Coordinated Receding-Horizon Control of Battery Electric Vehicle Speed and Gearshift Using Relaxed Mixed Integer Nonlinear Programming
aa r X i v : . [ ee ss . S Y ] F e b Coordinated Receding-Horizon Control of BatteryElectric Vehicle Speed and Gearshift Using RelaxedMixed Integer Nonlinear Programming
Nan Li, Kyoungseok Han ∗ , Ilya Kolmanovsky, and Anouck Girard Abstract —In this paper, we propose an approach to coordi-nated receding-horizon control of vehicle speed and transmissiongearshift for automated battery electric vehicles (BEVs) toachieve improved energy efficiency. The introduction of multi-speed transmissions in BEVs creates an opportunity to ma-nipulate the operating point of electric motors under givenvehicle speed and acceleration command, thus providing thepotential to further improve the energy efficiency. However, co-optimization of vehicle speed and transmission gearshift leads toa mixed integer nonlinear program (MINLP), solving which canbe computationally very challenging. In this paper, we proposea novel continuous relaxation technique to treat such MINLPsthat makes it possible to compute solutions with conventionalnonlinear programming solvers. After analyzing its theoreticalproperties, we use it to solve the optimization problem involvedin coordinated receding-horizon control of BEV speed andgearshift. Through simulation studies, we show that co-optimizingvehicle speed and transmission gearshift can achieve considerablygreater energy efficiency than optimizing them sequentially,and the proposed relaxation technique can reduce the onlinecomputational cost to a level that is comparable to the timeavailable for real-time implementation.
Index Terms —Battery Electric Vehicle, Energy Efficiency,Mixed Integer Optimization.
I. I
NTRODUCTION W ITH the market penetration of Battery Electric Vehicles(BEVs) projected to increase, there has been a growinginterest in approaches to improving energy efficiency of BEVs.On the one hand, the increasing levels of connectivity andautomation have opened up new opportunities to optimizethe vehicle-level energy management [1]–[5]. For instance,these technologies enable the vehicle to predict future roadand traffic conditions, so the vehicle can take this informationinto account to plan its speed profile in a way that minimizesthe energy consumption. On the other hand, further energyefficiency improvement may be achieved by adding compo-nents, such as multi-speed transmissions, to the existing BEVpowertrain architecture. Traditionally, BEV powertrain onlyused a single reduction gear for forward driving [6]. How-ever, in recent years, multi-speed transmissions for electric
This research was supported by the National Science Foundation AwardECCS 1931738.Nan Li, Ilya Kolmanovsky, and Anouck Girard are with the Departmentof Aerospace Engineering, University of Michigan, Ann Arbor, MI, 48109,USA (email: [email protected], [email protected], [email protected])Kyoungseok Han (Corresponding Author) is with the School of MechanicalEngineering, Kyungpook National University, Daegu 41566, Republic ofKorea (email:[email protected]) vehicles have also been considered [7]–[10]. With a multi-speed transmission, the operating point of the electric motorcan be adjusted under given vehicle speed and accelerationcommand so that improved energy efficiency can be achievedat the powertrain level.In this paper, motivated by potential synergies in vehicleand powertrain-level energy management optimization, weconsider co-optimization of vehicle speed and gearshift forBEVs equipped with multi-speed transmissions to maximizethe energy efficiency.The problem of energy management for BEVs has beenstudied by several researchers in the literature. Dynamic pro-gramming (DP) is a popular tool for determining the globallyoptimal trajectory offline when the entire trip is assumed to beknown a priori [11]–[13]. However, it has been revealed in [14]that prediction errors of the future speed trajectory may havea significant impact on the energy consumption. Therefore,a practical solution is to repeatedly update the predictionsbased on the latest available traffic information to mitigatesuch errors, and also update the control correspondingly inreal time. However, the intensive computations required byDP typically forbid its real-time implementation. Alternatively,Pontryagin’s minimum principle (PMP) has been exploited forvehicle speed planning in [15], [16]. A PMP-based energymanagement strategy for hybrid electric vehicles has alsobeen proposed in [8], which assumes a given speed profileand includes a gearshift strategy. Although PMP requires lesscomputational effort than DP, the two-point boundary valueproblem associated with PMP conditions may still be difficultto handle numerically.Using short-term preview information instead of predictionof the entire trip and optimizing vehicle speed based on areceding-horizon optimization/model predictive control frame-work has been considered in [17]–[20]. On the one hand,previewing road and traffic information over a short timehorizon may be easier and also more accurate than predictionover a long horizon or over the entire trip [21]–[23]. On theother hand, the online optimization is reduced to a finite-dimensional mathematical programming problem, which ishandled in real time.In this paper, we also consider a receding-horizon optimiza-tion strategy for improving BEV energy efficiency. In partic-ular, we develop an approach to coordinated receding-horizoncontrol of BEV speed and