Fuel Minimization of the Electric Engine Cooling System With Active Grille Shutter by Iterative Quadratic Programming

The electric engine cooling system with the active grille shutter requires intelligent and predictive control to reach its full benefits on fuel economy and thermal management. Conventional control methods regulate the coolant temperature to a fixed value but do not directly minimize the vehicle's fuel/energy consumption. By contrast, we design a fuel minimization controller through solving constraint nonlinear optimization problems, whose cost function is the total fuel consumption and constraints are the vehicle's physical limits. To achieve high computational efficiency and sufficient accuracy, the optimization problem is solved by iterative convex quadratic programming and quasilinearization. The advantages of the proposed control method on both fuel economy and engine thermal management are demonstrated by simulations.


I. INTRODUCTION
T HE engine cooling system regulates the temperature of the internal combustion engine and is vital for the engine's thermal safety, performance, fuel economy, emission characteristics, etc. A conventional engine cooling system contains a coolant pump and a radiator fan that are both mechanically coupled to the engine crankshaft through belts, which have little flexibility on the output speed and low energy efficiency. The electric engine cooling system uses electric motors to drive the coolant pump and/or the radiator fan. The higher energy efficiency and better controllability of the electric motors achieve less auxiliary fuel consumption and better engine thermal management [1]- [9]. Another new technology beneficial for both fuel economy and engine thermal management is the active grille/radiator shutter (AGS) [10]- [14]. If the shutter is not fully open, part of the front airflow is blocked from entering the engine compartment and circumvents to sides and undercarriage. Such airflow creates less aerodynamic drag than the same amount of airflow entering the engine compartment and hence reduces fuel consumption. On the thermal aspect, reduced radiator airflow is favorable when the engine temperature is low [15], but detrimental when its temperature is high. Therefore, the shutter control must accommodate the contradictory objectives of reducing aerodynamic drag and improving the cooling system's performance [16]- [18].
Because of the tight coupling of the shutter and the cooling system, the two must be controlled together as an integrated system to reach optimal fuel economy, better thermal management, and the satisfaction of all constraints. Simple feedback controllers using rules and lookup tables are reported in [19]- [22]. These controllers determine the shutter position according to engine oil temperature, coolant temperature, environment temperature, the vehicle speed, etc. While demonstrating the benefits of cooling systems with the active grille shutter, these simple controllers neither directly minimize the total fuel/energy consumption of the vehicle nor explicitly satisfy all constraints.
The predictive [23] and optimal control strategies are more effective for the two objectives. Karnik et al. [16] and Bonkoski et al. [17] formulate a model predictive controller (MPC) for temperature regulation and constraint satisfaction for advanced cooling systems with a continuous grille shutter, but the method does not directly minimize the energy consumption. By contrast, Cho et al. [18] formulate an optimal control approach where the power consumption by the aerodynamic drag is the cost function and the radiator heat rejection rate is the constraint; however, the method neither considers the energy consumption of the cooling actuators nor explicitly enforces the boundary constraint on coolant temperature. Moreover, the optimization is solved by a greedy algorithm that minimizes the power consumption at every step.
To the best of our knowledge, there is no optimal control method for the electric cooling system with the active grille shutter that directly minimizes total fuel consumption, explicitly satisfies all constraints, and exploits the coolant and battery as energy buffers. The smart usage of the battery largely contributes to fuel reduction. When the battery provides electricity to the cooling actuators, the engine has less auxiliary load. When the battery is recharged, the energy may come from either the engine or the vehicle kinetic energy through the regenerative This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ brake [24]. In the latter case, the generated electricity is free and can significantly reduce the fuel consumption of the electric cooling system. Energy buffers of the electric cooling system are elaborated in Section 1.2 of [9].
The paper has two contributions. One is an optimal control approach that uses the accumulated fuel consumption caused by the cooling actuators and the aerodynamic drag along the entire drive cycle as the cost function and all applicable limits as hard constraints. To achieve high computational efficiency and sufficient accuracy, the original nonlinear cost function is approximated as a convex quadratic function and the non-convex state update model and constraints are approximated as linear time-varying (LTV) models. Then the optimization problem is solved by iterative convex quadratic programming (QP) [25] and quasilinearization [26]. The other is to systematically study the properties of the iterative QP method, including convergence, robustness, optimality, and scalability.
Simulation studies with trucks of different weights on two long and realistic drive cycles show the improved fuel efficiency. Compared to a rule-based baseline controller, the proposed method can reduce the total fuel consumption by 0.39% to 0.70% and the alternator fuel consumption by 32.67% to 58.6%. The reduction on the total consumption appears small, because the percentage of the energy consumption by the cooling system over that by the complete vehicle is less than 1.2%. Compared to the same truck and the same cooling system without the grille shutter, the active shutter can reduce the total fuel consumption by 0.13% to 0.25%. The mean value of the coolant temperature is increased by 5.5 • C to 8.1 • C.
Owing to limited resource, the paper has a few limitations. First, the optimal trajectories of control inputs are computed for the entire drive cycle. Real-time MPC implementation will be developed in future work. Second, the paper investigates only the fuel saving potential by reducing the engine torque through the control of the electric cooling system. The possible benefit of higher coolant temperature, which may increase combustion efficiency in the engine and reduce friction of the lubricant oil [27], is not studied owing to lack of measurement data for the experimental truck. Third, the influence of our controller on tailpipe exhaust emission is not quantitatively studied. We will integrate emission control and fuel reduction control [28]- [30] in the future.
Section II introduces the models of the electronic cooling system with the active grille shutter. Section III formalizes the optimal control problem and proposes an efficient solution based on linear time-varying model approximation and quadratic programming. To improve the accuracy of the solution, Section IV uses quasilinearization to iteratively find better solutions. Section V verifies the advantages of the proposed approach through simulations. Section VI concludes the paper and suggests future works.

