A comparative analysis of real-time power optimization for organic Rankine cycle waste heat recovery systems

A comparative analysis of real-time power optimization for organic Rankine cycle waste heat recovery systems

Applied Thermal Engineering 164 (2020) 114442 Contents lists available at ScienceDirect Applied Thermal Engineering journal homepage: www.elsevier.c...

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Applied Thermal Engineering 164 (2020) 114442

Contents lists available at ScienceDirect

Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

A comparative analysis of real-time power optimization for organic Rankine cycle waste heat recovery systems

T



Bin Xu, Dhruvang Rathod , Adamu Yebi, Zoran Filipi Clemson University, Department of Automotive Engineering, 4 Research Dr., Greenville, SC 29607, USA

H I GH L IG H T S

three real-time optimization control strategies. • Compares dynamic programming strategy shows the best performance. • Real-time • Nonlinear model predictive control is the best when considering system physics.

A R T I C LE I N FO

A B S T R A C T

Keywords: Waste heat recovery Organic Rankine cycle Power optimization Real-time implementation Internal combustion engine

Organic Rankine cycle waste heat recovery technology has been gaining more and more attention in recent years. Real-time power optimization is crucial to the system performance. Even though individual real-time optimization methods exist, literature rarely investigate comparison of different real-time power optimization methods. This paper first time compares three real-time implementable power optimization methods for an organic Rankine cycle waste heat recovery system. Three optimization methods include Proportional-IntegralDerivative rule-based method, nonlinear model predictive control and dynamic programming. In the Proportional-Integral-Derivative rule-based method, rule-based method defines the optimal working fluid vapor temperature trajectory and Proportional-Integral-Derivative controller manipulates the pump speed to follow that trajectory. In the nonlinear model predictive control method, reference vapor temperature is defined close to the saturation temperature and then the model predictive control minimizes the vapor temperature tracking error by controlling the pump speed. In the Dynamic Programming method, the net power produced by the organic Rankine cycle system is defined in the cost function, which is maximized by controlling the pump speed. A Random Forest machine learning model is utilized to extract the rules for Dynamic Programming and then implemented in real-time. All the three methods are implemented in the same experimentally validated plant model for the comparison analysis. The comparison results show that the Dynamic Programming Random Forest method has similar performance with nonlinear model predictive control method and outperforms the Proportional-Integral-Derivative rule-based method by 9.9% in net power production. Dynamic Programming Random Forest method can be an alternative to the nonlinear model predictive control for its low computation cost and high net power production.

1. Introduction Driven by the strict emission regulations and fuel economy requirements, researchers keep pushing the internal combustion engine efficiency to its limit. For the gasoline and diesel engine, more than 40% of fuel energy are wasted as heat via exhaust gas and cooling system [1,2]. The large amount of waste heat makes the waste heat recovery (WHR) technologies attractive to the internal combustion (IC) engine field. Among other IC engine WHR technologies, such as



thermoelectric generator [3] and turbo-compounding [4], and organic Rankine cycle (ORC). Thermoelectric generator has the merit of compact and light. However, its efficiency is extremely low and the cost is relatively high [5]. Turbo-compounding is also compact, while its efficiency is limited [6,7]. Among the three WHR technologies, ORC has the highest efficiency, even though it has some limits, such as high complexity [8], safety [9,10], and durability [11,12]. In literature, ORC systems showed promising experimental results using turbine expander. For

Corresponding author. E-mail address: [email protected] (D. Rathod).

https://doi.org/10.1016/j.applthermaleng.2019.114442 Received 29 May 2019; Received in revised form 30 August 2019; Accepted 24 September 2019 Available online 24 September 2019 1359-4311/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

a, b A b cp cd d h J k L m ṁ N O p P r s t T u U w x x̂ y z η γ ρ τ ∂

NMPC ORC PID POD RB RC RF SVD TP UKF WHR

constant coefficients area [m2] constant coefficient heat capacity [J/kg·K] discharge coefficient diameter [m] enthalpy [J/kg] cost k th time step phase length in evaporator mass [kg] mass flow rate [kg/s] revolution speed [rpm] valve opening [%] pressure [Pa] power [w] reference entropy [J/K] time [s] temperature [K]/ prediction time [s] control inputs heat transfer coefficient [J/kg·K] disturbance input states of dynamic model estimated states measurement measured parameter in estimator efficiency specific heat ratio density [kg/m3] time variable partial derivative operator

Subscripts and superscripts

air val c d eg end f g HPP i int is lb ub o p ref sat sh turb TP vap vlv w

Abbreviations EKF FV IC MB

model predictive control organic Rankine cycle proportional integral derivative proper orthogonal decomposition rule-based Rankine cycle random forest singular value decomposition tail pipe unscented Kalman filter waste heat recovery

extended Kalman filter finite volume internal combustion moving boundary

0

ambient condition ambient condition control discharge electric generator end of the transient operating condition working fluid exhaust gas high pressure pump inlet/ith time step intermediate isentropic lower boundary upper boundary outlet pressure/prediction reference working fluid saturation condition superheat turbine tail pipe working fluid vapor condition at evaporator exit Valve wall separating the working fluid the exhaust gas in evaporator reference