gearshift. Due to the discrete-valued gear ratios of a multi-speed transmission, the onlineproblem representing the speed and gearshift co-optimization is a mixed integer nonlinear programming (MINLP) prob-lem. To exactly solve such an MINLP problem is compu-tationally very demanding [24]. For instance, conventionalapproaches based on exhaustive-search or branch-and-bound[25] that either brute-force or systematically enumerate allsolution candidates for the discrete variables have worst-casecombinatorial complexity [24]. Therefore, optimizations ofvehicle speed and transmission gearshift are often treated ina sequential/hierarchical manner in previous literature. In [9],the speed trajectory is optimized first based on a cost functionthat approximately represents the energy consumption but doesnot involve powertrain variables. The most energy efficientgear is then selected for the obtained speed trajectory. In ourprevious work [26], such a sequential procedure is augmentedby an additional step, which refines the speed trajectory afterthe gearshift trajectory has been determined in the second step.Although these sequential optimization approaches appear tobe effective in improving BEV energy efficiency based onsimulation studies, optimality of the obtained trajectories isnot guaranteed.In contrast, in this paper we treat the BEV speed andgearshift co-optimization problem in a receding-horizon con-trol setting by first transforming it to an MINLP problem in aspecific form, then proposing a novel continuous relaxation ofthis MINLP problem, and finally approximately solving theoriginal MINLP problem through solving its correspondingcontinuous relaxation.The contributions of this paper are as follows:1) We propose a coordinated control strategy for BEV speedand gearshift based on receding-horizon optimization withspecific cost and constraints.2) We propose a novel continuous relaxation to the for-mulated MINLP that represents the speed and gearshift co-optimization problem. We first show that the relaxed problemis a nonlinear programming (NLP) problem with continuouslydifferentiable cost and constraint functions and can be treatedby off-the-shelf NLP solvers. We then show that the originalMINLP problem and the relaxed NLP problem have nofeasibility and optimality gaps at their global minimizers.Moreover, we also characterize the relationship between thelocal minimizers of the two problems and further address thisthrough numerical studies.3) We show based on a comprehensive set of simulationcase studies that considerably greater energy efficiency canbe achieved through co-optimization of vehicle speed andtransmission gearshift than by optimizing them separately. Wealso show that the proposed relaxation technique can reducethe online computational cost to a level that is comparable tothe time available for real-time implementation. Our approachthus opens up a possibility for real-time coordinated receding-horizon control of vehicle speed and transmission gearshift.The rest of the paper is organized as follows: Section IIformulates the BEV speed and gearshift receding-horizonco-optimization problem and transforms the problem intoan MINLP with a specific form. Section III introduces theproposed continuous relaxation technique to the transformedMINLP problem and discusses its theoretical properties. Sec-tion IV describes several other energy efficiency optimization approaches for BEV speed and gearshift control to compareversus the proposed receding-horizon co-optimization strategy.A comprehensive set of simulation results is presented inSection V. Finally, Section VI concludes the paper.II. B
ATTERY E LECTRIC V EHICLE S PEED AND G EARSHIFT C O -O PTIMIZATION
In this section, we formulate the BEV speed and gearshiftco-optimization problem for improved energy efficiency. Wefirst introduce the models for vehicle longitudinal motion, bat-tery state-of-charge (
SOC ) and transmission gear dynamics,and then define the co-optimization problem.
A. Vehicle and Battery Models
The longitudinal motion of the vehicle is modeled asfollows: ˙ s = v, (1) ˙ v = T w r w m eff − m ρA f C d v − g (sin θ + µ cos θ ) , (2)where s is the vehicle travel distance, v is the vehicle speed, m is the vehicle mass, m eff is the vehicle effective massaccounting for both static mass and rotational inertia effects, T w is the wheel torque, r w is the tire radius, ρ is the airdensity, A f is the vehicle frontal area, C d is the aerodynamicdrag coefficient, g is the gravitational constant, θ is the roadinclination, and µ is the rolling resistance coefficient.The wheel torque T w is determined by the motor torque T m ,friction brake torque T b , reduction gear ratio i g , and final driveratio i as follows: T w = T m i g i − T b . (3)Ideally, friction brakes are used only when the maximumtorque that can be provided by the motor is not sufficient toachieve the required braking. In this paper, we assume thatsuch cases are not occurring, i.e., T b = 0 .The evolution of battery SOC is modeled as [10]: ˙ SOC = − I b C = − V oc − p V oc − R b P b CR b , (4)where I b is the battery current, C is the battery capacity, V oc = V oc ( SOC ) is the open-circuit voltage in series with the batteryresistance R b = R b ( SOC ) , both of which depend on battery SOC , and P b is the battery power determined as: P b = P m η + b = T m w m η + b η m when T m ≥ , P m η − b = T m w m η − b η m when T m < , (5)where η + b ∈ (0 , is the battery-depletion efficiency, η − b > is the battery-recharge efficiency, w m is the motor speeddetermined as: w m = vr w i g i , (6)and η m is the motor efficiency depending on the motoroperating point, i.e., η m = η m ( T m , w m ) . The maps V oc ( SOC ) , R b ( SOC ) and η m ( T m , w m ) are typically estimated based onexperimental data and provided as lookup tables. We discretize the continuous-time models (1)-(6) using theforward Euler method with a sampling period ∆ t , and combinethem into a discrete-time model in the form of ξ t +1 = Φ( ξ t , γ t ) , (7)where ξ = [ s, v, SOC ] ⊤ is the state vector, γ = [ T m , i g ] ⊤ isthe input vector, the subscript t represents the discrete timeinstant, and the function Φ is determined by (1)-(6) and thesampling period ∆ t . B. Transmission Model
