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Data reconciliation for steam turbine on-line performance monitoring
Xiaolong Jiang , Pei Liu , Zheng Li
PII: S1359-4311(14)00355-X
DOI: 10.1016/j.applthermaleng.2014.05.007
Reference: ATE 5605
To appear in: Applied Thermal Engineering
Received Date: 1 December 2013
Revised Date: 10 March 2014
Accepted Date: 2 May 2014
Please cite this article as: X. Jiang, P. Liu, Z. Li, Data reconciliation for steam turbineon-line performance monitoring, Applied Thermal Engineering (2014), doi: 10.1016/j.applthermaleng.2014.05.007.
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Data reconciliation for steam turbine on-line performance monitoring
Xiaolong Jianga, Pei Liua,*, Zheng Lia
aState Key Lab of Power Systems, Department of Thermal Engineering, Tsinghua University, Beijing, China 100084
* Corresponding author. Email: [email protected]; Tel+86 10 62795734(333); Fax. +86 10 62795736
Abstract
In a coal-fired power generation unit, steam turbine performance monitoring relies on steam cycle heat
balance analysis, in which the primary flow measurement accuracy is critical. However, in a steam turbine
on-line measurement system, the accuracy of most flow metering devices is not satisfactory, and using data
measured with these devices could lead to high uncertainty in the heat balance analysis and performance
monitoring results. In this work, we propose a data reconciliation approach in steam turbine on-line
performance monitoring, with the aim of reducing uncertainty of the primary flow measurements and steam
turbine heat rate. The proposed method is based on the establishment of a first-principle mathematical model
of a steam turbine unit and optimization of the reconciled values of measurements. Results of a case study
carried out on a real-life 1000 MW ultra-supercritical unit show that, after data reconciliation, uncertainties of
measured values of the outlet flow rate of the #1 feed water heater, the outlet flow rate of the feed water pump,
and the inlet flow rate of condensate water in the deaerator can be reduced by 72.1 %39.4 % and 21.4 %, and
that uncertainty of the steam turbine heat rate can be reduced by 17.9 %18.8 % and 18.8 %, when the unit
operates at 100 %75 % and 50 % of its design load.
Key words: Steam turbine, Performance monitoring, Data reconciliation, Uncertainty
1. Introduction
Performance monitoring is the process of continuously evaluating the production capability and efficiency
of a unit over time using on-line measured plant data. For steam turbine performance monitoring, heat rate is
an important efficiency indicator, and its calculation relies on steam turbine heat balance analysis [1].
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For steam turbine heat balance analysis, measuring instrument accuracy is of great importance, because
measurement errors are propagated into the uncertainty of the calculated results. Among all the measurements,
primary flow measurement is especially important and can have significant impact on the heat balance analysis
and heat rate calculation results [2]. Usually, the flow rate of either feed water or condensate water is needed to
serve as the primary flow measurement [3]. However, these flow rates are usually measured with low accuracy
instrument in the on-line measurement system [4] and can introduce great uncertainty to monitoring results [5].
Although only one of the feed water and condensate water flow rates is needed as primary flow
measurement in the heat balance analysis, both of them are measured in the on-line measurement system. As a
result, there exists redundant primary flow measurement. For redundant measurements, data reconciliation
method can be applied to give maximum likelihood estimation for the redundant measured data and let the
estimated data satisfy the system constraints. Data reconciliation is a data preprocessing method for improving
the accuracy of measurements by reducing the effect of random errors in the measured data. The principal
difference between data reconciliation and other methods is that data reconciliation explicitly makes use of
system constraints and obtains estimates of system parameters by adjusting system measurements so that the
estimates satisfy the constraints [6].
In order for data reconciliation to be effective, there should be no gross errors in the measured data. As a
result, gross error detection is necessary to validate data reconciliation calculation. Usually, gross error
detection and data reconciliation can be carried out iteratively and many statistical test methods can be used for
gross error detection, including the global test and the measurement test [7], the constraint test [8], the
generalized likelihood ratio test [9] and the principal component test [10] etc. Under some conditions,
equivalence of these gross error detection methods can be proved [11]. Besides, there are also methods which
can solve the problem of data reconciliation and gross error detection simultaneously [12].
Since 1960s, a number of studies have been carried out to develop theories and methods for data
reconciliation and gross error detection [13], but most of them focus on applications of data reconciliation in
the chemical industry. Ishiyama et al. [14] apply data reconciliation in oil refinery distillation preheat trains to
provide fouling rate parameters of heat exchangers for further desalter inlet temperature control. Ijaz et al. [15]
proposed a new algorithm to reconcile the energy balances as a follow-on step to material balance
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reconciliation thus greatly simplifying data reconciliation for large scale heat exchanger network models.
Maradiaga et al. applied data reconciliation to the steady-state operational data of a single-effect
ammonia/water absorption chiller and use the modified iterative measurement test method for gross error
detection [16]. Besides, data reconciliation has also been applied to an hydrogen plant [17], a pyrolysis reactor
[18], an ammonia plant [19], an coke-oven-gas purification process [20] and an polypropylene reactor [21] etc.
