneration plant powered by a natural gas, internal combustion engine via the use of explicit/multiparametric model predictive control. More specifically, we take advantage of the natural division of a combined heat and power (CHP) cogeneration system into two distinct but interoperable subsystems, namely, the power generation subsystem and the heat recovery subsystem, in order to derive a decentralized, two-mode model predictive control scheme that specifically targets the production of either electrical power or usable heat at a given time. We follow our recently developed PAROC framework for the design of the controllers, and we apply it in a decentralized manner. We show how the CHP system can efficiently operate in both modes of operation through closedloop validation of the control scheme against a high-fidelity CHP process model.

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INTRODUCTION The provision of affordable electrical and thermal power to the domestic and residential sector has been a challenging problem for decades. From the perspective of the end-user it is easily understood that the two main operational objectives are the minimum cost of power and minimum outage. To date, the provision of power in the residential sector takes place in a centralized manner in which a central grid also provides power based on the needs of the households, upon demand. The provision of thermal power happens in a similar way where a central natural gas grid provides the fuel needed to cover the thermal need of the households independently. Individual boilers are then utilized to transform the chemical power of the fuel into heat in the form of hot water, suitable for space heating and hot utility usage. The concept of the centralized approach is depicted in Figure 1.

such as domestic cogeneration units, as well as renewable resources, such as solar, photovoltaic, and wind power, for covering the heat and electrical power demand (Figure 2). Although the aforementioned resources act as a supplement to the central power and natural gas grids, they provide not only a more environmentally friendly alternative but also the potential for a more autonomous, efficient, and cost-effective power provision. Among the technologies presented in Figure 2, combined heat and power (CHP) systems have the potential to replace the conventional processes used to date for the production of usable heat and electricity. CHP systems utilize the same amount of fuel for the generation of both electrical power and usable heat, emitting consequently lower amounts of exhaust gases. Therefore, owing to their environmentally friendly and cost-effective nature, CHP systems can play a dominant role in emission reduction strategies.1 Several works have been published on the investigation of the operation and supply related aspects of the CHP operation on a domestic level. In Table 1 we present examples of recent works of CHP modeling and CHP supply problems. A variety of technologies that can potentially be applied in a residential scale cogeneration system are discussed. To identify and decide on the desired cogeneration technology, factors such as (a) the prime mover of the cogeneration plant, (b) its most usual application, (c) its reliability, (d) maintenance cost, (e) fuel type, as well as general advantages of its application

Figure 1. Centralized approach to the provision of electricity and hot water in the domestic sector.

Although the aforementioned approach provides a robust way of providing heat and power to the domestic sector in terms of power and heat shortages, its potential drawbacks, namely, (a) the environmetal impact from the use of fossil fuels and (b) the end user cost, have led to the consideration of alternative solutions. These include alternative power sources, © XXXX American Chemical Society

Special Issue: Sustainable Manufacturing Received: September 7, 2015 Revised: October 20, 2015 Accepted: October 26, 2015

A

DOI: 10.1021/acs.iecr.5b03335 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

Figure 2. Decentralized approach to the provision of electricity and hot water in the domestic sector.

design structures and algorithms were developed.16−20 After the 1990s, because of the advances in convex optimization along with other computational techniques, interest in decentralized MPC control again increased. Many efforts were then devoted to developing design methods guaranteeing stability and performance. Examples of methods developed include those based on optimization,21,22 overlapping decompositions,23−25 vector Lyapunov functions,26 and sequential design.15,27 Even though the problem of decentralized MPC can just be solved with regular MPC algorithms neglecting any interaction between subsystems, a small number of decentralized MPC algorithms have been developed to date that can guarantee stability and performance. A stabilizing decentralized MPC algorithm for nonlinear discrete time systems was presented in ref 28. This algorithm obtains closed-loop stability, relying on the inclusion of contractive constraint in the formulation of the MPC problem. Other approaches developed to stabilize DMPC controllers for nonlinear discrete-time systems treat the interactions between subprocesses as disturbances.29 In our work, we take advantage of the inherent dual nature of the domestic CHP system in order to partition it into two subsystems based on its operational and modeling principles, which are discussed in the next section. The rest of the paper discusses the decentralized explicit control approach and framework that was applied to the system. The results are presented for the two distinct operating modes.