For a BEV with a multi-speed transmission, the gear ratio i g can take a finite number of different values. In particular, for agiven transmission model with η max gears, i g is determined bythe gear position η g , i.e., i g = i g ( η g ) , with η g ∈ { , · · · , η max } .To avoid gear skipping, we model gear changes as [10]: η g ,t +1 = η g ,t + ζ t , (8)where ζ t ∈ {− , , } is the gearshift signal, with − and representing, respectively, the down- and up- shift signals, and representing maintaining the current gear position. In thispaper, a three-speed transmission is assumed to be employed,i.e., η max = 3 . C. Speed and Gearshift Receding-Horizon Co-Optimization
We pursue co-optimization of vehicle speed and trans-mission gearshift to maximize energy efficiency. Consideringan automated vehicle control system, the reference vehiclespeed v r and the corresponding reference travel distance s r are typically available over a short time horizon, however,deviations from these reference trajectories are permissiblewithin prescribed bounds. Such a reference speed and distancepreview can be informed by short-term prediction of thespeed(s) of the vehicle(s) driving in front [21]–[23] or by anautomated vehicle planning module [27]. In this paper, weassume v r corresponds to the predicted speed of the vehicledriving immediately in front of and being followed by the egoBEV.The speed and gearshift co-optimization is achieved bysolving the following minimization problem repeatedly in a receding-horizon manner: min J t = − SOC N | t + N − X k =0 (cid:0) w ( v k +1 | t − v r ,k +1 | t ) + w ( T w ,k | t − T w ,k − | t ) (cid:1) , (9a)s.t. ξ k +1 | t = Φ( ξ k | t , γ k | t ) , (9b) τ min ( v k +1 | t + δ ) ≤ s r ,k +1 | t − s k +1 | t ≤ τ max ( v k +1 | t + δ ) , (9c) | v k +1 | t − v r ,k +1 | t | ≤ max( ε v r ,k +1 | t , δ ) , (9d) T min ( w m ,k | t ) ≤ T m ,k | t ≤ T max ( w m ,k | t ) , (9e) i g ,k | t = i g ( η g ,k | t ) , (9f) η g ,k +1 | t = η g ,k | t + ζ k | t (9g) η g ,k +1 | t ∈ { , · · · , η max } , (9h) ζ k | t ∈ {− , , } , (9i) ∀ k = 0 , · · · , N − , N − X k =0 | ζ k | t | ≤ ζ max , (9j)with respect to the decision variables u k | t = [ T m ,k | t , ζ k | t ] ⊤ , k = 0 , · · · , N − , where the notation ( · ) k | t designates apredicted value of the variable ( · ) t + k with the prediction madeat the current time instant t .The motor torque T m and the gearshift signal ζ are chosen asthe decision variables because the values of all other variables,including the vehicle speed v and the battery SOC , can beuniquely determined by T m and ζ based on the models (7)and (8), which are treated as equality constraints in (9b) and(9g). The term − SOC N | t in the cost function (9a) is forminimizing energy consumption. The terms ( v k +1 | t − v r ,k +1 | t ) and ( T w ,k | t − T w ,k − | t ) = ( T m ,k | t i g ,k | t − T m ,k − | t i g ,k − | t ) i in (9a) are for penalizing deviations of the actual speeds fromthe reference speeds and for penalizing changes in wheeltorques, respectively, to improve safety and comfort (by re-ducing jerk). The constraint (9c) represents the requirement ofkeeping the ego BEV’s time-headway to its preceding vehiclewithin the range [ τ min , τ max ] to avoid rear-end collisions andcut-ins by other vehicles, where the predicted travel distancesof the preceding vehicle s r ,k +1 | t are determined according tothe dynamic equation s r ,k +1 | t = s r ,k | t + v r ,k | t ∆ t based onthe current distance s r , | t = s r ,t and the predicted speeds v r ,k | t . The constraint (9d) represents prescribed bounds on themaximum deviations of the actual speeds from the referencespeeds. The constraint (9e) represents the range of torques, [ T min , T max ] , that can be provided by the motor at the speed w m ,k | t . The constraints (9f)-(9i) correspond to the transmissionmodel introduced in Section II-B. And finally, the constraint(9j) requires the number of gearshifts over the planninghorizon to be upper bounded by ζ max , to avoid overly frequentgearshifts.At every discrete time instant t , after solving the optimiza-tion problem (9), the ego BEV applies the obtained T m , | t and ζ | t over one sampling period ∆ t to update its states, thenrepeats this procedure at the next time instant t + 1 . Due to the fact that T m ,k | t takes continuous values and ζ k | t takes values in the discrete set {− , , } , the problem (9) isa mixed integer problem with many constraints. Note that inpractice the number of gearshifts over a short time period (e.g., ∼ seconds) is typically small [8]–[10]. This means thatthe maximum number of gearshifts ζ max in (9j) can be chosenas a small positive integer. For such a case, we introduce atransformation of (9) that has fewer decision variables andconstraints in what follows. D. Problem Transformation
