**4. Data reconciliation**

#### **4.1. The basic problems of data reconciliation**

Just as mentioned in section 3, the data reconciliation problem is expressed as the solution of the constrained least-squares problems.However, the assumptions, the constraint equations, and the weighted parameter matrix have to be discussed.

#### *4.1.1. On the assumptions of data reconciliation in the present application*

For the application of the steam network, there are four latent assumptions. They are:

1. The process is at the steady state or approximately steady state.

Suppose that the electric valves are fixed at a certain position, and the mass flow rate in each steam has been close to a constant for a period of time. So the constraint equation (54) can be written out on the basis of the balance for all the nodes (including real nodes and pseudonode) and the solution of the problem can be in accordance with the actual state.

	- 2. The measurement data are serial uncorrelated. The assumption makes it easier to estimate the deviations of the measurements. Although this is usually not the real case, the serial data can be processed to reduce the confliction [17].
	- 3. The Gross Errors have been detected and removed. If there's gross error in the measurement data, the procedure of data reconciliation will propagate the errors.
	- 4. The constrained equations are linear. The style of equation (54) is linear; however the true constrained equation will be related with many other environmental variables and obviously not linear. So the assumption is to be approximately satisfied. Only these four assumptions are nearly satisfied, can the solution of the problem (equation (58)) be closer to the true value.

#### *4.1.2. On the constraint equation*

Need to note equations (51) and (52) are based on the assumptions of steady state and linear constraints, no pipeline leakage or condensate water loss. But it cannot avoid the condition of the pipeline leakage and condensate water. So the equations (51)(52)should be written as:

$$\begin{cases} X1 - X11 - X12 - \delta 1 = 0 \\ X2 - X21 - X22 - \delta 2 = 0 \\ X3 - X31 - X32 - \delta 3 = 0 \\ X4 - X41 - X42 - \delta 4 = 0 \\ X5 + X6 - X51 - X52 - \delta 5 = 0 \end{cases} \tag{66}$$

Data Processing Approaches for the Measurements of Steam Pipe Networks in Iron and Steel Enterprises 267

2

<sup>1</sup> ( ) ,(1 8) <sup>1</sup>

δ<sup>−</sup> = − − (71)

≈ = − ≤≤ <sup>−</sup> (72)

*Y X X Rand* =+ ×× 10% . (18,1) (73)

ˆ \* 1 ( )( ) *T T X Y QA AQA AY*

The selection of Q directly influences the result of the data reconciliation. In theory Q is recommended as the equation (57). However, the deviations are usually unknown or may vary as the time of instruments being used getting long. The deviation of each measurement

1

If a small number of high-precision instrument are applied,, the corresponding elements in the matrix of Q with smaller value, then the quality of data will be greatly improved. Though some literatures [18] [19] recommended the methods to determine or adjust the

The simulation results of data reconciliation for the present application are shown in table 3.

The standard deviations are set to be 0.5 percent of the true values. In the simulation, the pipe network loss is not considered. As shown in table 3, most of the rectified data are much

X01 X02 X1 X11 X12 X2 X21 X22 X3

X31 X32 X4 X41 X42 X5 X51 X52 X6

closer to the true values. The simulation testifies the efficiency of data reconciliation.

true value 5 5 50 25 25 60 30 30 30

true value 15 15 40 20 20 20 25 25 30

measured value 15.735 15.668 42.585 21.419 21.509 20.552 26.699 26.638 32.488

measured value 5.347 5.159 54.751 25.086 26.097 62.289 32.297 32.386 30.561 variance 0.0006 0.0006 0.0625 0.0156 0.0156 0.0900 0.0225 0.0225 0.0225 rectified value 5.347 5.150 52.228 25.090 25.514 62.178 31.946 31.730 31.174

=

*N i i ij i j s XX i*

*N*

Where, *Xij* is the *<sup>j</sup>* th sample of the variable *Xi* , and *Xi* is the average of *Xi* .

matrix Q, the methods or theories for general application is not available.

The solution to the least-squares problem of reconciliation is:

*4.1.3. On selection of the weighted parameter matrix Q* 

σ

The measured data is presumed according to (73)

*4.1.4. Simulation results* 

can be estimated by the standard deviation of the sample data.

$$X5 + X6 + X4 + X3 - X1 - X2 - X01 - X02 - \delta 0 = 0 \tag{67}$$

Here:

$$
\delta \mathbf{i} = f\_i(\mathbf{r}, D\_{i'}l\_{i'}T\_{i'}P\_{i'}X\_i) \qquad \text{( $\mathbf{i} = 0, 1, \dots, 5$ )}\tag{68}
$$

It represents the vector for the condensate water and leakage loss amount of each constraint equation. Each element has relations with the environmental temperature τ, pipe diameter D, pipe length l, steam temperature T, pressure P and the flow rates of main pipes.

