There are three basic algorithms for subspace identification, namely MOESP (Multivariable out-put Error State space), N4SID (Numerical algorithm for Subspace State space System Identification) and CVA (Canonical Variate Analysis), and they have different features. In this paper, N4SID [10] [11] open-loop subspace identification method is used to identify the supercritical thermal power unit system model. Subspace identification method adopts the form of state space to describe the system to be identified. Hankle matrices are constructed and calculated according to the input and output data. The core idea is to decompose the system into a deterministic part and a stochastic part by using the nature of oblique projection so as to eliminate the noise [12].

Consider the following linear time-invariant system:

x ( k + 1 ) = A x ( k ) + B u ( k ) + w ( k ) (1)

y ( k ) = C x ( k ) + D u ( k ) + v ( k ) (2)

where x ( k ) ∈ R n is the system state at time k, u ( k ) ∈ R p , y ( k ) ∈ R m are the input and output signals at time k respectively. w ( k ) and v ( k ) indicate the process and measurement noises respectively. Without the noise terms, a deterministic system is described.

From (1) and (2) while considering the deterministic case, one can easily get

[ y ( k ) y ( k + 1 ) ⋮ y ( k + i ) ] = [ C C A ⋮ C A i ] x ( k ) + [ D 0 0 0 C B D 0 0 ⋮ ⋮ ⋱ 0 C A i − 1 B C A i − 2 B ⋯ D ] [ u ( k ) u ( k + 1 ) ⋮ u ( k + i ) ] (3)

According to the standard expression of subspace identification, Equation (3) can be decomposed into past and future parts, namely

X f = A i X p + Δ i U p + Δ i s E p (4)

Y p = Γ i X p + H i d U p + H i s E p (5)

Y f = Γ i X f + H i d U f + H i s E f (6)

where

U p = ( u ( 0 ) u ( 1 ) ⋯ u ( j − 1 ) u ( 1 ) u ( 2 ) ⋯ u ( j ) ⋮ ⋮ ⋱ ⋮ u ( i − 1 ) u ( i ) ⋯ u ( i + j − 2 ) ) ∈ R p i × j

U f = ( u ( i ) u ( i + 1 ) ⋯ u ( i + j − 1 ) u ( i + 1 ) u ( i + 2 ) ⋯ u ( i + j ) ⋮ ⋮ ⋱ ⋮ u ( 2 i − 1 ) u ( 2 i ) ⋯ u ( 2 i + j − 2 ) ) ∈ R p i × j

Y p and Y f are constructed on the same way as U P and Y f .

The observability matrix Γ i , the lower triangle Toeplitz matrix H i d and other matrices are defined as follows:

Γ i = [ C C A C A 2 ⋮ C A i ] ,   H i d = [ D C B D ⋮ ⋱ ⋱ C A i − 2 B C A i − 3 B ⋯ D ] ,   H i s = [ 0 C 0 ⋮ ⋱ ⋱ C A i − 2 C A i − 3 ⋯ 0 ]

Δ i = [ A i − 1 B A i − 2 B ⋯ B ] ,     Δ i s = [ A i − 1 A i − 2 ⋯ I ]

First, we project Y f onto the orthogonal complement subspace U f ⊥ of U f , and obtain

Y f / U f ⊥ = Γ i X f / U f ⊥ + H i d U f / U f ⊥ + H i s E f / U f ⊥ (7)

Multiply the weighting matrices W 1 and W 2 on both sides of Equation (7), and through the correlation and orthogonality, the above formula can be simplified as

W 1 Y f / U f ⊥ W 2 = W 1 Γ i X f / U f ⊥ W 2 + W 1 H i s E f W 2 (8)

The weighting matrices W 1 and W 2 should meet the following conditions:

1) r a n k ( W 1 Γ i ) = r a n k ( Γ i ) ;

2) r a n k ( X f / U f ⊥ W 2 ) = r a n k ( X f ) ;

3) W 1 H i s E f W 2 = 0 .

The third condition requires that W 2 is independent of E f , so Equation (8) may be further simplified as

Φ = W 1 Y f / U f ⊥ W 2 = W 1 Γ i X f / U f ⊥ W 2 (9)

Through SVD decomposition, Equation (9) can be denoted as

Φ = ( U 1 U 2 ) ( S 1 0 0 0 ) ( V 1 T V 2 T ) (10)

The order of the system is determined according to the order of matrix S 1 .

