CN110397495B - Performance estimation method of engine post-processing device based on data driving - Google Patents

Performance estimation method of engine post-processing device based on data driving Download PDF

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CN110397495B
CN110397495B CN201910590191.4A CN201910590191A CN110397495B CN 110397495 B CN110397495 B CN 110397495B CN 201910590191 A CN201910590191 A CN 201910590191A CN 110397495 B CN110397495 B CN 110397495B
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CN110397495A (en
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胡杰
廖健雄
颜伏伍
蔡之洲
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Wuhan University of Technology WUT
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F01—MACHINES OR ENGINES IN GENERAL; ENGINE PLANTS IN GENERAL; STEAM ENGINES
    • F01N—GAS-FLOW SILENCERS OR EXHAUST APPARATUS FOR MACHINES OR ENGINES IN GENERAL; GAS-FLOW SILENCERS OR EXHAUST APPARATUS FOR INTERNAL COMBUSTION ENGINES
    • F01N11/00—Monitoring or diagnostic devices for exhaust-gas treatment apparatus, e.g. for catalytic activity
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F01—MACHINES OR ENGINES IN GENERAL; ENGINE PLANTS IN GENERAL; STEAM ENGINES
    • F01N—GAS-FLOW SILENCERS OR EXHAUST APPARATUS FOR MACHINES OR ENGINES IN GENERAL; GAS-FLOW SILENCERS OR EXHAUST APPARATUS FOR INTERNAL COMBUSTION ENGINES
    • F01N11/00—Monitoring or diagnostic devices for exhaust-gas treatment apparatus, e.g. for catalytic activity
    • F01N11/002—Monitoring or diagnostic devices for exhaust-gas treatment apparatus, e.g. for catalytic activity the diagnostic devices measuring or estimating temperature or pressure in, or downstream of the exhaust apparatus
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F01—MACHINES OR ENGINES IN GENERAL; ENGINE PLANTS IN GENERAL; STEAM ENGINES
    • F01N—GAS-FLOW SILENCERS OR EXHAUST APPARATUS FOR MACHINES OR ENGINES IN GENERAL; GAS-FLOW SILENCERS OR EXHAUST APPARATUS FOR INTERNAL COMBUSTION ENGINES
    • F01N11/00—Monitoring or diagnostic devices for exhaust-gas treatment apparatus, e.g. for catalytic activity
    • F01N11/002—Monitoring or diagnostic devices for exhaust-gas treatment apparatus, e.g. for catalytic activity the diagnostic devices measuring or estimating temperature or pressure in, or downstream of the exhaust apparatus
    • F01N11/005—Monitoring or diagnostic devices for exhaust-gas treatment apparatus, e.g. for catalytic activity the diagnostic devices measuring or estimating temperature or pressure in, or downstream of the exhaust apparatus the temperature or pressure being estimated, e.g. by means of a theoretical model
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T10/00—Road transport of goods or passengers
    • Y02T10/10—Internal combustion engine [ICE] based vehicles
    • Y02T10/40—Engine management systems

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  • Chemical & Material Sciences (AREA)
  • Chemical Kinetics & Catalysis (AREA)
  • Combustion & Propulsion (AREA)
  • Mechanical Engineering (AREA)
  • General Engineering & Computer Science (AREA)
  • Combined Controls Of Internal Combustion Engines (AREA)
  • Exhaust Gas After Treatment (AREA)

Abstract

The invention provides a performance estimation method of an engine post-processing device based on data driving, wherein a remote monitoring system based on the engine emission control post-processing device acquires real-time temperature values of an inlet and an outlet of the engine post-processing device in the using process; acquiring performance characteristics such as a temperature difference accumulated value, a normalization coefficient, temperature signal delay time and the like of the post-processing device by a data driving method based on the acquired real-time temperature big data; the performance of the post-processing device can be accurately identified and the fault prediction can be carried out through the corresponding relation between different performance characteristic values and the performance. The performance estimation method of the engine post-processing device based on data driving can quickly and accurately predict the performance of the engine post-processing device by fully utilizing real-time state data and historical data, diagnose the fault form and degree, is less influenced by working conditions, and can be better applied to actual post-processing diagnosis and performance estimation.