II. MODELS OF THE ELECTRIC COOLING SYSTEM
The electric cooling system contains a coolant circuit, an electric coolant pump, a number of parallel installed electric radiator fans, a wax-based thermostat, and a shutter, as illustrated in Fig. 1. This section briefly introduces control-oriented models of these components. More details on these models are provided in [9], [13], [14].

A. Coolant Temperature
The model of the coolant temperature is determined bẏ where T e is coolant temperature at the engine outlet (or coolant temperature for simplicity), T a the ambient temperature, Q in the heat power transmitted from the engine to the coolant, ν r coolant volumetric flow rate through the radiator, m r air mass flow rate through the radiator. Parameters c 1 , c 2 , and c 3 are constants obtained by model identification. The formula is simplified from those in [7], [31], [32]. The justification, parameter identification and verification of (1) are elaborated in [9]. The radiator coolant volumetric flow rate ν r is determined by the pump coolant volumetric flow rate ν p and the thermostat opening ratio θ m ∈ [0, 1]. The pump coolant flow rate is determined by its speed ω p and the thermostat ratio determined by the coolant temperature T e according to (2).
In summary, the value of ν r is determined by T e and ω p as follows.
The air mass flow rate through the radiator m r is determined by the shutter position s ∈ [0, 1], fan speed ω f , and vehicle speed v. When there is no shutter, the radiator air mass flow rate is a function of ω f and v, denoted as m a (ω f , v). The shutter decreases the airflow by covering part of the radiator. When the shutter is fully closed (s = 0), the ratio between the uncovered radiator area and the total area is θ s0 < 1. When the shutter is fully open (s = 1), the ratio between the uncovered radiator area and the total area is θ s1 . It must be true that 0 ≤ θ s0 < θ s1 ≤ 1. For an arbitrary s ∈ [0, 1], the area-reduction factor is estimated by the linear function Therefore, the radiator air mass flow rate is estimated as follows.
The determination of m a (ω f , v) is presented in Section II-C.

B. The Electric Coolant Pump
The models of the electric pump are built from measurement data on the coolant volumetric flow rate and electrical power consumption. The volumetric flow rate is approximately a linear function of the pump speed: where ν p (liter/min) is the coolant volumetric flow rate through the pump, ω p (rpm) is the pump speed in the range [ω p_min , ω p_max ] and λ 1 and λ 0 are constant coefficients calculated by curve fitting. The electrical power consumption dependent on the pump speed is approximated as a quadratic function: where P pmp is the electrical power consumption of the coolant pump and p 2 , p 1 and p 0 are constant coefficients calculated by curve fitting [9].

C. The Electric Radiator Fan
The cooling system of the studied truck consists of multiple identical electric fans in parallel. The model here is for the overall effect of all fans. Without the shutter, the total air mass flow rate (in kg/s) through the radiator is a function of fan speed ω f (rpm) and vehicle speed v (m/s). The function is measured as The relation of the electrical power consumption and fan speed is approximately a quadratic function obtained from measurement data.
where P fan is the electrical power consumption of the fans, and f 2 , f 1 , f 0 are constant coefficients [9].