and cooling circuit pump power consumption, maximize the expander power output [25,26], and extend the expander operation duration during transient operating conditions [27]. Most of optimization work in literature address the problems in the ORC-WHR development phase, while the literature in operation phase are still lacking. Therefore, this paper focuses on addressing the gap in optimization work during operation phase. The real-time power optimization fully explores the potential of the power generation capability of the ORC-WHR system. It is reported that the optimization could improve the ORC-WHR net power generation by up to 12–17% [28,29]. The real-time power optimization can be conducted in many ways differentiated by real-time, offline, steady state and transient. The most common optimization method is offline optimization in steady state engine operating conditions. After the optimization, correlations are built between the heat sources parameters and working fluid operating conditions. The derived correlations are then implemented real-time in the control as references. The most adopted controllers in the ORC-WHR applications are Proportional-IntegralDerivative (PID) and model predictive control (MPC). Quoilin et al. correlated the working fluid evaporation temperature with heat source temperature, working fluid mass flow rate and condensation

instance, Cummins claimed 5–10% Heavy-Duty Engine fuel economy improvement achieved by ORC system [13]. AVL achieved 3–5% fuel economy improvement [1,14]. Bosch reported 2–9 kW power generation from Heavy-Duty Engine ORC system [15]. ORC technology is selected in this paper for its high efficiency, mature technologies and wide industrial applications such as solar, coal, and geothermal power plants [16,17]. For the ORC-WHR system, optimization is critical in development phase and operation phase. In the system development phase, optimization is utilized to minimize the components size/weight/cost [18,19], maximize the components efficiency/life/reliability [20], choose working fluid [21], evaluate heat sources [22], and choose components configuration [23]. After the system development, optimization is critical to the power production in real-time operation. The thermal inertial from heat exchangers challenges the real-time temperature controls and optimization. According to [24], the temperature response time triples as the mass of wall increases. The long response time is generally captured in model, thus model-based controls have the potential to deliver efficient temperature control performance. Optimization searches the optimal actuator positions (e.g. pump speed, expander speed) and/ or state trajectories (e.g. working fluid evaporation pressure, vapor temperature), which minimize the working fluid pump 2

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evaporates to vapor phase inside the evaporator. The vapor working fluid then passes through the turbine valve, pushes turbine and outputs power. The turbine bypass valve is open during the ORC system warmup or when liquid working fluid presents during the warm condition, which prevents liquid working fluid entering turbine to avoid turbine damage. Turbine valve combines with the turbine bypass valve for the turbine protection. In addition, the turbine bypass opens as a pressure relief valve when the system high pressure passes the pressure limit (40 bar). An electric generator is connected with the turbine to harvest the power. After the turbine, the vapor working fluid enters the condenser, releases heat and changes back to liquid phase. Expansion tank acts as a buffer in the ORC system. It stores the excessive working fluid, resulting from the vapor expansion in the warm operating condition. A low pressure pump supplies working fluid to the high pressure pump and avoids cavitation phenomenon in the high pressure pump.

temperature based on steady state optimization results [25]. The trajectory was implemented to PID controllers. Feru et al. extracted the rules of working fluid vapor quality by optimizing the ORC-WHR system in steady state heat source conditions and MPC controller was considered to follow the optimal trajectory [30]. Real-time optimization utilizing optimal control such as MPC is another real-time optimization method. Hernandez et al. designed and experimentally validated a MPC for a small-scale pilot plant [28,29]. Working fluid pump speed was optimized in MPC to follow the optimal saturation temperature trajectory. Feru et al. developed a MPC for the heavy-duty diesel engine ORC-WHR system [30]. The MPC followed the optimal working fluid vapor quality at the exit of evaporators by optimizing the working fluid pump returning line valve opening. Esposito et al. defined the net power production in the NMPC cost function and optimized the net power real-time [31]. The total number of states were 16, which made the NMPC computation intensive and compromise its real-time implementation. Petr et al. utilized NMPC to optimized the ORC-WHR system net power and the optimal control inputs were realtime implemented in sophisticated plant model [32]. Some researchers use offline net power optimization for the ORCWHR system over transient operating conditions. Xu et al. considered three working fluid vapor temperature reference strategies in ORCWHR net power optimization over a transient driving cycle [26]. The waste power dependent reference temperature strategy turned out to be the optimal among the three strategies. Peralez et al. offline optimized the ORC-WHR system over transient engine condition utilizing dynamic programming (DP) [33]. The rules behind the DP results were discussed, but it is not extracted and implemented in real-time. Even though different real-time power optimization strategies have been published, few publications compared different optimization strategies. This paper compares three real-time implementable ORC-WHR power optimization methods including NMPC, DP and PID. In the ORC-WHR system, NMPC is usually considered to replace PID and reduce the working fluid temperature tracking error. DP offline generates benchmark. In this paper, a random forest (RF) machine learning model learns the DP offline optimization rules and later implements the rules in real-time operation. After the development, all three optimization methods are real-time implemented in the same plant model over a transient driving cycle. As the ORC system performance heavily depends on expander, expander selection is very critical. Based on the characteristic of expansion ratio, expanders can be classified into volumetric expanders (fixed expansion ratio) and turbine expanders (varying expansion ratio). In general, volumetric expanders operate at low expansion ratios and not sensitive to the working fluid vapor quality, while turbine expanders operate at high expansion ratios and are vulnerable to the working fluid droplets [24]. Turbine expanders have higher efficiency thanks to its high expansion ratios, which is the main reason this study utilizes turbine expander. More importantly, expander selection should undergo a systematic analysis and it is not the focus of this paper.