It's difficult to set up the mathematical models of these loss amounts. However, by the first order of Taylor Expansion at the point of ( 0 00 0 0 0 , ,, , , *DlTPX ii i i i* τ):

$$\begin{split} \delta \mathbf{i} &= f\_i(\mathbf{r}\_0, D\_{i0}, l\_{i0}, T\_{i0}, P\_{i0}, X\_{i0}) + \frac{\partial f\_i}{\partial \mathbf{r}\_0}(\mathbf{r} - \mathbf{r}\_0) + \frac{\partial f\_i}{\partial T\_0}(T - T\_{i0}) + \frac{\partial f\_i}{\partial P\_{i0}}(P\_i - P\_{i0}) + \frac{\partial f\_i}{\partial X\_{i0}}(X\_i - X\_{i0}) \\ &= \delta\_{i0} + k\_{i\mathbf{r}}(\mathbf{r} - \mathbf{r}\_0) + k\_{i\mathbf{T}}(T - T\_0) + k\_{i\mathbf{P}}(P\_i - P\_{i0}) + k\_{i\mathbf{r}}(X\_i - X\_{i0}) \end{split} \tag{69}$$

The constants in equation (69) can be determined by the method of multiple linear regression with a certain number of history data. As the constraint equation changed into:

$$AX = \mathcal{S} \tag{70}$$

The solution to the least-squares problem of reconciliation is:

$$\hat{X}^\* = Y - QA^T (AQA^T)^{-1} (AY - \delta) \tag{71}$$

#### *4.1.3. On selection of the weighted parameter matrix Q*

The selection of Q directly influences the result of the data reconciliation. In theory Q is recommended as the equation (57). However, the deviations are usually unknown or may vary as the time of instruments being used getting long. The deviation of each measurement can be estimated by the standard deviation of the sample data.

$$\sigma\_i = s\_i = \sqrt{\frac{1}{N-1} \sum\_{j=1}^{N} (X\_{ij} - \overline{X}\_i)^2}, \text{( $1 \le i \le 8$ )}\tag{72}$$

Where, *Xij* is the *<sup>j</sup>* th sample of the variable *Xi* , and *Xi* is the average of *Xi* .

If a small number of high-precision instrument are applied,, the corresponding elements in the matrix of Q with smaller value, then the quality of data will be greatly improved. Though some literatures [18] [19] recommended the methods to determine or adjust the matrix Q, the methods or theories for general application is not available.

#### *4.1.4. Simulation results*

266 Energy Efficiency – The Innovative Ways for Smart Energy, the Future Towards Modern Utilities

the serial data can be processed to reduce the confliction [17].

(equation (58)) be closer to the true value.

 (, ,, , , ) *i ii i i i* δ

*4.1.2. On the constraint equation* 

Here:

δ

τ

 ττ

 τ

δ

2. The measurement data are serial uncorrelated. The assumption makes it easier to estimate the deviations of the measurements. Although this is usually not the real case,

3. The Gross Errors have been detected and removed. If there's gross error in the measurement data, the procedure of data reconciliation will propagate the errors.

4. The constrained equations are linear. The style of equation (54) is linear; however the true constrained equation will be related with many other environmental variables and

Need to note equations (51) and (52) are based on the assumptions of steady state and linear constraints, no pipeline leakage or condensate water loss. But it cannot avoid the condition of the pipeline leakage and condensate water. So the equations (51)(52)should be written as:

> 1 11 12 1 0 2 21 22 2 0 3 31 32 3 0 4 41 42 4 0 5 6 51 52 5 0

*XXX XXXX X* 5 6 4 3 1 2 01 02 0 0 + + + −− − − −=

It represents the vector for the condensate water and leakage loss amount of each constraint equation. Each element has relations with the environmental temperature τ, pipe diameter

It's difficult to set up the mathematical models of these loss amounts. However, by the first

0 00 0 0 0 0 0 0 0

(, ,, , , ) ( ) ( ) ( ) ( )

*i ii i i i i ii i i*

The constants in equation (69) can be determined by the method of multiple linear regression with a certain number of history data. As the constraint equation changed into:

> *AX* = δ

):

∂∂ ∂ ∂

00 0 0

*TP X*

*ii i i*

D, pipe length l, steam temperature T, pressure P and the flow rates of main pipes.

τ

*ff f f i f DlTPX TT P P X X*

∂∂ ∂ ∂ <sup>=</sup> + −+ − + − + −

− − −= − − −=

 − − −= − − −= + − − −=

δ

δ

δ

δ

δ

δ

*i i*

(70)

*i f DlTPX* = ( ) i 0,1, ,5 = … (68)

(66)

(67)

(69)

*XX X XX X XX X XX X XXX X*

 τ

order of Taylor Expansion at the point of ( 0 00 0 0 0 , ,, , , *DlTPX ii i i i*

00 0 0 0

*k kTT k P P kX X*

τ

*i i iT iP i i ix i i*

=+ −+ −+ − + −

( )( )( )( )

τ τ

Only these four assumptions are nearly satisfied, can the solution of the problem

obviously not linear. So the assumption is to be approximately satisfied.

The simulation results of data reconciliation for the present application are shown in table 3. The measured data is presumed according to (73)

$$Y = X + 10\% \times X .\times \text{Rand} (18, 1) \tag{73}$$

The standard deviations are set to be 0.5 percent of the true values. In the simulation, the pipe network loss is not considered. As shown in table 3, most of the rectified data are much closer to the true values. The simulation testifies the efficiency of data reconciliation.



Data Processing Approaches for the Measurements of Steam Pipe Networks in Iron and Steel Enterprises 269

This work is supported by the Knowledge Innovation Project of Chinese Academy of Science (No.KGCX2-EW-104-3), National Nature Science Foundation of China (No.61064013) and Natural Science Foundation of Jiangxi Province,China (No.

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**Table 3.** The Result of Data Rectification