D. The Battery and the Alternator
The alternator converts the mechanical energy of the engine into electricity and is the primary power source for the electrical system. The battery is in parallel connection with the alternator output and serves as an energy buffer. A detailed description of the electrical system is presented in Section 2.5 of [9]. Battery model is a complex research topic. Our prior article [9] adopts a simple battery model based on the measurement data of the studied battery. Although the nominal voltage of the battery is 24 V, the measurement data show that the battery voltage quickly surges to a high value around 29 V during fast charging and slowly declines to a lower value around 25 V during normal operations. Then we simply estimate the battery output voltage U b by two distinct values depending on whether the regenerative brake is applied, which is determined by the method in Section II-E.
Denote the current through the battery as I b (A). If I b is positive/negative, the battery is charged/discharged. Let Q b be the nominal capacity of the battery's electric charge. Then the change rate of SOC isṠ where η c is the coulombic efficiency of the battery and estimated from the measurement data. Its value is typically 1 when discharging and less than 1 when charging. The output power of the battery is The alternator electrical power P alt is calculated as where P oth is additional power consumption of other electrical loads including lights and ECUs. By (5), (6) and (8), the alternator electrical power is a quadratic function of ω p , ω f , and I b , The alternator current I alt is I alt = P alt U b . The maximal current supply of the alternator is a function of the engine speed and the alternator current cannot be negative. Therefore The alternator is connected to the engine's crankshaft via belts and adds additional torque to the engine crankshaft. The alternator's average power efficiency is dependent on engine speed, denoted as η a (ω e ). Owing to the friction of the alternator and the belt transmission, we consider a minimal mechanical power loss of the alternator, denoted as M alt (ω e ). Finally the alternator's additional torque on the engine crankshaft is For the purpose of reducing fuel consumption, there is no benefit to have P alt less than M alt (ω e )η a (ω e ). Therefore we replace (10) by an equation and an inequality: The combination of (9) and (12) is equal to the constraint By (11) we have a further constraint on the alternator torque where

E. Driveline Torque
The engine's crankshaft is connected to the gearbox through a clutch. The torque applied at the input shaft of the gearbox is the driveline torque τ d . Assume that all relevant vehicle parameters and the drive cycle are known, including the trajectories of speed and slope angle. Under the common assumption that all vehicle components are stiff and the tire slip ratio is zero, the traction force F t between the driving wheels and the ground can be computed by the standard approach [33], [34]. The aerodynamic drag F a (v, s) is a term of F t and dependent on both vehicle speed and the shutter position.
where ρ a is the air density, The traction force F t is realized by the driveline torque τ d and the brake torque τ b through the transmission. Let n be the gear number, i t (n) the total transmission ratio including both the gearbox and the final gear, η t (n) the total transmission efficiency, and r w the radius of the driving wheels.

F. The Engine Model
The engine is characterized by the fuel rate map and the heat release map from measurement. Both are 2D lookup tables dependent on the engine's torque and speed. The engine fuel rate mapṁ f is approximated by a quadratic function of the engine torque at different engine speeds, i.e., The map of heat transfer to the coolant is approximated by linear functions at different engine speeds.
The speed-dependent coefficients in (17) and (18) are obtained by curve-fitting methods elaborated in [9]. The engine speed is calculated from the vehicle speed and the transmission ratio.
The torque value τ e is the effective torque at the engine crankshaft. Owing to internal friction, τ e may be negative. The minimal engine torque τ e_min < 0 is equivalent to the internal friction torque at which the engine fuel rate is 0, and the value is dependent on the engine speed. The effective engine torque is equal to the sum of the driveline torque and alternator torque.
where τ a is calculated from the three control variables ω p , ω f , I b via (10) and always positive. When τ d (s) ≥ 0, the engine provides torque to both the driveline and the alternator. When τ d (s) < 0, the driveline requires brake torque at the input shaft of the gearbox. When the negative driveline torque is within the range τ d (s) + τ a ≥ τ e_min , the brake torque is fulfilled by the combination of engine friction and alternator load. This suggests that vehicle brake torque may drive the alternator for generating electricity, enabling regenerative brake for energy recovery. When the negative driveline torque τ d (s) is so small that τ d (s) + τ a < τ e_min , the friction brakes at the wheels must be engaged. On the other hand, the sum of the driveline torque and the alternator torque cannot exceed the maximal engine torque: The inequality above is replaced by the following stronger constraint.
The alternator torque is also bounded by its own physical limit as in (13). The two inequalities both constrain the upper bound of the alternator torque. Denote the actual upper bound of the alternator torque as Then the constraint of (13) on the alternator torque becomes