Table 1. Small-Scale Cogeneration Modelling and Power Supply Review authors (year)

contributions

Wu and Wang (2006)2 Onovwiona and Ugursal (2006)3 Savola and Keppo (2005)4 Videla and Lie (2006)5 Konstantinidis et al. (2010)6 Diangelakis et al. (2014)7 Savola and Fogelholm (2007)8 Menon et al. (2013)9 Fazlollahi et al. (2012)10

review on combined cooling, heat, and power cogeneration systems technology review on heat and power cogeneration systems

Mehleri et al. (2011, 2012)11,12 Ondeck et al. (2015)13

small-scale CHP model development for the investigation of its thermodynamic operation in part loads modeling and dynamic simulation of small-scale internal combustion-based CHP systems multiparametric model predictive control of small-scale CHP systems suitable for domestic and residential use first principle modeling and design optimization of an internal combustion-based residential scale CHP system MINLP formulation for increasing the power production of small-scale biomass-fueled CHP systems optimal design of polygeneration systems, including CHP technologies, under optimal control assumptions multiobjective optimization techniques for the optimal design of complex energy systems, including CHP technologies distributed energy systems optimization, based on superstructure approach optimal operation of a residential CHP and PV for electricity, cooling, and heating

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RESIDENTIAL CHP SYSTEM Any cogeneration system can be normally viewed as composed of two distinct subsystems, (a) the power generation subsystem and (b) the heat recovery subsystem, which continuously interact with each other to produce usable heat and power through a single fuel source (Figures 4). The power generation subsystem of a typical residential CHP system consists of the throttle valve, the inlet and exhaust manifolds, and the internal combustion engine which is coupled to an electrical generator through a crankshaft. The heat recovery subsystem involves two levels that facilitate the

related to the specific of its application need to be taken into account.7 We present a framework for the design of decentralized advanced model predictive control scheme based on the model developed in our earlier works.7 Decentralized Model Predictive Control. Figure 3 presents helpful control problem structures and decentralization policies for model predictive control (MPC) problems. Earlier works on decentralization were mainly focused on the stability of decentralized linear control of large-scale systems with interactions between subsystems, and various control B


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Figure 3. Different types of control problem structures: centralized, decentralized, distributed, and hierarchical structure14,15

Figure 4. Concept of cogeneration.

(ii) Inlet water flow rate: The inlet water mass flow rate is the amount of water that runs through the heat recovery subsystem at any given time throughout the operation of the process. It affects (a) the amount of hot water produced and (b) the temperature of the produced water. More specifically, as the electrical power production of the system changes in time, the flow of the water through the heat recovery subsystem determines the amount of byproduct heat that is recovered through the heat exchangers; therefore, both the temperature of the water and its flow rate are affected. Domestic cogeneration systems can be considered as multiproduct processes. More specifically, the process cannot produce at the same time (a) electrical power at a desired level and (b) hot water of certain temperature and flow rate. The reason behind this is that the operation of the system is restricted by the electrical and heat efficiency of the prime mover, in this case the internal combustion engine. Figure 6 represents the ratio between the electrical power and the usable heat throughout the system’s operation. In other words, consider the following example assuming the following:

exchange of heat from the combustion between (a) the engine coolant and (b) the hot exhaust gases with an external cold water stream. An overview of the model developed for the process is presented in Table 2, while the full model description and its simulation results can be found in ref 7 (for nomenclature, see Table S1). The modeling issues of the CHP plant are revisited (e.g., the assumption of perfect combustion, the composition of exhaust gases and their thermodynamic behavior, as well as the operation of the modified logarithmic mean temperature difference heat exchangers) and a comprehensive study on the open-loop operation of the CHP system is presented. Figure 5 graphically represents the CHP system at hand. The full model is implemented in PSE’s gPROMS30 and consists of a total of 379 equations. The complex differential algebraic equation (DAE) system has two dynamic degrees of freedom: (i) Throttle valve: The open area of the throttle valve determines the amount of air and fuel that enters the system. Through the position of the throttle valve, the open area is affected, consequently the produced electrical power is manipulated. The opening of the throttle valve also determines the amount of heat that is produced through combustion. The former is one of the main products of the process, while the latter is a byproduct that cannot be directly measured but is correlated only with the produced power level. The throttle valve position is the sole degree of freedom of the power generation subsystem.

• The system operates at 4000 rpm, • At this level a 0.5 kW electical power demand is satisfied, • Water needs to be heated to 70 °C. Because of the restrictive power to heat ratio (in this case 20%−80%), 2 kW of the fuel’s power is transformed into usable heat. The desired temperature difference can be satisfied only for a maximum of about 9 g/s of water flow rate, assuming C


