A Bayesian Estimation Model for Transient Engine Exhaust

[12] Protea, “Protea FTIR: Frequenty Asked Questions,” 2016. [Online]. Available: https://www.protea.ltd.uk/pdfs/protea-ftir-faqs.pdf. [13] H. Li,...
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A Bayesian Estimation Model for Transient Engine Exhaust Characterization Using Fourier Transform Infrared Spectroscopy David Wilson and Casey Allen* Department of Mechanical Engineering, Marquette University, 1637 West Wisconsin Avenue, Milwaukee, Wisconsin 53233, United States ABSTRACT: Comprehensive emissions models extensively use engine exhaust data from vehicle experiments to represent the relationship between fuel composition and pollutants. However, the predicted emissions from these models often neglect the effects of transients and speed-load history. Fourier transform infrared (FTIR) spectroscopy is a high frequency measurement technique capable of comprehensive speciation. However, due to long residence times of exhaust within a FTIR spectrometer gas cell, FTIR measurements are contaminated by the effects of historical emissions, precluding the attainment of time-resolved engine exhaust data. This work presents a Bayesian estimation model for processing FTIR measurements to obtain accurate estimations of instantaneous engine exhaust composition. The Bayesian model utilizes a simple model of the mixing dynamics of the gas cell and measurement noise statistics to estimate the composition of exhaust entering the FTIR gas cell during a measurement period. To validate the estimation model, synthetic FTIR measurements are generated from simulated engine exhaust data from the Federal Test Procedure driving cycle. These synthetic measurements are processed by the estimation model, which is shown to yield improved estimations of instantaneous composition as compared to the raw FTIR measurements, although the degree of improvement depends on the magnitude of measurement noise and flow rate through the FTIR gas cell. For a measurement noise standard deviation of 0.5% of the maximum measurement, the estimation model improved estimates of instantaneous NO emission by approximately 42.5% on average, while about a 7.5% improvement was achieved for a measurement noise standard deviation of 2% of the maximum measurement for a FTIR flow rate of 10 L/min. For a flow rate of 25 L/min, improvements of approximately 41.5% and 6% were achieved for measurement noise standard deviations of 0.5% and 2% of the maximum measurements, respectively. The application of the model in this work is to generate time-resolved emissions estimates to further elucidate the relationship between fuel composition and engine emissions.

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INTRODUCTION Volatile organic compounds (VOCs) are key precursors to the formation of tropospheric ozone and other pollutants that contribute to the greenhouse effect and pose risks for human health.1 According to the National Emissions Inventory, vehicles were responsible for over 23% of anthropogenic VOCs emitted in the U.S. in 2014.2 To reduce anthropogenic VOCs emitted by vehicles, improvements will need to be made in future engine designs, control strategies, and fuel composition optimization techniques. This task requires the elucidation of the relationships between engine conditions, fuel composition, and VOC emissions. Previous research has shown that VOC emissions are fuel dependent,3 and that instantaneous and integrated engine emissions are a function of the speed-load history of an engine for real-world driving conditions,4,5 which are largely transient.6 Quasi-steady engine maps, which estimate the emissions of an engine for a given speed/load from steady-state data, neglect these historical effects and provide an overly simplified and inaccurate portrayal of the relationships between speed/load, fuel, and emissions. A key step to building more accurate engine models is to conduct engine experiments involving real-world speed/load profiles. However, to properly capture transient and historical effects from these experiments, a measurement technique with high response time and measurement frequency is required. One measurement technique capable of characterizing VOCs, in addition to a wide variety of other species, is Fourier transform infrared (FTIR) spectroscopy. In FTIR spectroscopy, © XXXX American Chemical Society

broadband IR radiation is transmitted toward a beam splitter, where it is split into two beams of equal intensity. To create constructive/destructive interference between the two beams, one beam is directed to a moving mirror, and the other to a stationary mirror. The two beams reflect off their respective mirrors, coalesce at the beam splitter, and are redirected toward a gas cell containing the sample to be analyzed. After making multiple passes through the gas cell, the coalesced beam reaches a detector, where the intensity is measured. A computer performs a Fourier transform to convert the measured intensity to a spectrum. This spectrum is compared to a background spectrum to determine the wavelengths absorbed by the sample, which are indicative of the chemical composition. A general schematic of an FTIR is shown in Figure 1. FTIR spectroscopy presents many advantages for engine exhaust characterization over other conventional techniques. First, each measurable species has a distinguished spectrum, allowing many species to be measured at once. FTIRs can therefore differentiate between different VOCs. On the other hand, individual VOCs can only be measured via flame ionization detection (FID) if the FID is integrated with a complex setup (such as using gas chromatography for separating hydrocarbons). Nonmethane hydrocarbons Received: July 29, 2017 Revised: September 5, 2017 Published: September 6, 2017 A