3. ORC-WHR system modeling The heat exchanger is the key component in the system as it has complex phase change dynamics. Different evaporator heat exchanger modeling methods are chosen for different purposes in this paper. In NMPC, moving boundary (MB) heat exchanger modeling method is chosen for its low state-dimension and satisfactory accuracy [34]. In DP formulation, Proper Orthogonal Decomposition (POD) Galerkin projection heat exchanger modeling method is chosen for its extremely low state dimension and moderate accuracy [35]. Different optimization methods are finally validated over the plant model. As the plant model, the accuracy should be high and the computation time restriction is not strict. Thus, finite volume (FV) heat exchanger modeling method is selected as the plant model for its high accuracy [34]. MB model, POD Galerkin, and FV heat exchanger models utilized in this paper can be found [27,36,37], respectively. Besides the heat exchanger model, valve model, pump model, and turbine expander model are presented in this section. In Fig. 1, there are two components possibly resulting in pressure drop during the normal turbine operation conditions. They are heat exchangers (i.e., TP evaporator and condenser), and pipes. Based on the four steady state test data points from [1], average pressure drop in evaporator and condenser are 0.075 bar and 0.038 bar, respectively. The average pressure drop in the pipes between pump and evaporator is 0.11 bar. Even though the aforementioned pressure drops are not very large, it could be substantial if the system development does not consider pressure drop carefully. The pressure drop is closely related to the inner diameter of heat exchangers, pipes and junctions, and working fluid mass flow rate. In this study, the pressure drops across components are not considered. Further details of the components modeling, identification, validation and system model integration can be found in [37]. In the ORC-WHR system, the engine model is built in Gamma

2. System configuration The ORC-WHR system considered in this paper is a typical heavyduty diesel engine ORC-WHR system as shown in Fig. 1. The engine is a 13L 2014 model year heavy-duty diesel engine. The engine interacts with the Rankine cycle system through the tail pipe (TP) evaporator located downstream of the aftertreatment system. A TP bypass valve is installed to bypass TP exhaust gas when the engine waste heat is over the maximum capacity of the condenser heat exchanger, which protects the working fluid from overheating and degradation. Based on the operating pressure, the Rankine cycle system is divided into two regions: (1) high pressure region (20–40 bar) and (2) low pressure region (1–5 bar). A positive displacement high pressure pump creates the high pressure and pumps the liquid working fluid to the TP evaporator. The liquid working fluid absorbs heat from the TP exhaust gas and

Fig. 1. Schematic of ORC-WHR system. 3

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Technology Power (GT-POWER) software. The details of engine modeling can be found in [37].

To, pump = Ti, pump +

3.1. Valve

Ppump =

TP exhaust gas bypass valve, turbine inlet valve and turbine bypass valve mass flow rate are modelled according to the valve pressure expansion ratio. When the expansion ratio is greater than working fluid mass flow rate is expressed as follows: 2

ṁ val = Aval Cd

⎡ po, val γ 2γ ⎞ ⎛ po, val ⎞ pi, val ρi ⎢ ⎜⎛ ⎟ − ⎜ ⎟ γ−1 ⎢ ⎝ pi, val ⎠ ⎝ pi, val ⎠ ⎣

( )

follows:

2 ⎞ ṁ val = Aval Cd ⎜⎛ ⎟ 1⎠ γ + ⎝

( )

ρi, pump ηis, pump

(9a)

3

ṁ pump ⎞ log ⎜⎛ ⎟ ̇ m ⎝ pump, max ⎠ (1)

(9b)

where Ti, pump, To, pump represent pump inlet and outlet working fluid temperature, respectively, ηis, pump is isentropic efficiency and is expressed as a function of pump mass flow rate. The empirical expression and coefficients can be found in [25,38], Ppump represents pump power consumption, cp represents working fluid specific heat capacity, pi, pump , po, pump represent pump inlet and outlet pressure, respectively.

γ

γ−1 2 , γ+1

ṁ pump (po, pump − pi, pump )

ṁ pump ⎞ ⎛ ṁ pump ⎞ − 0.06 = 0.93 − 0.11log ⎜⎛ ⎟ − 0.2log ⎜ ⎟ ̇ m pump max , ⎝ ⎠ ⎝ ṁ pump, max ⎠

where ṁ represents mass flow rate, Aval represents opening area of the valve, Cd represents valve discharge coefficient, γ represents specific heat capacity ratio, pi, val , po, val represents inlet and outlet pressure, respectively, ρi, val represents inlet working fluid density. When the expansion ratio is less than

(8)