In power industry, most previous studies on data reconciliation are for nuclear power plants, where
requirements on data accuracy and operation safety are much higher than other power plants. Langenstein and
Jansky [22] applied data reconciliation method to calculate the thermal reactor power in a nuclear power plant
with an uncertainty of less than ±0.5 %. Langenstein [23] pointed out that data reconciliation could help
avoiding producing losses in nuclear power plants by holding the actual thermal reactor power in a narrow
range. Valdetaro and Schirru [24] developed a method which can perform model parameter tuning, data
reconciliation and gross error detection simultaneously in thermal reactor power calculation.
Application of data reconciliation in gas turbine and combined cycle power generation units are also
introduced in the literature. Chen and Andersen [5] applied data reconciliation method to eliminate the
common contradictions among the balance equations and make the daily operation optimization of a gas
turbine more confident. Gülen and Smith [25] determined the gas turbine power output and inlet airflow in a
single-shaft combined cycle with an uncertainty of ±0.5 % and ±1.15 % through data reconciliation.
Martini et al. [26] applied the data reconciliation method to a micro-turbine based test rig and showed the
ability of data reconciliation for accuracy improvement and gross error detection.
Recently, data reconciliation is also used in coal-fired power plants. Liu et al. [27] used the boiler mass and
energy balance constraints to reconcile the feed water mass flow rate, heat value and flow rate of coal to
improve the data accuracy and detect gross errors. Fuchs [28] investigated the impact of data reconciliation on
the calculated value of steam turbine exhausting steam enthalpy and steam cycle heat rate based on thermal
acceptance test data. Zhou et al. [29] applied a simultaneous data reconciliation and gross error detection
method to a boiler spray water system. Harter et al. [30] embedded data reconciliation method into a
commercial heat balance software and reduced the effort of constraints formulation for data reconciliation.
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For the steam turbine system in a coal-fired power generation unit, heat rate is the key performance
monitoring indicator and primary flow measurements have significant impact on the heat rate monitoring
results. However, research on the effect of data reconciliation for primary flow and heat rate uncertainty
reducing is still rather limited. Moreover, unlike a chemical process, it is common for a coal-fired power
generation unit to operate at off-design loads, and previous research on data reconciliation in off-design
situations is also rather limited.
In this work, data reconciliation method was applied for steam turbine on-line performance monitoring in a
real-life 1000 MW ultra-supercritical coal-fired power generation unit. A global test was carried out to make
sure that there exists no gross error in the operational data and thus validate the data reconciliation calculation.
The uncertainty of the measured and reconciled value for primary flow measurements were compared to
investigate the effect of data reconciliation for reducing the uncertainties of the estimated values of primary
flow rates. We also evaluated the effect of data reconciliation for reducing the uncertainty of the calculated
value of steam turbine heat rate with the unit operating at the design load and off-design loads. Finally, we
evaluated the impact of data reconciliation on the heat rate sensitivities with respect to different measurement
parameters to give a better understanding on the data reconciliation effect.
2. System description
2.1. System equipment and heat rate calculation
It shows in Figure 1 an illustration of a steam turbine on-line performance monitoring system in a 1000 MW
ultra-supercritical coal-fired power generation unit. Key equipment in Figure 1 includes, a generator (Gen), a
boiler (Boiler), the first and second stage group of high pressure steam turbine cylinder (HP1 and HP2), the
three high pressure feed water heater (HPFW1, HPFW2 and HPFW3), a feed water pump (FWP) and a
deaerator (DA).
In the deaerator, air dissolved in the condensate water is removed in a mixing process by steam extracted
from the steam turbine and drain water from the high pressure feed water heater. The deaerator outlet water is
then pressurized by the feed water pump and heated as feed water in the three high pressure feed water heaters
by extracted steam from the steam turbine. After heating the feed water, extracted steam becomes sub-cooled
drain water and is sent to the next feed water heater and finally enters the deaerator. The boiler heats the feed
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water to produce main steam, which expands to do work in the first and second stage group of high pressure
steam turbine cylinder and becomes cold reheat steam. The cold reheat steam is heated again in the boiler to
produce hot reheat steam and further expand to do work in the following intermediate pressure and low
pressure steam turbine cylinders (not shown in Figure 1). The generator is connected to the steam turbine rotor
and driven by the steam turbine to generate power. To help regulate the hot reheat steam temperature, part of
the feed water is extracted at the feed water pump and injected into boiler reheater as reheater spray water
(RHS). Besides these main flows, there also exist leakage flows, including main steam valve stem leakage
(MLK1), high pressure cylinder front side gland sealing steam leakage (MLK2), high pressure cylinder back
side gland sealing steam leakage (MLK3) and intermediate pressure cylinder cooling steam (MLK4).
Figure 1. An illustration of a steam turbine on-line performance monitoring system
The efficiency of this steam turbine system is usually indicated by heat rate, i.e., the power produced by the
generator divided by the heat absorbed by feed water, cold reheat steam and reheater spray water in the boiler
[1], as follows:
( ) /HR MMS HMS MFFW HFFW MHR HHR MCR HCR MRHS HRHS Power= ⋅ − ⋅ + ⋅ − ⋅ − ⋅ (1)
Where MMS, MFFW, MHR, MCR and MRHS represent the flow rates of the main steam, final feed water, hot
reheat steam, cold reheat steam and reheater spray water; HMS, HFFW, HHR, HCR and HRHS represent the
specific enthalpy of the main steam, final feed water, hot reheat steam, cold reheat steam and reheater spray
water; Power represents the power generated by the steam turbine.