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technique uses a priori knowledge of the underlying system dynamics to correct for errors in state estimations caused by measurement noise. Because an FTIR gas cell can be modeled as a well-mixed system, the problem of estimating the composition of sample entering a FTIR gas cell is well-suited for Bayesian estimation. In this Article, a BEM for estimating the composition of exhaust entering a FTIR gas cell is developed. Synthetic FTIR data are used to show that this model improves instantaneous engine exhaust estimations as compared to the well-mixed model with no filter and raw FTIR measurements. Computational fluid dynamics (CFD) and mixing network (MN) simulations are used to generate the synthetic FTIR data. In these simulations, total gas cell composition is calculated for specific inlet composition profiles and volume flow rates. White Gaussian noise is added to the total gas cell composition calculated from these simulations to produce the measurements. The CFD simulations are conducted for cases where tracer gas within nitrogen is flowed through the gas cell. These cases are ultimately used to tune the Bayesian filter and develop the MN simulations. MN simulations are used to calculate total gas cell composition for cases where simulated exhaust from a Federal Test Procedure (FTP) driving cycle is flowed through the gas cell. CFD simulations would be intractable for these cases, because the FTP cycle is longer than 1800 s. This Article is outlined as follows. First, the Bayesian filter equations are presented. This is followed by a presentation of the state-space equations used in the BEM, which describe inlet and total gas cell composition. Details regarding the CFD and MN simulations are then discussed. The Results begins with a comparison between total gas cell composition calculated from CFD and the well-mixed model for different tracer inlet composition profiles. These results show that the well-mixed model reasonably represents the dynamics within a FTIR gas cell. This is followed by a presentation and discussion of the results from the BEM applied to the synthetic FTIR data. Finally, conclusions are given and future work is discussed.

Figure 1. General schematic of an FTIR spectrometer.

(NMHCs) are also characterized by FTIR spectroscopy with relative ease, although a correction factor must be applied to account for heavier hydrocarbons that may not be detected.7 However, NMHC characterization with FID requires a downstream catalyst and an additional FID to isolate and measure methane. Furthermore, this method is susceptible to large errors if there is an abundance of methane in the exhaust.8 Individual VOCs can also be characterized with offline methods such as gas chromatography/mass spectrometry (GC/MS)9 and high performance liquid chromatography (by analyzing the stable products of the reaction between the VOCs and an acid solution of 2,4 dinitrophenylhydrazine).10 However, for offline methods, the exhaust gas from an entire test cycle is collected into one sample, preventing transient analysis. FTIRs, on the other hand, sample continuously and in real-time. Despite the advantages of FTIR spectroscopy, its suitability for studying engine transients is limited. A FTIR gas cell must be of sufficient volume to allow multiple passes of the IR beam through the sample. Because of its large volume, residence times within a gas cell can be significantly greater than the measurement period of a FTIR. Consequently, FTIR data are obscured by historical emissions, precluding time-resolved analysis. This issue is not necessarily mitigated by simply increasing the sample flow rate through the gas cell to decrease residence times, as this can cause dramatic changes in total gas cell composition during a scan and increase turbulence within the gas cell, yielding erroneous spectra.11,12 FTIR flow rates for engine exhaust characterization in the literature vary, but are typically between 1 and 12 L/min.7,13−20 One way to obtain time-resolved composition estimates is by estimating composition of exhaust entering the FTIR gas cell during a measurement period.16,21 The exhaust entering the gas cell comes from fewer engine cycles than the total sample within the gas cell. This method assumes that the FTIR gas cell behaves as a well-mixed system. The composition of exhaust entering the gas cell during a measurement period is backcalculated according to the well-mixed model and the difference in consecutive FTIR measurements. While this method improves the overall trend in estimated instantaneous engine exhaust as compared to raw measurements, noise from the FTIR measurements greatly exacerbates the noise of the estimated inlet composition, yielding considerable errors in instantaneous emissions estimates.16 However, this issue can be mitigated by filtering the FTIR data with a Bayesian estimation model (BEM). BEMs calculate the most statistically probable values for system states (i.e., composition of sample entering an FTIR gas cell) using a system model and known/assumed statistics regarding measurement noise.22 This estimation

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BAYESIAN ESTIMATION MODEL DESCRIPTION In this section, the BEM for determining the composition of sample entering a FTIR gas cell is developed. Three main components of the model will be discussed in detail: the Bayesian filter, state-space model, and measurement model. The purpose of a Bayesian filter is to calculate the most statistically probable system states by appropriately weighing state prediction and system measurements according to model uncertainty and measurement noise, respectively. Every Bayesian filter consists of four main steps: state prediction, measurement prediction, gain, and update. The state prediction step utilizes the state-space model to predict the current state values, using the previous state values as initial conditions. Current measurement predictions are then calculated using the current state predictions and measurement model. The innovation matrix is then calculated, which consists of the predicted measurements subtracted from the actual measurements. A gain is then calculated, which is multiplied by the innovation and then added to the initial state estimation, yielding the final state estimation. The Bayesian filter chosen for the BEM is the unscented Kalman filter (UKF). This filter is chosen because it is accurate to the second order and was designed specifically for nonlinear systems.22,23 The UKF and its equations are discussed below. B

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Energy & Fuels Unscented Kalman Filter. The UKF was developed on the basis of the idea that approximating a Gaussian distribution is easier than approximating a nonlinear transformation.24 This statistical approximation is accomplished by deterministically sampling from the state distribution. The sampling points, called sigma points, are selected to capture the true means and covariances of the state estimations. These sigma points are nonlinearly propagated throughout the prediction, gain, and update steps of the filter, ultimately leading to an updated estimation for the system states and covariances. The first set of sigma points (X) are generated using the state (x̂) and covariance (P̃) estimations from the previous time (t). If this is the first iteration of the filter, these state and covariance values are initial conditions. Otherwise, they are calculated by the filter at the previous time step. X 0 = x(̂ t − 1|t − 1)

(1)

Xi = x(̂ t − 1|t − 1) + ( (Nx + κ )P(̃ t − 1|t − 1) )i

(2)