III. THE OPTIMAL CONTROL PROBLEM
The control objective is to minimize the total fuel consumption for a given drive cycle, which contains timed signals of vehicle speed v(t) and road slope angle θ(t), where 0 ≤ t ≤ T and T is the traveling time of the drive cycle. This paper assumes that the truck follows the driving cycle without any deviation. Let Δt be the sampling time and N the length of the drive cycle, i.e., T = N Δt. The discrete-time state space model of the electric cooling system and the battery is the following: where k = 0, 1, . . . , N − 1 andṪ e (k) is calculated by (1) at time kΔt. Let the state vector be Eq. (20) is simply denoted by The optimal control problem is formalized as follows.
subjec to (20) and the following constraints In (21b), is a small positive number. The problem is solvable by offline global optimal control methods including dynamic programming (DP) [35]- [37] and Pontryagin's minimum principle (PMP) [24], [38], [39]. Since the offline methods rely on accurate dynamic model and the full knowledge of the drive cycle, and are expensive in computation time and memory consumption, they are not suitable for real-time feedback control with embedded microprocessors. In the following, we simplify the problem to benefit from more efficient optimization solvers.
The objective in (21a) is to minimize the total fuel consumption for both vehicle traction and auxiliaries. Given the assumption that the vehicle exactly follows the drive cycle and the torque demand of the auxiliaries does not alter gear selection, the fuel consumption for vehicle traction is invariant of the control demands of the cooling system, i.e., pump speed, fan speed, and battery current. Furthermore the default position of the shutter is open. Decreasing its opening position only slightly decreases the driveline torque. Consequently the objective function can be changed to the fuel consumption variation caused by the cooling system and the shutter.
Let Δṁ f (k) be the fuel rate variation at time kΔt. For simplicity we skip the argument k in the following discussion.
where α f = 2r 2 (ω e )(τ d1 + τ a_min ) + r 1 (ω e ) is the derivative of fuel rate over engine torque. Its value is independent of the control input. If τ d1 + τ a_max ≤ τ e_min , the brake torque from the driveline can produce the maximal alternator torque and still keep the engine fuel rate to be zero. All energy to drive the cooling system and charge the battery comes from the kinetic energy and hence does not consume any fuel. α f is then refined as Δṁ f is linear in the driveline torque τ d (s) and the alternator torque τ a . The first term of (22) is identical to our previous result on estimating the fuel consumption of the cooling system [9]. The second term of (22) estimates the fuel consumption reduction owing to the reduced aerodynamic drag.
According to Section II-D, τ a is the quadratic function In the expression of α f (τ a − τ a_min ), p 0 , f 0 , P oth , and τ a_min are irrelevant of the control input u. We then remove them and simplify the first term of Δṁ f to a quadratic cost function.
Note that α is dependent on the driveline torque and the engine speed, and hence a function of time. Ignoring the brake torque τ b in (16), we calculate the driveline torque demand τ d (s) as follows. where is the conversion ratio from the traction force at the wheels to the torque at the gearbox input. The equation shows the dependency of β t on the control input s. For example, it is possible that when s = 1, F t (1) ≥ 0 but when s = 0, F t (0) < 0. The switching possibility complicates the calculation of τ d (s) and prevents us from solving this optimization problem by QP. In quantity, the difference between F t (1) and F t (0) is very small. We can simply replace F t (s) by F t (1) (or F t (0)) to evaluate β t , i.e., To obtain F t (s), only F a (v, s) is related to the control input s by (14). The derivative of F t (s) over s is Ignoring parameters irrelevant of the control input, we have the cost function of the second term of (22) as Note that β is dependent on the driveline torque, engine speed, gear number, and vehicle speed. It is hence a function of time. The total cost function is then Then the cost function at step k ∈ {0, 1, . . . , N − 1} becomes Finally the optimal control problem is reduced to the new quadratic format.
subject to the state update equation in (20) and all constraints in (21b)-(21i). Its solution is presented in the next section.

IV. ITERATIVE QP BASED ON QUASILINEARIZATION
The optimal control problem in (29) cannot yet be solved by a QP solver because the state update function in (20) and the constraint in (21i) are nonlinear. This section presents a quasilinearization method [35] to approximate the nonlinear equations by a sequence of linear equations and solve each linear approximation by QP.