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flow rate of hot water of a certain temperature is guaranteed. During this mode of operation the power production level is not guaranteed. The main product in this case is the hot water. It is clear from the above that a control policy that takes into account only one of the aforementioned operating modes is unable to capture the full production potential of the residential CHP plant. Furthermore, a CHP plant that ignores the ability to produce one of two products at a time, at a given set point, restricts the economic advantages that the process could have, i.e., at times of high electrical power demand, the operation of the domestic CHP plant could be such that electrical power production would partially cover the demand. The byproduct hot water would be, in this case, stored in sufficiently large insulated tanks in order to cover the heating demands at a later time. This leads to reduced electricity costs from the electricity grid at times of high demand, as well as reduced costs of water heating in the near future when hot water is required. In contrast, during times of high heating demand the operation of the plant can be switched to mode 2, where hot water production is the driving force. The byproduct electrical power can be immediately traded to the electrical grid, thus providing a small profit from the CHP usage. At this point, it should be noted that the problem of switching from one mode of operation to another for a single domestic CHP plant (or for a series of them) can be considered as a supply scheduling problem with economic evaluation criteria. More specifically, optimality in terms of operation of the plant during each operating mode is the objective of the control scheme presented in this work, while the economically optimal mode of the operation which inherently addresses the question of “which operating mode should be used and when” is among the objectives of a scheduling policy. The issue of the interactions between the control of the system and its scheduling will be visited at a later stage based on the principles presented in ref 31. It has become clear from the development of the process model that any CHP system can be considered as the coupling of two subsystems. Furthermore, the system can be viewed as a two-product process, according to the mode of two modes of operation (dual-operation) being used at every point in time. Based on those inherent characteristics of the system we derive a decentralized dual control approach, performed in an explicit manner through multiparametric programming which is discussed in the next section.

Table 2. Residential CHP Model Overview piece of equipment: description

equations

throttle valve: fuel and air manipulation

⎛P ⎞ Pab d ψ ⎜ ab ⎟ mth = cdA th dt RβTab ⎝ Pmn ⎠

manifolds: pressure difference driven flow for the inlet air and exhaust gases

d mmn = ṁ mn,in − ṁ mn,out dt d Vmn,out = cpf(Pmn − Pmn,out) dt d d d d Vmn E = mmn,inhmn,in − mmn,out hmn,out dt dt dt dt

internal combustion engine: energy and mass balances

V P d mex = mn d ηvl ωen RβTmn 4π dt d ° mmn,out (hmn,out + ∑ (xair, ihf,air, i)) dt i = aircomp. d ° + mϕ(hϕ + ∑ (xϕ , jhf,fuel, j)) dt j = fuelcomp. d ° ∑ (xex,khf,ex, − mex (hex + k )) dt k = exhaustcomp. d ̇ = Q f + Q̇ cg → cw + Wċ + Wen dt

crankshaft: torque generation

Toen =

PmebVd 4π

generator: power generation through torque

d 1 Tocl = (Toen − Tocl ) dt Fl Pec = ηenTocl ωen

engine cooling system: energy balances

Q̇ in, i − Q̇ out, i d Ti = mic p, i dt ∀ i ∈ engine cooling system components d = TCabAab(Ta − Tb) Q dt a → b ∀ a , b ∈ engine cooling system components

heat exchangers: energy balances

Q = UAΔTmean ΔTmean =

ΔTin − ΔTout ⎛ ΔT ⎞ log⎜ ΔT in ⎟ ⎝ out ⎠

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standard liquid water heat capacity of 4.18 J/g·K and inlet water temperature of 15 °C. Equivalently, a desired flow rate of hot water of a certain temperature dictates the operating level of the CHP system; therefore, the power production level cannot be determined independently. In most domestic applications, the temperature of hot water produced via the use of electrical or thermal boilers is fixed to temperatures close to 70 °C. Based on the capacity of the boilers, the flow rate of the water that can be heated to such temperatures is determined. We have followed a similar approach where we have identified the two products of the CHP system as well as the two modes of operation of the system (also presented graphically in Figure 7): Operation mode 1: The power production driven operation denotes the operation during which a certain electrical output is guaranteed but the flow rate of the water is such that the produced hot water is of a certain temperature. In this mode of operation the main product is the electrical power. Operation mode 2: The heat recovery driven operation corresponds to the mode of operation during which a certain

DUAL SUBSYSTEM CONTROL APPROACH Based on (i) the partitioning of the system into two distinct subsystems and (ii) the two modes of operation, here we present a decentralized explicit model predictive control strategy that enables either the operation of the plant via considering the electrical power as the main product and the hot water as the byproduct (mode 1) or vice versa (mode 2). The proposed decentralized control study is shown in Figure 8 and discussed next. Decentralized Model Predictive Control in Mode 1. In mode 1, the control scheme attempts to (i) cover the electrical power demand and (ii) produce water of a predefined temperature, regardless of the flow rate. For this to happen, the controller of the power generation subsystem attempts to manipulate the position of the throttle valve in such a way that a predefined electrical output is covered. In this case, the power generation subsystem is treated as a single input−single output (SISO) system and is not aware of the existence of a second D


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Figure 5. Graphical representation of the CHP system within gPROMS modeling environment.