For a given gear position at the current time instant, η g , | t = η g ,t ∈ { , · · · , η max } , there are a finite number of dis-tinct gear position sequences, π t = { η g , | t , η g , | t , · · · , η g ,N | t } ,that satisfy both the gear dynamics (9g)-(9i) and the bound(9j) on the number of gearshifts. We denote the set of allsuch sequences as Π( η g ,t ) , called the set of admissible gearposition sequences. We have that 1) π t takes values in Π( η g ,t ) ,and 2) Π( η g ,t ) ∈ (cid:8) Π(1) , · · · , Π( η max ) (cid:9) , where the sets Π(1) , · · · , Π( η max ) can be constructed offline and stored foronline use.Then, we can transform (9) into the following problem: min (9a) (10a)s.t. (9b) − (9f) (10b) { η g , | t , η g , | t , · · · , η g ,N | t } = π t ∈ Π( η g ,t ) , (10c)with respect to the decision variables T m ,k | t , k = 0 , · · · , N − ,and π t .Moreover, after indexing the admissible gear position se-quences π t in Π( η g ,t ) by natural numbers , , · · · , we canwrite the problem (10) into the abstract form (11). We remarkthat to achieve such an abstraction, the intermediate variables ξ k | t and i g ,k | t governed by the equality constraints (9b) and(9f) need to be considered as deterministic functions of thedecision variables T m ,k | t and π t . This way, not only theequality constraints (9b) and (9f) can be dropped from theproblem definition, but also the inequality constraints (9c)-(9e)can be treated as conditions directly constraining the decisionvariables T m ,k | t and π t (i.e., by substituting the expressionsof ξ k | t and i g ,k | t as functions of T m ,k | t and π t into them).Furthermore, the constraint (9d) needs to be expressed astwo inequalities so that each of them involves a continuouslydifferentiable function.In the next section, we deal with the problem (10) bylooking at its abstract, condensed form (11).III. A C ONTINUOUS R ELAXATION TO A M IXED I NTEGER O PTIMIZATION P ROBLEM
In Section II-D, we transformed the BEV speed andgearshift co-optimization problem into the following form,which is an MINLP: min u,v f ( u, v ) , (11a)s.t. u ∈ U ( v ) = { u ∈ R n u : g ( u, v ) ≤ m } ,v ∈ V = { , · · · , n v } , (11b) where the cost function f ( u, v ) : R n u × N → R is assumed tobe continuously differentiable in u . The continuous variable u takes values in a set U ( v ) , which depends on v and ischaracterized by the inequalities g ( u, v ) ≤ m with g : R n u × N → R m being continuously differentiable in u . The integervariable v takes values in a finite set V = { , · · · , n v } ⊂ N .In general, MINLP problems are difficult to solve exactly.Continuous relaxation techniques may be exploited to obtainapproximate solutions [28]. They typically transform the orig-inal MINLP problem into a nonlinear programming problemwith purely continuous variables (NLP). For instance, for someproblems, the integrality constraint v ∈ V = { , · · · , n v } maybe replaced with v ∈ ¯ V = [1 , n v ] . Then, one can use off-the-shelf NLP solvers, such as the interior-point method [29] andthe sequential quadratic programming (SQP) method [30], tocompute solutions to the transformed problem.However, in our BEV speed and gearshift co-optimizationproblem, neither a gear ratio i g ( η g ) with a non-integer gearposition η g (see (9f)) nor a gear position sequence π t ∈ Π( η g ,t ) with a non-integer index (see (10c)) are defined. This means amodel that represents the powertrain response with such non-integer settings is unavailable. In this case, the above relaxationwhere v ∈ V = { , · · · , n v } is replaced with v ∈ ¯ V = [1 , n v ] is not applicable to our problem. Therefore, in what followswe introduce another continuous relaxation and also discussits theoretical properties.Firstly, it is easy to see that the problem (11) is equivalentto the following MINLP problem: min u,p p ⊤ f ( u ) , (12a)s.t. g ( u ) p ≤ mn v ,p ∈ Ω , (12b)where f ( u ) = f ( u, ... f ( u, n v ) , g ( u ) = g ( u, . . . g ( u, n v ) , (13)and Ω = (cid:8) p ∈ { , } n v | p ⊤ n v = 1 (cid:9) . (14)The set Ω defined in (14) represents the set of vertices ofan ( n v − -dimensional standard simplex, and the constraint p ∈ Ω ensures that the vector p has precisely one entry to be and all others to be . In particular, the index of the entry corresponds to the value of v in (11).We consider the following continuous relaxation to (12): min u,p p ⊤ f ( u ) , (15a)s.t. g ( u ) p ≤ mn v ,p ∈ ¯Ω , (15b)where ¯Ω = (cid:8) p ∈ [0 , n v | p ⊤ n v = 1 (cid:9) . (16)It can be seen that (15) is transformed from (12) by replacingthe vertex set Ω with its convex hull ¯Ω . We now discuss severaltheoretical properties of the relaxed problem (15). Proposition 1:
The cost and constraint functions of (15) arecontinuously differentiable in the decision variables ( u, p ) . Proof:
This follows from the expressions of the cost andconstraints in (15) and (16), and our assumptions that f and g are continuously differentiable in u made when problem (11)is defined. (cid:4) The significance of Proposition 1 is that the continuousdifferentiability of the cost and constraint functions enablesus to use derivative information to characterize minimizers of(15), e.g., through the Karush-Kuhn-Tucker conditions. Thisalso implies that many off-the-shelf NLP solvers can be usedto solve (15).We are interested in characterizing the feasibility and op-timality gaps between the original MINLP problem (11) andthe relaxed problem (15). The following two propositions arededicated to such properties.