In Equation (1), the power is directly measured at the generator and the specific enthalpies of the flows can
be determined with corresponding pressure and temperature measurements. However, the flow rates needed for
heat rate calculation is usually not directly measured and need to be determined through heat balance analysis.
In this heat balance analysis, there are three measured flow parameters which can serve as the primary flow
measurement, including the outlet flow rate of the #1 feed water heater (MFFW), the outlet flow rate of the
feed water pump (MFWP), and the inlet flow rate of condensate water in the deaerator (MCW). With a
selected primary flow measurement, other flow rates for the extracted steams, main steam and reheat steam can
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be determined with mass and energy balance equations for the feed water heaters, the boiler and the steam
turbine stage groups.
Since only one of these three measured flow parameters is required for heat balance analysis as primary flow
measurement, two of them can be regarded as redundant measurements. This redundancy can be used through
data reconciliation to reduce the uncertainty of heat balance analysis and heat rate calculation results.
2.2. Measured data and measurement uncertainty
In this work, data reconciliation is carried out with measured data collected from the database of the on-line
monitoring system, with a collection interval of one minute. Since this work mainly focuses on steady state
data reconciliation, only the operational data collected at steady operational status were used. A sliding time
window method was used to collect steady state operational data from the on-line monitoring system, as
follows [31]:
( )2
,1
1 10.01
1
t
SSC SSC k SSCk t NSSC t
S x xx N = − +
= − <− ∑ (2)
Where SSSC represents the relative standard deviation of the steady state criteria parameter SSCx ; tN
represents the length of the time window; SSCx represents the average of the steady state criteria parameter in
the time window.
The power generated by the steam turbine is taken as the steady state criteria parameter and the length of the
time window is set to be 30 min. A set of steady state measured data at unit design load is shown in Table 1.
For those parameters measured with multiple sensors, the average of the multiple measurements is reported.
Table 1 Values and uncertainties of the measured parameters
In order to apply data reconciliation method, the uncertainty of the measured data needs to be evaluated. In
this work, the measurement uncertainty is expressed as standard deviation [32] and we mainly focus on the
uncertainty caused by measuring instruments.
An error caused by a measuring instrument is usually a systematic error with a constant value. However, in
practice this value is always unknown and we only know the permissible error iξ for the measuring
instrument of a certain accuracy grade. In this work, we assume that the value of error caused by a measuring
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instrument follows normal distribution with zero mean, and the permissible error iξ corresponds to the
confidence interval with 95 % confidence probability [27]. Then the uncertainty of the measured parameter
( )ixσ can be evaluated as follows:
( )1.96
iix
ξσ = (3)
For parameters measured with multiple sensors, the uncertainty of the average data can be evaluated as
follows [32]:
( ) ( )ii
s
xx
N
σσ = (4)
Where sN represents the number of sensors used to measure the same parameter.
For the valve stem leakage flows and gland sealing steam leakage flows, there is no actual measuring
instrument and these flow rates are estimated based on the design materials provided by the manufacturer. In
this case, the uncertainties of the estimated values are assumed to be 100 % of the design values according to
the VDI standard [33].
2.3. Constraint equations
In this work, constraint equations for data reconciliation mainly include the steady state mass and energy
balance equations and the heat rate calculation equation, as shown in Table 2.
Table 2 System constraint equations
In these constraint equations, ( ),h p T represents the specific enthalpy of water or steam calculated
according to pressure p and temperature T ; ( ),sat wh p represents the saturated water specific enthalpy
calculated according to pressure p . According to the ASME standard [3], the specific enthalpy of drain water
was calculated with corresponding extraction steam pressure and drain water temperature, and the specific
enthalpy of feed water was calculated with corresponding feed water temperature and outlet pressure of feed
water pump. The water and steam thermodynamic properties are calculated according to the IAPWS IF97
standard [34].
The unmeasured parameters included in these constraint equations are described in Table 3.
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Table 3 Descriptions of the unmeasured parameters
3. Methodology
3.1. Data reconciliation problem formulation and solution
Data reconciliation problem is usually formulated as follows [6]:
( ) ( )( )
1ˆ ˆmin
ˆ. . ,
T
s t
ϕ −= − ∑ −
=
x x x x
f x y 0 (5)
Where 1n×x and 1ˆ n×x represent the measured and reconciled value for measured parameters; n n×∑
represents the covariance matrix describing the measurement uncertainties; 1m×f represents the system
constraint equations; 1p×y represents the calculated value for the unmeasured parameters.
An algorithm based on the successive linear data reconciliation method [6, 15] is used to solve the data
reconciliation problem, shown as follows.
Step 1: Start with the measured values x as initial estimates for the reconciled value x̂ and initial
estimates for the unmeasured parameters y are obtained through heat balance analysis with MFWP as
primary flow measurement.
Step 2: Evaluate the Jacobian matrices of the constraint equations with respect to variablex̂ , y .