Figure 6. CHP system heat-to-power ratio.

power generation subsystem. More specifically, the controller of the heat recovery subsystem treats the operating level of the power generation subsystem as an uncertain parameter. This uncertainty is bounded between the minimum and maximum power generation output of the power generation subsystem. The controller manipulates the water flow rate inlet of the heat recovery subsystem in order to ensure that the output temperature will converge to a predefined set point. The task of the heat recovery controller becomes less challenging when the prognosis of the power generation output is taken into account. In this way, the heat recovery subsystem controller is less likely to produce actions that will cause a deviation from the set point and assists the entire process to converge to the desired values more efficiently because future information is derived as the system propagates. Because the procedure

subsystem and controller. According to the standard MPC and mp-MPC formulation and its rolling horizon approach, the optimal actions for the entire control horizon are calculated but only the action for the first time step is used. The actions for all future steps of the control horizon are discarded. In our approach though, the future, already computed actions are utilized. The optimal actions for the rest of the control horizon of the power generation subsystem controller are utilized and treated as potential future actions. The future actions are utilized to simulate the power generation subsystem, thus providing a prognosis of the power generation output of the system for as many time steps as one less than the control horizon. On the other hand, the heat recovery controller is aware of the inherent uncertainty provided by the operating level of the E


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Figure 7. Residential CHP modes of operation.

Figure 8. Decentralized model predictive control strategy for the residential CHP system.

subsystem merely provides a set point for the power generation output. The approach described in remark 1 allows for the application of the same procedure for future information such as the one of mode 1. In this case though, future inlet flow rates provide a better set point for the operating level of the power generation subsystem. According to the second remark, the set point for the operating level of the power generation subsystem is provided via the solution of the control problem. The set point is then treated by the controller of the power generation subsystem. The later operates in a way similar to what was described in mode 1 operation (i.e., the manipulation of the position of the throttle valve based on a SISO operation). To summarize, the heat recovery subsystem can be though as a multiple input−multiple output (MIMO) system. In operation mode 1 only one output is of interest (temperature) while the other (water flow rate outlet) is not among the degrees of freedom of the system, thus making the problem a multiple input−single output (MISO) problem. Furthermore, during mode 1 operation, the water flow rate inlet is treated as a manipulating variable while the operating level of the power generation subsystem is treated as a disturbance. In operation mode 2, both outputs of the MIMO are of interest. However, a desired outlet flow rate determines the inlet flow rate. The

described utilizes only already existing information and does not compute new data, it does not have an increased additional computational burden in terms of optimization, offline or otherwise. Decentralized Multiparametric Model Predictive Control in Mode 2. In mode 2, the control scheme attempts to produce hot water of (i) a predefined flow rate and (ii) temperature, regardless of the operating level of the power generation subsystem. In order for this to happen, the controller of the heat recovery subsystem treats the outlet water flow rate and temperature as set points which are attempted to be met via the manipulation of the subsystem’s inlet water flow rate and the operating level of the power generation subsystem. At this point two remarks should be made: Remark 1: The water inlet flow rate and the operating level of the power generation are treated in complete opposition compared to mode 1. The operating level of the power generation subsystem becomes the manipulating variable while the water inlet flow rate is merely a disturbance the value of which is dictated by the desired water outlet flow rate. Remark 2: The operating level of the power generation subsystem cannot be directly manipulated. Therefore, in this mode of operation, the controller of the heat recovery F


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Industrial & Engineering Chemistry Research manipulating variable in this case is the power generation output that targets to reject the uncertainty introduced by the inlet flow rate fluctuation. The following sections focus on the development of the multiparametric/explicit model predictive controllers for both schemes as well as their application.