Description

Performance estimation method of engine post-processing device based on data driving
Technical Field
The invention belongs to the technical field of engines, and particularly relates to a performance estimation method of an engine post-processing device based on data driving.
Background
The engine is one of high-efficiency power sources, and is widely applied to the fields of transportation, engineering machinery, agricultural machinery, ships and the like. With the high-speed increase of the engine holding capacity, the problem of engine emission pollution is increasingly serious, becomes a main contributor of air pollution emission, becomes one of main factors restricting the development of the engine, and needs to be solved urgently. In order to meet the national VI and stricter emission standards, the addition of an aftertreatment device is a necessary choice for controlling the emission of the engine, and the aftertreatment device can effectively reduce the emission of the engine while ensuring the performance of the engine. The exhaust emission components of the gasoline engine and the diesel engine have difference due to different working principles. Aftertreatment devices for different exemplary engine types are shown in table 1. The working principle of different after-treatment devices is also different, the three-way catalytic converter TWC mainly passes NOXCO and HC react with each other to reduce exhaust emission, the particulate matter trap GPF and the DPF trap particulate matter by adopting mutually staggered closed pore passages, the oxidation type catalyst DOC achieves the aim of removing HC and CO through a catalyst coated on the surface of a carrier, and the selective reduction catalysis technology SCR is adoptedThe NOx emission values are reduced by injecting a urea solution to react with the NOx.
TABLE 1 aftertreatment devices for different engine types
Figure GDA0002186793590000011
Because of the wide breadth of our country, factors such as fuel quality, lubricating oil quality, road conditions, legal supervision and the like, and the fact that the performance of an after-treatment device is reduced or deteriorated due to reasons such as thermal failure, chemical poisoning (DOC), blockage/crystallization/fouling, mechanical Damage (DPF) and the like in the actual use process, the engine emission is rapidly deteriorated, the real-time monitoring management and estimation of the working state and the performance of the after-treatment device are very necessary, and currently, for the performance estimation of the after-treatment device, three types of methods are mainly adopted at home and abroad: (1) direct sensor-based measurements, such as PM sensors, radio frequency sensors, etc.; (2) model-based estimation, such as a pressure drop model, a temperature gradient model, etc.; (3) based on sensor signals, such as differential pressure sensors, temperature sensors, etc. The method is mainly used for estimating the performance based on the working condition of the engine, the post-processing temperature, the pressure drop and the like, so that a real-time state signal (such as temperature and pressure drop) acquisition device of the post-processing device needs to be additionally arranged to monitor the post-processing state and estimate the performance.
At present, a vehicle-mounted diagnosis system has good real-time performance in the aspect of automobile emission control fault diagnosis, can detect corresponding faults immediately and remind a driver through a fault indicator lamp, but when sub-systems or components related to an emission control system are in fault, normal work of an engine is possibly not influenced, so that timely processing and maintenance of the faults are not mandatory, a vehicle is still in a state that emission seriously exceeds the standard, and the expected target of reducing the vehicle emission control cannot be achieved.
With the rapid development of technologies such as 5G communication, cloud computing, big data and artificial intelligence, the car networking technology is widely applied, can conveniently acquire running state data of various systems or subsystems or components in the running process of a car, and how to generate values of the data is a problem to be faced by people. In recent years, data-driven-based methods have been rapidly developed in other fields to fully correlate and apply the large data generated on the engineering machinery without establishing an accurate mathematical model.
Disclosure of Invention
The technical problem to be solved by the present invention is to provide a performance estimation method of an engine post-processing device based on data driving, which analyzes and reuses real-time and historical big data to obtain the performance change of an engine.
