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One of the main challenges during the development of operating strategies for modern diesel engines is the reduction of the CO_{2} emissions, while complying with ever more stringent limits for the pollutant emissions. The inherent trade-off between the emissions of CO_{2} and pollutants renders a simultaneous reduction difficult. Therefore, an optimal operating strategy is sought that yields minimal CO_{2} emissions, while holding the cumulative pollutant emissions at the allowed level. Such an operating strategy can be obtained offline by solving a constrained optimal control problem. However, the final-value constraint on the cumulated pollutant emissions prevents this approach from being adopted for causal control. This paper proposes a framework for causal optimal control of diesel engines. The optimization problem can be solved online when the constrained minimization of the CO_{2} emissions is reformulated as an unconstrained minimization of the CO_{2} emissions and the weighted pollutant emissions (i.e., equivalent emissions). However, the weighting factors are not known a priori. A method for the online calculation of these weighting factors is proposed. It is based on the Hamilton-Jacobi-Bellman (HJB) equation and a physically motivated approximation of the optimal cost-to-go. A case study shows that the causal control strategy defined by the online calculation of the equivalence factor and the minimization of the equivalent emissions is only slightly inferior to the non-causal offline optimization, while being applicable to online control.

Today, almost 17% of the carbon dioxide (CO_{2}) emissions are caused by road transportation [_{x}) emissions, such as is the common practice with spark ignition engines. Furthermore, a large part of the combustion is diffusion combustion, which leads to the formation of particulate matter (PM). Since these pollutants are harmful to both humans and the environment, emission legislation is becoming increasingly stringent with the introduction of ever lower emission limits. These emission limits are usually defined as cumulative or cycle-averaged values for a given test cycle [

However, a reduction of the engine-out NO_{x} emissions usually increases the CO_{2} emissions, because the fast, high-temperature combustion required for a high combustion efficiency is also favorable for the production of NO_{x} [_{x} emissions using exhaust-gas aftertreatment also increases the CO_{2} emissions. For example, lean NO_{x} traps require periods of rich exhaust gas for regeneration. Selective catalytic reduction (SCR) does not increase the CO_{2} emissions directly, but urea consumption and fuel consumption (i.e., CO_{2} emissions) are equivalent in light of operating costs. Generally, there is a trade-off between lowering the CO_{2} emissions and lowering the NO_{x} emissions of a diesel engine. At the same time, a similar trade-off exists between the CO_{2} emissions and the PM emissions. For example, high injection pressures and post injections can be used to reduce the engine-out PM emissions. However, they may increase the fuel consumption. The use of exhaust-gas aftertreatment, such as diesel particulate filters, also increases the fuel consumption, due to the requirement of high exhaust-gas temperatures for regeneration. As a result of these trade-offs, the optimal operating strategy is the one that achieves the lowest possible CO_{2} emissions, while staying below the upper limit for the respective cumulated pollutant emissions on a given cycle.

Optimal control theory [_{x} emissions are included in an optimization of the operating costs of a diesel engine. There, the weighting factor is considered a tuning parameter. Alternatively, the optimal weighting factor can be obtained by solving the non-causal optimal control problem. However, in both cases, the control system is sensitive to disturbances and model errors. Furthermore, the driving cycle used for tuning or optimization may not accurately represent real driving situations. In order to comply with the pollutant emission limit, even in the presence of disturbances, model errors and situations not covered by the reference cycle, the weighting factor needs to be adapted online. So far, no method for the online adaptation of the weighting factor has been proposed.

This paper proposes a generic framework for the causal minimization of the CO_{2} emissions, while keeping the pollutant emissions on a certain level. It is based on the minimization of the equivalent emissions, which are defined as the sum of the CO_{2} emissions and the weighted pollutant emissions. A simple, yet effective method for the online adaptation of the weighting factors is derived. Using these weighting factors, the equivalent emissions can be minimized online to realize a causal optimal operating strategy. The approach is demonstrated in a case study, where its performance is compared with that of the non-causal optimal solution. The proposed causal approach is shown to achieve CO_{2} emissions that are only marginally larger than the non-causal optimum value, while keeping the pollutant emissions very close to the legislative limit.

