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

T

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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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