,ˆk k
k k
∂ ∂= =∂ ∂
f fA B
x y (6)
The constraint equation ( )ˆ,f x y can then be linearized through Taylor series method with only the first
order derivative term retained, as follows.
ˆk k k+ =A x B y C (7)
Where ( )ˆ ˆ ,k k k k k k k= + −C A x B y f x y
Step 3: Perform QR factorization of the Jacobian matrix kB as follows.
,, ,[ ] k
k k k k k
= =
11 2
RB Q R Q Q
0 (8)
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Where ,k1R is a nonsingular p p× upper triangular matrix; kQ is a m m× orthogonal matrix; ,k1Q is
the first p basis vector for matrix kQ ; 2,kQ is the second m p− basis vector for matrix kQ .
QR factorization decomposes matrix kB to a product of an orthogonal matrix kQ and an upper triangular
matrix kR and detailed description of QR factorization methods can also be seen in Ref. [35].
Step 4: Since kQ is orthogonal, the matrix ,kT2Q has the following property.
, ,2, 2, , ,[ ] [ ]k kT T
k k k k k
= = =
1 11 2
R RQ B Q Q Q 0 I 0
0 0 (9)
Multiply the linearized constraint equations shown in Equation (7) with matrix ,kT2Q and obtain constraint
equations with only measured parameter as follows.
,k k k= T2M x Q C (10)
Where ,k k k= T2M Q A
Step 5: Solve the data reconciliation problem with constraints shown in Equation (10) through the Lagrange
multiplier method. Then we can update the estimates for the reconciled valuesx̂ .
( ),ˆ Tk k k k k= −∑ −T
2x x M K Q A x C (11)
Where ( ) 1Tk k k
−= ∑K M M
Step 6: Since matrix kQ is orthogonal, it also has the propertyTk k =Q Q I . Multiplying Equation (7) with
matrix TkQ , we can obtain:
ˆT Tk k k k k+ =Q A x R y Q C (12)
Using Equation (8), we can express matrix TkQ and kR as block matrix and obtain:
,1, 1,
2, 2,
ˆT T
kk kk kT T
k k
+ =
1RQ QA x y C
0Q Q (13)
Then we can obtain updated estimates of the unmeasured variablesy .
( )11, 1, ˆT
k k k k−= − −y R Q A x C (14)
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Step 7: Stop if the new estimates are not significantly different from those obtained in the preceding
iteration. Otherwise, repeat the procedure from Step 2 with the new estimates for x̂ and y .
( )11, 1, ˆT
k k k k−= − −y R Q A x C (15)
After the iterative algorithm converges, evaluate the Jacobian and other related matrices with the converged
estimates ̂x for reconciled data and y for unmeasured data, and the relationship between the reconciled
values x̂ and measured values x can be expressed in the following linearized form:
( )ˆ T= −∑ − T2x x M K Mx Q C (16)
Calculated values y for the unmeasured parameters obtained through data reconciliation can also be
expressed as follows:
( )11 1 ˆT−= − −y R Q Ax C (17)
3.2. Gross error detection
In this work, the global test is carried out after data reconciliation to check whether there exist gross errors in
the measured data, so as to validate the data reconciliation calculation. The global test uses the test statistic
given by [6]:
( ) ( )TTRϕ = = − −T T
2 2r Kr Mx Q C K Mx Q C (18)
When there exist no gross errors in the measured data, Rϕ follows a chi-squared distribution with r
degrees of freedom, where r is the rank of the matrix M . Then the test criterion is usually chosen as 21 ,rαχ − ,
which is the critical value of chi-squared distribution at the chosen α level of significance. After data
reconciliation, if the inequality 21 ,R rαϕ χ −> is satisfied, it indicates that there exist gross errors. The choice of
the test criterion ensures that when there exist no gross errors, the possibility of 21 ,R rαϕ χ −> is less than or
equal to the α level of significance. It can be proven that Rϕ is equal to the minimum objective function
value of the data reconciliation problem [6].
3.3. Uncertainty evaluation for data reconciliation results
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In this work, the measurement errors caused by measuring instruments are independent and the uncertainty
of the reconciled value ̂x for measured parameters and calculated value y for unmeasured parameters can
be evaluated according to the uncertainty combination method for uncorrelated input [32]. As a result,
according to Equation (16), Equation (17), the uncertainties of the reconciled value x̂ and calculated value
y can be evaluated as follows:
( ) ( ) ( )2
2 2 2,
1 1
ˆˆ
n ni
i j i j jj jj
xx x G x
xσ σ σ
= =
∂= = ∂ ∑ ∑ (19)
( ) ( ) ( )2
2 2 2,
1 1
n ni
i j i j jj jj
yy x H x
xσ σ σ
= =
∂= = ∂ ∑ ∑ (20)
Where ,i jG is the element of matrix ( )T= −∑G I M KM ; ,i jH is the element of matrix
( )11 1
T T−= − −∑H R Q A I M KM .
Matrix G and H are obtained through Equation (16) and Equation (17). The partial derivatives ,i jG
and ,i jH are called sensitivity coefficients in uncertainty evaluation, because ,i jG describe how the
reconciled value ̂ ix varies with changes in the measured value jx , and ,i jH describe how the calculated
value iy varies with changes in the measured value jx .
