The proposed offline RMPC approach for MST system is tested by control simulations in this section. A zonotope-type uncertain model for controller design is achieved firstly in a recursive way, and then proposed control strategy for MST system is demonstrated via comparison with PID and other MPC methods.
5.1. Establishment of the Zonotope-Type Uncertain Model for MST System
In this section, on-site data of an ultra-supercritical CFPP located in in Shanghai, China is recorded to establish the prediction model for controller design through a closed-loop identification method for uncertain system, and the principles of algorithm realization are referred to reference [
31].
Figure 5 shows 3000 groups of routine operating data of MST system around unit load 700 MW with sampling interval of 5 s. The last 2000 groups of I/O data are utilized to identify uncertain model for MST system and the first 1000 groups of I/O data are used as verification data set to demonstrate the effectiveness of the identified model.
The model structure, which is compatible with identification data set, can be achieved by the asymptotic method [
38], i.e.,
, there are two parameters to be identified, and the order of zonotope is 3, thus, the zonotope-type uncertain model at time instant
k is
, in which
means an 3-order unitary box, let the model with center parameter
be the nominal model of
, and
illustrates variation range of the parameters
and
. Initial nominal parameter vector can be estimated via the output error identification method [
39], and
must be set large enough to cover the true parameter, to this end, the initial zonotope-type uncertain model is shown as (28).
where the notion “⊕” denotes the Minkowski summation of two sets
and
, i.e.,
.
Essential parameters of closed-loop identification method [
31] are set in
Table 1.
The final uncertain model is achieved by applying the recursive closed-loop identification method [
31] shown as (29).
where
are vertexes of the convex hull listed as
Table 2.
Model validation is conducted via the verification data set, the result is shown as
Figure 6. We see that the nominal output and the output bound of the uncertain model has the similar variation trend to the one of the real output. However, the difference from the ideal simulation experiment is that the output of uncertain model identified with the on-site operating I/O data cannot cover the whole real output, because complicated uncertain disturbances unavoidably influence MST system, there is information deviation between identification data set and verification data set, moreover, in the identification and verification period MST system show different nonlinearity. Nonetheless, compared with the model with sole parameter, parameters of the uncertain model belong to specific model set causes the output to cover the real output, which illustrates the uncertain model has improved descriptive ability for MST system.
Step response test of the identified uncertain model (44) is shown as
Figure 7, the upper part of
Figure 7 illustrates the valve position (input of model (44)) variation, and the lower part illustrates the MST (output of model (44)) variation with the valve position varying. At
t = 50 s, the valve position steps from 47.73% to 52.73%, the output decreases slowly from 600 °C, the nominal model output approaches to 599.3 °C at
t = 1600 s, while the uncertain output is a range between 598.7 °C and 599.7 °C with a transient time of 1400–2000 s, the dynamic characteristics shows with a large thermal inertia. We can see that the identified uncertain model with a varying static gain and transient time is essentially nonlinear, with which it is effective to describe the dynamic features of MST system that the static gain and transient time will keep changing with the unit load following the command from the power grids.
5.2. Control Simulation for MST System
In this section, the proposed OFAERMPC method is utilized to design controller for MST system of 1000 MW ultra-supercritical power plant, the control objective is that MST is kept at 600 °C with less fluctuation under complicated operating conditions while guaranteeing the stability by adjusting the valve position of 2nd attemperator.
Simulation platform: matlab2015b with yalmip solver [
40]; computer hardware: CPU—Intel Core i5-2410M, RAM-DDR3 6 GB.
The corresponding parameters of OFAERMPC and standard AEMPC are set as follows: initial state space
;
,
,
; weighting coefficients
and
; compensation coefficient
; sampling interval
= 5 s; amplitude and variation rate constraints of manipulated variable are
and
. The offline computation time and number of partitioned subspaces are listed in
Table 3.
In the offline control law design stage, OFAERMPC consumes much less time than standard AEMPC, and obtains a state space partition with fewer subspaces. We can see that standard AEMPC achieves a poor convergence when designing MST controller, which is actually an SISO system. However, when applying the basic idea of AEMPC to design RMPC controller for the MIMO system, the control algorithm may suffer from falling into endless loop.
Three case studies are considered to demonstrate the effectiveness of the proposed OFAERMPC method.
Case 1. Power plant unit load varies
During the routine operation of power plant, to hold the power grid frequency constant the unit load must trace the load demand from the grid, therefore, the unit load daily varies according to the demand of electricity users. However, the steam temperature of attemperator inlet and dynamic property of MST system will change correspondingly when the unit load W varies, the valve position of attemperator u must be regulated to hold MST constant with less fluctuation. Objective of this control simulation is to test control performance of the proposed OFAERMPC during unit load variation.
Simulation condition: initial MST and valve position of attemperator is 600 °C and 29.4%, the on-site operating data of unit load
W and steam temperature of attemperator inlet
, which is produced at the period of increasing unit load from 500 MW to 950 MW, is utilized as the input variables of the simulation system, shown as
Figure 8.
Four control strategies are employed in this control simulation.
the proposed OFAERMPC;
incremental model predictive controller (IMPC) based on the nominal model of the identified zonotope (29) with weighting coefficients and , control horizon 5, prediction horizon 500 and sampling interval = 5 s;
digital PI controller with proportional coefficient 5.26 and integral time 292.22 (design by matlab PID controller tuning modular);
standard AEMPC.