Proposition 2:
Suppose (¯ u, ¯ p ) is a global minimizer of (15).Let ˆ p ∈ Ω be such that ˆ p i = 1 for some i satisfying ¯ p i > . Then, (¯ u, ˆ p ) is necessarily a global minimizer of (12). Inturn, (¯ u, i ) is a global minimizer of (11). Moreover, we have ¯ p ⊤ f (¯ u ) = ˆ p ⊤ f (¯ u ) = f (¯ u, i ) . Proof:
Let V ′ = { j ∈ V : g (¯ u, j ) ≤ m } . Since (¯ u, ¯ p ) isfeasible for (15), i.e., g (¯ u, k ) ≤ m for all k ∈ { j ∈ V : ¯ p j > } , the set V ′ is non-empty. Let us rename the integers in V ′ as , , · · · , m v , m v + 1 with ≤ m v ≤ n v − .Let Σ = (cid:8) q ∈ [0 , m v | q ⊤ m v ≤ (cid:9) and ¯Ω ′ = ( q − q ⊤ m v n v − m v − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ∈ Σ ) ⊆ ¯Ω . By construction, (¯ u, p ′ ) isfeasible for (15) if and only if p ′ ∈ ¯Ω ′ . In particular, ¯ p ∈ ¯Ω ′ .Let us write ¯ p as ¯ q − ¯ q ⊤ m v n v − m v − with ¯ q ∈ Σ .Let us now consider h : R m v → R defined by h ( q ) = m v X k =1 q k f (¯ u, k ) + (1 − q ⊤ m v ) f (¯ u, m v + 1) , (17)which is a linear function on R m v . Since (¯ u, ¯ p ) is a globalminimizer of (15), ¯ q is a global minimizer of h on Σ , whichis a compact subset of R m v .Since h is linear, either ¯ q locates on the boundary of Σ or h is constant on Σ . For the former case, either ¯ q j ∈ { , } for some j ∈ { , · · · , m v } or ¯ q ⊤ m v = 1 .If ¯ q j = 1 , then ¯ p satisfies ¯ p j = 1 and ¯ p k = 0 for all k = j . In this case, the ˆ p defined in the proposition statementis identical to ¯ p , and (¯ u, ˆ p ) = (¯ u, ¯ p ) is feasible for both theMINLP problem (12) and the relaxed problem (15). Moreover,we have ˆ p ⊤ f (¯ u ) = ¯ p ⊤ f (¯ u ) .If ¯ q j = 0 which implies ¯ p j = 0 or ¯ q ⊤ m v = 1 whichimplies ¯ p m v +1 = 0 , then we exclude j or m v + 1 from V ′ , re-name the remaining integers in V ′ as , , · · · , m ′ v , m ′ v +1 with m ′ v = m v − , and repeat the above arguments. By iteratingthis procedure, we will eventually fall into the case where h isconstant on Σ . Specifically, we will end up with a maximum This follows from the fact that linear functions are harmonic and themaximum principle for harmonic functions [31]. set of integers V ′′ = { , · · · , m ′′ v , m ′′ v + 1 } and its correspond-ing ¯Ω ′′ = ( q − q ⊤ m ′′ v n v − m ′′ v − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ∈ [0 , m ′′ v , q ⊤ m ′′ v ≤ ) such that 1) f (¯ u, j ) = f (¯ u, k ) for j, k ∈ V ′′ , and 2) f (¯ u, j ) ≤ f (¯ u, k ) for j ∈ V ′′ and k ∈ V ′ \ V ′′ . In particular, ¯ p ∈ ¯Ω ′′ .The above 1) and 2) show that when the continuous variable u is fixed at the given value ¯ u , every ˆ p ∈ Ω such that ˆ p i = 1 only if ¯ p i > is a global minimizer of the induced integerprogram. Note that by construction, such ˆ p ’s must belong to ¯Ω ′′ . Moreover, we have that ¯ p ⊤ f (¯ u ) , as a convex combinationof identical f (¯ u, j ) ’s with j ∈ V ′′ , satisfies ¯ p ⊤ f (¯ u ) = ˆ p ⊤ f (¯ u ) for every such ˆ p .Since the admissible set defined by (12b), Ξ , is a subset ofthat defined by (15b), ¯Ξ , and ˆ p ⊤ f (¯ u ) = ¯ p ⊤ f (¯ u ) ≤ p ⊤ f ( u ) for all ( u, p ) ∈ ¯Ξ , it must hold that ˆ p ⊤ f (¯ u ) ≤ p ⊤ f ( u ) for all ( u, p ) ∈ Ξ ⊂ ¯Ξ , i.e., (¯ u, ˆ p ) is a global minimizer of (12).The remaining part of Proposition 2 follows from theequivalence of (11) and (12). (cid:4) Proposition 2 says that the original MINLP problem (11)and the relaxed problem (15) have no optimality gap at theirglobal minimizers. Furthermore, if a global minimizer (¯ u, ¯ p ) tothe relaxed problem (15) has been found, it is straightforwardto derive a global minimizer (¯ u, i ) to the original MINLP prob-lem (11). In practice, it may not be easy to identify a globalminimizer to a non-convex NLP problem such as (15). Instead,typical NLP solvers, especially the ones exploiting derivativeinformation, compute only local minimizers. Therefore, we arealso interested in estimating the gap between (11) and (15)at their local minimizers. The following proposition presentssuch a result. Proposition 3:
Suppose (¯ u, ¯ p ) is a local minimizer of (15).Let ˆ p ∈ Ω be such that ˆ p i = 1 for some i satisfying ¯ p i > . Then, (i) (¯ u, ˆ p ) and (¯ u, i ) are guaranteed to be feasiblepoints of (12) and (11), respectively, and, ¯ p ⊤ f (¯ u ) = ˆ p ⊤ f (¯ u ) = f (¯ u, i ) . (ii) If ¯ p ∈ Ω , then we have ˆ p = ¯ p , and (¯ u, ˆ p ) and (¯ u, i ) are guaranteed to be local minimizers of (12) and (11),respectively. Proof:
The proof for part (i) follows the same steps asthose in the proof of Proposition 2 except for the secondlast paragraph. If ¯ p is itself an integer vector (i.e., ¯ p ∈ Ω ),then ˆ p must be identical to ¯ p by its definition. Moreover,the fact that (¯ u, ¯ p ) is a local minimizer of (15) ensures ¯ u to be a local minimizer of the NLP problem with respect to u that is induced from (15) by fixing p = ¯ p . The part (ii) ofProposition 3 thus follows. (cid:4) The significance of Proposition 3 is that it implies one canfind a feasible solution (¯ u, i ) to the MINLP problem (11) byrounding a solution (¯ u, ¯ p ) to the NLP problem (15), which canbe relatively easily solved for (e.g., using off-the-shelf NLPsolvers). Moreover, the quality of such a rounded solution canbe monitored via the cost value ¯ p ⊤ f (¯ u ) of the solution to theNLP problem. In addition, the following two remarks discussthe optimality of the rounded solution (¯ u, i ) . Remark 1:
In general, the rounded solutions (¯ u, ˆ p ) and (¯ u, i ) are not necessarily local minimizers of (12) and (11). This can be seen from the following example: min u,p ,p ( p + p ) u, (18)s.t. (cid:20) − p u − p ( u + 1) (cid:21) ≤ ,p , p ∈ { , } , p + p = 1 , which has the following continuous relaxation, min u,p ,p ( p + p ) u, (19)s.t. (cid:20) − p u − p ( u + 1) (cid:21) ≤ ,p , p ∈ [0 , , p + p = 1 . It can be easily checked that (¯ u, ¯ p , ¯ p ) = (0 , . , . isa local minimizer of (19), but one of its rounded solutions, (0 , , , is not a local minimizer of (18). Indeed, ( − , , is the global minimizer of both (18) and (19), consistent withour theoretical result of Proposition 2. Remark 2:
In the proof of Proposition 2, we have alsoshown that for (¯ u, ¯ p ) with a non-integer ¯ p to be a (globalor local) minimizer of (15), there must exist distinct vertices i, j ∈ V of Ω such that f (¯ u, i ) = f (¯ u, j ) . This can rarelybe encountered in many practical problems. For instance, wewill show in Section V that the numerical solutions to therelaxed version of our speed and gearshift receding-horizonco-optimization problem (10) have integer ¯ p ’s at the majorityof the time instants. Proposition 3 ensures that these solutionswith integer ¯ p ’s are guaranteed to be local minimizers of theoriginal MINLP problem (10).On the basis of Propositions 1–3, we approximately solvethe BEV speed and gearshift co-optimization problem (10)through solving its continuous relaxation with the form (12).In particular, we determine the value of the integer variableaccording to ¯ v = arg max i ∈ V ¯ p i .IV. A LTERNATIVE E NERGY E FFICIENCY O PTIMIZATION A PPROACHES FOR C OMPARISON
To evaluate performance of the proposed vehicle speed andtransmission gearshift coordinated receding-horizon controlstrategy for improving BEV energy efficiency, in this sectionwe describe several other energy efficiency optimization ap-proaches for comparison.Firstly, we consider the globally optimal solution for vehi-cle speed and transmission gearshift trajectories in terms ofminimizing battery
SOC consumption. It is computed as thesolution to the following optimization problem: min J dp = − SOC t f , (20a)s.t. (9b) − (9i) , (20b)with respect to u t = [ T m ,t , ζ t ] ⊤ , t = 0 , · · · , t f − , where t f corresponds to the entire duration of the trip.We remark that to compute such a globally optimal solutionbased on (20), the reference speed v r over the entire tripmust be known a priori. This may not be as practical as theassumption of being able to predict v r for a short time horizonin the proposed receding-horizon optimization strategy (9). In addition, the solution to (20) concerns itself only with the SOC consumption, unlike the proposed strategy (9) which alsoaccounts for passenger comfort (characterized by the secondand third terms in the cost function (9a)) and avoids overlyfrequent gearshifts (through the constraint (9j)). The reasonfor considering such a global
SOC consumption minimizationsolution is that it quantifies an upper limit on the energyefficiency performance, which will be used as a benchmarkfor evaluating performance of the other approaches.We use a dynamic programming (DP) algorithm [32] tosolve (20), where the constraints (9c)-(9e) are handled throughpenalties.Next, to highlight the benefit of employing multi-speedtransmissions in BEVs for improving energy efficiency, weconsider a BEV with a single reduction gear of ratio i sg forforward driving. Its speed trajectory is optimized throughrepeatedly solving the following problem in a receding-horizonmanner: min J sg t = N − X k =0 (cid:0) w ′ ( v k +1 | t − v r ,k +1 | t ) + w ′ ( T w ,k | t − T w ,k − | t ) (cid:1) , (21a)s.t. (9b) − (9e) , (21b)with respect to T w ,k | t , k = 0 , · · · , N − , where T w ,k | t = T m ,k | t i sg instead of following (3). The cost function (21a) ismotivated by the observation that energy efficiency can beimproved by smoothing speed profiles and reducing batterycurrent spikes, which has been shown by the results in [26],[33], [34].The advantage of minimizing (21a) is that battery SOC dynamics (4) are not involved, so the number of states isreduced and estimating V oc , R b and η m values through lookuptables + interpolations is not needed, and thus, computationsare simplified. It is shown through simulation case studies inSection V that a speed trajectory determined based on (21)can achieve, depending on driving cycles, . ∼ . energy savings compared to following v r exactly. The resultscorresponding to (21) and a single reduction gear are referredto as Optimized speed & Single gear , and those correspondingto following v r exactly and a single gear are referred to as Baseline .Finally, for BEVs equipped with multi-speed transmissions,we show the benefit of optimizing vehicle speed and gearshifttrajectories simultaneously using our proposed co-optimizationstrategy versus optimizing them separately. For the latter, wefirst optimize a static gear shift map offline following theapproach of [35] (the obtained shift map is shown in Fig. 1),and then optimize the vehicle speed/wheel torque online basedon the receding-horizon optimization (21). Specifically, afterthe pair of vehicle speed and desired wheel torque ( v t , T w ,t ) has been determined through (21), the gear position η g ,t isselected according to the shift map and the motor torqueis computed as T m ,t = T w ,t i g ( η g ,t ) i . The results correspondingto such a separate optimization procedure are referred to as Optimized speed & Shift map ( Map ). UpshiftDownshift
Fig. 1. Gear shift map optimized through the approach of [35].