The sigma points are propagated through the measurement model (c) to yield a new set of sigma points (Υ). These new sigma points are multiplied by their corresponding weights, and then added together to yield the measurement prediction (ŷ). Υi(t |t − 1) = c[X̂i (t |t − 1)]

(11)

2Nx

y (̂ t |t − 1) =

∑ Wi Υi(t |t − 1)

(12)

i=0

The predicted measurement residual covariance (Rξξ) is then calculated according to the spread of the sigma points and the measurement noise. Rvv is the measurement noise covariance matrix. Measurement noise is assumed to be Gaussian and white. ξi(t |t − 1) = Υi(t |t − 1) − y (̂ t |t − 1)

(13)

2Nx

R ξξ(t |t − 1) =

Xi + Nx = x(̂ t − 1|t − 1) − ( (Nx + κ )P(̃ t − 1|t − 1) )i

i=0

(3)

(14)

Nx is the number of states, κ is a scalar tuning parameter, and ( (Nx + κ )P̃ )i is the ith column/row of the square root of the state error covariance matrix multiplied by (Nx + κ). As recommended in ref 22, κ is set to Nx − 3, which is zero for this particular model. Each of these sigma points is assigned a corresponding weight (W), which will be used throughout the remainder of the algorithm. κ W0 = (Nx + κ ) (4) 1 Wi = 2(Nx + κ )

(5)

1 2(Nx + κ )

(6)

Wi =

The cross-covariance between the predicted state and measurement regressions is then calculated. This is then multiplied by the inverse of the measurement residual covariance to yield the gain. 2Nx

R X̃ ξ(t |t − 1) =

(7)

∑ Wi Xi(t |t − 1) i=0

(8)

The initial state error covariance prediction is then calculated according to the spread of the sigma points and the uncertainty of the state transition model. The uncertainty of the state transition model is quantified in the process noise covariance matrix (Rww). Here, Rww is a diagonal matrix whose values are tuned for optimal performance. The tuning of Rww is discussed in the Filtered Tracer Data. X̃ i (t |t − 1) = Xi(t |t − 1) − x(̂ t |t − 1)

(9)

2Nx

P(̃ t |t − 1) =

i=0

(15)

Κ(t ) = R X̃ ξ(t |t − 1)R ξξ −1(t |t − 1)

(16)

e(t ) = y(t ) − y (̂ t |t − 1)

(17)

x(̂ t |t ) = x(̂ t |t − 1) + Κ(t )e(t )

(18)

P(̃ t |t ) = P(̃ t |t − 1) − Κ(t )R ξξ(t |t − 1)Κ′(t )

(19)

The state transition model (a in eq 7) will now be introduced. State Transition Model. The state transition model propagates the state values through time according to the systems dynamics. It is a critical element of the BEM, as it provides the initial prediction for the current state values. The system is described by three states: composition entering the FTIR gas cell (Zin), total gas cell composition (Z), and mass flow rate (ṁ ). Total gas cell composition is modeled using the well-mixed assumption. Mass flow through the gas cell is assumed to be quasi-steady. The initial predictions for inlet composition and mass flow rate are the values from the previous time step. These states are modeled this way because their values are influenced by factors outside of the system. Essentially, these states are updated within the filter according to the measurements and total gas cell composition model. The state-space equations are given below.

2Nx

x(̂ t |t − 1) =

∑ Wi X̃i (t |t − 1)ξi′(t |t − 1)

The estimated measurement is then subtracted from the actual measurement (y) at the current time. This quantity is multiplied by the gain and added to the initial state prediction to yield the final state estimation at the current time. The state error covariance matrix is updated as well.

The sigma points are then propagated through the nonlinear state transition model (a) to generate a new set of sigma points. These new sigma points are multiplied by their corresponding weights and then added together to give the initial state prediction at the current time step. Xi(t |t − 1) = a[Xi(t − 1|t − 1)] + b[u(t − 1)]

∑ Wi ξi(t |t − 1)ξi′(t |t − 1) + R vv(t )

∑ Wi X̃i (t |t − 1)X̃i ′(t |t − 1) + R ww(t − 1) i=0

(10) C

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Energy & Fuels ⎡ Z (t |t − 1)⎤ ⎢ in ⎥ x(̂ t |t − 1) = ⎢ Z(t |t − 1) ⎥ = ⎢ ⎥ ⎢⎣ ṁ (t |t − 1) ⎥⎦ ⎤ ⎡ Z (t − 1|t − 1) ⎥ ⎢ in ⎢ Z (t − 1|t − 1)(1 − e−Δt / τ(t − 1|t − 1)) + Z(t − 1|t − 1)⎥ ⎥ ⎢ in −Δt / τ(t − 1|t − 1) ⎥ ⎢ e ⎥ ⎢ ṁ (t − 1|t − 1) ⎦ ⎣