We further introduce the impact factors ˆ ,ix je and ,iy je as follows:
( )( )
2 2,
ˆ , 2 ˆi
i j j
x ji
G xe
x
σσ
= (21)
( )( )
2 2,
, 2i
i j j
y ji
H xe
y
σσ
= (22)
Impact factors ˆ ,ix je and ,iy je describe the contribution of measured parameters to the uncertainty of the
reconciled value ̂ ix and the estimated value iy , respectively.
4. Results and discussion
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In this section, data reconciliation problem was formulated based on the measured data, measurement
uncertainties and constraint equations described in Section 2 and solved with the method described in Section
3. Results and discussions are as follows.
4.1. Gross error detection
In this subsection, we use the global test method to validate the data reconciliation calculation. For the data
reconciliation problem formulated in this work, the rank of matrix M is two. As a result, the test criterion for
the global test equals to21 5%,2 6.0χ − = , with a 5 % level of significance.
After solving the data reconciliation problem, we obtained the gross error test statistic 0.5Rϕ = . Since the
test statistic is smaller than the test criterion, this global test result indicates that no gross error is detected in
the on-line measured data and the data reconciliation calculation is validated.
4.2. Uncertainty comparisons for the measured and reconciled primary flow rates
After data reconciliation and uncertainty evaluation, the values and uncertainties of the reconciled primary
flow rates, i.e., MFFW, MFWP and MCW, is obtained and shown in Table 4 together with their measured
value and uncertainties.
Table 4 Comparisons of the measured and reconciled primary flow rates
From Table 4, it can be seen that the three measured primary flow rates are corrected through data
reconciliation to obtain the reconciled data. And for the measured primary flow rates with higher uncertainty,
e.g. MFFW, the correction value is also bigger.
Table 4 also shows that, although the measured primary flow rates have different uncertainties, the
uncertainties for the three reconciled values are smaller compared to those for the measured values. For MFFW,
MFWP and MCW, the uncertainties for the measured values are reduced by 72.1 %, 39.4 % and 21.4 %. As a
result, data reconciliation has efficiently reduced the uncertainty for the estimated value of primary flow rates.
4.3. Effect of data reconciliation for reducing heat rate uncertainty
4.3.1. Heat rate uncertainty reducing effect at the design load
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In this subsection, we firstly evaluate the effect of data reconciliation for reducing heat rate uncertainty,
when the power generation unit operates at the design load. For convenience, we introduce three base cases
called Case A, Case B and Case C. In the three base cases, only one of the MFFW, MFWP or MCW
measurements serves as primary flow measurement for heat balance analysis and corresponding steam turbine
heat rate calculation.
For comparison, we further introduce Case D. In this case, only MCW serves as primary flow measurements
for heat balance analysis, but its measuring instrument accuracy is assumed to be changed from 0.8 % to 0.2 %
and leads to a reduction of measurement uncertainty for MCW from 4.8 kg/s to 1.0 kg/s. This case corresponds
to changing the MCW measuring instrument from low accuracy on-line measuring instrument to high accuracy
thermal acceptance measuring instrument [36].
In Case E, data reconciliation was carried out and the three existing flow measurements including MFFW,
MFWP and MCW, all serve as primary flow measurements for heat balance analysis and corresponding heat
rate calculation.
The calculated steam turbine heat rates and corresponding uncertainties for these five cases are shown in
Figure 2.
Figure 2. Comparisons of heat rates and uncertainties at design load
From Figure 2, we can see that among the base cases, Case C leads to the smallest heat rate uncertainty of
70.3 kJ / kWh. This is mainly because that the measuring instrument for MCW is more accurate than those for
MFFW and MFWP. The measurement uncertainty for MCW is only 0.8 % of its measured value, however for
MFFW and MFWP the measurement uncertainties are 2.2 % and 1.0 % of their measured values.
In Case D, the uncertainty of heat rate can be reduced to be 38.2 kJ / kWh, which is 45.8 % smaller than the
best result of 70.3 kJ / kWh for the three base cases. However, this case only happens in the thermal
acceptance test, when a special high accuracy flow measuring instrument is used.
In Case E, the uncertainty of heat rate is reduced to be 57.7 kJ / kWh, which is 18.0 % smaller than the best
result of 70.3 kJ / kWh for the three base cases. Compared to the thermal acceptance test Case D, data
reconciliation has also reduced the uncertainty of heat rate obviously. What’s more, this result is obtained
based on existing measurements, without any extra measuring instrument investment. As a result, data
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reconciliation is an efficient and inexpensive method for reducing the uncertainty of steam turbine heat rate
and is beneficial for steam turbine on-line performance monitoring systems.
4.3.2. Heat rate uncertainty reducing effect at off-design loads
We further evaluate the effect of data reconciliation for reducing heat rate uncertainty, when the power
generation unit operates at 100 %, 75 % and 50 % load rates. As an example, we mainly compare the heat rate
uncertainties in Case C, which takes MCW as primary flow measurement and has the smallest heat rate
uncertainty for the base cases, and Case E, the data reconciliation case. The calculated steam turbine heat rates
and corresponding uncertainties with different unit load rates are shown in Figure 3.