V. R
ESULTS
In this section, we evaluate performance of the proposedBEV speed and gearshift coordinated receding-horizon controlstrategy for improving energy efficiency through a compre-hensive set of simulation case studies, and also compare itwith the alternative energy efficiency optimization approachesdescribed in Section IV.Table I summarizes the parameter values used for generatingthe results in this section. The values of parameters related tothe vehicle and battery models (1)-(6), as well as the mapsfor V oc ( SOC ) , R b ( SOC ) and η m ( T m , w m ) , are extracted fromthe high-fidelity powertrain simulation model ADVISOR [36].We use standard driving cycles to represent profiles of thereference speed v r for different driving conditions. The drivingcycle is revealed gradually to the ego BEV as it drives forward,to model the real-time prediction of v r for a short time horizon. TABLE IM
ODEL P ARAMETERS .Symbol Value [Unit] m, m eff r w ρ ] A f ] C d g ] θ µ C
55 [Ah] η + b , η − b { } [-] ∆ t i g ( η g ) { } i , i sg { } [-] τ min , τ max {
1, 2 } [s] δ , δ , ε {
5, 2, 0.1 } [-] ζ max w , w , w ′ , w ′ { × − , . × − , , − } Fig. 2 illustrates the results for the Urban DynamometerDriving Schedule (UDDS) cycle. Fig. 2(a) plots the obtainedvehicle speed trajectories with different approaches, includ-ing the solution to (20) computed using DP, our proposed receding-horizon co-optimization solution, and the solutioncorresponding to separate optimization based on (21) and theshift map in Fig. 1. Note that the speed constraints (9d), shownby the black dashed lines, are satisfied by all the approaches.Note also that the distance constraints (9c) guarantee that thetotal travel distances corresponding to different approaches areclose to each other (the difference is within meters). Thisensures the SOC consumption results of different approachesto be comparable. Fig. 2(b) – (d) display, respectively, thecorresponding motor torque, gear position, and battery
SOC trajectories.It can be observed from Fig. 2(d) that the DP solutionconsumes the least
SOC , followed by our proposed solution(referred to as
Speed-and-gearshift Co-optimization ), and thesolution of
Optimized speed & Shift map consumes the most
SOC . Note that the DP solution relies on the assumption that v r over the entire trip is known a priori, which is hard toenforce in practice, while the other two approaches rely onlyon short-term predictions of v r , which has been shown to bepossible [21]–[23]. Moreover, the DP solution allows largewheel torque changes and arbitrarily frequent gearshifts inorder to minimize SOC consumption. As a result, we canobserve significant spikes in the motor torque trajectory and ahigh frequency of gearshifts in the gear position trajectory ofthe DP solution compared to the other two approaches.Our proposed solution consumes considerably less
SOC than the solution of
Optimized speed & Shift map , corre-sponding to . improvement, and consumes only slightlymore than the DP solution. This shows the effectiveness of ourproposed receding-horizon control strategy for improving BEVenergy efficiency and the superiority of speed and gearshiftco-optimization over separate optimization. Specifically, forgenerating this result, we use ζ max = 1 and N = 8 , i.e.,allow at most gear change over a planning horizon of [s].We choose ζ max = 1 to balance the tradeoff between energysavings performance and computational complexity. For largervalues of ζ max , further improvements in energy efficiency arenot significant. Note that allowing at most gear change overthe planning horizon where the change can take place at anytime instant of the horizon is more flexible than the assumptionof constant gear position over the horizon in [9]. The latterrestricts the place over the planning horizon where the gearcan be shifted to the beginning of the horizon.In addition to the UDDS cycle, we also consider three otherdriving cycles and summarize the SOC consumption results ofall the approaches described in this paper for those drivingcycles in Table II. All simulations are conducted with theinitial
SOC = 80% .As expected, the DP solutions consume the least
SOC for allof the four driving cycles. Our proposed
Speed-and-gearshiftCo-optimization strategy is the second best, with, dependingon the cycles, . ∼ . improvements compared to Baseline and . ∼ . better than Optimized speed & Shift map . In particular, except for the Worldwide HarmonizedLight-duty Vehicles Test Cycles – Class 3 (WLTC),
Speed-and-gearshift Co-optimization with a shorter planning horizonof N = 5 even outperforms the other two approaches, Optimized speed & Single gear and
Optimized speed & Shift (a)
DP Proposed Map (b) (c)
Time (sec) (d)
Fig. 2. Simulation results for the UDDS driving cycle with planning horizon N = 8 : (a) vehicle speed trajectories, (b) motor torque trajectories, (c) gearposition trajectories, and (d) battery SOC trajectories. TABLE IIB
ATTERY
SOC
CONSUMPTION ( % ) AND IMPROVEMENT COMPARED TO
Baseline