The technical scheme adopted by the invention for solving the technical problems is as follows: a method for estimating performance of a data-driven engine aftertreatment device, comprising the steps of:
s1) acquiring real-time temperature values of front and rear measuring points of the engine exhaust control post-processing device based on a monitoring system of the engine exhaust control post-processing device;
s2) obtaining a temperature difference accumulated value fitting line and the slope of the fitting line by using a time series model and a fitting method according to the inlet and outlet temperature values of the current engine post-processing device;
s3) calculating and obtaining a covariance function, a normalized correlation coefficient and delay time of inlet and outlet temperature signals of the engine post-processing device by using a temperature correlation-based method;
s4) analyzing the obtained performance characteristic values, and combining all the performance characteristic values to predict and correct the performance, if the engine post-processing device is in a normal state, returning to the step I, and if the engine post-processing device is out of a normal range, utilizing the performance characteristic values to identify and diagnose faults.
According to the scheme, the calculation formula of the temperature difference accumulated value in the step S2) is as follows:
Tsum(n)＝Tsum(n-1)+(T2(n)-T1(n))
wherein n is a sequence of samples, Tsum(n) is the temperature difference integrated value of the sampling point n, T1And T2Respectively, the aftertreatment device inlet temperature and the outlet temperature.
According to the above scheme, the covariance function in step S3) is defined as follows:
Figure GDA0002186793590000021
wherein, N is the length of the signal,
Figure GDA0002186793590000022
and
Figure GDA0002186793590000023
is the average of x (n) and y (n), x (n) and y (n) are two signals of equal and finite length,
the dimensionless normalized correlation coefficient can be expressed as:
Figure GDA0002186793590000031
wherein, cxxAnd cyyThe variances of the sequences x (n) and y (n), respectively; rhoxyIs a normalized correlation function of variables with a variation range of [ -1,1 [)]For calculating where signal y (n) has the greatest similarity to signal x (n), y (n) is shifted by k forward or backward in time sequence, and then the cross-correlation function between x (n) and y (n + k) is calculated, where ρxyThe lag time of two sets of temperature signals as a function of time shift k may be determined by a cross-correlation function ρxy(k) At p, atxyThe time shift k when taking the maximum value is the delay time, and the formula is as follows:
Figure GDA0002186793590000032
Figure GDA0002186793590000033
according to the scheme, the fault identification and diagnosis by using the performance characteristic value in the step S4) includes the following steps: the temperature difference accumulated values corresponding to different fault types and fault degrees are obviously different, the difference is more obvious along with the increase of the sampling point n, the fitted temperature difference accumulated value is divided into a plurality of areas according to the slopes of the fitted data lines of different fault types and fault degrees, and the performance of the post-processing device is estimated according to the slope of the fitted line of the temperature difference accumulated values of the post-processing device with unknown faults and the area where the fitted line is located; the covariance of the two temperature signals is calculated by utilizing the inlet temperature sensor and the outlet temperature sensor of the post-processing device to represent the variation trend difference of the two temperature signals, the similarity degree of the two different signals is evaluated only through the value of the covariance and is not accurate enough, and the covariance function is standardized by introducing a standardized correlation coefficient, so that the influence caused by the amplitude of the sequence to be compared is avoided; in the process of transmitting the heat of the exhaust gas of the engine from the inlet to the outlet of the aftertreatment device, the temperature signal is correspondingly delayed, and the delay time is different under different fault conditions, so the performance of the aftertreatment device of the engine is estimated according to the corresponding relation between the delay time and the performance state of the engine and the temperature difference accumulation curve.
The invention has the beneficial effects that: the invention provides a performance estimation method of an engine post-processing device based on data driving, the influence of performance state change such as the fault occurrence of the post-processing device on the inlet and outlet temperatures is low-frequency, slow in occurrence and long-term, and a small number of sample points can not clearly represent the working state of the post-processing device; the method fully utilizes a large amount of real-time temperature data generated in the operation process of the post-processing device, and reuses the historical data of the post-processing device again, so that whether the post-processing device fails or not and the failure form and degree can be known in time, the post-processing device can normally work in the whole period, and the method can be widely applied to performance prediction and fault identification and diagnosis of various engine post-processing devices.
Drawings
FIG. 1 is a schematic view of a monitoring system for an engine aftertreatment device in accordance with one embodiment of the invention.
FIG. 2 is a flow chart of one embodiment of the present invention.
FIG. 3 is a flow chart of performance estimation for a diesel particulate trap (DPF) according to one embodiment of the present invention.