2

the

γ+1 γ ⎤

⎥ ⎥ ⎦

ṁ pump cp

ηis, pump

γ

γ−1 2 , γ+1

(1 − ηis, pump ) Ppump

the mass flow rate is expressed as

4. Optimization methodologies γ+1 2(γ − 1)

γpi, val ρi

4.1. PID rule-based method (2) The PID rule-based (PID-RB) method integrates the RB method with PID controller. RB method real-time selects the working fluid vapor temperature reference for the PID controller based on exhaust gas conditions and working fluid conditions. The rule of RB method is optimized in the off-line optimization using transient engine driving cycle and the details of PID-RB method can be found in author’s previous work [26]. PID outputs the working fluid pump speed to the ORCWHR plant model. The schematic of the PID-RB optimization method is shown in Fig. 3. The exhaust gas mass flow rate and temperature are expressed as the measured disturbance inputs. Working fluid vapor temperature is the controller reference and working fluid pump speed is the control input to the ORC-WHR plant model. PID controller is tuned based on Z-N tuning method [39]. The RB method selects different working fluid vapor temperature reference based on the real-time TP exhaust gas waste power level. The general principle is that the optimal working fluid superheat temperature is different at different engine operating conditions. In the ORCWHR community, many researchers concluded that the optimal working fluid superheat temperature is close to zero [40,41]. However, most of those conclusions were from the steady state analysis. In the real application, the ORC-WHR system operates in transient conditions

3.2. Turbine expander Different from positive displacement expander (e.g. piston expander), turbine expander has varying expansion ratio. The turbine nozzle is designed in small diameter so that it undergoes choke flow even at low inlet pressure. Based on this design, the turbine mass flow rate is modelled as a linear function of inlet pressure as follows:

ṁ turb = aturb pi, turb + bturb

(3)

where ṁ turb is turbine working fluid mass flow rate, pi, turb is turbine inlet pressure, aturb , bturb are the coefficients. Turbine outlet temperature is calculated based on thermodynamic tables as follows:

To, turb = f (ho, turb, po, turb )

(4)

where po, turb is turbine outlet pressure and ho, turb is outlet working fluid enthalpy. The thermodynamic table is shown in Fig. 2. Outlet enthalpy is calculated as follows:

ho, turb = hi, turb − ηis, turb (hi, turb − ho, is, turb)

(5a)

pi, turb , Ti, turb⎞⎟ ηis, turb = f ⎜⎛Nturb, p o , turb ⎝ ⎠

(5b)

where ηis, turb is turbine isentropic efficiency during the expansion process, the efficiency map is from turbine manufacturer and is confidential, hi, turb is inlet working fluid enthalpy, ho, is, turb is outlet working fluid isentropic enthalpy. Turbine power is calculated as follows:

Pturb = ηeg ṁ turb (hi, turb − ho, turb)

(6)

where ηeg is electric generator efficiency, which is assumed to be 0.9. 3.3. Pump The ORC system contains two pumps, both of which are displacement pumps. The mass flow rate is correlated with pump speed with a linear relation as follows:

ṁ pump = apump Npump + bpump

(7)

where ṁ pump is pump working fluid mass flow rate, Npump is pump revolution speed, apump , bpump are two coefficients. Pump outlet temperature is calculated as follows:

Fig. 2. Thermodynamic map of ethanol temperature as a function of pressure and enthalpy. 4

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Fig. 3. Schematic of RB-PID optimization method real-time implementation.

Fig. 4. Diagram of NMPC optimization method real-time implementation.

most of time. Due to different vapor temperature, controllers has different performance in the transient conditions, the optimal superheat temperature calculated from steady state does not necessarily produce the most power. In some highly transient scenarios, the vapor temperature controller varies significantly and could touch the saturation line, thus interrupting the power production if the expander is turbine. This phenomenon cannot be simulated and hence it is not possible to be considered in steady state optimization. Therefore, in the real life ORCWHR system operation, the optimal working fluid vapor temperature varies with engine condition and possess additional challenges for vapor temperature controller. If the controller can track the reference with small error, the optimal vapor temperature reference can be setup close to the saturation line. However, if the controller has large reference tracking error, the optimal vapor temperature reference has to increase and keep a certain amount of safety margin to allow certain vapor temperature variation. The rule of the RB method for the PID control is expressed as follows:

Tf , o, ref

∗ ⎧ Tsh,1 + Tf , sat , ⎪ ⎪ ∗ Tsh, j + Tf , sat , = ⎨ ⎪ ∗ ⎪Tsh + Tf , sat , ⎩ ,n

4.2. Nonlinear model predictive control method In this section, NMPC algorithm is formulated as reference tracking controller. The control target and input are the vapor temperature and working fluid pump speed, which are the same as PID-RB method. MB model is chosen as the control model. The MB modeling and NMPC control design were conducted in authors’ previous published papers, both in simulation [27,42] and experiments [43,44]. The state vector of the MB model is [L1, Tw1, L2 , Tw2, hf , o, Tw,3 ]T , where L1 and L2 are the length of working fluid liquid and mixed phase, Tw1, Tw2, Tw3 are mean wall temperature in liquid, mixed and vapor phase respectively, hf , o is the working fluid outlet enthalpy. Among the six states, only working fluid outlet enthalpy can be indirectly measured from the measured working fluid outlet temperature and outlet pressure. The working fluid outlet enthalpy then can be calculated utilizing the thermodynamic function - hf , o = f (pf , o , Tf , o) . The rest of the five states cannot be measured in the heat exchanger. Thus, an Extended Kalman Filter (EKF) is utilized to estimate the five states. EKF is chosen for its lower computation cost compared with Unscented Kalman Filter (UKF). The schematic of NMPC is shown in Fig. 4. The exhaust gas mass flow rate and temperature are considered as the measured disturbance. The cost function of NMPC accounts for the normalized working fluid vapor temperature tracking error and the normalized control input as shown in Eq. (12).