Section 2 gives an overview of the optimal control problem and its solution using optimal control theory. In Section 3, the method used for the adaptation of the weighting factor is derived. Section 4 presents the case study, and Section 5 concludes the paper.

This section describes the optimal control problem and its solution using Pontryagin's minimum principle. Section 2.1 describes the structure of the class of systems considered here. In Section 2.2, the optimization problem is defined, and the properties of the optimal solution are analyzed in Section 2.3.

The engine is assumed to be described by a system of first-order ordinary differential equations (ODE):
_{eng} is the state vector and

Because the cumulated mass of the pollutant emissions is restricted by legislation, the engine model is extended by a set of artificial state variables, _{emis} to track the cumulative emissions:

The emission state vector, _{emis}, consists of _{emis} should be defined. The legislative emission limit is often defined in terms of brake-specific emissions [_{emis}, and the total amount of mechanical work delivered to the crankshaft, _{emis}. Alternatively, the engine model can be extended by state variables, which represent the absolute (i.e., not brake-specific) mass of the respective emissions, _{emis}. The time derivative of such state variables is:

This difference is crucial. Choosing _{emis} = _{emis} is preferable, because then, the dynamics of _{emis} are independent of _{emis}. The advantage provided by this property will become apparent in the next section. The state variables of the extended system thus are:

The engine model was deliberately chosen to be very general. The approach described in this paper is applicable to a wide range of systems, which is reflected by the choice of the model.

The control task is to minimize the CO_{2} emissions, while staying below a cumulative limit for the pollutant emissions, _{emis,lim}. Technically, CO_{2} emissions are also pollutant emissions. However, in this paper, the term “pollutant emissions” and the emission state variables, _{emis}, refer to the emission species, which are limited by legislation (e.g., NO_{x} and PM). Adopting the notation introduced in [

The engine state variables, _{eng}, are assumed to be unconstrained.

The properties of the optimal solution are analyzed using Pontryagin's minimum principle [

According to Pontryagin's minimum principle, the following conditions must hold for the optimal solution, which is indicated by the superscript “^{o}

For the co-states, the following equation must hold:

The unconstrained final state of the engine state variables is represented by the condition:

For all admissible inputs

The dynamics of the co-states

Because neither the system dynamics nor the CO_{2} emissions depend on the cumulated pollutant emissions, the following relations hold:

The dynamics of the co-states are thus:

The optimal co-state vector corresponding to the cumulated emissions,
_{emis} = _{emis}. If _{emis} = _{emis} had been chosen instead, the term,

Using

The time dependency of the inputs, _{2} emissions and the weighted pollutant emissions with a free final state:

In this case, the vector, _{emis} ≥ 0, is not a co-state vector, but a set of parameters. They can be interpreted as the equivalence factors, which are used to calculate the equivalent CO_{2} emissions for a certain amount of the respective pollutant emissions. A strategy that solves _{2} emissions

This reformulation of the problem is similar to the Karush–Kuhn–Tucker conditions for static optimization [

When a causal solution of the optimal control problem is considered, the dual formulation has a crucial advantage. For model predictive control (MPC), a rule of thumb is to choose the length of the prediction horizon to be equal to approximately five times the relevant time constant of the system [

The last section showed that a constrained minimization of the CO_{2} emissions can be reformulated to an unconstrained minimization of the equivalent emissions. This reformulation enables the online solution of the dual optimal control problem with a limited time horizon. However, the equivalence factors, which yield the correct cumulated emissions, are not known

In Section 3.1, a method for the online calculation of the equivalence factors is derived. It is based on the feedback control of the pollutant emissions. Section 3.2 describes the calculation of the corresponding reference values. Section 3.3 discusses the similarities between this approach and the one used for the causal energy management of hybrid electric vehicles.

