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Uncertainty Propagation in Device Characteristic Based Virtual Sensors.


In current heating, ventilating and air-conditioning (HVAC) industry practice, control and performance monitoring of equipment and systems are often limited by the number of sensors installed. Hardware and installation costs are usually cited as the reason for not installing more sensors. Virtual sensors may provide a lower-cost alternative to physical sensors.

The use of virtual sensors was first explored for HVAC in 2003, when Joo et al. [2003] developed a virtual fan airflow station for measuring both supply air and return air volumetric flow rate for controlling the return air flow rate based on tracking supply air flow rate. In the study, Joo and colleagues used fan head, fan speed and the fan curve to determine the total air-flow rate without the need for direct physical measurements. In 2004, Lee and Dexter [2004] developed a fuzzy sensor for determining the average temperature of air leaving the mixing-box of an air-handling unit (AHU). This virtual sensor corrected measurements from conventional mixed-air temperature sensors. They used a fuzzy relationship between the measured temperature from the conventional sensor and the average mixed-air temperature determined through computational dynamic simulations of the mixing chamber. Several additional studies of virtual sensors for HVAC have followed since (e.g., Wichman and Braun [2008], Huang et all. [2008], Li and Braun, [2008], Li and Braun [2009a, 2009b] and [Yu et al., 2011], but still no well-documented procedure to quantify the uncertainty of virtual sensors exists.

In this paper, a procedure is described for determining the uncertainty of virtual sensors that use equipment characteristic curves and other sensed inputs as the basis for virtual measurements. For example, measurements of the supply air-flow rate from air handling units (AHUs), the outdoor air-flow rate into air handlers, the temperature of the mixed-air leaving the AHU mixing box, and the chilled water flow rate through cooling coils are usually not available or not accurate enough for HVAC control and fault detection. However, all of these variables are affected by the movements of the mixed-air damper, the supply-air fan, the chilled water valve, and the chilled water pumps. As a result, curves that relate the behavior of these devices to the desired virtual values and commonly measured conditions can be used as the basis for virtual sensors, which we call device characteristic curve based (DCCB) virtual sensors. They use measureable operation conditions of devices such as damper position, valve position, fan/pump speed, or fan/pump head to determine virtual measurements. Device characteristic curves can be determined empirically using measurements or analytically from first principles. The desired virtual values can then be calculated using measured values of variables for which sensors are commonly installed.

The uncertainty of DCCB virtual sensors is created by two factors: 1) the uncertainty of measurements from sensors and propagation of this uncertainty through the empirical and/or analytic relationships from which the virtual sensor values are determined, and 2) uncertainty in the device characteristic curves. Understanding and quantifying this uncertainty is critical to validate the accuracy and precision of virtual sensors. Awareness of the accuracy of the virtual sensors is critical in evaluating them for specific applications. Furthermore, by identifying and managing the uncertainty in dominant variables that contribute the greatest uncertainty to virtual measurements, the total uncertainty in virtual sensors can be minimized.

This paper presents a procedure for assessing the uncertainty in virtual measurements for DCCB virtual sensors. The sources of uncertainty are identified; then an appropriate approach for quantifying the uncertainty caused by each source is described. The paper concludes with two example assessments of uncertainty in virtual sensors, a virtual chilled water flow-rate meter based on the DCC of the chilled water valve and a virtual pump flow meter based on the DCC of the chilled water pump.


Device characteristic curve based (DCCB) virtual sensors use device characteristic curves to relate the virtual measurements (consequential output of the device movements) to the input sensor measurements, which quantify device operation, as shown in Equation (1).

Y= f([a.sub.1],[a.sub.2],[a.sub.3],... [a.sub.m], [X.sub.1],[X.sub.2],[X.sub.3]... [X.sub.n]) (1)

Here, Y represents the virtual measurement, which is not obtainable by direct measurements because a sensor for it is not present; [X.sub.1],[X.sub.2],[X.sub.3] ...[X.sub.n]represent the sensed independent device operation variables; [a.sub.1], [a.sub.2], [a.sub.3],... [a.sub.m]represent constants appearing in the function f which are determined empirically or analytically from first principles; m is the total number of constants; n is the number of independent variables, [X.sub.1],[X.sub.3] ...[X.sub.n]. The function, including the values of the constants, can be determined empirically by fitting a curve (e.g., using regression or an artificial neural network) through measured points (Y, [X.sub.1], [X.sub.2], [X.sub.3]... [X.sub.n]) over a full range of operating conditions. Alternatively, the function can be derived from first principles and knowledge of the device.