∗ PTP, g, max ≥ PTP, g ≥ PTP , g ,1

⋮ ∗ ∗ PTP , g , j − 1 ≥ PTP , g ≥ PTP , g , j ⋮ ∗ PTP , g , n − 1 ≥ PTP , g ≥ PTP , g , min

PTP, g = ṁ TP, g Cp, TP, g (TTP, g − Tair )

(10a)

J= (10b)

b=

⎧ 0, ⎨ 1, ⎩

2 duf2 ⎫ ⎧ (Td − Tf , o, ref ) + (I − c ) 2 c dτ 2 ⎨ Tf , max uf , max ⎬ ⎭ ⎩

(12)

The NMPC problem formulation is given as follows:

minJ (x (ti ),u (.))

(13)

u (∙), x (∙)

x ̇ (τ ) = f1 (x (τ ), z (τ ), u (τ ), w (ti )), s. t: ⎧ ⎨ ⎩ y (τ ) = f2 (x (τ ), z (τ ), u (τ ), w (ti )) x lb ≤ x (τ ) ≤ x ub ,

∀ τ ∈ [ti, ti + Tp]

y lb ≤ y (τ ) ≤ y ub ,

∀ τ ∈ [ti, ti + Tp]

ulb ≤ u (τ ) ≤ uub,

∀ τ ∈ [ti, ti + Tc ]

u (τ ) = u (ti + Tc ),

δulb ≤ δu (τ ) ≤ δuub,

∀ τ ∈ [ti + Tc , ti + Tp]

∀ τ ∈ [ti, ti + Tc ]

x = [L1, L2 , Tw1, Tw2, Tw3, hf , o ]T , u = Npump

(11a)

where J is the cost function, x ̇ = f1 (.) represents the system dynamics and y = f2 (.) represents system algebraic equations, the superscripts lbandub indicate the lower and upper bounds of the constrained variables, Tp and Tc denote the prediction and control horizon, respectively with Tp ≥ Tc , c is the weight of temperature error term, Tf , max and ṁ f , max are maximum working fluid temperature and mass flow rate,

∗ PTP, g, max ≥ PTP, g ≥ PTP , g ,1 ∗ PTP , g ,1 > PTP , g ≥ PTP , g , min

ti + Tp

i

∗ th working fluid superheat and it could be different at where Tsh , i is i ∗ different exhaust gas waste power levels, Tsh , i ∈ [10, 120], PTP ,g, min and PTP, g, max are minimum and maximum waste heat power and they are predefined based on the engine operating range, P ∗j is the optimal exhaust gas power level threshold P ∗j ∈ [PTP, g, min, PTP , g, max ], PTP,g , ṁ TP,g , TTP, g , Cp, TP, g are TP exhaust gas power, mass flow rate, temperature and heat capacity, respectively, and Tair is ambient air temperature. In this paper, the parameter n in Eq. (10) is chosen to be 2 for the saving of ∗ computation cost and the PTP , g ,1 is set as the average of maximum and ∗ ∗ , Tsh,2 minimum TP exhaust gas power. Two constants Tsh,1 are optimized in the off-line simulation given constant speed variable load heavy-duty diesel engine driving cycle and the results are 60 and 50 respectively [26]. The engine conditions are transient, which could result in high frequency switch between two exhaust waste power. Thus a low pass filter is utilized to smooth the switch. The expression of the filter is as follows:

∗ ∗ Tf , o, ref (k ) = a1 [(1 − b) Tsh ,1 + bTsh,2] + a2 Tf , o, ref (k − 1)

∫t

(11b)

5

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respectively. The lower boundary (y lb ) is the working fluid saturation temperature to prevent liquid working fluid passing through the turbine expander and reducing turbine life. The upper boundary of the working fluid temperature (y ub ) is set to be the working fluid decomposition temperature, above which the working fluid properties staring to change. The input limits (ulb, uub ) are set by the minimum and maximum physical HP pump speed. The pump speed change rate (δulb, δuub ) is constrained by the physical deceleration and acceleration of the HP pump measured in experiments.