Detailed Description
For a better understanding of the present invention, reference is made to the following description taken in conjunction with the accompanying drawings and examples.
As shown in fig. 1 and 2, a performance estimation method of an engine after-treatment device based on data driving comprises the following steps:
step S10, acquiring real-time values of inlet temperature and outlet temperature in the engine post-processing device through a monitoring system of the automobile emission control post-processing device;
step S20, calculating the temperature difference accumulation value of the inlet and outlet temperature of the post-processing device through the time series model, wherein the calculation formula of the temperature difference accumulation value is as follows:
Tsum(n)＝Tsum(n-1)+(T2(n)-T1(n))
wherein n is a sequence of samples, Tsum(n) is the temperature difference integrated value of the sampling point n, T1And T2Respectively, the aftertreatment device inlet temperature and the outlet temperature. The temperature difference value of the normal post-processing device presents a more stable trend, the fitted temperature difference cumulative value is divided into a plurality of areas according to the obtained fitted data line slopes of different fault types and fault degrees, and the performance of the post-processing device is estimated according to the slope of the fitted line of the temperature difference cumulative value of the post-processing device with unknown fault and the area where the fitted line is located;
in step S30, the covariance of the two temperature signals is calculated by the post-processing device inlet and outlet temperature sensors to represent the difference in the trend of the two temperature signals, and the covariance function is defined as follows:
Figure GDA0002186793590000041
where N is the length of the signal,
Figure GDA0002186793590000042
and
Figure GDA0002186793590000043
is the average of x (n) and y (n), x (n) and y (n) being two signals of equal and finite length.
The normalized covariance function is then used to eliminate the effect of the sequence amplitudes to be compared. The dimensionless normalized correlation coefficient can be expressed as:
Figure GDA0002186793590000044
wherein, cxxAnd cyyThe variances of the sequences x (n) and y (n), respectively; rhoxyIs a normalized correlation function of variables with a variation range of [ -1,1 [)]。
Finally, shifting y (n) forward or backward in time sequence by k, and calculating the cross-correlation function of x (n) and y (n + k) when rhoxyThe lag time of two sets of temperature signals as a function of time shift k may be determined by a cross-correlation function ρxy(k) At p, atxyThe time shift k when taking the maximum value is the delay time. The formula is as follows:
Figure GDA0002186793590000051
Figure GDA0002186793590000052
the correlation coefficient and the delay time of the inlet temperature and the outlet temperature of the normal post-processing device keep constant values along with the increase of sampling points, however, the delay time of the post-processing device in different fault states has larger difference, so the performance of the post-processing device can be monitored according to different fault forms and the corresponding relation between the fault degree and the delay time;
and step S40, finally judging the real-time performance, failure fault form and degree of the diesel engine post-processing device by combining the performance characteristic values obtained by utilizing and mining the real-time data and the historical data.
Example one
As shown in fig. 3, applying the performance estimation method of the data-driven engine aftertreatment device to the performance estimation of a Diesel Particulate Filter (DPF) includes:
(1) acquiring real-time values of DPF inlet temperature and outlet temperature in an automobile emission control post-processing device monitoring system;
(2) calculating the temperature difference accumulation value of the DPF inlet and outlet temperatures in the diesel engine post-treatment device through a time series model, wherein the calculation formula of the temperature difference accumulation value is as follows:
Tsum(n)＝Tsum(n-1)+(T2(n)-T1(n))
wherein n is a sequence of samples, Tsum(n) is the temperature difference integrated value of the sampling point n, T1And T2DPF inlet and outlet temperatures, respectively. The fitting curves of the temperature difference accumulated value corresponding to different fault types and fault degrees are obviously different, and the difference is more obvious as the sampling point n is increased. DPF plugging failure can result in a drop in DPF outlet temperature and therefore exhibit a decreasing trend (increasing absolute value) in its difference from the DPF inlet temperature. Failure to break can cause the DPF outlet temperature to rise and thus its difference from the DPF inlet temperature exhibits an increasing trend (decreasing absolute value). And dividing the fitted temperature difference accumulated value into a plurality of regions according to the slopes of the fitted data lines of different fault types and fault degrees, and estimating the performance of the post-processing device according to the slope of the fitted line of the temperature difference accumulated value of the post-processing device with unknown fault and the region where the fitted line is located.