According to Equation (1), uncertainty of the virtual measurements is caused by errors from device coefficients such as [a.sub.1], [a.sub.2], [a.sub.3],... [a.sub.m] and measurable device operation variables such as [X.sub.1],[X.sub.2],[X.sub.3] ...[X.sub.n]. Therefore, uncertainty results from two types of sources: (1) uncertainty associating with generation of the device characteristic curve; (2) uncertainty propagated through the analysis originating from the uncertainty in the measured inputs ([X.sub.1],[X.sub.2],[X.sub.3] ...[X.sub.n]). The uncertainty in the measured inputs is from sensor bias and limitations on precision, which are usually provided by sensor manufacturers. Users may also introduce errors in measurements, but these are more difficult to characterize.

For errors in generating characteristic curve, even though most of devices, such as pumps, valves, fans and dampers, have a manufacturer's characteristic curve, it is usually inadequate so that an in-situ curve should be generated after the devices are installed in the system to improve accuracy. Curves given by manufacturers are generated based on data for limited specific conditions, which might not match the actual device operating conditions. The deviation between a manufacturer's curve and in-situ curve can cause significant errors for virtual measurements. Generation of an in-situ characteristic curve requires physical measurements of the dependent variable and all independent variables, Y,[X.sub.1],[X.sub.2],[X.sub.3] ...[X.sub.n] over the full range of expected operating conditions. A curve can then be generated by regression, yielding the coefficients [a.sub.1], [a.sub.2], [a.sub.3],... [a.sub.m]In general, the regression equation will not (and should not) fit every measured data point. Some deviation will exist between the most measured data points and the regression line. The line represents a best fit line through the data points, which are scattered about the line. Least squares regression, however, will ensure that the line of the function form chosen minimizes the sum of the squared deviations of the points about the line. The standard deviation needs to be estimated to describe the precision of using the regression equation to represent the underlying functional relationship between the independent and dependent variables (Figliola and Beasley, 2000). The uncertainty in the curve fit and its precision are defined in Equation (2) for a two-dimensional relation.

[u.sub.curve fit] = [+ or -][square root of (S X [[t.sub.v,p].sup.2])] (2)


[mathematical expression not reproducible]

[f.sub.m,i] and [f.sub.r,i] represent the measured value of device characteristic and the value of the characteristics from the regression curve at each measurement point, respectively, [nu] represents the degrees of freedom of the regression and is defined as [nu] = N - (m + 1), N is the number of measurement points, and m is the number of the regression coefficients.

In addition to the curve fitting precision, the uncertainty in the in-situ characteristic curve is also caused by uncertainty in each of the physical sensors as well. In addition to the uncertainty of measurements from the sensor used for output variable (Y), often the same set of sensors is used to collect data for the independent variables ([X.sub.1],[X.sub.2],[X.sub.3] ...[X.sub.n]) for use in generating the in-situ characteristic curve and for measuring inputs for the virtual measurements. Therefore, the same sensor uncertainties are propagated into the virtual measurements twice, once in generating the characteristic curve and again in using the sensors to make measurements of inputs to the virtual sensor. The analysis of uncertainty for a generic two-dimensional problem is illustrated in Figure 1. The overall uncertainty caused by characteristic curve regression error is summarized in Equation (3).

[u.sub.TYPE1] = [+ or -][square root of [([u.sub.curve fit]).sup.2] + [([u.sub.x]).sup.2] + [([DELTA] X/[DELTA]X1[u.sub.x1]).sup.2] (3)

The second source of uncertainty is associated with the uncertainty of sensor measurements of [X.sub.1],[X.sub.2],[X.sub.3] ...[X.sub.n] propagated through equations to the virtual measurements. Measurements of [X.sub.1],[X.sub.2],[X.sub.3] ...[X.sub.n] are usually required for basic HVAC control or their procurement, installation, and maintenance are relatively inexpensive (compared to the cost of a physical sensors for the parameter for which the virtual sensor is created). The propagation of errors through different independent measured variables to the result is defined by Equation (4),

[mathematical expression not reproducible] (4)

Therefore, based on Equations (3) and (4), the total uncertainty of a DCCB virtual sensor is given by Equation (5).

[mathematical expression not reproducible] (5)


In this section, a virtual chilled water flow meter is introduced as an example. The virtual water flow meter uses a control valve as a measurement device. The water flow rate through the valve is calculated using measurements of the differential pressure across the valve and its associated coil, the corresponding valve command, and the valve characteristic curve obtained by regression (see Figure 2). Theoretically, virtual water flow measurements can be calculated from two measurable inputs, the valve stem position (x) and the differential pressure across the loop ([DELTA][P.sub.L]), and three other inputs, the valve inherent characteristic (f(x)), the valve flow coefficient ([C.sub.v]), and the valve authority (N), as presented in Equation (6) [Swamy et al., 2011].