Flow rates of 10 and 25 L/min are used for both kinds of simulations. The lower flow rate is within the typical range used for FTIR experiments involving engine exhaust characterization. However, simulations with a flow rate of 25 L/min were also conducted to quantify the improvements in inlet composition estimation when flow rate is maximized. At higher flow rates, sample within the FTIR gas cell is displaced more quickly, yielding a total gas cell composition that more closely resembles the inlet composition. However, it should be reiterated that high flow rates can increase measurement noise, limiting the advantages of increasing flow rate beyond a certain point. The details for the CFD and MN simulations will now be discussed. Computational Fluid Dynamics. The computational domain used for the CFD simulations is based on the gas cell of the MKS 2030 HS FTIR, which has a volume of approximately 200 mL and inlet/outlet pipes with diameters of 3/8”. Unsteady, compressible, RNG k−ε turbulence simulations were performed using Converge CFD software. The RNG k−ε model was chosen over the standard k−ε model, as the RNG k−ε outperforms the standard model for cases involving impingement,26 which occurs within the gas cell on the opposite wall from the inlet. A temperature of 191 °C is enforced at the domain boundaries and inlet, because the MKS 2030 is controlled to maintain this temperature. Near-boundary velocity is calculated using the law of the wall. Flow through the gas cell is induced through pressure boundary conditions set at the inlet and outlet. For the 10 and 25 L/min cases, approximately 70 000 and 250 000 cells were used, respectively. Grid refinement was applied within the inlet jet and near the boundaries of the gas cell and surrounding plumbing to resolve composition and velocity gradients. Mesh independence was verified by increasing the number of computational cells until gas cell tracer composition changed by no more than 1% at any time during the simulation. To allow the inlet flow to adequately develop, 2 in. of the inlet pipe was included in the computational domain. Results from an additional simulation with a 4-in. inlet pipe at a flow rate of 25 L/min suggest that 2 in. is an adequate length, as the total gas cell tracer composition changed no more than 1% at any time. 3D and 2D diagrams of the computational domain are shown in Figure 2. Mixing Network. The MN model is a combination of plugflow and well-mixed systems, which collectively represent the FTIR gas cell. This model is designed to provide a simple yet accurate portrayal of the gas cell mixing dynamics and their effect on total gas cell composition. Ultimately, the MN calculates total gas cell composition for a given inlet composition profile and flow rate by summing the composition of each system normalized by its respective size. Figure 3 shows the layout of the MN. As shown, the model consists of a wellmixed system, followed by two plug-flow systems in parallel. An additional well-mixed system follows one of the plug-flow systems. The initial well-mixed system 1 represents an arbitrary volume immediately adjacent to the gas cell inlet. As shown in Figure 2b, a well-mixed system is appropriate here, because the inlet sample quickly mixes with the surrounding sample, as indicated by the steep gradients in tracer composition near the inlet. The plug-flow system 2 accounts for sample from the well-mixed system that “short-circuits”, or exits the gas cell without thoroughly mixing with the remaining gas cell contents. Mixing effects following system 1 are modeled with an additional well-mixed system 4. The preceding plug flow

(20)

Here, τ represents the effective time constant the gas cell, or the mass divided by mass flow rate. Mass within the gas cell is assumed to remain constant in this model. It should be noted that total gas cell mass could be included as a state variable, because FTIRs measure temperature and pressure within the gas cell, and total gas cell molecular weight could be deduced from Z. Mass could therefore be calculated via the ideal gas law. However, because FTIRs control temperature and pressure to remain constant, and the main combustion products usually remain relatively constant, mass within the gas cell should remain nearly steady. The measurement model describes the relationship between system states and measurement output. For this system, the measurement model is straightforward. Because FTIRs are calibrated to yield mole fractions, the model for FTIR measurements is simply the conversion from total gas cell mass fraction to mole fraction. Mass flow rate is measured directly. The equations for the measurement model are given below. ⎡ MW ⎤ ⎥ ⎢ Z(t |t − 1) · MWmix ⎥ y(t ) = ⎢ ⎥⎦ ⎢⎣ ṁ (t |t − 1)

(21)

MW and MWmix represent the molecular weights of the species of interest and the molecular weight of the total gas cell sample, respectively.

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FTIR SIMULATIONS Simulations of FTIR experiments were performed. In these simulations, sample with time-varying composition flows into the FTIR gas cell at a constant flow rate. Two kinds of simulations were performed: CFD and MN. In the CFD simulations, the sample consists of nitrogen mixed with a tracer component with identical properties (density and viscosity). The CFD simulations are used to verify the accuracy of the well-mixed model by comparing total gas cell composition calculated by the two models for identical inlet composition profiles. Synthetic FTIR measurements are also generated from the total gas cell composition calculations from CFD, which are used to tune the process noise covariance of the filter. In the MN simulations, simulated engine exhaust from an FTP driving cycle flows through the FTIR gas cell. The total gas cell nitric oxide (NO) composition calculated in these simulations is used to produce synthetic FTIR NO measurements. MN simulations are used for cases involving simulated exhaust, because the FTP driving cycle lasts over 1800 s, a simulation time that would be computationally prohibitive for CFD. Engine-out NO emissions are simulated using the model from ref 25, which determines emission levels for a given vehicle speed and acceleration. The vehicle parameters used for this study are the same as those in ref 25. D

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Table 1. Optimized System Masses and Plug-Flow Pathway Flow Rates for the MN Model V̇ (L/min)