Figure 3. Comparisons of heat rates and uncertainties at off-design load
From Figure 3, we can see that as the unit load decreases from its design value, the steam turbine heat rate is
increased, which indicates decreased steam turbine efficiencies at off-design load. At the same, heat rate
uncertainties for both Case C and Case E, are obviously increased, due to the increase of the heat rate values.
However, when the unit operates at 100 %, 75 % and 50 % load rates, the heat rate uncertainties in Case E are
smaller than those in Case C by 17.9 %, 18.7 % and 18.8 %. This result indicates that data reconciliation
method can reduce the heat rate uncertainties efficiently, even when the unit operates at off-design loads.
4.4. Impact of data reconciliation on heat rate sensitivity
In this subsection, we study the impact of data reconciliation on heat rate sensitivity coefficients with respect
to different measured parameters. Heat rate sensitivity coefficients describe how the calculated heat rate value
varies with changes in the measured values and thus indicate the sensitivity of heat rate results with respect to
the measurement errors. As an example, we mainly compare the sensitivity coefficients with respect to
different measured parameters in Case C and Case E with the unit operating at design load. Comparison results
are shown in Table 5.
Table 5 Comparisons of heat rate sensitivity coefficients
Table 5 has mainly included the measured parameters, which have significantly contribute to the heat rate
uncertainties and have impact factors for the heat rate uncertainty larger than 1 %. From Table 5, we can see
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that the sensitivity coefficients of the flow measurements are significantly influenced by data reconciliation. In
Case C, MCW is the only primary flow measurement, with a heat rate sensitivity coefficient of 12.5. However,
in Case E, because the three measurements of MFFW, MFWP and MCW are all taken as primary flow
measurements through data reconciliation, the sensitivity coefficient of MCW is reduced to be only 7.4. This
result indicates that the impact of the measurements MCW on the calculated heat rate results has been
significant reduced through data reconciliation. For example, if a measurement error causes the measured
value of MCW to be 1 % smaller than its measured value of 628.3 kg/s, the calculated heat rate will be reduced
by 78.5 kJ/kWh in Case C, which is 1.0 % of its original value 7832.5 kJ/kWh. However, this measurement
error will only reduce the calculated heat rate by 46.4 kJ/kWh in Case E, which is only 0.6 % of its original
value 7773.2 kJ/kWh.
From Table 5, we can also see that besides the primary flow measurements, the absolute values of heat rate
sensitivity coefficients with respect to the temperature of feed water at feed water pump outlet (TFWP) and the
temperature of condensate water at deaerator inlet (TCW) have also been reduced by 41.3 % and 41.3 %. This
result indicates the impact of these two measurements on the calculated heat rate have also been significantly
reduced through data reconciliation.
From this study, we can see that data reconciliation has significantly reduced the heat rate sensitivity
coefficients with respect to the related measurements, thus can reduce the impact of measurement errors
corresponding to these measurements on the calculated heat rate results and finally reduce the heat rate
uncertainties.
5. Conclusions
In this work, data reconciliation method was applied for steam turbine on-line heat balance analysis and
performance monitoring in a real-life 1000 MW ultra-supercritical coal-fired power generation unit. The global
test method was used to detect gross errors in the measured data thus validate the data reconciliation
calculation. Research results are as follows.
Through data reconciliation, the measured values of redundant primary flow measurements were corrected
according to their measurement uncertainties to obtain reconciled data. At the same time, the uncertainties of
the measured values of the outlet flow rate of the #1 feed water heater (MFFW), the outlet flow rate of the feed
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water pump (MFWP) and the inlet flow rate of condensate water in the deaerator (MCW) were reduced by
72.1 %, 39.4 % and 21.4 %. As a result, data reconciliation has efficiently reduced the uncertainties of the
estimated value of the primary flow rates.
Case study results also show that when the unit operates at 100 %, 75% and 50 % of its design load, the
uncertainties of the key system performance indicator, heat rate, can be reduced by 17.9 %, 18.8 % and 18.8 %.
It illustrates that data reconciliation can be an efficient method for reducing the uncertainty of steam turbine
heat rate at various unit load rates and is applicable in steam turbine on-line performance monitoring systems
to give heat rate monitoring results with smaller uncertainties.
Finally, we studied the impact of data reconciliation on the heat rate sensitivity coefficients with respect to
different measured parameters. Case study results show that the heat rate sensitivity coefficient with respect to
the primary flow measurement of MCW, the temperature of feed water at feed water pump outlet (TFWP) and
the temperature of condensate water at deaerator inlet (TCW) have been reduced by 40.9 %, 41.3 % and
41.3 % through data reconciliation. This result indicates that data reconciliation has significantly reduced the
heat rate sensitivity coefficients with respect to these measurements, thus can reduce the impact of
measurement errors corresponding to these measurements on the calculated heat rate results and finally reduce
the heat rate uncertainties.
ACKNOWLEDGMENT
The authors gratefully acknowledge the financial support from National Natural Science Foundation of
China (project No. 51106080), from the IRSES ESE Project of FP7 (contract No:
PIRSES-GA-2011-294987), and from BP company in the scope of the Phase II Collaboration between BP and
Tsinghua University.