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MULTIPARAMETRIC MODEL PREDICTIVE CONTROL DEVELOPMENT The development of multiparametric controllers has been an active area of research for over a decade.32 A variety of applications has been presented in the literature throughout the years (see ref 33 and references therein). Seldom have the multiparametric model predictive controllers been used in a decentralized manner,34,35 the main reason being the division of the system to be controlled into subsystems and the efficient monitoring of interactions. Because the system at hand can be naturally divided into two subsystems and their interactions are measurable with inexpensive means, we proceed to an application of the PAROC framework33 for each subsystem independently while preserving the interactions between them. The PAROC Framework. PAROC is a comprehensive framework that enables the representation and solution of demanding model-based operational optimization and control problems following an integrated procedure featuring highfidelity modeling, approximation techniques, and optimizationbased strategies, including multiparametric programming. A step-by-step description of the framework is provided below while the full description of the framework and its principles are presented in detail in ref 33. Step 1: “High Fidelity” Dynamic Modeling. For the development of the “high-fidelity model”, its quality and robustness determine the validity of the framework. The modeling of the system takes place in gPROMS.30 Step 2: Model Approximation. For the resulting highly complex dynamic models of the subsystems of the first step (most commonly DAE or PDAE programs), although sufficiently accurate compared to the real process, are not directly suitable for multiparametric programming studies. Hence, reduction techniques36,37 and identification methods (System Identification Toolbox of MATLAB) are employed to (i) reduce the model complexity while (ii) preserving the model accuracy. Step 3: Design of the Multiparametric Model Predictive Controllers. The design of the controllers is based on the validated procedure described in refs 38 and 32. The resulting multiparametric program is solved via the POP toolbox in MATLAB, thus acquiring the map of optimal control actions. Step 4: Closed-Loop Validation. The procedure is validated through a closed-loop procedure in which the controllers are tested against the original model of step 1. This can happen either via the interoperability between software tools such as gPROMS and MATLAB via gO:MATLAB or via the straight implementation of the controllers in the gPROMS simulation via the use of C++ programming and the creation of dynamic link libraries. Figure 9 presents the proposed decentralized PAROC framework application followed in the next section. Approximate Models. The next step is the reduction of the complexity of the model via the model approximation. Two approximate, simplified models are developed for the two subsystems based on input/output (I/O). The data are transferred between gPROMS, the simulation platform used

Figure 9. Decentralized PAROC framework approach. Actions within the gray area happen once and offline.

for this study and MATLAB, used for reduction and control purposes, through gO:MATLAB. The System Identification Toolbox of the latter is used to fit a linear state-space model to the I/O data of each subsystem. The next two subsections discuss the specifics of the approximation process for each subsystem. Power Generation Subsystem Model Approximation. The power generation subsystem is treated as a SISO system in which the throttle valve position is the input variable that determines the amount of generated electrical power. The interactions of the power generation subsystem with the heat recovery subsystem are preserved via monitoring the output of the former. The identification process yields the state-space model of eq 1. xk + 1 = 0.9913xk + 0.0044uk yk = 3.5927xk Ts = 0.1s

(1)

Despite the high complexity of the high-fidelity model of the power generation subsystem, the System Identification Toolbox manages to provide a highly accurate approximation with only a single identified state variable. Note that because of the identification process the state variables in this case have no physical meaning correlated to the high-fidelity model. The input variable uk in this case is the throttle valve position. The continuous value range of the variable is from 0 to 1, 0 meaning a closed valve, while 1 denotes the fully open valve. Panels a and b of Figure 10 show the output mismatch between the models and the step and impulse response of the approximate model, respectively. The perturbation of the state-space model that results in Figure 10b corresponds to a 10% change in the throttle valve opening. This value is the maximum allowable step change between two consecutive control actions. It is shown that the system is able to return to steady state in 60 s after the perturbation. The mismatch between the high-fidelity G


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Figure 10. Model approximation results for the power generation subsystem.

Figure 11. Model approximation results for the heat recovery subsystem.

model and the approximate model is as low as 9.08% according to the results of the System Identification Toolbox.

The low dimensionality of the approximate model is a result of the limited effect of the nonlinearity of the original highH


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Industrial & Engineering Chemistry Research fidelity model to (i) the power generation subsystem as well as (ii) the design of the system. In the power generation subsystem, the greatest source of nonlinearity comes from the correlations of (a) its operating level and (b) its design with the electrical and heating power efficiency. Because the design of the plant (i.e., the size of the internal combustion engine) is fixed, a major source of nonlinearity is lifted. Furthermore, the assumption for perfect combustion within the internal combustion engine helps toward that direction as reaction rates have not been used throughout the model. Heat Recovery Subsystem Model Approximation. Similarly to the power generation subsystem, the heat recovery subsystem is subjected to the same procedure. Based on the subsystem’s operation, this is a MIMO system. The inputs of the subsystem are (i) the operating level of the power generation subsystem, which is the power generation (i.e., the output of the power generation subsystem SISO model) and (ii) the inlet water flow rate of the heat recovery subsystem. Both inputs affect the first of two outputs of the subsystem, which is the temperature of the water at the outlet, but only the latter affects the second, which is outlet water flow rate of the heat recovery subsystem. Furthermore, the correlation of the inlet flow rate and the outlet flow rate is a simple mass balance. Therefore, the heat recovery subsystem is treated as a MISO system, monitoring only the temperature while a single mass balance equation correlates the inlet and outlet water flow rates. For the latter, an identification procedure is not needed as the mass balance is already linear and consists of a single state equation. The continuous representation of the mass balance equation is discretized approximately using finite differences and a discretization time that agrees with that of the MISO heat recovery subsystem approximate model. The interactions between the subsystems are preserved because at this point it is obvious that the output of the SISO power generation subsystem approximate model is one of the inputs of the MISO heat recovery subsystem approximate model. The I/O databased identification procedure yields the state-space model of eq 2.