ṁ pf/ṁ

m1/m

m2/m

m3/m

m4/m

10 25

0.512 0.483

0.168 0.152

0.107 0.106

0.237 0.227

0.488 0.515

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RESULTS This section is outlined as follows. First, results from CFD, MN, and the well-mixed model simulations are presented. Total gas cell composition for the step and impulse tracer profiles described earlier are compared for each simulation type. Next, inlet tracer composition profiles estimated by the BEM for the step change profile are presented, demonstrating how the performance of the BEM varies with respect to measurement noise and inlet composition process noise. Estimated inlet tracer composition profiles for step and sine wave profiles of different frequencies are then presented. These results illustrate the limitations of the BEM in detecting instantaneous and gradual composition changes. Next, estimations calculated by the BEM of synthetic NO emissions for the FTP driving cycle are presented. The improvement in the estimation of instantaneous NO emitted from an engine as compared to raw FTIR measurements for different measurement noise levels is demonstrated. Finally, the NO emission estimations from the BEM are compared to estimations from the well-mixed model uncoupled from a filter, illustrating the importance of using a filter for mitigating measurement noise effects. Evaluation of Well-Mixed Model. For the BEM to perform well, the well-mixed model must be an accurate representation for how total gas cell composition changes with respect to time. To verify the fidelity of the well-mixed model, total gas cell composition calculated by the well-mixed model is compared to composition calculated by CFD simulations for different inlet profiles and flow rates. Figure 4 shows the calculated total gas cell composition for the step and impulse inlet tracer composition profiles for flow rates of 10 and 25 L/ min. It is shown that there is reasonable agreement between the well-mixed model and the CFD results in every case. The maximum percent error for both step cases is about 11%, while the maximum errors for the impulse case for 10 and 25 L/min are about 16% and 14%, respectively. Maximum discrepancy between the two models for 10 and 25 L/min occurs about 300 and 100 ms after the tracer is introduced, respectively, for both the impulse and the step cases. In every case, the well-mixed model underestimates total gas cell composition. Discrepancy between the two models is explained by the plug-flow-like behavior predicted from the CFD. The wellmixed model assumes that sample entering the gas cell mixes instantaneously with the remaining contents, which causes

Figure 2. (a) 3D schematic of the FTIR gas cell and surrounding plumbing, used for CFD simulations. (b) Top-down slice of the CFD computational grid, colored according to tracer mass fraction. The snapshot is taken 0.15 s after tracer is introduced at the gas cell inlet for the 25 L/min case.

system 3 creates a time delay for when sample exits system 1 and mixes into sample 4, which is necessary to prevent inlet sample from immediately exiting the gas cell. To achieve accurate total gas cell composition calculations by the MN for a given inlet composition profile, the masses and flow rates of each system were optimized. Specifically, the optimized parameters were the masses occupied by systems 1− 4 (m1, m2, m3, and m4), and the mass flow rate that passes through system 2 (ṁ pf). Optimization was achieved with a genetic algorithm (GA). The GA optimized these parameters on the basis of total gas cell tracer composition profiles from CFD simulations. Fitness for a given population of parameters was calculated by summing the absolute deviation of the total gas cell composition between the MN and CFD simulations for two different inlet profiles. The two inlet composition profiles were a step change from nitrogen to tracer, and a tracer “impulse”, where tracer is introduced at the inlet for 0.2 s. Population sizes of 100 with an elitism replacement and mutation rate of 0.03 and 0.04 were utilized in the GA, respectively. A total of 2000 generations were performed. The optimal parameters for each flow rate are given in Table 1.

Figure 3. MN model used to determine the composition inside the FTIR gas cell. E

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CFD results after the transient, the overshoot of inlet composition estimation is short-lived. Furthermore, the overshoot issue is mitigated by adding process noise to the wellmixed model in the processor, which damps the response of the Bayesian estimation to measurement changes. As will be shown in the remainder of this Article, the well-mixed model is sufficient for the BEM. Also shown in Figure 4 is the total gas cell tracer composition calculated by the MN model. The MN profile closely follows that of the CFD in every case. Differences between the two profiles are imperceptible in the figure. This shows that the MN model accurately calculates total gas cell composition, and is suitable for generating synthetic FTIR data for a given emissions profile. Filtered Tracer Data. To explore the effects of process and measurement noise on the response of the BEM, synthetic FTIR data were generated from the CFD calculations for the 10 L/min, step inlet profile case. The measurements were produced by adding white Gaussian noise to the calculated total gas cell tracer composition at intervals of 200 ms, which is one measurement period of the MKS 2030 HS FTIR. These synthetic measurements were processed with the BEM, which produced estimates for the inlet tracer composition. Figure 5 shows the estimated inlet tracer composition for various inlet composition process noise (wZin ) and FTIR measurement noise (vZ) values. The inlet composition process noise term determines how readily the estimate for inlet composition can be adjusted by the BEM. A higher value means that the inlet composition is more sensitive to changes in measured gas cell composition. The covariance of the inlet composition process noise is the value in the first row/column of the process noise covariance matrix in eq 10. FTIR measurement noise covariance is the value in the first row of the measurement noise matrix in eq 14. In this analysis, it is assumed that measurement noise covariance is known, and the covariances in eq 14 are equal to the covariances of the Gaussian noise used to

Figure 4. Calculated FTIR gas cell tracer composition for different inlet composition profiles and flow rates: (a,b) step profile, (c,d) “impulse” profile.

some of the entering tracer to immediately escape the gas cell. However, the CFD results indicate that a finite minimum residence time exists within the gas cell. This phenomenon is apparent in the impulse cases in Figure 4, where the well-mixed model calculates a sharp decrease in tracer fraction immediately after the impulse completes, while the tracer fraction from the CFD plateaus briefly before decreasing. Despite their differences early on, the two profiles fall back into alignment within 400 ms of the time of maximum discrepancy. Because the wellmixed model initially under-predicts the magnitude of total gas cell composition changes, the BEM initially overestimates the magnitude of inlet composition changes. However, because the well-mixed model quickly falls back into agreement with the