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NOMENCLATURE
kA Jacobian matrix of the constraint equations with respect to reconciled parameter at
the kth iteration
kB Jacobian matrix of the constraint equations with respect to unmeasured parameter at
the kth iteration
kC Constant term matrix in the linearized constraint equations at the kth iteration
ˆ ,ix je Impact factor of the jth measured parameter for the ith reconciled parameter
,iy je Impact factor of the jth measured parameter for the ith unmeasured parameter
1m×f System constraint equations
G Sensitivity coefficient matrix of the measured parameters for the reconciled data
,i jG Sensitivity coefficient of the jth measured parameter for the ith reconciled data
H Sensitivity coefficient matrix of the measured parameters for the unmeasured
parameter
,i jH Sensitivity coefficient of the jth measured parameter for the ith unmeasured parameter
kK Intermediate matrix in the data reconciliation algorithm
kM Coefficient matrix for constraint equations with only measured parameters
Ns Number of sensors for the same parameter
Nt Length of the time window
kQ Matrix Q obtained through QR factorization of matrix kB
,k1Q First p basis vector in matrix kQ
2,kQ Second m p− basis vector in matrix kQ
kR Matrix R obtained through QR factorization of matrix kB
1,kR Nonsingular p p× upper triangular matrix in matrix kR
r Vector of constraint equation residuals
r System redundancy
SSCx Steady state criteria parameter
1n×x Vector of measured values for measured parameters
ix Measured value of the ith measured parameter
1ˆ n×x Vector of reconciled values for measured parameters
ˆix Reconciled value of the ith measured parameter
1p×y Vector of calculated value for the unmeasured parameters
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iy Calculated value of the ith unmeasured parameter
n n×∑ Covariance matrix describing measurements uncertainties
Greek letters
α Level of significance for the global test
iξ Permissible error of the ith measuring instrument
( )ixσ Uncertainty of the ith measurement with single sensor
( )ixσ Uncertainty of the ith measurement with multiple sensors
ϕ Objective function of data reconciliation problem
minϕ Gross error test statistic
21 ,rαχ −
Critical value of chi-squared distribution at the chosen level of significance
Superscripts
T Matrix transposition
-1 Matrix inversion
Subscripts
k Iteration step in the data reconciliation solution algorithm
i Index of parameters
j Index of parameters
min Minimum value
m Number of constraint equations
n Number of measured parameters
p Number of unmeasured parameters
SSC Steady state criteria
s Sensor
sat,w Saturated water
t Time
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Table 1 Values and uncertainties of the measured parameters
Parameter Description and location Unit Value Uncertainty
Power Power produce by generator MW 1003.8 ±2.4
MFFW Outlet flow rate of the #1 feed water heater kg/s 803.5 ±18.3
MFWP Outlet flow rate of the feed water pump kg/s 808.9 ±8.4
MCW Inlet flow rate of condensate water in the deaerator kg/s 628.3 ±4.8
MRHS Flow rate of reheater spray water kg/s 3.9 ±0.5
MLK1 Flow rate of valve stem leakage kg/s 0.7 ±0.7
MLK2 Flow rate of high pressure cylinder front side gland sealing steam
leakage
kg/s 7.2 ±7.2
MLK4 Flow rate of intermediate pressure cylinder cooling steam kg/s 2.0 ±2.0
MLK3 Flow rate of high pressure cylinder back side gland sealing steam
leakage
kg/s 2.9 ±2.9
PMS Pressure of main steam at boiler outlet bar 254.6 ±0.4
PHR Pressure of hot reheat steam at boiler outlet bar 44.2 ±0.1
PCR Pressure of cold reheat steam at boiler inlet bar 47.5 ±0.2
PES1 Pressure of #1 feed water heater extracted steam bar 85.2 ±0.3
PES2 Pressure of #2 feed water heater extracted steam bar 47.5 ±0.2
PES3 Pressure of #3 feed water heater extracted steam bar 23.2 ±0.1
PES4 Pressure of deaerator extracted steam bar 11.4 ±0.1
PFWP Pressure of feed water at feed water pump outlet bar 302.8 ±1.0
PRHS Pressure of reheater spray water bar 118.0 ±0.8
PCW Pressure of condensate water at deaerator inlet bar 14.4 ±0.2
TMS Temperature of main steam at Boiler outlet °C 589.4 ±2.4
THR Temperature of hot reheat steam at boiler outlet °C 597.2 ±2.4