(3) In the process of transmitting engine exhaust from the DPF inlet to the DPF outlet, corresponding delay occurs in the temperature signals at the DPF inlet and the DPF outlet, the delay time is different under different fault conditions, and the covariance of the two temperature signals is calculated by using the temperature sensors at the DPF inlet and the outlet to represent the variation trend difference of the two temperature signals, and the covariance function is defined as follows:
Figure GDA0002186793590000053
where N is the length of the signal,
Figure GDA0002186793590000054
and
Figure GDA0002186793590000055
is the average of x (n) and y (n), x (n) and y (n) being two signals of equal and finite length.
Evaluating the similarity of two different signals by the value of the covariance alone is not accurate enough. The normalized covariance function can be used to avoid the effect of the sequence amplitudes to be compared. The dimensionless normalized correlation coefficient can be expressed as:
Figure GDA0002186793590000061
wherein, cxxAnd cyyThe variances of the sequences x (n) and y (n), respectively; rhoxyIs a normalized correlation function of variables with a variation range of [ -1,1 [)]。
For calculating where signal y (n) has the greatest degree of similarity to signal x (n), y (n) is shifted by k forward or backward in time sequence, and then the cross-correlation function between y (n) and y (n + k) is calculated, where pxyAs a function of the time shift k. The lag time of two sets of temperature signals can be determined by the cross-correlation function ρxy(k) At p, atxyThe time shift k when taking the maximum value is the delay time. The formula is as follows:
Figure GDA0002186793590000062
Figure GDA0002186793590000063
when the filter body is broken and has a fault, the DPF inlet temperature and the DPF outlet temperature can obtain a larger cross correlation coefficient on a smaller translation phase k, and when the filter body is blocked and has a fault, the DPF inlet temperature and the DPF outlet temperature can obtain the largest cross correlation coefficient on a larger translation phase. The correlation coefficient of the DPF inlet temperature and outlet temperature increases along with the increase of the damage degree, and the delay time is reduced; the correlation coefficient of the DPF inlet temperature and outlet temperature decreases as the degree of clogging increases, and the delay time increases. It is therefore possible to set limits according to the delay times for different fault forms and fault degrees, in order to monitor the performance of the aftertreatment device.
(4) And finally judging the real-time performance, the failure fault form and the failure fault degree of the diesel engine post-processing device by combining the performance characteristic values obtained by utilizing and mining the real-time data and the historical data of the DPF.
As a fault diagnosis and performance estimation technology, the method has high fault identification accuracy and does not need to establish an accurate mathematical model, and the data-driven method is rapidly developed in other fields. The data driving mainly adopts various data mining technologies to obtain useful information implied in online data and offline data, the useful information represents normal and fault states of a current system, and finally fault detection and diagnosis are realized. Therefore, the invention utilizes the engine emission control post-processing monitoring system to acquire the signals of the temperature sensor, and utilizes the data driving method to acquire the characteristic quantity representing the real-time performance of the engine post-processing device. In the post-processing system, the inlet and outlet temperature differences corresponding to a normal post-processing device and post-processing devices with different fault types and fault degrees are obviously different, meanwhile, in the process of transmitting the heat of engine exhaust from the inlet to the outlet of the post-processing device, corresponding delay occurs to a temperature signal, and the delay time is different under different fault conditions, so that a time sequence method is used for obtaining a temperature difference accumulated value, a temperature correlation coefficient and the delay time as characteristic quantities reflecting the performance of the post-processing device, a temperature difference accumulated value fitting curve is obtained by fitting, different performance characteristic quantities and the temperature difference accumulated value fitting curve can reflect different real-time performances (such as normality, damage types, damage degrees and the like) of the engine, and the performance of the post-processing device is estimated according to the corresponding relation between the characteristic quantities and the performance of the post-processing device of the engine.
The above description is only for the preferred embodiment of the present invention and is not intended to limit the scope of the present invention, and all equivalent structural changes made by using the contents of the present specification and the drawings, or any other related technical fields, are intended to be covered by the scope of the present invention.