Figure 5. Inlet tracer composition estimations by the BEM for various process and measurement noise values. V̇ = 10 L/min. F

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Figure 6. Inlet tracer composition estimations by the BEM for inlet composition step wave profiles at various frequencies and various flow rates. The standard deviation of the measurement noise (vZ) is 0.01. (a) V̇ = 10 L/min, f = 1.25 Hz; (b) V̇ = 10 L/min, f = 2.5 Hz; (c) V̇ = 25 L/min, f = 1.25 Hz; (d) V̇ = 25 L/min, f = 2.5 Hz.

generate synthetic measurements. Intuitively speaking, higher measurement noise results in the BEM “trusting” the measurements less, and therefore state estimations are less sensitive to measurement changes. In Figure 5, the standard deviation of the inlet composition process noise is varied between three values − 0.02, 0.07, and 0.12. The standard deviations of the measurement noise values are varied between 0.01, 0.025, and 0.05. Preliminary analysis showed that a satisfactory process noise standard deviation for total gas cell composition was 1/7th of the process noise standard deviation for inlet composition. This value adequately damped inlet composition estimation to prevent overshoot without significantly decreasing response time. Also, the standard deviation of the noise for mass flow rate is assumed to be 2.5% of the maximum flow rate in this analysis. The previously discussed trends associated with changes in inlet composition process noise and FTIR measurement noise are apparent in Figure 5. As the inlet composition process noise increases, the response time of the inlet composition estimations decreases. This is apparent in Figure 5d and g, where the estimated inlet composition reaches the actual inlet composition within three measurement periods (600 ms) after the step change. However, too much process noise results in a noisy estimated inlet composition profile, as shown in Figure 5g. The effect of measurement noise is apparent from comparing the estimated inlet composition on a column-bycolumn basis. Plots toward the right, which have higher measurement noise, have a slower response. The case with the highest measurement and lowest process noise (Figure 5c) results in an inlet composition estimation that is further from the actual inlet composition than the FTIR measurements for most of the data. These results indicate that a process noise with a standard deviation of 7% of the range of inlet composition values yields a good combination of response time and stability, and is used as the process noise for the remainder of the results presented.

To quantify the limits of the BEM in terms of response time, synthetic FTIR data were generated for step changes in inlet tracer composition at varying frequencies. These measurements were processed with the BEM, and the results are shown in Figure 6. Figure 6a and c shows estimated inlet tracer composition for a step wave with a frequency of 1.25 Hz for flow rates of 10 and 25 L/min, respectively. This frequency is half the highest frequency that can be accurately sampled by the FTIR, according to the Nyquist Theorem. At this frequency, the BEM estimates a reasonably accurate profile for the inlet composition, although greater accuracy is achieved for the higher flow rate. The reason better estimations are attained at higher flow rates is explained by the rate at which sample enters/exits the gas cell. For example, at lower flow rates, less sample enters/exits the gas cell during a measurement period. Therefore, substantial changes in inlet composition result in relatively modest changes in total gas cell composition, and, therefore, measured composition by the FTIR. Consequently, changes in FTIR measurements induced by changing inlet composition can be of comparable magnitude as measurement noise, preventing the BEM from detecting the inlet composition change. Figure 6b and d shows the estimated inlet tracer composition for step changes at the Nyquist limit (2.5 Hz) for flow rates of 10 and 25 L/min, respectively. As shown, the BEM does a poor job of estimating the inlet composition at the Nyquist limit regardless of flow rate. These results illustrate that the BEM requires at least two measurement periods (400 ms) to respond to a change in inlet composition. Dramatic changes with duration less than 400 ms may not be accurately captured by the BEM. So far, only Bayesian estimation results for instantaneous changes in composition have been presented. Because composition profiles are likely to contain gradual changes, the BEM was applied to synthetic data generated from sine wave tracer composition profiles with frequencies of 1.25 and 0.4 Hz. The results are shown in Figure 7. In all cases, the estimated G

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Figure 7. Estimated inlet tracer composition for inlet composition sine wave profiles at various frequencies and various flow rates. The standard deviation of the measurement noise (vZ) is 0.01. (a) V̇ = 10 L/min, f = 1.25 Hz; (b) V̇ = 10 L/min, f = 0.4 Hz; (c) V̇ = 25 L/min, f = 1.25 Hz; (d) V̇ = 25 L/min, f = 0.4 Hz.

Figure 8. Estimated NO emissions from an FTP driving cycle from 20 to 40 s for a FTIR flow rate of 10 L/min.

times. This can be seen by comparing the first column and second columns of plots in Figure 7, which have inlet composition frequencies of 1.25 and 0.4 Hz, respectively. Just as in the step wave cases, better estimations are achieved for higher flow rates, which can be seen by comparing the plots on a row-by-row basis. Despite the lag in estimated inlet composition, nearly every estimation is closer to the true inlet composition than the raw FTIR measurement. Moreover,

inlet composition lags behind the actual inlet composition. This lag is due to the quasi-steady nature of the BEM. Because the profile of the inlet composition between measurements is unknown, the BEM assumes that inlet composition remains constant during a measurement period and calculates its estimation accordingly. The lag of the estimation is more pronounced for sine waves with higher frequency, because the inlet composition changes more rapidly between measurement H

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Figure 9. Estimated NO emissions from an FTP driving cycle from 20 to 40 s for a FTIR flow rate of 25 L/min.