inlet of the heat recovery subsystem (uk,2 of eq 2 and uk of eq 4). xk + 1 = Axk + Buk + Cdk yk = Dxk Ts = τs

(3)

Panels a and b of Figure 11 show the output mismatch between the models and the step and impulse response of the approximate model, respectively. The perturbation of the state-space model that results in Figure 11b corresponds to a 5 kW change in the power output and 0.01 kg/s change in the water flow rate. Note that the two perturbations happen independently. These values are the maximum allowable step changes between two consecutive control actions for each mode of operation. It is shown that the system is able to return to steady state in 20 s after the perturbation. The mismatch between the high-fidelity model and the approximate model is 10.01% according to the results of the System Identification Toolbox. The dimensionality of the heat recovery subsystem shows a greater dependence of the subsystem to effects and phenomena that have been nonlinearly described within the high-fidelity model. The operating range of the heat recovery subsystem, though, aids toward an identified model of a low number of states. A CHP system is employed to produce hot water of around (if not exactly) a certain temperature, while at the same time ensuring that the latter will not exceed its boiling point. It is therefore sufficient for the approximate model to be able to capture the behavior of the high-fidelity model for a certain operating range in terms of temperature (i.e., for a range of 45 °C, from 55 to 100 °C). The discrete model that correlates the water inlet flow rate with the outlet flow rate is described by eq 4. In this case, the input uk is the inlet flow rate while the state variable and output yk is the outlet flow rate. Note that the linear state-space model presented in eq 4 is the discretized form of the exact, continuous high-fidelity model; therefore, there is no mismatch between this and the original process.

⎡ 1 ⎡ 9 e − 5 2e − 5 ⎤ 0.0016 2e − 5 ⎤ ⎥ ⎢ ⎥ ⎢ xk + 1 = ⎢−0.0027 0.9127 −0.0696 ⎥xk + ⎢−0.0076 −0.0233⎥uk , i ⎢⎣−0.0012 −0.0113⎥⎦ ⎢⎣ −0.0003 −0.0063 0.9649 ⎥⎦

xk + 1 = 0.99xk + 0.01uk

yk = [3991 2.411 −0.0436]xk

yk = 1xk

Ts = 0.1s

Ts = 0.1s

i = {1, 2}

(2)

(4)

The approximation results for the two subsystems show a small deviation between the high-fidelity model and its approximate counterparts. Given the purpose of the approximation, which is the development of explicit rolling horizon policies, and their recurring nature, we can at this point assume that the approximate model is sufficiently accurate for such application.The next step of the procedure focuses on the application of multiparametric programming on the approximate models for the development of explicit controllers. Multiparametric Model Predictive Control. Following the complexity reduction of the original high-fidelity model to linear discrete state-space models and the preservation of the interactions between the subsystems, the model predictive control problems can be formulated. The standard generic representation of the model predictive control problem is used (eq 5).

In the case of the heat recovery subsystem, a linear state-space model with three states is needed to sufficiently represent the interactions between the inlet flow rate, the power generation levels, and the temperature of the water at the outlet of the system. In contrast to the scalar nature of the input of the previous case, the input is a vector. The first uk,1 is the operating level of the power generation subsystem (i.e., the power generation output). uk,2 is the flow rate of the water at the inlet of the heat recovery subsystem. In terms of the system operation, one of the inputs is treated as a measured disturbance and the state-space model takes the general form of eq 3. In mode 1 operation the input that is treated as a disturbance is the operating level of the power generation subsystem (uk,1 of eq 2). In mode 2 operation the input that is treated as a measured disturbance is the water flow rate at the I


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Industrial & Engineering Chemistry Research N−1

min J = xNT PxN + x ,y,u

M−1

+

∑ (xkTQ kxk + (yk

Table 3. mpMPC Formulation Data for the Power Generation Subsystem

− ykR )T QR k(yk − ykR ))

k=1

∑ ((uk − ukR)T Rk(uk − ukR) + ΔukT R1k Δuk) k=0

s.t.