TCR Temperature of cold reheat steam at boiler inlet °C 338.9 ±1.0
TES1 Temperature of #1 feed water heater extracted steam °C 420.7 ±1.5
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TES2 Temperature of #2 feed water heater extracted steam °C 338.9 ±1.0
TES3 Temperature of #3 feed water heater extracted steam °C 507.1 ±1.5
TES4 Temperature of deaerator extracted steam °C 392.8 ±1.2
TFFW Temperature of feed water at #1 feed water heater outlet °C 299.2 ±0.9
TDW1 Temperature of #1 feed water heater drainage water °C 265.4 ±0.9
TFW2 Temperature of feed water at #2 feed water heater outlet °C 258.3 ±0.8
TDW2 Temperature of #2 feed water heater drainage water °C 227.5 ±0.8
TFW3 Temperature of feed water at #3 feed water heater outlet °C 220.1 ±0.8
TDW3 Temperature of #3 feed water heater drainage water °C 197.6 ±0.8
TFWP Temperature of feed water at feed water pump outlet °C 190.5 ±0.8
TRHS Temperature of reheater spray water °C 184.3 ±0.8
TCW Temperature of condensate water at deaerator inlet °C 159.9 ±0.7
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Table 2 System constraint equations
Equipment Constraint equations Equation type
Boiler
MMS-MFFW=0 Mass balance
MHR-MCR-MRHS=0 Mass balance
HP1 MMS-MHP2-MES1-MLK1-MLK2-MLK4=0 Mass balance
HP2 MHP2-MCR-MES2-MLK3=0 Mass balance
HPFW1
MES1-MDW1=0 Mass balance
MFFW·h(PFWP,TFW2)+MES1·h(PES1,TES1)-MFFW·h(PFWP,TFW)-
MDW1·h(PES1,TDW1)=0
Energy balance
HPFW2
MES2+MDW1-MDW2=0 Mass balance
MFFW·h(PFWP,TFW3)+MES2·h(PES2,TES2)+MDW1·h(PES1,TDW1)-
MFFW·h(PFWP,TFW2)- MDW2·h(PES2,TDW2)=0
Energy balance
HPFW3
MES3+MDW2-MDW3=0 Mass balance
MFFW·h(PFWP,TFWP)+MES3·h(PES3,TES3)+MDW2·h(PES2,TDW2)-
MFFW·h(PFWP,TFW3)- MDW3·h(PES3,TDW3)=0
Energy balance
FWP
MFFW-MFWP=0 Mass balance
MDA-MFWP-MRHS=0 Mass balance
DA
MES4+MCW+MDW3-MDA=0 Mass balance
MES4·h(PES4,TES4)+ MCW·h(PCW,TCW)+
MDW3·h(PES3,TDW3)-MDA·hsat,w(PES4)=0
Energy balance
Heat rate
HR-[ MMS·h(PMS,TMS)+MHR·h(PHR,THR)-MFFW·h(PFWP,TFW)-
MCR·h(PCR,TCR)-MRHS·h(PRHS,TRHS) ]/Power=0
Heat rate calculation
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Table 3 Descriptions of the unmeasured parameters
Parameter Description and location
MMS Flow rate of main steam at boiler outlet
MHR Flow rate of hot reheat steam at boiler outlet
MHP2 Flow rate of steam at #2 stage group of high pressure cylinder inlet
MCR Flow rate of cold reheat steam at boiler inlet
MES1 Flow rate of #1 feed water heater extracted steam
MES2 Flow rate of #2 feed water heater extracted steam
MES3 Flow rate of #3 feed water heater extracted steam
MES4 Flow rate of deaerator extracted steam
MDW1 Flow rate of #1 feed water heater drainage water
MDW2 Flow rate of #2 feed water heater drainage water
MDW3 Flow rate of #3 feed water heater drainage water
MDA Flow rate of deaerator outlet water
HR Heat rate of steam turbine
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Table 4 Comparisons of the measured and reconciled primary flow rates
Parameter Unit Measured data Reconciled
data
Correction
data
MFFW
Value kg/s 803.5 816.4 12.9
Uncertainty kg/s ±18.3 ±5.1
MFWP
Value kg/s 808.9 816.4 7.5
Uncertainty kg/s ±8.4 ±5.1
MCW
Value kg/s 628.3 623.9 -4.4
Uncertainty kg/s ±4.8 ±3.8
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Table 5 Comparisons of heat rate sensitivity coefficients
Parameters Unit Measured value Measurement
uncertainty
Sensitivity
coefficient
Impact factor
(%)
Case C
MCW kg/s 628.3 4.8 12.5 73.6
TMS °C 589.4 2.4 8.9 9.0
Power MW 1003.8 2.4 -7.8 7.0
THR °C 597.2 2.4 5.6 3.5
TFWP °C 190.5 0.8 -15.8 3.0
TCW °C 159.9 0.7 -13.3 1.5
Case E
MCW kg/s 628.3 4.8 7.4 38.2
MFWP kg/s 808.9 8.4 3.2 21.8
TMS °C 590.4 2.4 8.9 13.1
Power MW 1003.8 2.4 -7.7 10.3
THR °C 597.2 2.4 5.6 5.2
MFFW kg/s 803.5 18.3 0.7 4.6
TFWP °C 190.5 0.8 -9.3 1.5
TCR °C 338.9 1.0 -6.4 1.3
TFFW °C 299.2 0.9 -6.8 1.1
TCW °C 159.9 0.6 -7.8 0.7
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Highlights
� Data reconciliation application to steam turbine on-line performance monitoring systems.
� Uncertainties reduced for primary flow measurements through data reconciliation.
� Uncertainties reduced for steam turbine heat rate with the unit operating at design and
off-deisgn load rates.
� Impact of data reconciliation on heat rate sensitivity coefficients analyzed.