measurement noise levels, because noise levels vary according to a wide range of factors such as spectral resolution, and the absorptivity and quantity of the species of interest.11 It is seen from the figures that the BEM improves the estimation of the simulated NO emissions as compared to raw FTIR measurements, although this improvement becomes more modest as measurement noise increases. As shown in the top subplots of both figures, when the standard deviation of the measurement noise is 0.5% of the greatest NO measurement, the estimated NO profile closely follows the actual profile and accurately captures transients. However, for a measurement noise standard deviation of 4% of the greatest NO measurement, the estimated inlet composition is closer to the raw measurements than it is to the actual NO profile, as shown in the bottom subplots. The BEM provides greater improvement from raw measurements for the lower flow rate. This is due to residence times within the gas cell being shorter at higher flow rates, reducing the effect of historic emissions on the total gas cell composition. Therefore, measurements at higher flow rates have less room for improvement. However, some improvement in NO estimation is still seen for the higher flow rate case, especially for smaller measurement noise values. The improvement in emissions estimation from the BEM is quantified in terms of instantaneous and integrated emissions in Figures 10 and 11, respectively. Figure 10 shows the average error in instantaneous NO emissions for the Bayesian estimation and the raw FTIR measurement for the entire driving cycle. As shown in the figure, the BEM improves the estimation of the NO emissions by more than 40% as compared to raw FTIR measurements for the lowest measurement noise presented. As measurement noise increases, the improvement in instantaneous NO estimation becomes more modest. For a measurement noise standard deviation of 4% of the maximum measured NO composition, the BEM

the overall profile of the estimated inlet composition resembles the actual inlet composition more closely than the profile of the FTIR measurements in every case. Filtered Synthetic Emissions Data. To evaluate how the BEM may perform when processing emissions data, synthetic NO emissions for the FTP driving cycle were generated using the emissions model from ref 25. This is a calibrated model that estimates emissions for a limited number of species given vehicle speed, acceleration, and parameters (such as mass and drag coefficient). The vehicle parameters used to produce the NO data are identical to the parameters used to generate the example data in ref 25. MN simulations were conducted with the synthetic NO emissions data used as the gas cell inlet profile. Synthetic FTIR measurements were produced by adding white Gaussian noise to the total gas cell NO composition calculated from the MN simulations. It should be noted that the well-mixed model was preferred over the MN model as the state-transition model for the BEM, because the system masses and pathway flow rates of the MN model were optimized for specific total flow rates, and are not generally applicable. Furthermore, using the relatively complicated MN model as the state transition model would significantly increase the number of states and require a greater quantity of process noise parameters to be tuned. Another key point to be noted is that the BEM is equally applicable to other species that are measurable by an FTIR. NO was selected as the species of interest simply due to its ease of calculation from the emissions model. Figures 8 and 9 show a snippet of the NO emissions profiles estimated by the BEM for sample flow rates of 10 and 25 L/ min, respectively. The snippet of the driving cycle from 20 to 40 s is presented because it contains a combination of gradual and abrupt transients, as well as a steady-state portion. Estimated NO emissions are presented for a range of I

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an adaptive process noise covariance matrix will be the focus of future work. Figure 11 shows the error in integrated NO emissions over the entire driving cycle for the Bayesian estimation and the raw FTIR measurement. The BEM yields significantly improved integrated emissions calculations as compared to raw data, especially for low measurement noise levels. However, the integrated emissions calculated from FTIR measurements are also accurate, with the greatest percent error being less than 3%. The lower errors for integrated emissions as compared to instantaneous emissions are due to the cancellation of positive and negative errors in instantaneous emissions during integration. To prove the necessity of a Bayesian filter for accurately estimating emissions, NO estimations calculated from the BEM are compared to estimations from the well-mixed model uncoupled from a filter for the FTP driving cycle. The uncoupled well-mixed model estimates the inlet composition solely by deducing the inlet composition from measurement changes, and takes no account of measurement noise. Comparisons of estimated NO composition between the two models for two different flow rates and measurement noise values are shown in Figure 12. It is seen from the figure that the uncoupled well-mixed model yields noisy estimates, which oscillate above and below the true NO emissions, although the degree of estimation noise depends on flow rate and measurement noise. The noisiest estimations occur for the high noise, low flow rate case, while the most stable estimations occur for the low noise, high flow rate case. In every case, the BEM yields estimations with greater stability and accuracy than the uncoupled well-mixed model. However, it is clear that the BEM becomes more valuable as measurement noise increases and flow rate decreases. To quantify the performance differences between the BEM and the well-mixed model uncoupled from a filter, errors in instantaneous and integrated NO emissions from each model are presented. Figure 13 shows the average error in instantaneous NO emissions for the entire FTP driving cycle for different flow rates and measurement noise values. As shown in the figure, the average errors in instantaneous NO emissions are significantly larger for the uncoupled well-mixed model as compared to the BEM for a flow rate of 10 L/min. For noise standard deviations of 1% and 2% of the maximum measurement, the average errors of the uncoupled well-mixed model calculations are nearly double and triple the errors of the BEM calculations, respectively. However, the difference in instantaneous NO emissions error between the BEM and uncoupled well-mixed model is not as significant at the higher flow rate (25 L/min). Again, the favorable results at higher flow rates are due to the low residence times within the gas cell, which makes inlet composition estimates less sensitive to measurement noise. It should be reiterated that while higher flow rates mask the effects of measurement noise in these numerical results, higher flow rates can yield erroneous spectra and increase measurement noise in reality, as discussed earlier. The purpose of presenting results for a higher flow rate is to demonstrate the benefit of maximizing flow rate of a FTIR up until the introduction of these additional sources of error. Figure 14 shows the error in integrated emissions from the BEM and the uncoupled well-mixed model. Interestingly, the integrated emissions from the uncoupled well-mixed model are more accurate in every case, despite the instantaneous emissions estimates from the well-mixed model being worse

Figure 10. Average percent error of instantaneous NO emissions calculated from estimations from the BEM (red) and raw synthetic FTIR measurements (black) for different flow rates and measurement noise levels.