OF DIFFERENT CONTROL APPROACHES FOR DIFFERENT DRIVINGCYCLES .Baseline DP N Optimized speed& Single gear Optimized speed& Shift map Proposed ∆ SOC (%) Computation time (sec)average worstUDDS 7.47 5.83(21.95%) 5 7.07 (5.35%) 6.83 (8.57%) 6.37 (14.73%) 0.73 1.398 6.78 (9.24%) 6.55 (12.32%) 6.08 (18.61%) 1.58 2.70WLTC 17.55 15.0(14.53%) 5 17.1 (2.56%) 16.72 (4.73%) 16.17 (7.86%) 0.63 1.148 16.3 (7.12%) 16.01 (8.77%) 15.74 (10.31%) 1.52 3.23LA92 12.86 10.12(21.31%) 5 11.79 (8.32%) 11.43 (11.12%) 10.72 (16.64%) 0.64 1.498 11.52 (10.42%) 11.17 (13.14%) 10.48 (18.51%) 1.51 2.85US06HWY 9.43 8.28(12.2%) 5 9.01 (4.45%) 8.88 (5.83%) 8.74 (7.32%) 0.71 3.078 8.91 (5.51%) 8.78 (6.89%) 8.68 (7.95%) 1.85 3.17 map , with a longer planning horizon of N = 8 .Among the four driving cycles, the benefit of employingmulti-speed transmissions for improving energy efficiency ismost significant for the UDDS cycle. This is because theUDDS cycle represents urban driving conditions, includingmany speed changes and several stop & go maneuvers.For the UDDS cycle, our proposed Speed-and-gearshift Co-optimization strategy achieves, respectively, . and . more energy savings when N = 5 , and . and . moreenergy savings when N = 8 , than the other two approaches. We also remark that because different approaches rely ondifferent cost functions, for a fair comparison, when generatingthe results in Table II, the weights for different terms of theircost functions have been tuned with trial and error to achievetheir best energy efficiency performance.To demonstrate the potential of our vehicle speed andtransmission gearshift coordinated receding-horizon controlapproach based on the proposed relaxation technique forreal-time implementation, we plot the computation time forobtaining the numerical solution at each time instant over the UDDS cycle in Fig. 3, and summarize the average andworst computation times for the other cycles in Table II.The computations are performed on the MATLAB R2018aplatform running on an Intel Xeon E3-1246 3.50-GHz PCwith 16.0-GB RAM. The NLP problems are solved using theMATLAB fmincon function with the SQP method [30]. Thecomputation times are calculated using the MATLAB tic-toc command. It can be seen that the computational cost is at alevel that is comparable to the time available for real-time im-plementation. We remark that there are various ways to furtherreduce the computation times, for instance, by implementingthe computations in C [37], replacing fmincon with moreefficient or tailored NLP solvers, exploiting inexact and real-time iteration solution strategies [38], [39] as well as symbolicand software optimization techniques [40]. The investigationinto these methods for further reducing the computational costof our approach is left to future work. Fig. 3. The computation time for obtaining our numerical solution at eachtime instant over the UDDS driving cycle.
Lastly, in Remark 2 we claimed that for the majority oftime instants the numerical solutions to the relaxed version ofour speed and gearshift co-optimization problem (10) shouldhave integer ¯ p ’s. Proposition 3 ensures these solutions to be atleast local minimizers of the original problem (10). To confirmRemark 2, we plot the largest entry of the vector ¯ p , max(¯ p ) ,of our numerical solution at each time instant over the UDDScycle in Fig. 4. In particular, Fig. 4(a) shows the max(¯ p ) whenthe maximum number of SQP iterations is specified as .This is the setting for generating the above SOC consumptionand computation time results. In this case, . data pointshave max(¯ p ) ≈ , where we categorize max(¯ p ) ≈ if max(¯ p ) ∈ (0 . , . Indeed, the deviations from of manydata points are due to numerical errors. This is verified byFig. 4(b), which shows the max(¯ p ) when the maximum num-ber of SQP iterations is specified as . In this case, . data points have max(¯ p ) ∈ (0 . , . Such an observationconfirms our claim of Remark 2. We also remark that whenthe maximum number of SQP iterations is increased to ,the computation time is also significantly increased, but withnegligible SOC consumption improvement. Therefore, isrecognized as sufficient, and is also recommended, as themaximum number of SQP iterations for practical purpose. Fig. 4. The largest entry of the auxiliary vector p of our numerical solutionat each time instant over the UDDS driving cycle: (a) when the maximumnumber of SQP iterations is ; (b) when the maximum number of SQPiterations is . VI. C
ONCLUSIONS
In this paper, we proposed a receding-horizon control strat-egy that simultaneously optimized vehicle speed and trans-mission gearshift trajectories for BEVs to achieve improvedenergy efficiency.The speed and gearshift co-optimization problem was for-mulated as an MINLP problem. To handle this MINLP prob-lem, we proposed a novel continuous relaxation technique,transforming the original MINLP problem to a continuousoptimization problem, so that approximate solutions to theoriginal problem were obtained through solving the relaxedproblem using off-the-shelf NLP solvers. Several theoreticalresults with respect to the feasibility and optimality corre-spondences between the original MINLP and the proposedrelaxation have been discussed.We applied the proposed continuous relaxation techniqueto solving the speed and gearshift receding-horizon co-optimization problem. Through a comprehensive set of simula-tion case studies and comparisons to several other energy effi-ciency optimization approaches, we showed that co-optimizingspeed and gearshift could achieve considerably greater energyefficiency than optimizing them separately. We also showedthat the proposed relaxation technique could reduce the onlinecomputational cost to a level that had the potential for real-time implementation.The proposed continuous relaxation technique may also beapplied to other control-related problems involving systemswith discrete modes, such as vehicle speed and transmissiongearshift coordinated control for conventional internal com- bustion engine vehicles, vehicle speed and operation modecoordinated control for hybrid electric vehicles, etc. These areleft as topics for future research.R EFERENCES[1] A. Vahidi and A. Sciarretta, “Energy saving potentials of connectedand automated vehicles,”
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