detrimental to the turbine operation and waste power harvesting. Another method to implement DP results in real-time is to extract the relationship between the exhaust gas conditions and working fluid pump speed as shown in Fig. 6. In this case, the extracted rule can be directly implemented to the actuator using the real-time engine operating condition without extra controller. The second DP real-time implementation method has two advantages over the first method: (1) less effort required in development process by avoiding the extra controller, and (2) the response time of the developed correlation will no longer be dependent on the limitations of the extra controller. The challenge of the second method is the high accuracy requirement of the correlation connecting the engine operating conditions and working fluid pump speed. The evaporator model contains highly nonlinear phase change physics as well as thermal inertia, which increases the challenges of the correlation. This paper chooses the second DP real-time implementation method for its less effort requirement in development phase and no response time limitation compared with the first DP real-time implementation method. In order to address the challenge of the nonlinear physics in evaporator model, RF model [45], a nonlinear machine learning modeling method, is utilized to correlate the engine conditions and optimal pump speed from DP results. The nonlinearity of RF model can be changed by varying the number of trees, the depth of each tree and the number of leaves at each node of the tree. A schematic of RF model in correlating the exhaust mass flow rate and temperature to the working fluid pump speed is shown in Fig. 7. The RF model is written as ensemble center, which ensembles the results of N trees. In each tree, the path is selected based on the input values (value of exhaust mass flow rate and temperature). The target is the working fluid pump speed. During the RF model training process, the judgement condition threshold is calibrated so that the inputs can finally connect to right target as recorded in the training dataset through the green path shown in the Fig. 6. After the RF model is trained, there still exists certain error in the RF model. The trained model shows 1.26% working fluid temperature error. This model error increases the possibility of working fluid saturation and interrupts the turbine operation. Therefore, a rule-based method is added on top of the trained RF model, such that the pump speed is decreased when the rule-based detects a high possibility of working fluid saturation event at the evaporator exit (or turbine inlet). The rule-based choose the pump speed reduction based on the real-time superheat temperature as expressed as follows:

4.3. Dynamic programming method DP is a global optimization method, which generates best-case benchmark for real-time control performance evaluation and creates optimal state/ inputs trajectories for the real-time controllers to follow. DP as the global optimizer is limited by the dimensionality computation issues. Therefore, it is utilized in offline optimization. Even though DP simulation is conducted offline, the state dimension of DP model should be as small as possible with satisfactory accuracy to reduce computation time from days/months to hours and even to minutes. As mentioned in the beginning of Section 3, a POD reduced evaporator model is chosen as DP model, which has only one state, hf , related to the working fluid enthalpy. The details of POD model can be found in [36]. DP optimizes the ORC-WHR system net power production, which is the power generated by the turbine expander minus the working fluid pump power consumption. The expression of the cost function is as follows:

J=

∫t

ti + Tp

i

− (Pturb (τ ) − Ppump (τ )) dτ

(14)

The negative sign express the cost function as a minimization problem. The DP formulation is given below:

minJ (x (ti ),u (.))

(15)

u (∙), x (∙)

x ̇ (τ ) = f3 (x (τ ), u (τ ), w (ti )), s. t: ⎧ ⎨ ⎩ y (τ ) = f4 (x (τ ), u (τ ), w (ti )) x lb ≤ x (τ ) ≤ x ub ,

∀ τ ∈ [0, Tend]

y lb ≤ y (τ ) ≤ y ub ,

∀ τ ∈ [0, Tend]

ulb ≤ u (τ ) ≤ uub,

∀ τ ∈ [0, Tend]

δulb ≤ δu (τ ) ≤ δuub,

∀ τ ∈ [0, Tend]

x = hf , u = Npump where Pturb is turbine power generation from Eq. (6), Ppump is working fluid pump power consumption from Eq. (9), x ̇ = f3 (.) represents the system dynamics and y = f4 (.) represents system algebraic equations, Tend represents the time duration of the transient engine profiles. The rest of variables definition are the same as Eq. (13). Fig. 5 showed the DP optimal vapor temperature and pump speed over 1200 s transient engine operating condition. From the subplot (a), it shows that the optimal vapor temperature trajectory is very close to the saturation temperature with slightly offset. The results reveal the optimal vapor temperature should be as low as possible within the constraint boundaries. So far, the DP optimization is conducted offline. There are different methods to implement the DP results real-time. One method is to extract the states/inputs trajectories from DP results and implement them as the reference for real-time controller. In this case, the trajectories can be the working fluid superheat, which is 5–10 °C in the DP results. However, the trajectory is hardly implemented in the ORC-WHR controller. The reason is that the ORC-WHR controller has large working fluid vapor temperature tracking variation, which is greater than the window of 5–10 °C. Thus during the implementation of DP rules, the working fluid temperature saturates a lot, which is

Fig. 5. DP optimization results over 1200 s transient engine operating condition. 6

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The plant model was identified and validated using experimental data. The details of the plant model can be found in [37]. Among the three comparison simulation, the exhaust gas conditions, initial conditions are the same. The only difference is the working fluid pump speed, which are the output of the optimization methodologies. TP exhaust gas mass flow rate and temperature are generated from GT-POWER engine model simulation as shown Fig. 9. The engine speed has little variation, while torque varies from 800 Nm to 1400 Nm. This engine operating condition mimics the conditions of long-haul truck in high way transportation, in which the vehicle speed is close to 60 mph and vehicle traction force varies based on the road slope. The exhaust gas mass flow rate varies between 0.24 kg/s and 0.32 kg/s and exhaust gas temperature varies between 328 °C and 314 °C. Exhaust gas mass flow and temperature are interpolated from tables in the simulation. All three simulations are conducted in Matlab/Simulink. The working fluid pressure at turbine outlet is fixed at 1.5 bar. Turbine speed is real-time optimized given working fluid inlet temperature and inlet/outlet pressure as the inputs. Condenser cooling power consumption and feed pump power consumption are not considered in the optimization. In the plant model, the simulation time step is set as 0.025 s. In the NMPC simulation, the MB model time step is set as 0.6 s and prediction horizon is set as 100 steps, which results in 60 s prediction time window. In the DP simulation, the POD reduced model time step is set as 1 s.