[Q.sub.x] = [C.sub.v] X [square root of [DELTA][P.sub.L,x]N/N+[f.sup.2](x)(1-N)] (6)

Because the two constants, [C.sub.v] and N, and the inherent valve characteristic curve, f(x), are determined when the system is designed and they are considered as known constants, the relationship for the water flow rate can be formulated as Equation (7).

[Q.sub.x] = [F.sub.L,x](x)[square root of [DELTA][P.sub.L,x]] (7)

[F.sub.LX] is defined by Equation (8), as the installed valve characteristic at the valve opening position of x.

[F.sub.L,x](x) = [square root of [([C.sub.v] X f(x)).sup.2]N/[N+[f.sup.2](x)(1-N)] (8)

Empirically Determining the Valve Characteristic Curve

To obtain the installed valve characteristic curve ([F.sub.l.x]), the valve characteristics should be measured both while opening (ascending) and closing (descending). Results for an example air-handler chilled water valve are presented in Figure 3 along with the regressed installed valve characteristic curve. The circles represent ascending points and the exes represent descending points. Each point show in Figure 3 is an average of 10 measurements at each valve position. The ascending and descending points appear two separate curves that overlap except in the range of 50% to 65% for the valve opening position. As shown in Figure 3, deviations exist between measured points and regressed curve. The scatter of the measured points indicates the uncertainty with which the single line represents the true values of FL,x, and the standard deviation of the points about the line can be used as a quantitative indicator of this uncertainty.

Other sources of uncertainty

According to Equation (7), two measured variables, the valve command (x) and differential pressure ([DELTA][P.sub.l,x]), and the installed valve characteristic curve obtained by regression (with the regression coefficients), are used in a functional correlation to determine the resultant water flow rate ([Q.sub.x]). Thus, the total uncertainty in the calculated water flow rate results from the propagation of uncertainties in the valve command, differential pressure sensor measurements, and uncertainty associated with the regression itself. For simplicity, we assume that the valve responds perfectly to signals to the valve actuator; otherwise, another source of uncertainty would need to be accounted for. The sources of uncertainty are summarized in Table 1.

The total uncertainty in the virtual water flow rate consists of components from the input variable measurements, from the direct measurements, and from curve fitting. Therefore,

[mathematical expression not reproducible] (9)

where u represents the uncertainty of each input and the output variable, [theta] represents the sensitivity index for each direct measured variables, subscript curve fit represents the standard deviation from the regressed curve, and the other subscripts L and x represent water pipe loop and valve open position respectively.

Empirical Results

Application of the virtual flow meter to the example air handler valve showed overall uncertainty in the virtual flow meter in a range from 0.25% to 1.46% with the maximum uncertainty for valve positions of 50% to 70% open. Curve fitting is the dominant source of uncertainty, contributing from 0.19% to 1.43%. The second most dominant factor source of uncertainty is the components propagated from the differential pressure sensor, which ranges from 0.01% to 0.25% and increases as the valve opens. The error propagated from the ultrasonic meter used for collected flow rate data for development of the in situ characteristic curve is approximately 0.16% throughout the full range of valve opening. The error propagated from the valve stem position is minimum. The total uncertainty is well within a usable range for flow rate monitoring and fault detection.


As a second example, the uncertainty analysis is applied in this section to a virtual pump flow meter, which uses the measured pump head, the pump speed, and the pump characteristic curve, to determine the flow rate through the pump. The water flow rate is given by Equation (10) (Song et al., 2012), which is based on the pump hydraulic characteristics.

[mathematical expression not reproducible] (10)

where H is the pump head [in ft (m) water column] at the full pump speed,

Q' is a water flow rate [gal/min, l/s] at full pump speed.

[omega][??] is the pump speed ratio, which is defined as the pump operating speed (in Hz) divided by the full (100%) pump speed (60HZ).

[a.sub.0], [a.sub.1] and [a.sub.2] are pump curve coefficients of a 2nd order polynomial regression.

Sources of uncertainty include measurements of the pump differential pressure, the pump speed, the regression coefficients for the pump characteristic curve, and the control system constants. The total uncertainties of the virtual water flow rate readings are presented in Figure 4. As the system curve moves from Line C to Line A, i.e. the slope of the pump curve changed from -5.0 ft water/kGPM (0.24kPa/L/s) to -17ft water/kGPM (0.81kPa/L/s), the total error of the virtual flow meter reduced by almost a factor of three (from 0.6355 kGPM to 0.2238 kGPM at full speed). Line B is the design curve, Line A represents overloaded conditions, and Line B represents underloaded conditions. The results also show that running the pump at close to the design system curve (defined by Line B) is always preferred, not only for higher pump efficiency, but also for more accurate virtual pump flow meter applications. In addition, the conclusions is also applied generally for judging if the pump curve is stiff enough for the virtual flow meter application. For example, to achieve less than 3% full scale error (10k GPM (631 L/s) at full scale, a minimum slope of -10 ft water/kGPM (0.47 kPa/L/s) is required.