Figure 11. Percent error of integrated NO emissions calculated from estimations from the BEM (red) and raw synthetic FTIR measurements (black) for different flow rates and measurement noise levels.

provides little benefit in instantaneous emission estimates for either flow rate. Decreased performance of the BEM for high measurement noise values is explained by the contribution of noise to measurement error, and the fact that measurement noise desensitizes the NO emission estimates to measurement changes. A significant contributor to the average instantaneous NO estimation error is the noisiness of the estimation during steady-state. In fact, the raw FTIR measurements are often significantly closer to the actual NO emissions during steadystate than the estimations provided by the BEM. The noisiness of the estimations during steady-state is due to the high process noise for the inlet composition used in the filter. It is expected that the high process noise would cause the estimations during a mild transient (such as a long ramp) to be noisy as well. However, high process noise is necessary, as it allows NO estimates to be properly adjusted during steep transients. Further improvement in performance of the BEM could be achieved by using an adaptive process noise covariance matrix, where process noise would increase during transients and decrease during steady-state.27,28 This would allow the BEM to readily adjust emissions estimates during transients, while providing stability during steady-state. The implementation of J

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Figure 12. Estimated NO emissions from an FTP driving cycle from 20 to 40 s from the BEM and the well-mixed model uncoupled from a filter for various noise levels and FTIR flow rates.

Figure 13. Average percent error of instantaneous NO emissions calculated from estimations from the BEM (red) and the well-mixed model uncoupled from a filter (green) for different flow rates and measurement noise levels.

Figure 14. Percent error of integrated NO emissions calculated from estimations from the BEM (red) and the well-mixed model uncoupled from a filter (green) for different flow rates and measurement noise levels.

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than the estimates from the BEM. Again, the improvement in integrated emission from the well-mixed model is due to the cancellation of positive and negative errors during the integration of instantaneous emissions. This improvement could be partly explained by the fact that the estimations from the uncoupled well-mixed model contain the unaltered effects of measurement noise. Because the mean of measurement noise is essentially zero over many measurement points, the positive/negative errors induced by measurement noise are nearly canceled during integration.

CONCLUSIONS A BEM for filtering FTIR measurements of engine exhaust to improve instantaneous emissions estimates was presented. The BEM utilizes an unscented Kalman filter and a well-mixed model for the FTIR gas cell to estimate the composition of sample entering the gas cell during a measurement period. Synthetic FTIR data were generated using data from CFD and MN simulations of sample with time-varying composition flowing through a FTIR gas cell. These data were processed K

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v = measurement noise vZ = FTIR species measurement noise W = weight w = process noise wZin = gas cell inlet composition process noise x = system state y = measurement Z = total FTIR gas cell composition for various species Zin = FTIR gas cell inlet composition for various species κ = tuning parameter ξ = predicted measurement residual τ = effective FTIR gas cell time constant Υ = measurement sigma point X = state sigma point

with the BEM, which was shown to yield improved estimates of instantaneous engine-out emissions, especially for transient composition profiles. Among the synthetic FTIR data were measurements generated from synthetic NO emissions produced by an emissions model for the FTP driving cycle. The BEM yielded improved instantaneous and integrated NO emissions for every case, demonstrating its suitability for analyzing engine exhaust for real-world speed/load profiles. Last, NO emissions estimations from the BEM were compared to estimations from the well-mixed model uncoupled from a filter. It was shown that the filter is necessary for reducing the effects of measurement noise in the estimation. Future work will focus on experimentally validating the BEM. Experiments will involve flowing a sample with a known composition profile through the FTIR, and comparing the known composition with that estimated by the BEM. Another future endeavor is to add an adaptive process noise covariance algorithm, which will improve gas cell inlet composition estimations during steadystate.

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^ = model prediction ∼ = error

REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*Tel.: (312) 237-1675. E-mail: [email protected]. ORCID

David Wilson: 0000-0002-2682-5292 Casey Allen: 0000-0003-3684-5796 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge the helpful input and support received from Steve Wright and Sylvie Bosch-Charpenay of MKS Instruments. We additionally thank Dr. Matti Maricq of Ford Motor Co. for sharing his FTIR applications knowledge.

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NOMENCLATURE BEM = Bayesian estimation model CFD = computational fluid dynamics FTIR = Fourier transform infrared spectroscopy FTP = Federal Test Procedure MN = mixing network NO = nitric oxide UKF = unscented Kalman filter VOC = volatile organic compounds a = state transition model b = forcing function c = measurement model e = innovation f = frequency Κ = Kalman gain m = mass ṁ = mass flow rate ṁ pf = mass flow rate through plug flow pathway of the mixing network MW = molecular weights of species of interest MWmix = molecular weight of total gas cell sample Nx = number of states P̃ = state error covariance matrix R = covariance matrix t = time u = input V̇ = volume flow rate L

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