The optimal equivalence factors, λ_{emis,}_{i}

The starting point for this approach is the original optimization problem _{emis}, from the reference value, _{ref},
_{i}_{norm,}_{i}

The additional penalty term of the extended Hamiltonian does not depend explicitly on the control input, ^{o}

Using the optimal cost-to-go function:

The optimal cost-to-go function,
^{o}

As indicated, the first two terms correspond to the CO_{2} emissions in the extended cost function

Nominal cost
_{nom}(_{2} emissions caused by driving the rest of the cycle with optimal instantaneous pollutant emissions.

Emissions-saving cost
_{emis}(_{2} emissions caused by bringing the cumulated pollutant emissions to their respective reference level.

Penalty cost
_{pen}(

The following sections will describe the three terms in detail.

The term,
_{nom}(_{2} emissions that are produced by running the rest of the cycle with optimal instantaneous pollutant emissions. This term is independent of the current cumulated pollutant emissions,

Since we are only interested in the derivative of the cost with respect to the state variables, _{emis} _{nom}(

The term,
_{emis}(_{2} emissions that are produced in addition to the nominal CO_{2} emissions described by
_{nom}(_{2} emissions. It can thus be assumed that the term,
_{emis}(_{emis}. Assuming that for the horizon considered, the average value, _{i}_{2} emissions caused by the saving of the respective pollutant emissions is known, the CO_{2} penalty is:

The term,
_{pen}(_{h,i}

The total penalty for the deviation of the actual values from the reference values can be calculated by the integration of the future trajectories.

The approximated (suboptimal) cost-to-go function is:

The average additional CO_{2} emissions for a given saving of a pollutant (_{i}_{i}_{i}_{int} can be used to achieve this goal,

Assuming a quadratic penalty for the pollutant control error (_{i}

The method for calculating the equivalence factors _{e}, and torque

These reference values can be stored in a lookup table. They may represent the steady-state calibration of the engine. If a non-causal optimal solution has been obtained for the reference cycle, the pollutant emissions of the optimal solution can also be used to parametrize the lookup table.

Alternatively, a simpler approach is to consider a constant brake-specific reference value, _{emis,ref,}_{i}

For passenger-car applications, where it is common to specify the emission limit in ^{g}/_{km}, the integral in

Clearly, the assumption of a constant _{emis,ref,}_{i}

The equivalent emission minimization strategy (EEMS) presented in Sections 2.3 and 3.1 is similar to the equivalent consumption minimization strategy (ECMS) used for the online optimal energy management of hybrid electric drivetrains [_{2} emissions, while maintaining a certain pollutant emission level. The goal of optimal energy management is the minimization of the fuel consumption, while sustaining the battery charge. For both problems, the final state constraint(s) can be represented by an extension of the performance criterion. This approach results in a minimization of the equivalent emissions and the equivalent consumption, respectively. In both cases, the unknown optimal equivalence factor(s) is/are constant, and an estimate can be calculated using feedback in order to allow a causal online optimization. Due to the similarity of the two problems, the calculation of the equivalence factors using the HJB equation and the approximated cost-to-go (Section 3.1) was inspired by [

The similarity between the two problems is also likely to lead to future combinations of the EEMS approach with the ECMS approach. Clearly, the NO_{x} and PM emissions are a concern for diesel electric hybrid drivetrains. In [_{x} emissions are included in the performance criterion of the ECMS problem. However, there, the equivalence factor for the NO_{x} emissions is only a tuning parameter, and it is not adapted actively in order to maintain a certain NO_{x} emission level, as is proposed in this paper.

This section presents a case study in which the performance of the EEMS on a driving cycle is studied. The optimal solution obtained with dynamic programming (DP) is used as a benchmark for various versions of EEMS. The case study is carried out in simulation using a validated model of a diesel engine. Section 4.1 defines the control problem and describes the different controllers to be compared. Section 4.2 presents and discusses the results of the comparison. Due to reasons of confidentiality, all emissions data have been normalized.