The control simulation results are shown in
Figure 9 and
Figure 10, and control performance is listed in
Table 4, where the performance index is mean square of the dynamic error. OFAERMPC has achieved the best performance index, and fluctuation of MST is the smallest among four controllers. Standard AEMPC obtains a satisfactory dynamic error, however, when dynamic process approaches the steady state (7000–1000 s), owing to the set point of control variable being unknown, there exists steady-state deviation that is unacceptable for control application. Both PI and IMPC can realize the offset-free control for MST system. The control action of IMPC is faster and lack of robust stability guarantee, we can find from
Figure 9 and
Figure 10 that the attemperator valve position of IMPC varies sharply, causing strong fluctuations of MST, especially when the regulation approaches the steady state. Since the variation of attemperator valve position is smooth, the poor cooling effect of the PI controller results in the largest dynamic deviation. It also indicates that all the four controllers can meet the requirement for control variable restraint. On control performance: OFAERMPC is the best, PI controller has the largest dynamic deviation and IMPC shows a poor stability when process approaches the steady state, standard AEMPC is unfit for MST regulation system design. From the perspective of online computational burden: the total simulation time is shown in
Table 4, the one of PI controller is the least, and OFAERMPC is better than IMPC and standard AEMPC. In summary, the control simulation illustrates the proposed OFAERMPC achieves a satisfied control performance for a unit load variation, an excellent applicability for real-time control is demonstrated because its online simulation time is a little more than that of PI and a stronger robustness is shown.
Case 2. Unpredictable disturbance occurs
Since complicated coal combustion process is involved in the boiler, unpredictable combustion status may change arising from coal sources variation, the heat exchange is thus disturbed. The steam temperature of attemperator inlet and dynamic property of MST system will change when unmeasurable disturbance exists, the valve position of attemperator u must be regulated to hold MST constant with less fluctuation. Objective of this control simulation is to test disturbance rejection performance of the proposed OFAERMPC.
Simulation condition: initial MST and valve position of attemperator is 600 °C and 75.5%, at
t = 0 s unpredictable disturbance occurs that results in steam temperature of attemperator inlet
varying frequently, the on-site operating data of unit load
W and
are used as the input variables of the simulation system, shown as
Figure 11.
In this control simulation, OFAERMPC, PI and IMPC control strategies are employed, the corresponding control parameters are set the same as those in Case 1 The control simulation results are shown in
Figure 12 and
Figure 13 and
Table 5. It can be seen that control action of IMPC is faster and PI is slower compared with OFAERMPC; the dynamic deviation of OFAERMPC is the lowest; the dynamic deviation of IMPC is medium, variation of MST fluctuates sharply, oscillation occurs when process approaching the steady state; the variation of PI controller output is smooth and dynamic deviation is the largest. All of PI, IMPC and OFAERMPC can meet the requirement for control variable restraint when unpredictable disturbance occurs. On control performance: OFAERMPC is the best, PI controller achieves the largest dynamic deviation and a smooth control result, the dynamic deviation is satisfied for IMPC controller, but a poor stability is obtained when process approaches the steady state. From the perspective of online computational burden: the required calculation time of OFAERMPC is larger than that of the PI controller, while much less than the one of IMPC. In summary, this control simulation illustrates the proposed OFAERMPC achieves a satisfied control performance and applicability for real-time control when unpredictable disturbance occurs.
Case 3. Plant behavior changes
The real dynamic property of MST system may deviate from the designed one due to aging of equipment, which may cause performance degradation of the control system. The objective of this control simulation is to test the control performance of OFAERMPC strategy when an unknown change of dynamic property of controlled system occurs.
Simulation condition: initial MST and valve position of attemperator is 600 °C and 75.5%, at
t = 2000 s unpredictable disturbance occurs, the on-site operating data of unit load
W and
are used as the input variables of the simulation system, shown as
Figure 11.
OFAERMPC, PI and IMPC control strategies are employed in this simulation, the corresponding control parameters are set the same as those in Case 1.
The control simulation results are shown in
Figure 14 and
Figure 15 and
Table 6. They indicate that control performance of all the three control strategies get worse in case of plant behavior change. Among them, the performance degradation of OFAERMPC is smaller than that of PI after
t = 2000 s. Compared with the lines in
Figure 12 and
Figure 13, due to the restriction of control action getting worse of OFAERMPC regulation is more serious than that of PI regulation when
t = 5000–6000 s and
t = 7500–8500 s, the robustness of OFAERMPC is better for the other time; the trend of getting worse for IMPC is obvious, and oscillation aggravates when the process approaches the steady state. PI, IMPC and OFAERMPC can meet the requirement for control variable restraint when dynamic property of MST system changes. Regarding the control performance, OFAERMPC has the similar robustness to PI, and robustness of IMPC is the poorest. From the perspective of online computational burden: the required calculation time of OFAERMPC is larger than that of the PI controller, but much less than that of IMPC. In summary, this control simulation illustrates the proposed OFAERMPC achieves a satisfied control performance and applicability for real-time control when strong plant behavior change occurs.