A. Linear Time-Varying Model
The nonlinear discrete-time state space model is presented in (20). A fundamental assumption in this work is that the control on the electric cooling system and the radiator shutter does not alter the vehicle speed trajectory and the vehicle exactly follows the drive cycle. This means that the trajectory of the disturbance vectors w(k) (k = 0, . . . , N − 1) is deterministic for the entire drive cycle. We first linearize the nonlinear model at an arbitrary pointp = (x,û,ŵ).
Since it always holds that w =ŵ, the last term is eliminated.
x k+1 = f (x k ,û k , w(k)) , k = 0, 1, . . . , N − 1 In real executions, the actual trajectory of control inputs {u(k)|k = 0, 1, . . . , N − 1} often deviates from the prediction trajectory. Consequently the actual trajectory of states {x(k)|k = 0, 1, . . . , N} is also different from the prediction trajectory of states. The relationship of the actual trajectories of states and inputs is approximately the following. For k = 0, 1, . . . , N − 1 In this equation, A k and B k are simple representations of matrices A (x k ,û k ,w k ) and B (x k ,û k ,w k ) as defined in (31).

B. Boundary Constraints on Torque
To use standard QP solvers, all constraints must be linear. The constraints below (21a) are all simple boundary constraints on state and control input, except the inequality constraint (21i) on alternator torque. This is because τ a (k) is unfortunately quadratic in the control input by (23). Consequently, τ a (k) is linearized at the estimated control vectorû k .
This is a linear function of the control input u(k). Hence the inequality constraint on the alternator torque in (21i) becomes

C. Solution by Quadratic Programming
The objective function of (29) is a quadratic function of the sequence of control inputs. We formalize it as a QP problem and solve it by a QP solver. All unknown variables are organized into the vector where k = 1, . . . , N − 1. Its length is 6 N . The total cost becomes a quadratic function of the vector z.
where H is a 6 N × 6 N square matrix such that for k = 0, . . . , N − 1, H(6k + 1 : 6k + 4, 6k + 1 : 6k + 4) = H k and other elements are 0. The vector g is a 6 N vector such that g(6k + 1 : 6k + 4) = g k and other elements are 0. Recall that H k and g k are defined in (28). The linear state update function of (32) is also reformulated as the matrix format.
where A eq is a 2 N × 6 N matrix and b eq is a 2 N vector. Their elements are given below. I 2 denotes the 2 × 2 identity matrix.
where A in is a 2N × 6 N matrix and b in is a 2 N vector.
All boundary constraints in (21b)-(21h) are formalized as boundary vectors of z. The lower and upper bound vectors of z are L b and U b , respectively. They are both 6 N vectors. For k = 0, . . . , N − 1 (29) is formalized as a QP problem of vector z. z T Hz + g T z subject to (36), (37), and L b ≤ z ≤ U b . This is a typical QP problem with linear constraints. Since the Hessian matrix H is diagonal and all diagonal elements are non-negative, the Hessian matrix is positive semi-definite. The QP problem is hence convex and can be efficiently solved by the interior point method. This paper solves the QP problem by the quadprog function from MATLAB Optimization Toolbox and chooses the interiorpoint-convex algorithm.

D. The Quasilinearization Algorithm
Owing to linearization, the QP solution to (29) is dependent on the estimated trajectories of control inputs {û k |k = 0, . . . , N − 1}. To alleviate the dependency, this paper uses quasilinearization method to iteratively find the best solution [26], [35], [40]. The principle is to iteratively linearize the nonlinear state update equations and the constraints and find the corresponding QP solutions. The optimal control sequence obtained at one iteration is used as the estimated control sequence for the next iteration. The initial estimate is obtained from a nonpredictive simple controller and the iterations continue until a convergence criterion is satisfied or a maximal iteration limit is reached.
Algorithm 1 shows the quasilinearization method. For simplicity, the range of the time step k is ignored in Algorithm 1, because the ranges of the control inputs and state vectors are always k = 0, . . . , N − 1 and k = 0, . . . , N, respectively. The superscript l enclosed by parentheses indicates values obtained at the lth iteration. Variable l * indicates the optimal solution satisfying all state constraints and achieving the smallest fuel consumption. Parameter maxIter at line 5 is a positive integer to limit the maximal number of iterations. The sequence {x (l) k } at line 7 is the state sequence estimated by the linear approximation model in (32) and is obtained together with {u (l) k } through solving the QP problem. Line 10 computes the RMSE of the two states between the nonlinear model and the linear model with the same control sequence. The QP solution is accurate only if the RMSE of the two states are less than the tolerances ξ i (i = 1, 2). Parameter δ at line 13 is a small number indicating the convergence of the quasilinearization algorithm.
Theorem 1: Algorithm 1 terminates within maxIter iterations. If the state sequence {x (0) k } obtained by the initial estimate of the control sequence {û k } satisfies all constraints of (21b), (21c), and (21d), the output of the optimal state sequence {x * (k)} must also satisfy all these constraints and fc * ≤ fc (0) .
Proof: Since the for-loop at line 5 specifies an iteration upper bound, the algorithm must terminate with at most maxIter iterations.
At line 1, l * is initiated as 0. When the algorithm terminates, two cases may occur: l * = 0 or l * > 0. If l * = 0, then x * (k) = x A conventional truck VOLVO FH 460HP 6 × 2 Tractor with the electric cooling system is studied in this paper. Its maximal gross weight is 60 tonnes and emission level EURO 6. The engine is 13 liter with the maximal power 460 hp at 1400-1800 rpm and the maximal torque 2300 Nm at 1400 rpm. The same truck is employed in [8], [9], [13], [14]. The accuracy of the cooling system model in (1) is verified in [9] and the normalized RMSE is less than 3%. In the following simulation studies, the weights of the simulated truck are 40 or 60 tonnes, and the ambient temperature is 35 • C.