xk + 1 = Axk + Buk + Cdk yk = Dxk + Euk + e umin ≤ uk ≤ umax Δumin ≤ Δuk ≤ Δumax xmin ≤ xk ≤ xmax ymin ≤ yk ≤ ymax (5)

where xk are the state variables; uk and uRk are the control variables and their respective set points; Δuk denotes the difference between two consecutive control actions; yk and yRk are the outputs and their respective set points; dk are the measured disturbances; Qk, Rk, R1k, and QRk are their corresponding weights in the quadratic objective function; P is the stabilizing term derived from the Riccatti equation for discrete systems; N and M are the output horizon and control horizon, respectively; k is the time step; A, B, C, D, and E are the matrices of the discrete linear state-space model; and e denotes the mismatch between the actual system output and the predicted output at initial time. This quadratic problem is reformulated into a multiparametric quadratic programming (mpQP) problem according to the principles presented in ref 32. The states at the initial time (x0), set points (uRk and yRk ), initial output mismatch, previous control actions in Δuk, and the disturbances (dk) are treated as uncertain parameters denoted by the parameter vector θ. The general form of the mpQP is presented in eq 6.

mpMPC design parameters

value

N M QRk, ∀k ∈{1,...,N} Rk, ∀k ∈{1,...,M} R1k, ∀k ∈{1,...,M} xmin xmax umin umax ymin ymax Δumin Δumax

10 10 1000 0.1 10 0 0.7 0 1 0 2.1 0.1 −0.1

The bounds on the output variable y have been set according to the high-fidelity model simulation results and are a function of the system’s configuration and size. Furthermore, because of the identification process, the state variables lose their physical meaning. The bounds on those variables are consistent with the bounds on the outputs and their correlation with them via the approximate state-space representation. The problem is treated in a multiparametric fashion and is solved for a total of four parameters, namely the initial state variable, the previous step’s control variable, the output set point, and the output set point mismatch at initial time. The result consists of affine functions of 10 control variables, one for every step of the control horizon. The solution to this problem is presented in the form of Critical Regions (CRs) as a twodimensional (2D) projection of a four-dimensional space. For this purpose the previous control action has been set to 0.3 and the output set point has been set to 0.6 after the solution of the problem (Figure 12). The parameter space is divided in a total of 2252 fragments, of which 50 appear in the 2D projection.

min J = uT Qu + uT FT θ u,θ

s.t.

Au ≤ b + Bθ u ∈ U = {u ∈ n|umin ≤ u ≤ umax , ∀ n} θ ∈ Θ = {θ ∈ q|θmin ≤ θ ≤ θmax , ∀ q}

(6)

It should be noted that in this case u is a vector that consists of M control actions, hence the lack of k to denote the time step. The letters n and q in eq 6 denote the size of the control and parameter vector, respectively, and consequently the size of the parametric solution and parametric space of the problem. The parametric solution of the problem is an affine function of the uncertainty. Therefore, upon the realization of the uncertainty associated with the control problem, an affine evaluation suffices in order to acquire the solution to the MPC problem. In the next sections, the formulation and solution of the multiparametric MPC problems of the subsystems derived in Approximate Models are presented. Power Generation Subsystem mpMPC. The controller for the power generation subsystem is computed using the identified linear state-space model of eq 1. For both modes of operation, the formulation of the controller of this subsystem remains the same. Its objective is to regulate the produced electrical power via manipulating the throttle valve opening. Table 3 provides the parameters of the formulation as those were presented in eq 5.

Figure 12. Power subsystem mpMPC.

Every critical region corresponds to a different parametric expression of the optimal control action. Heat Recovery Subsystem mpMPC. The procedure for the heat recovery subsystem mpMPC is similar. The major difference is the fact that the heat recovery subsystem control needs to account for two distinct modes of operation. As was discussed previously, in the first mode of operation the inlet water flow rate is treated as a control variable while at the second mode of operation it is treated as a disturbance. The form of this section will follow that of Power Generation J


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Industrial & Engineering Chemistry Research Table 4. mpMPC Formulation Data for the Heat Recovery Subsystem mode 1

mode 2


value


value

N M

3 3

QRk, ∀k ∈{1,...,N} Rk, ∀k ∈{1,...,M} R1k, ∀k ∈{1,...,M} xmin xmax umin umax dmin dmax ymin ymax Δumin Δumax

100 6 70 100·[−1, − 1, − 1]T 100·[1, 1, 1]T 0 5 0 2.1 0 100 0.5 −0.5

N M Qk, ∀k ∈{1,...,N} QRk, ∀k ∈{1,...,N} Rk, ∀k ∈{1,...,M} R1k, ∀k ∈{1,...,M} xmin xmax umin umax dmin dmax ymin ymax Δumin Δumax

2 2 10 5 0.01 480 100·[−1, − 1, − 1]T 100·[1, 1, 1]T 0 2.1 0 5 0 100 0.01 −0.01

Figure 13. Heat recovery subsystem mpMPC.

Figure 14. Closed-loop validation results for mode 1 operation (clockwise from top left: water temperature, water flow rate, throttle valve position, and power generation).