Using MATLAB notation, the matrix C can be directly derived from Γ i as follows:

C = Γ i ( 1 : m , : ) (11)

According to the following property

Γ i _ ⋅ A = Γ i ¯ (12)

where Γ i _ is the matrix Γ i deleting the last i rows, and Γ i ¯ is the matrix Γ i deleting the first i rows. By left multiplying Γ i _ † of Equation (12), we can obtain matrix A as follows:

A = Γ i _ † Γ i ¯ (13)

If both sides of Equation (6) multiply Γ i † by left and U f † by right, we can get

Γ i † Y f U f † = Γ i † H i d (14)

Let the left side of the above equation be denoted as P and Γ i † be denoted as L , then

( P 1 P 2 ⋯ P i ) = ( L 1 L 2 ⋯ L i ) × [ D 0 0 0 C B D 0 0 ⋮ ⋮ ⋱ 0 C A i − 2 B C A i − 3 B ⋯ D ] (15)

Equation (15) constitutes a set of overdetermined linear equations, which can be solved by the least square method to obtain matrices B and D .

The main controlled variables of a supercritical thermal power unit are unit load, main steam pressure and main steam temperature. A large number of simulation experiments show that based on the subspace identification method discussed in the previous subsection, the system load and main steam pressure can be identified satisfactorily. However, if the input and output data are used to directly identify the main steam temperature, the effect is poor. A typical set of direct identification results is shown in Figure 1. The data are normalized and the sampling period is 5 s, which applies to all the figures in this paper.

To overcome the above problem in direct identification, we have made a detailed analysis on the process and propose the following indirect identification method. Firstly, the mathematical model between the middle point temperature and the main control inputs is identified by using the fuel feed, the water supply and the opening degree of the steam turbine valve. Then the main steam temperature of the system is obtained by the steam flow at the middle point passing through the multi-stage superheater.

The main steam temperature model obtained by the indirect identification method can meet the requirements of system control. Figure 2 shows the main

steam temperature identifying results obtained by the proposed indirect identification method. All-day operation data of a power plant are used in the system identification.

Taking a thermal power plant in Henan Province as an example, the basic process and results based on the indirect identification method are given below. The model structure is shown in Figure 3.

Firstly, according to the actual operation data of the system, the subsystem from the turbine valve opening, total fuel quantity, and total water supply to the system load and main steam pressure are identified. It is a module with three inputs and two outputs, as shown in Figure 3. The related system matrices are as follows:

A 1 2 = [ 0.7773 − 0.1209 − 0.1516 0.8871 ] B 1 2 = [ − 0.001713 − 0.00007509 − 0.001384 − 0.001267 − 0.0005914 − 0.001289 ] C 1 2 = [ 29.96 − 72.14 40.14 − 77.42 ] D 1 2 = [ 0 0 0 0 0 0 ]

Secondly, the subsystem from the turbine valve opening, total fuel feed, total water supply to the intermediate point temperature is identified. It is a module with three inputs and one output, as shown in Figure 3. The obtained system matrices are as follows:

A 03 = [ 0.9619 − 0.02362 − 0.06153 0.8526 ] B 03 = [ − 0.0009411 − 0.0007184 − 0.0004354 − 0.004808 − 0.002631 − 0.002633 ] C 03 = [ 16.21 − 12.66 ] D 03 = [ 0 0 0 ]

Thirdly, the subsystem from the intermediate point temperature, air-coal ratio, desuperheating water flow to the main steam temperature is identified. It is a module with three inputs and one output in Figure 3. The identified system matrices are as follows:

A 3 = [ 0.8611 − 0.02323 0.1741 0.9902 ] B 3 = [ − 0.0003134 − 0.0006446 − 0.0004444 − 0.0005305 − 0.000952 − 0.0006654 ] C 3 = [ 16.41 20.84 ] D 3 = [ 0 0 0 ]

The above separated models are repeatedly verified by using different operation data, which include almost all the characteristics of the normal operation. Based on the above results, the total system model of the thermal power unit is obtained as follows:

A = [ 0.9862 − 0.0064 0 0 0 0 − 0.01069 0.9911 0 0 0 0 0 0 0.9996 − 0.045 0 0 0 0 0.02414 0.919 0 0 0 0 − 0 .000602 1 .02 e − 05 0.9955 − 0.05659 0 0 0 .00169 − 2 .87 e − 05 0.05911 0.8846 ]

B = [ 0 0.0001118 5.615 e − 05 4.623 e − 06 0 0 0.0001065 4.608 e − 05 2 .23 e − 06 0 0 − 0.0001031 − 0.0001032 − 0 .000118 0 0 − 0 .000166 − 0.0004942 8 .7 e − 05 0 0.0001053 0 0 0 − 9 .55 e − 06 0.0002139 0 0 0 4 .85 e − 05 ]

C = [ 106.5 3.506 0 0 0 0 70.69 68.54 0 0 0 0 0 0 0 0 31.04 − 0.6756 ]

D = [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]

It is seen that the overall identification model of the unit has five inputs and three outputs. Based on the above model, a lot of experiments are carried out. The actual operation data of the unit in different time periods are used to verify the model and very satisfactory results are obtained. Figure 4 shows a set of actual output and the model output.