A. Drive Cycles
Two drive cycles are chosen for simulation. One is a real route of around 90 km recording a round trip between Borås and Landvetter in Sweden. The route is called BLB. The other is an extension of the standard drive cycle ACEA Long Haul published by the European Association of Car Manufacturers (ACEA) [41]. The route is designed for long haul trucks for goods deliveries and consists of mainly highway operations and

B. Convergence of the Quasilinearization Method
This section verifies the convergence property of the quasilinearization algorithm in Algorithm 1. The algorithm starts with an initial estimate of the control sequence, which is determined by a baseline controller in this paper. The baseline controller simply determines the speeds of the coolant pump and the radiator fans and the shutter position by coolant temperature according to three 1D lookup tables illustrated in Fig. 3. The battery current is constant 0 A.
The plots show the percentages of the control demands. The maximal speeds of the pump and the fans are 3100 and 3400 rpm, respectively. The minimal pump speed is 500 rpm (16.13%). The thermostat is closed when the coolant temperature is below 82 • C and fully open when the temperature becomes 92 • C. Afterward, when the coolant temperature rises, the pump speed increases and reaches the maximal speed at 102 • C. The shutter is closed when the coolant temperature is less than 92 • C and becomes fully open when the coolant temperature reaches 95 • C. After the shutter is fully open, the fans speed starts to increase   when the coolant temperature is larger than 95 • C and reaches its maximum at 105 • C.
This section elaborates the converging process of Algorithm 1 for a 40 tonne truck on the BLB cycle. The time step is Δt = 2 seconds. Then the total number of sampling steps is N = 1886. According to (34), the number of optimization variables is 6 N = 11316. Let maxIter = 5 in Algorithm 1. Fig. 4(a) visualizes that the maximal coolant temperature is safe in all iterations except the third one. Fig. 4(b) shows that the final SOC converges to acceptable values from the second iteration. The plots in Fig. 5 show the RMSE of coolant temperature and SOC between the linear and the nonlinear models. The differences between the two models quickly decrease to acceptable small values.
The termination criterion of Algorithm 1, as in line 13, is the normalized difference between the fuel consumptions of two consecutive iterations. The values of five iterations are plotted in Fig. 6(a). The fuel consumptions of the truck during the iterations of Algorithm 1 are plotted in Fig. 6(b). Compared with other plots in this subsection, the iteration axis starts at 0, which indicates the fuel consumption by the baseline controller shown in Fig. 3. The fuel consumptions of iterations 1-5 are obtained by the QP method presented in Section IV. Fig. 6 shows that both termination criterion and fuel consumption converge after 3 iterations.
The outcome of Algorithm 1 is dependent on several tuning parameters. If at line 11 the maximal difference between the final SOC and the initial SOC is 0.001, the maximal coolant temperature is 105 • C, the maximal RMSE of the coolant temperature and the SOC are ξ 1 = 0.2 • C and ξ 2 = 0.001, and termination threshold at line 13 is δ = 10 −4 , then Algorithm 1 terminates after 4 iterations and the minimal fuel consumption at this case is 30.653 liters. Although Fig. 6(b) shows that the fuel consumption at iteration 1 is minimal, Fig. 4(b) shows the final SOC at iteration 1 violates the constraint.
Similar converging properties are observable for all other test cases: 40/60 tonne trucks on the BLB/ACEA cycles. There are all together four test cases. Owing to space limit, plots of the other three cases are skipped.