Fig. 6. Diagram of DP-RF optimization method training process.

6. Results The comparison results are shown in Fig. 10. In Fig. 10(a), PID-RB produces the highest working fluid vapor temperature along the 300 s simulation, followed by DP-RF and NMPC. Due to the saturation temperature of the three simulations are similar, only saturation temperature from PID-RB is plotted in Fig. 10(a) to increase the figure readability. Only NMPC and DP-RF methodologies maintain working fluid vapor state during the entire simulation, which ensures the consistent turbine power production without interruption. In PID-RB simulation, working fluid vapor temperature saturates at around 140 s. According to the Fig. 9(a) and (d), the saturation is caused by the temperature overshoot at 100 s. After the PID-RB control detects the overshoot, it dramatically increases the working fluid mass flow rate to compensate the high temperature, which later results in the working fluid saturation at 140 s. Different from PID-RB, both DP-RF and NMPC maintain certain level of vapor temperature and increase the turbine operation safety margin (i.e. keep the vapor temperature stay away from the saturation temperature and prevent turbine power production interruption). In terms of turbine operation safety, NMPC has the best performance among the three methodologies. It maintains certain amount of superheat, while DP-RF and PID-RB have some moments that vapor temperatures are very close to the saturation temperature. Thus, PID-RB and NMPC has higher possibility of saturation compared with DP-RF. DP-RF has less vapor temperature variation than NMPC and NMPC has less variation than PID-RB. Working fluid pressures at pump outlet are shown in Fig. 11(a). The maximum design pressure of the ethanol

Fig. 7. Schematic of RF model in DP rule extraction.

⎧ ⎪ ⎪ NRF − RB =

∗ NRF , Tvap − Tsat ≥ Tsh ,1 ∗ ∗ NRF − ΔN1∗, Tsh ,1 > Tvap − Tsat ≥ Tsh,2

⋮ ⎨ ∗ ∗ ∗ ⎪ NRF − ΔNn − 1, Tsh, n − 1 > Tvap − Tsat ≥ Tsh, n ⎪ ∗ ∗ NRF − ΔNn , Tvap − Tsat < Tsh, n ⎩

(16)

where NRF − RB is the optimal working fluid pump speed of RF-RB method, NRF is the optimal pump speed of RF model, ΔNi∗ is the pump ∗ speed reduction at ith working fluid vapor superheat level, Tsh , i is the working fluid superheat constant, n is the number of superheat levels. To save the computation time, n is chosen as 2 in this paper and ∗ ΔN∗, Tsh are optimized using particle swarm optimization. The results ∗ ∗ are ΔN1∗ = 15, ΔN∗2 = 30, Tsh 1 = 5, Tsh2 = 10 . The schematic of DP-RF optimization method is shown in Fig. 8. Npump, int represents the intermediate pump speed. The ORC-WHR plant model is the same as PID-RB method and NMPC optimization method. The RF model is modeled and trained in Python and the DP-RF simulation is conducted with the Matlab/ Python co-simulation. The details of DP-RF model training and inputs selection can be found in [46]. 5. Comparison of simulation setup All the three optimization methodologies are implemented in the same ORC-WHR system plant model in Matlab/Simulink environment.

Fig. 8. Diagram of DP-RF optimization method real-time implementation. 7

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Fig. 9. Engine operating condition for the optimization algorithms comparison: (a) engine speed and torque, and (b) TP exhaust gas mass flow rate and temperature at the engine aftertreatment system outlet.

system is 30 bar [37], the pressure safety valve only opens when the system maximum pressure reaches 30 bar. During the transient operation, all the three controls produce safe pressure levels. From Fig. 10(a) and (b), it is observed that the lower the vapor temperature is, the higher the net power is generated. Compared with PID-RB, both NMPC and DP-RF have the lower vapor temperature and produce the higher net power most of the time. This vapor temperature and net power relationship has been mentioned in several literatures [40,41]. DP-RF has similar mean vapor temperature with NMPC and produces similar power with NMPC. Given the heat source conditions, the vapor temperature can be indirectly represented by the working fluid mass flow rate. In Fig. 10(d), PID-RB shows the least mean working fluid mass flow rate. The more mass flow is pumped through the evaporator, the lower vapor temperature becomes at the exit of evaporator. More mass flow is beneficiary for the turbine power generation according to Eq. (6). Even though the more mass flow rate consumes more pump power, the increase of pump power is less than the power gain in the turbine. Thus, low temperature is connected with higher net power. In Fig. 10(c), the cumulative energy reveals the net power generation capability of three optimization methodologies at the end of the simulation. PID is left behind almost from the beginning until the end. NMPC closely follows the DP-RF and only produces 0.4% less cumulative energy than the DP-RF at the end of simulation. However, the PID-RB produces 9.9% (or 9.5%) less cumulative energy than the DP-RF (or NMPC) at the end of simulation. The results from Fig. 10 show the similar level in this field compared with references [32,47]. The NMPC shows better disturbance rejection than the PID, thus the vapor temperature shows less variation in NMPC. The 10% improvement of energy recovery by NMPC over PID is in the range of 7–15% from reference [32]. Different from the working fluid temperature, the exhaust gas temperatures show little discrepancy among three controls as shown in Fig. 11(b). In addition, the exhaust gas temperature variation in each control is less than 10 °C in Fig. 11(b), whereas the working fluid temperature variation is as large as large as 50 °C in Fig. 10(a). This is due to the larger fluid speed and smaller heat transfer coefficient in the exhaust gas flow than that in the working fluid flow. The PID-RB methodology shows the largest vapor temperature variation, which is the main reason of its lowest cumulative energy in the simulation. If the variation can be reduced, the PID-RB would control the vapor temperature in a lower level where the working fluid