We consider an 8.71 turbocharged, heavy-duty diesel engine running a driving cycle lasting 1,800 s. It consists of five repetitions of a section of the urban part of the World Harmonized Transient Cycle (WHTC). _{2} emissions, while keeping the NO_{x} emissions below the final value constraint, _{NOx,lim}. The control input is the deviation of the crank angle for the start of injection (SOI) from its nominal value:

In order to determine the optimal SOI and to simulate the engine on the driving cycle, an engine model is required that reproduces the influence of the SOI on the CO_{2} and NO_{x} emissions. Because we are considering time scales in the order of entire driving cycles, the dynamics of the engine can be neglected, and the engine may be represented by a steady-state model with the inputs, engine speed _{e}

The steady-state engine model,

The driving cycle prescribes the engine speed, _{e}, and the torque, _{NOx} are discretized using the Euler forward integration scheme, such that a discrete-time optimal control problem is obtained.

This is an optimization problem of the structure of

This optimization problem has the same structure as

In the following, three different versions of EEMS are compared to the optimal solution obtained by DP. All three versions solve the minimization of the instantaneous equivalent emissions

_{x} emissions. This equivalence factor can be obtained by an iterative simulation of the closed loop consisting of the engine model and the control law

_{x}_{x} emissions. The reference NO_{x} mass flow is calculated as a quadratic function of the operating point:
_{0} … _{5}, are obtained by a least-squares fit of the NO_{x} emissions of the optimal solution obtained by DP. _{x} emissions resulting from the DP and the fitted quadratic function. The quadratic model is a good approximation of the optimal values. However, this approach requires the availability of an optimal solution.

_{x} emissions. The reference value for the brake-specific NO_{x} emissions is equal to the average value, which was demanded for the DP solution:

The two causal EEMS approaches, OPdep and BSconst, use the adaptation law _{int}, for the adaptation of the equivalence factor. The tuning parameters were tuned manually.

_{x} emissions, while the rest of the model does not depend on that state variable.

The first plot in _{x} emissions of the different solutions. The NO_{x} mass has been normalized with the final value of the NO_{x} mass of the DP solution. The plot also shows the full range of attainable NO_{x} emissions, the upper limit corresponding to λ_{NOx} = 0 and the lower limit corresponding to λ_{NOx} = ∞. The cumulated NO_{x} emissions of all approaches are very similar, such that on the given scale, the respective lines cannot even be discerned. Therefore, the second plot shows the NO_{x} error:
_{x} mass of the DP solution, _{NOx},dp, and the respective EEMS solution, _{NOx}. Since the NO_{x} mass has been normalized, the NO_{x} error is normalized, as well.

As the results of the CEF approach show, the online solution of the dual problem _{x} emission level by following the reference value for the NO_{x} emissions using feedback control. The cumulated NO_{x} emissions are kept within less than one percent of the final value.

The initial value, _{0}, for the equivalence factor in

The third plot of _{2} error:
_{2} mass of the DP solution, _{CO2,dp}, and the respective EEMS solution, _{CO2.} The CO_{2} mass and the CO_{2} error have been normalized with the final CO_{2} mass of the DP solution. The plot clearly shows the trade-off between NO_{x} and CO_{2} emissions. When the NO_{x} error increases, the CO_{2} error decreases and _{2} error is generated during the first two repetitions of the section before the EEMS controller has reached its limit cycle. In the beginning of the cycle, the equivalence factor is very low, and the engine is operated with a very high fuel efficiency, thus emitting large amounts of NO_{x}. Then, the NO_{x} emissions have to be reduced at a later point when the CO_{2} penalty is larger than the amount saved in the beginning. However, once the controller has reached the limit cycle after two repetitions of the section, the CO_{2} penalty of the causal EEMS controllers is small.

The performance of the various strategies can be compared by considering the cumulative emissions at the end of the cycle. In order to obtain a fair comparison and to eliminate the influence of the initial value of the equivalence factor, only the last three sections shown in _{2} emissions than the DP solution. The NO_{x} emissions are even slightly lower for both strategies. The OPdep approach has both lower CO_{2} and NO_{x} emissions than the BSconst approach, because the more realistic reference value leads to an equivalence factor that is closer to the optimal value. Generally, the performance of the causal EEMS approaches is very close to the non-causal, theoretical optimum.

