Integrated Parameter Mapping and Real-Time Optimization for Load

Subscriber access provided by UNIVERSITY OF THE SUNSHINE COAST

Process Systems Engineering

Integrated Parameter Mapping and Real-Time Optimization for Load Changes in High Temperature Gas-Cooled Pebble Bed Reactors Cheng Yang, Kexin Wang, Zhijiang Shao, and Lorenz T. Biegler Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b05174 • Publication Date (Web): 21 Jun 2018 Downloaded from http://pubs.acs.org on June 26, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Industrial & Engineering Chemistry Research

Integrated Parameter Mapping and Real-Time Optimization for Load Changes in High Temperature Gas-Cooled Pebble Bed Reactors Cheng Yang,† Kexin Wang,† Zhijiang Shao,*,† Lorenz T. Biegler‡ † College of Control Science and Engineering, Zhejiang University, Hangzhou 310027, China ‡ Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, United States

Abstract High temperature reactors with pebble bed modules (HTR-PM) are designed to operate over wide conditions, which presents challenges to model-based optimization and control. The main difficulty lies in that an accurate model applicable to the full operating region is usually unavailable. In order to implement significant load changes for HTR-PM, iterative strategies are developed to deal with plant-model mismatch. The model, embedded with a parameter mapping from the plant status to the model characteristics, is determined not only by parameter updates on a refined subset which ensures reliable estimation, but also by parameter sensitivities to enhance the ability on prediction. Based on such a model, we design algorithms centered on trust-region concepts, so that input corrections allowed at each iteration are determined adaptively through rigorous optimization. Moreover, the underlying model is updated adaptively to maintain a sufficiently accurate approximation to the plant. The proposed optimization strategy is illustrated with the load change problem of HTR-PM, where both reactors undergo load changes synchronously. Keywords: trust region, plant-model mismatch, process optimization, parameter selection, HTR 1. Introduction Nuclear energy plays an important role in clean energy, as it reduces carbon emissions and meets the increasing energy demand. As one of the promising technologies in building the Generation IV nuclear energy systems, the modular high 1

ACS Paragon Plus Environment

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 36

temperature gas-cooled reactor (MHTGR) is characterized by its inherent safety, economic competitiveness, standardization and modularization, and potential broad applicability.1,2 With these advantages, MHTGR has attracted much attention in practice, such as the HTTR project and GTHTR300 project in Japan, the PBMR project in South Africa, the GT-MHR project in USA, and agreements between Saudi Arabia, South Africa and China on HTGR plant construction. Moreover, the world’s first multi-modular demonstration plant for high temperature gas-cooled reactor pebble bed module (HTR-PM) has been under construction since 2012 at Shidaowan, China,3 and is expected to start commercial operation in 2018. The HTR-PM considered in this paper consists of two nuclear steam supply system (NSSS) modules, each having a reactor of 250MWth and a steam generator, and being connected to a common steam turbine generator set. Both reactors are designed to have a wide operating range, from 30% to 100% reactor full power (RFP). They can work at different power levels and the modules interact with one another through the turbine in between. This enables the whole system to operate economically and flexibly with the ever-changing power demand, but at the same time poses challenges to operation and control. Safe, optimal HTR-PM operation is critical, despite a complex mechanism and little experience that can be borrowed from operation of its single reactor counterpart. Some key outputs, such as steam pressure and temperature can deviate far from their nominal values, and this impairs system efficiency and even causes physical damage. Therefore, an accurate model of the entire system is desirable. However, the HTR-PM models are usually not as accurate as required; simplifications and empirical formulations are indispensable in describing this complex system. Also, with operation of the system, some inherent characteristics may change due to high temperature, pressure, and radiation, leading to further disagreement between the model and the real process. Therefore, calibration of model-related parameters is necessary to improve model fidelity. Moreover, models of complex systems often suffer from over-parameterization, which gives rise to ill-conditioned parameter estimation problems. Hence the task of parameter 2


Page 3 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


estimation requires systematical treatment beyond a simple data-fitting problem. In order to guarantee reliable estimation, methods of conditioning analysis and regularization have been proposed.4-6 In this paper, we design a systematic parameter estimation approach for the steady-state HTR-PM model that applies to the case where large amounts of data sampling are impractical. Moreover, process models often apply to a certain operating range, and they lose accuracy over the full range. For instance, heat transfer coefficients of HTR-PM vary with the load, thus their estimated values for some loads will not apply for operations at other levels. Proper operation over a wide range calls for strategies capable of handling plant-model mismatch. This problem was considered in standard iterative two-step (ITS) method, where parameter estimation (PE) and real-time optimization (RTO) are performed iteratively. However, this method rarely converges in the presence of structural mismatch.7 To promote convergence, the integrated system optimization and parameter estimation (ISOPE) were then proposed8-10 to track the first-order optimality conditions of the plant. Later, this approach was refined with improved efficiency by eliminating the PE stage,11,12 but at the cost of losing knowledge of parameters with physical meaning. Instead, this paper develops an integrated strategy for parameter mapping and real-time optimization that addresses significant load changes for HTR-PM. According to this method, both the values and the structure of the parameters are considered in the course of optimization; the distinguishing feature is the trust-region strategy therein, which forces sufficient accuracy of the parameter mapping and drives convergence, despite the structural mismatch. The paper is organized as follows. The next section introduces briefly the model of HTR-PM. Section 3 designs a systematic approach to determine the estimable parameters of the model. To motivate the need for trust region in significant load change, Section 4 illustrates a simple but unsatisfactory implementation with iterative two-step method. The parameter mapping strategy is developed in Section 5. A trust-region method is designed to integrate parameter mapping and optimization, leading to adaptive load change steps during the full iteration process. Further, the 3