It is shown that the load deviation is about −10 - 10 MW, the main steam pressure deviation is about −0.2 - 0.2 MPa, and the main steam temperature deviation is about −5˚C - 5˚C. The accuracy completely meets the requirements of the model prediction control method. The effectiveness of the proposed indirect identification is well verified.

In the actual operation process of the system, there are two control strategies to increase or decrease the generation load. One is the “turbine following boiler” strategy. When the generating load changes, the boiler side of the unit will adjust the corresponding control inputs according to the generating demand. By increasing and decreasing the boiler energy storage, the purpose of matching the boiler capacity with the demand of the generating unit can be achieved. The other one is the “boiler following turbine” strategy. The generator unit directly adjust the opening of the steam turbine valve according to the actual power generation demand, which ensures that the generating capacity of the unit will increase or decrease rapidly. The control goal of the boiler side is to maintain the stability of the main steam temperature and main steam pressure in the boiler.

We have adopted the “turbine following boiler” strategy. By adjusting the actual energy storage at the boiler side to adjust the future load change of the system, i.e. using predictive control method, can quickly adjust the input of relevant fuel and auxiliary raw materials. Also, the opening of the turbine valve can be gradually adjusted. Thus the system energy output changes rapidly, and at the same time, the stability of the main steam temperature and the slow change of the main steam pressure in the system can be maintained.

In the case that the thermal power unit needs to increase the power rapidly, set

the load setting value of the power generation as large as possible, so as to show the ultimate capacity of the system to increase the load without violating the relevant constraints. Because the upper and lower bounds of each variable and the upper and lower bounds of the rate of change are limited, especially the constraints on the main steam temperature (±2˚C) and the main steam pressure (±10 MPa), the controller ensures that the system can maintain stable operation state when the load is increased rapidly.

In the following simulation, the unit controller adopts the constrained model predictive control method based on the identification model. The main control signals during load raising of the unit are shown in Figure 5, including valve opening (%), coal consumption (t/h), total water supply (t/h), air-coal ratio and desuperheating water (t/h), respectively.

Figure 5 shows that in the controlled input constraint range, in order to make full use of the energy storage and accelerate the initial load response speed of the unit, the valve opening gradually increases. The valve opening is the main means of rapid regulation of the system. Its steady-state value is maintained between 85% and 90%, and the valve opening returns to the original state after achieving the purpose of rapid load regulation. Coal and water are the main control inputs of the generating unit. When the unit is loaded, the coal feed and water supply increase rapidly. The coal feed and water supply determine the internal variable state when the load of the unit is finally stable. The air-coal ratio is an important parameter affecting the unit load. In order to ensure full combustion and less heat dissipation of coal, the air-coal ratio is set as an empirical constant. Desuperheating water is the last process of regulating the main steam temperature, and it is an auxiliary means. Considering its regulating effect on the main steam temperature, there is no strict restriction here.

The system outputs at load rise period are shown in Figure 6, including unit load (MW), main steam pressure (MPa) and main steam temperature (˚C).

It can be seen from Figure 6 that when the system load needs to increase rapidly, the main steam temperature and main steam pressure will change accordingly. It is not difficult to find that coal and water have a certain lag effect on the unit load. The variation range of main steam temperature and main steam pressure meet the given constraints. Under various constraint conditions, the maximum load change is a gradual rise processing, which reflects the load raising capacity of the unit.

In case of rapid power reduction, the setting value of the generation load is set to a downward step as large as possible. In order to facilitate the comparison with the results of load rise, the same constraints and processing methods are adopted for each variable during load decrease. The simulation results of the control inputs are shown in Figure 7. It can be seen that the control inputs vary within the constraint range. In order to reduce the unit load rapidly, the valve opening decreases rapidly and then gradually returns to the original state. The coal feed and water supply decrease rapidly, the air-coal ratio remains unchanged, and the desuperheating water has no strict restriction.

The controlled variables in load reduction are shown in Figure 8. When the system load drops rapidly, the change of main steam temperature and main steam pressure is kept within certain range. On the premise of satisfying the constraints of each input, the maximum load reduction is a gradual decline curve, which reflects the load reduction capacity of the unit.