Finally, the performance of the various approaches for different reference NO_{x} emission levels is compared. _{2} emissions and the cumulated NO_{x} emissions for the DP solution and the two causal EEMS approaches. The optimal trade-off can be obtained by varying the constant equivalence factor of the dual problem _{x} limit value of the original problem _{2} and pollutant emissions of a diesel engine.

This paper presents a generic framework for the minimization of the CO_{2} emissions of a diesel engine, while maintaining a certain pollutant emission level. It consists of an online adaptation of the equivalence factors in combination with a minimization of the equivalent emissions. First, it was shown that the constrained state variables associated with the pollutant emissions can be eliminated by minimizing the equivalent emissions instead. The optimal equivalence factors, which weigh pollutant emissions _{2} emissions, are constant for a given cycle and emission limit. A method for the online adaptation of the equivalence factors was then derived using the HJB equation and a physically motivated approximation of the optimal cost-to-go function. Finally, a case study combined this adaptation method with an online minimization of the equivalent emissions. The performance of the proposed causal control strategies was very close to the theoretical non-causal optimum, which was determined using dynamic programming. This result confirmed that the EEMS framework is well suited to causal optimal control of the CO_{2} and pollutant emissions of a diesel engine.

The case study considered a static minimization of the equivalent emissions in combination with an adaptation of the equivalence factor. The results are very encouraging. In a next step, the EEMS framework could be applied to a more complex optimization problem. For example, additional control inputs, such as the injection pressure or additional injections, could be considered. In that case, the steady-state engine model

Another topic worth pursuing is the realization of an EEMS with multiple pollutant emission limits. The implementation is straightforward; in addition to the weighted NO_{x} emissions, the dual formulation of the control problem

Furthermore, instead of the steady-state model

Finally, future research could aim at the combination of EEMS and ECMS for hybrid vehicles. As discussed in Section 3.3, the goal would be to achieve a fuel-optimal, charge-sustaining operation, while respecting the legislative limit for the pollutant emissions.

The authors would like to thank FPT Motorenforschung AG, Arbon, Switzerland, for providing the measurement data used to calibrate the engine model on which the case study was performed.

The authors declare no conflict of interest.

_{x}

_{x}emission potential investigation and trade-off of a hybrid electric vehicle based on dynamic programming

_{x}Emission Reduction Strategy for Lightweight Hybrid Electric Vehicles

_{2}and NO

_{x}emissions of a diesel hybrid passenger car

_{x}emissions of a diesel engine

Section of the World Harmonized Transient Cycle (WHTC) used in the case study.

Normalized NO_{x} emissions of the optimal solution (dotted) and the quadratic model

Normalized comparison of the optimal dynamic programming (DP) solution and the equivalent emission minimization strategy (EEMS) approaches. It shows the feasible range (light grey area) and the actual values of the cumulated NO_{x} emissions, _{NOx}, the NO_{x} error, ΔNO_{x}, the CO_{2} error, ΔCO_{2}, and the respective equivalence factor, λ_{NOx}, of the EEMS approaches. The vertical lines indicate the beginning of a new section. The terms CEF, OPdep, and BSconst refer to the EEMS strategies with a constant equivalence factor, operating-point dependent reference emissions, and constant brake-specific reference emissions, respectively.

Normalized Pareto front of the cumulated NO_{x} and CO_{2} emissions for the last three sections. Shown are the optimal DP solution and the two causal EEMS solutions. The solution shown in

Normalized difference Δ between the various strategies and the optimal DP strategy for the last three sections.

_{2} (%) |
_{x} (%) | |
---|---|---|

DP | 0 | 0 |

| ||

EEMS OPdep | +0.032 | −0.129 |

EEMS BSconst | +0.050 | −0.104 |