Subsystem mpMPC. Table 4 provides the design parameters of the mpMPC controllers of the heat recovery subsystem for both mode 1 and mode 2. In mode 1 the problem is solved for a total of nine parameters, namely the initial state variables (three according to the identified linear state-space model), the previous step’s control variable, the measured disturbance (one parameter per time step of output horizon considered), the output set point,

and the output set point mismatch at initial time. Equivalently, in mode 2 the problem is solved for eight parameters because the control and output horizons are smaller by one time step. The two-dimensional projections of the of the parametric solution space are presented in Figure 13. For the projections, the θ parameter space has been fixed as follows: Mode 1: The system is assumed to be in steady state at 70 °C hot water production, and the projected power generation K


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Figure 15. Closed-loop validation results for mode 2 operation (clockwise from top left: water temperature, controlled water flow rate, throttle valve position, and power generation).

Mode 2 Hot Water Production Driven Closed-Loop Validation. During this mode of operation, the controllers attempt to produce hot water of a predefined temperature (70 °C) and flow rate. The electrical power generation during this mode of operation is not of the essence. The power production levels are dictated by the desired water flow rate and temperature. For this reason, the power generation subsystem controller, the heat recovery subsystem controller of mode 2, as well as the flow rate controller are used. The results of the closed-loop simulation are presented in Figure 15. In operational mode 2, all three controllers behave as expected. The power generation subsystem controller follows the set point so closely that it is very difficult to notice a deviation after the startup. The flow rate controller manages to manipulate the outlet flow rate efficiently while the inlet flow rate disturbance is treated seamlessly by the heat generation subsystem controller. Note that the two control structures ensure operational optimality in their respective operational mode. Overall optimal operation of the plant depends on two additional considerations. The first is the design of the plant. A different engine size could result in a certain mode of operation being more favorable because of different dynamics, fuel consumption, initial capital cost, and capturing the operational objectives. The second is the transition between the two modes of operation based on the demand for hot water and electrical power throughout the operation of a plant. Therefore, the integration of process design, control, and scheduling is of great importance here as it combines operational optimality with process control, scheduling, and the consideration of process economics.39−41

of the power production subsystem is set at 12 kW. The previous control action uτ−1 as well as the real water temperature are the two free parameters. Mode 2: The system is assumed to be in steady state at 70 °C hot water production, and the projected water inlet flow rate is set to 0.3 kg/s. The previous control action uτ−1 as well as the real water temperature are the two free parameters. Note that another mpMPC controller is formulated to account for the inlet−outlet water flow rate of the heat generation subsystem. The controller is based on a single mass balance equation, and a similar procedure is followed. The simplicity of this particular state-space model though and the fact that model reduction techniques were not needed are the reasons we omit here a full description of that particular controller (see the Supporting Information for the description). The control scheme is put into action and tested against the original high-fidelity model to validate the procedure. Closed-Loop Validation and Results. The closed-loop validation of the decentralized control scheme of the domestic CHP system is performed in two modes, following the dual mode approach of the control scheme. In the first mode, the power generation driven approach is validated and presented, while in the second, the heat recovery driven approach is tested. The closed-loop validation happens through the interaction of MATLAB and gPROMS, as well as the software tool presented in ref 33. Mode 1 Power Production Driven Closed-Loop Validation. During this mode of operation, the controllers attempt to (i) meet a power output set point and simultaneously (ii) produce hot water of a predefined temperature. The latter was set to 70 °C. The flow rate of hot water is not of the essence during this mode of operation; therefore, only the power generation subsystem controller and the heat recovery subsystem controller of mode 1 are used. The results of the closed-loop simulation are presented in Figure 14. Both controllers follow the set points effectively and efficiently. The power generation subsystem controller shows an exceptional response even for large step changes, while the heat generation subsystem controller in mode 1 manages to reject the uncertainty of operation via the manipulation of the water flow rate in a good time span.

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CONCLUSIONS AND FUTURE WORK We presented a decentralized multiparametric model predictive control approach to efficiently operate a domestic CHP system. We took into consideration (i) a dual mode control scheme as well as (ii) the physical division of the CHP system into a power generation subsystem and a heat generation subsystem from the modeling starting point all the way to the control design and closed-loop validation via the PAROC framework. We showed how the dual control scheme can effectively L


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operate the system in a power generation driven mode and in a heat recovery system mode. In future work, the interactions between the two modes of operation will be discussed as part of the integration of the control policies with operational scheduling driven by economic objectives. Furthermore, the design of the system will be visited as part of a unified framework for the simultaneous solution of the design, control, and scheduling problem.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b03335. Nomenclature for the model overview of Table 2 and development of the input−output water flow rate mpMPC (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from EPSRC (EP/I014640) and from the European Commission (PIRSES, G.A. 294987) is gratefully acknowledged. Financial support from Texas A&M University is also gratefully acknowledged.

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REFERENCES

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