Claims (4)

1. A method for estimating performance of a data-driven engine aftertreatment device, comprising the steps of:
s1) acquiring real-time temperature values of front and rear measuring points of the engine exhaust control post-processing device based on a monitoring system of the engine exhaust control post-processing device;
s2) obtaining a temperature difference accumulated value fitting line and the slope of the fitting line by using a time series model and a fitting method according to the inlet and outlet temperature values of the current engine post-processing device;
s3) calculating and obtaining a covariance function, a normalized correlation coefficient and delay time of inlet and outlet temperature signals of the engine post-processing device by using a temperature correlation-based method;
s4) obtaining and analyzing the performance characteristic values of the temperature difference accumulated value, the temperature correlation coefficient and the delay time through the steps, fitting a curve by using the temperature difference accumulated value obtained through fitting, wherein different performance characteristic quantities and the curve fitted by the temperature difference accumulated value can reflect different real-time performances of the engine, estimating the performance of the post-processing device through the corresponding relation of the performance characteristic quantities and the performance of the post-processing device of the engine, returning to the step S1 if the post-processing device of the engine is in a normal range, and identifying and diagnosing faults by using the performance characteristic values if the post-processing device of the engine is out of the normal range.
2. The performance estimation method based on the data-driven engine post-processing device according to claim 1, characterized in that the calculation formula of the temperature difference integrated value in step S2) is:
Tsum(n)＝Tsum(n-1)+(T2(n)-T1(n))
wherein n is a sequence of samples, Tsum(n) is the temperature difference integrated value of the sampling point n, T1And T2Respectively, the aftertreatment device inlet temperature and the outlet temperature.
3. The performance estimation method based on the data-driven engine post-processing device according to claim 1 or 2, characterized in that the covariance function in step S3) is defined as follows:
Figure FDA0002933764390000011
wherein, N is the length of the signal,
Figure FDA0002933764390000012
and
Figure FDA0002933764390000013
is the average of x (n) and y (n), x (n) and y (n) are two signals of equal and finite length,
the dimensionless normalized correlation coefficient can be expressed as:
Figure FDA0002933764390000014
wherein, cxxAnd cyyThe variances of the sequences x (n) and y (n), respectively; rhoxyIs a normalized correlation function of variables with a variation range of [ -1,1 [)]，
For calculating where signal y (n) has the greatest degree of similarity to signal x (n), y (n) is shifted by k forward or backward in time sequence, and then the cross-correlation function between x (n) and y (n + k) is calculated, where pxyThe lag time of two sets of temperature signals as a function of time shift k may be determined by a cross-correlation function ρxy(k) At p, atxyThe time shift k when taking the maximum value is the delay time, and the formula is as follows:
Figure FDA0002933764390000021
Figure FDA0002933764390000022
4. the method for estimating the performance of the data-driven-based engine aftertreatment device according to claim 3, wherein the step S4) of identifying and diagnosing the fault by using the performance characteristic value comprises the following steps: the temperature difference accumulated values corresponding to different fault types and fault degrees are obviously different, the difference is more obvious along with the increase of the sampling point n, the fitted temperature difference accumulated value is divided into a plurality of areas according to the slopes of the fitted data lines of different fault types and fault degrees, and the performance of the post-processing device is estimated according to the slope of the fitted line of the temperature difference accumulated values of the post-processing device with unknown faults and the area where the fitted line is located; the covariance of the two temperature signals is calculated by utilizing the inlet temperature sensor and the outlet temperature sensor of the post-processing device to represent the variation trend difference of the two temperature signals, the similarity degree of the two different signals is evaluated only through the value of the covariance and is not accurate enough, and the covariance function is standardized by introducing a standardized correlation coefficient, so that the influence caused by the amplitude of the sequence to be compared is avoided; in the process of transmitting the heat of the exhaust gas of the engine from the inlet to the outlet of the aftertreatment device, the temperature signal is correspondingly delayed, and the delay time is different under different fault conditions, so the performance of the aftertreatment device of the engine is estimated according to the corresponding relation between the delay time and the performance state of the engine and the temperature difference accumulation curve.
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