In addition to Figure 4, the uncertainty analysis also shows that pump speed is the most influential factor in virtual pump flow meter applications. Higher controller bits provide better total accuracy for the virtual flow meter, and an 8-bit controller is not suitable for this application. Furthermore, while developing in-situ pump curves, a higher pump operating speed is desired for enhanced accuracy. Finally, the accuracy of the virtual flow meter is also increased by a larger pump curve slope. In general, slopes less than -10 ft water/kGPM (0.47kPa/L/s) is not preferred. In this section, all the uncertainty analysis is based on 95% confidence intervals.


Using an indirect calculated value, the uncertainty of virtual sensors is influenced by the uncertainty of many input variables. An uncertainty analysis procedure for device characteristic curve based virtual sensors is presented in this paper, and the procedure is also applied in two virtual sensor case studies. It was found that uncertainty analysis is critical for quantifying the expected overall uncertainty of virtual sensors and assessing their appropriateness for specific applications, identifying dominant variables which have most influence on the uncertainty of virtual sensors, and providing guidance on improving the accuracy of virtual sensors.


Atul Swamy, Gyujin Shim and Ik-seong Joo provided measurement data used in the paper.


Figliola, R. S., Beasley, D. E. 2000. Theory and Design for Mechanical Measurements, third edition, ISBN 0-471-35083-4. Page 109-128.

Joo, I., Liu, M., Wang, J., Hansen, K. 2003. Application of new technologies during continuous commissioning, Proceedings of International Conference on Enhanced Building Operation. Berkeley, CA, October 2003.

Joo, I., Liu, G., Liu, M. 2007. Application of fan airflow stations in air-handling units. Energy Engineering. Volume 104 (2): page 66-73.

Li, H., and Braun, J.E. 2008. Decoupling features for diagnosis of reversing and check vales faults in heat pumps. International Journal of Refrigeration, Volume 32(2):316-26.

Li, H., Braun, J.E. 2009a. Development, evaluation, and demonstration of a virtual refrigerant charge sensor, HVAC&R RESEARCH, Volume 15 (1): pages 117-136.

Li, H., Braun, J. E. 2009b. Virtual Refrigerant Pressure Sensors for Use in Monitoring and Fault Diagnosis of Vapor-Compression Equipment, HVAC&R RESEARCH, Volume 15 (3): Pages 597-616.

Song, L., Joo, I., Wang, G. 2012. Uncertainty analysis of a virtual water flow measurement in building energy consumption monitoring, HVAC&R RESEARCH, available online April 2012, DOI: 10.1080/10789669.2012.658137.

Swamy, A., Song, L., Wang, G. 2011. "A Virtual Chilled Water Flow Meter Development at Air Handling Unit Level", ASHRAE Transactions, Vol.118 part 1, CH-12-024.

Wichman, A., Braun. J.E. 2009, "A smart mixed-air temperature sensor", HVAC&R RESEARCH, Volume 15 (1): pages 101-115.

Yu, D., Li, H., Yu, Y. 2011. Virtual calibration of a supply air temperature sensor in rooftop air conditioning units, HVAC&R RESEARCH, Volume 17 (1): pages 31-50.

Li Song, PhD, PE


Gang Wang, PhD, PE


Li Song is an assistant professor in the School of Aerospace and Mechanical Engineering, Accredited State University, Norman OK. Michael Brambley is a research fellow at PNNL, Richland, WA. Gang Wang is an assistant professor in Department of Civil, Architectural and Environmental Engineering.
Table 1: Input error sources.

Error Input            Symbols

Differential pressure  [DELTA][P.sub.l,x]

Valve command          x

Deviation from curve

Error Input            Error sources

Differential pressure  Uncertainty by the differential pressure
                       transducer and the controller
Valve command          An output variable from the controller to
                       the valve, which in this example is assumed
                       identical to the valve position so the error
                       is associated with the 16 bit resolution of
                       the controller.
Deviation from curve   Propagated uncertainty originating from the
fit                    ultrasonic flow meter, and the differential
                       pressure sensor used for measurements in
                       developing the installed valve
                       characteristic curve
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Author:Song, Li; Wang, Gang
Publication:ASHRAE Conference Papers
Date:Dec 22, 2013
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