C. Robustness of the Quasilinearization Method
This section shows the robustness of Algorithm 1 against random initial control estimates via Monte Carlo simulations. The result also implies that the iterative QP method can find the real optimum of the nonlinear optimization problem (21a). Random perturbations are added to the baseline controller in Fig. 3 to generate distinct initial control sequences. The control demand to each of the three actuators is characterized by two switching temperature values. For example, the first switching temperature of the pump is 92 • C and the second is 102 • C. The six switching values in Fig. 3 are replaced by six random Every iteration of the Monte Carlo simulation generates a random sample of the six switching value. The sample then determines a perturbed baseline controller, which generates a control sequence {û k |k = 0, . . . , N − 1}, the corresponding state sequence {x k |k = 0, . . . , N}, and the fuel consumption fc (0) . To quantify the influence of the randomly perturbed baseline controller, we calculate the RMSE of the estimated temperature sequences obtained by the normal and the randomly perturbed baseline controllers, and the absolute difference of the fuel consumptions fc (0) between the two baseline controllers. Denote the two error values as RM SE_T e _in and err_fc_in. Algorithm 1 takes the control sequence as the input and then returns the optimal control and state sequences {u * k |k = 0, . . . , N − 1}, {x * k |k = 0, . . . , N} and the minimal fuel consumption fc * . We also calculate the RMSE of the optimal temperature sequences obtained by the normal and the randomly perturbed baseline controllers, and the absolute difference of the optimal fuel consumptions achieved by the normal and perturbed baseline controllers. Denote the two error values as RM SE_T e _out and err_fc_out. Fig. 7(a) shows the comparison between RM SE_T e _in and RM SE_T e _out for 10 random samples for the 40 tonne truck on the BLB cycle. Fig. 7

D. Computation Time Measurement
The computations are implemented by MATLAB 2018b on a PC with Intel Core i7-4810MQ@2.80 GHz and 16 GB RAM. The time step for all cases is Δt = 2 s. The numbers of optimization variables for all test cases are listed in Table I Table I. The data in parentheses represent (Mean, Standard Deviation).

E. Fuel Efficiency and Thermal Management
This section compares fuel consumptions of the iterative QP (IQP) method, the baseline controller, and the dynamic programming (DP) method [35], [36]. The baseline controller represents a non-predictive and simple state feedback control emulating the state-of-practice of vehicular cooling system control. The baseline controller emphasizes the thermal safety of the engine rather than the energy consumption of the cooling system. The DP method represents another extreme on finding the global optimal solution that both ensures thermal safety and minimizes energy consumption. DP is preferred to PMP, because the result of DP contains lookup tables that can be used as optimal state-based feedback controllers when the actual state value is different from the offline optimal state trajectory. A fundamental difference between the DP and the QP methods is that the DP method uses the original nonlinear model of the cooling system without any simplification. Table II lists the fuel consumptions and average coolant temperatures of the three comparison controllers for all test cases. For all comparison items, the baseline controller always has the largest fuel consumptions. The rows on reduced consumptions compute the relative difference in percentage between the iterative QP or the DP method and the baseline controller. For example, let m f _base and m f _qp be the total fuel consumptions of the baseline controller and the iterative QP method, respectively. Then, the reduction on the total fuel consumption (Reduc. Total FC) of the three-input method is calculated as follows: The first observation is that the iterative QP and the DP methods achieve very similar fuel consumptions, although the two methods employ completely distinct optimization approaches. The similarity confirms the correctness of the iterative QP method. Compared to the baseline controller, the two optimal controllers achieve around 0.5% reduction on the total fuel consumptions. The value seems marginal, but the optimal controllers mainly manage the cooling system, whose contribution to the fuel consumption is only around 1% for the baseline controller. If we compare only the auxiliary fuel consumptions of the alternator, the two optimal controllers achieve reductions from 33% to 59%. By contrast, the optimal controllers reduce only around 0.02% fuel consumptions of the driveline via the optimal control of the active shutter. The benefit of the active shutter is hence more on maintaining higher coolant temperature.
The plots in Figs. 8-12 illustrate the trajectories of states and control signals of the three comparison controllers for the 60 tonne truck on the ACEA cycle. The temperature trajectories of the two optimal controllers are very similar in Fig. 8(a). The optimal controllers use the coolant temperature as an energy buffer. Before the engine produces high power, e.g., at sections 10-18 km, 55-65 km, and 100-105 km, both optimal controllers decrease the coolant temperature to the minimum in advance. During the heavy load sections, the optimal controllers allow the coolant temperature to reach its maximum to reduce the auxiliary consumption of the cooling system. By contrast, the baseline controller does not exploit the coolant temperature as an energy buffer but tries to regulate the temperature around 91 • C regardless of the engine load.
The plots in Fig. 8(b) show the battery SOC of the three controllers. Since the baseline controller does not use battery electricity to drive the cooling system, the SOC of the baseline controller is constantly 0.8. The SOC trajectories of the two optimal controllers have noticeable difference, but both terminate at values close to the initial value of 0.8.  Figs. 9 and 10 compare the speed demands of the pump and the fans by the three controllers. Both the optimal controllers demand minimal speeds from the pump and the fans when the regenerative brake is not applied, except at the sections 10-20 km and 100-105 km. According to the drive cycle data in Fig. 2, the two sections are steep and long uphills. The engine produces a lot of heat while running the sections and large cooling effort is required to keep the coolant temperature safe, even though  the optimal controllers have proactively decreased the coolant temperature to the minimum before the uphill.
When the regenerative brake is applied during 0 to 60 km, the DP controller demands larger speeds from the pump and the fans than the IQP controller does. The coolant temperature under the DP controller, therefore, is generally lower than that under the QP controller, as illustrated in Fig. 8(a). The baseline controller does not take advantage of the regenerative brake and often requires larger pump speed than the two optimal controllers do when the regenerative brake is not applied. This increases the alternator fuel consumption. If the engine is already fully loaded by the driveline torque, the large pump speed creates large alternator torque and the total engine torque demand may be higher than its maximum. The two optimal controllers never violate the engine torque limit. Fig. 11 compares the shutter position requests by the three controllers. The baseline controller generally demands larger shutter position when the regenerative brake is not applied. This is a reason that the baseline controller achieves slightly larger driveline fuel consumption. Fig. 12 compares the battery current trajectories determined by the three controllers. As discussed before, the baseline controller does not use the battery electricity. When the regenerative brake is applied, both optimal controllers always demand the maximal charging current to recharge the battery by the free energy. When the brake is not applied, the two optimal controllers generally take around 40 A from the battery  to drive the electric cooling system. At the two steep uphills at 10-20 km and 100-105 km, the optimal controllers request large current from the battery to drive the electric cooling systems. The electrical energy from the battery greatly reduces the alternator fuel consumption during the heavy load sections.