Fig. 10. Optimization methodologies comparison results. Net power, cumulative energy and working fluid mass flow rate are normalized by the respective maximum values.

mass flow rate, net power and cumulative energy will increase. The large vapor temperature variation of PID has been revealed in several ORC-WHR publications where MPC is proposed for its less vapor temperature variation performance. The poor performance of PID is caused by the heat source contradictory dynamics in ORC-WHR system, which are the temperature delay caused by the TP evaporator thermal inertia (slow dynamics) and the large magnitude change of exhaust gas mass flow rate (fast dynamics). DP-RF has less vapor temperature variation than PID-RB because its natural advantages are able to reduce the impact of the aforementioned challenge. On one hand, TP exhaust gas temperature change rate is one of the four inputs to the RF model. This change rate signal indicates the exhaust gas temperature trend in the future 10–50 s, with which the RF model can take action ahead to compensate the future slow changing temperature impact. This is one of the main reason why temperature change rate ranks as the second important feature ahead of exhaust gas temperature itself and exhaust mass flow rate change rate. On the other hand, during the RF model training process, the RF model learns the pump speed relation with the magnitude of exhaust gas mass flow rate. Even though the exhaust gas mass flow rate magnitude varies significantly, the RF model can react accordingly without delay just like the DP. Therefore, DP-RF perform less vapor temperature variation than the PID-RB. Even though PID is upgraded with the help of RB method, RB method only helps PID 8

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7. Conclusion This paper compares three real-time power optimization methods for an organic Rankine cycle waste heat recovery system. Nonlinear model predictive control and Dynamic Programming Random Forest methods exhibit more power production than the Proportional-IntegralDerivative rule-based method. The main contribution of this paper is that the Dynamic Programming method can be indirectly implemented real-time and showing comparable results with the nonlinear model predictive control. Compared with Dynamic Programming Random Forest method, the nonlinear model predictive control development requires more effort to build estimators and requires more computation power in the real-time application. Therefore, in the product development, Dynamic Programming Random Forest method not only saves development time, but also saves hardware cost for the products. The nonlinear model predictive control method is preferred when system physics is the focus on the products. Even though Random Forest model is not physics-based, its extrapolation weakness can be assuaged by increasing the operating range of the training data. Thus, Dynamic Programming Random Forest method has the potential in the real-time implementation and could be an alternative to the nonlinear model predictive control in the organic Rankine cycle waste heat recovery applications. Some limitations of current methods will be addressed in future work: (i) the calculation conditions are narrow and should be expanded to wider ranges, (ii) the electric generator efficiency should be a function of torque and speed rather than constant value, (iii) pressure drops across components should be considered, and (iv) experimentally validated the three methods.

Fig. 11. . (a) Working fluid pressure at pump outlet and (b) exhaust gas temperature at TP evaporator outlet.

choose the best vapor temperature reference and does not reduce its vapor temperature variation. NMPC and DP-RF shows similar cumulative energy and net power. In terms of temperature variation, NMPC shows better performance as it can maintain the vapor temperature in a narrow window. DP-RF shows slightly more cumulative energy along the 300 s simulation. With the similar performance, the DP-RF has some advantages over NMPC in the development process as well as the implementation cost. In the NMPC development, an estimator is utilized, while the DP-RF does not need the estimator. In the implementation, NMPC method needs to execute state estimation and optimizes the inputs in the prediction horizon, while DP-RF method only needs to calculate input at given exhaust conditions with the trained RF model and rule-based method. Therefore, NMPC method consumes more computation power than the DP-RF method. Among the three methods, only NMPC requires online optimization and the rest two methods only look up the results optimized offline. Thus, NMPC consumes the longest calculation time, followed by DP-RF and PID-RF, respectively. Based on the experimental data showed in [48], NMPC required 87 ms to finish one round of calculation in dSPACE MicroAutobox hardware, which is shorter than the actuator position refresh period (200 ms). The calculation time of PID-RF and DP-RF are less than 10 ms. For the DP-RF method, RF model is black box machine learning method, while the NMPC method utilizes MB physics-based model. As the black box model, RF model is inferior to MB model in the extrapolation when operating conditions are beyond the range of training/identification operating conditions. This weakness of RF model can be assuaged if training operating conditions covers large range. Based on the above discussion, the main reasons of large difference among the results of three methods are consideration of thermal delays and physics representation by the models. The PID-RB method shows large temperature and mass flow rate in working fluid because of lacking the consideration of thermal delays, whereas the NMPC and DPRF consider thermal delays in the models. The difference of NMPC and DP-RF model is caused by the physics representation by the models. The MB model used by NMPC captures more thermal system physics, while the RF model in DP-RF method lacks physics representation.

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