F. Benefits of the Active Shutter
This section presents the benefits of the active grille shutter for the fuel consumptions and thermal management of automotive engine cooling systems through comparing the results of the iterative QP controller for the same electric cooling system with and without the active shutter. The latter controller is equivalent to the iterative QP controller if the shutter position is fixed to constant 1. Table III presents the comparison results. The rows labeled by "Mean&Std" show the mean and standard deviation of the coolant temperature and shutter position, respectively. The column labeled by "Diff." shows the difference between the values with and without the shutter. If the difference is the relative value in percentage, the denominator is the value without the shutter. The differences of coolant temperature and shutter position are only for the mean values. Table III shows that the active shutter can slightly decrease the total fuel consumption, because of the decreased aerodynamic drag. The table shows positive causal relationship between the reduction of the average shutter position and the reductions on both total and driveline fuel consumptions. The negative impact of the shutter is the increase of alternator fuel consumption, because the closed shutter decreases the radiator air flow and the coolant temperature becomes higher. The shutter increases the average coolant temperature but the corresponding controller still keeps the temperature within the safe limits. Higher coolant temperature may increase combustion efficiency, reduce friction of the lubricant oil, and increase the efficiency of exhaust catalysts [15], [27]. The radiator shutter hence has the benefits of increasing engine energy efficiency and decreasing pollutant emissions.

VI. CONCLUSION AND FUTURE WORK
The paper applies iterative quadratic programming and quasilinearization methods to the optimal control of the automotive electric engine cooling system with the active grille shutter. The coolant and the battery are exploited as energy buffers for improving engine efficiency and recuperating brake energy. The original formulation of the energy minimization problem is a complicated nonlinear optimization problem with nonlinear cost and constraints. Its direct solution is time consuming. A main contribution of this paper is to approximate it by a convex quadratic problem with linear constraints, which can be solved efficiently in large scale. To mitigate the discrepancy between the approximate and original models, the quasilinearization method is adopted to iteratively find approximations of the original models and solve QP problems until the answers are sufficiently accurate.
Compared with a rule-based baseline controller, the proposed optimal control method reduces the total fuel consumption by 0.39% to 0.69% and the alternator fuel consumption by 32.67% to 58.6%. The active grille shutter reduces the total fuel consumption by 0.13% to 0.25% and increases the average coolant temperature by 5.5 • C to 8.1 • C. This paper is limited to the global optimal control for the complete drive cycle, but the same method can be adapted to MPC, where the drive cycle data within a limited prediction horizon are estimated online [9]. We shall develop the real-time MPC controller and test it on a high-fidelity simulation model or an experimental truck in the future. The influences of higher coolant temperature on fuel efficiency and exhaust emission will also be studied.