Page 4 of 36

cost of parameter mapping is reduced by introducing an adaptive model updating mechanism. Section 6 summarizes this approach and concludes the paper. 2. Steady-State Model of HTR-PM The multi-modular structure of HTR-PM is illustrated in Figure 1. Each nuclear steam supply system (NSSS) includes a reactor, a steam generator and a helium blower. Steam from the two NSSS modules shares a common steam header, which promotes the turbine to generate electricity. This thermal-hydraulic process is essential to HTR-PM. The cold helium (about 523.15K) is pressurized by the blower and then enters the cold gas duct. This cools the side reflector when flowing through its channels from the bottom to top. As the helium reaches the reactor core, it passes through the pebble bed from the top to bottom, where it is heated to about 1023.15K. The hot helium flows through the hot gas duct into the primary side of the steam generator, transferring heat through the metal tube wall to the water flowing in the secondary side. After this heat exchange, the water turns into superheated steam and the helium is cooled back to 523.15K.

Reactor 1#

Helium Blower 1#

Reactor 2#

Steam Generator 1#

Feed Water Pump 1#

NSSS1

Steam Header

Steam Generator 2#

Turbine

Conde nsor

NSSS2 Helium Blower 2#

Feed Water Pump 2#

Figure 1. Schematic of HTR-PM.

Based on the helium flow path, the reactor model is divided into the core, reflector, lower plenum, lower header, riser, upper header, downcomer, and outlet 4


Page 5 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


header. Following the water/steam status on the secondary side, the model of the steam generator is partitioned into economizer, evaporator, and superheater along the length of the tube. The models of these units are the same for both NSSS modules as presented in Tables 1 and 2; these are the only parts of the steady-state model of HTR-PM that include parameters. All of the symbols therein are defined in the Notation section. For more details of the HTR-PM model we refer the readers to References 13,14. Table 1. Reactor Model Downcomer:

Lower plenum:

0

Lower header: Outlet header:

0 1 Riser:

Reflector: 0

0 Core: 0

Upper header:

Reactivity feedback: 0

Table 2. Steam Generator Model Economizer: /

Primary side: / 2

0

sin

2

2

/

/

2

2 5



Page 6 of 36

Metal tube: Evaporator:

/

/

0

sin

Heat transfer:

2 2 Superheater:

/

/

0

sin 2 2

The complete system model comprises 996 variables. The degrees of freedom for operation consist of external reactivity (

), and

) of both NSSS modules, and the valve opening prior to

feedwater flow rate ( the turbine (

), helium inlet flow rate (

). Some of plant outputs are characterized by design values, including

relative nuclear power (

), outlet helium core temperature (

temperature and pressure of the steam generator (

and

temperature and steam pressure just before the turbine (

), outlet steam

), as well as steam and

). Deviating far

from the design values would jeopardize safe, reliable and economical operation of the plant and thus is forbidden. Requirements on the plant inputs and outputs are listed in Table 3. Table 3. Restrictions on Inputs and Outputs Outputs

Inputs

NSSS

[0, 0.003]

30% to 100% (±3%)

[30, 100] kg/s

963.15K~1023.15K (±5K)

[30, 100] kg/s

844.15K±5K 13.9 MPa±0.5MPa

Steam header

[0, 1]

839.15K±3K 13.24 MPa±0.1MPa 6


Page 7 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Also, there are 33 measured outputs as specified in Table 4, including relative reactor power (

) and temperature ( ), pressure ( ) and flowrate ( ) of the helium,

water and steam. The parameters to be estimated include the helium leakage ratio ( ), ,

load-related heat transfer coefficients ( coefficients

and areas

(

,

,

,

), and the products of heat transfer ). Table 5 lists the parameters with

their physical ranges and their nominal values at 100% reactor full power (RFP); the last column lists the Euclidean norms of the sensitivity vectors of the outputs in Table 4 with respect to the parameters at the nominal values. Table 4. Measured Outputs NSSS Steam generator

Reactor

Steam header

Table 5. Parameters and Nominal Information at 100%RFP Parameter

Physical range [0.07, 0.13] [103253, 375648] W.K-1 [4101380, 14290289] W.K-1 [984, 1615] W.K-1 [520,1626] W.m-2.K-1 [643, 1885] W.m-2K-1 [459, 1581] W.m-2.K-1

Nominal value 0.1 313040 11908574 1346.50 1355.78 1571.05 1318.85

Sensitivity norm 0.2022 0.0053 0.0676 0.0091 0.1532 0.0544 0.0227

3. Systematic Parameter Estimation Approach At a given operating point, the model parameters are estimated from the plant data by the following parameter estimation problem, which is given by the following standard weighted least-squares form ∑

min . .

∈

and the measured outputs

the plant inputs and states;

0

(1)

, measures the goodness of fit between the model

Here the objective function, prediction

, , , 0.

∈

∈

;

∈

and

are parameters to be estimated; 7


∈

are

denotes the


plant model and

are constraints on the parameters;

Page 8 of 36

represents the weighting

factor. Reliable estimation from (1) is essential, as an unreliable parameter estimation will fail to identify the true characteristics of the plant and may lead to wrong decisions. For example, helium leakage ratio can be used to monitor the pressure boundary integrity, and unreliable estimation of this indicator may give rise to unsafe operations. Over-parameterization, data insufficiency and noise lead to ill-conditioned parameter estimation problems, whose results are very sensitive to perturbations in the problem data. Therefore, besides convergence of the data-fitting problem (1), attention should be paid to determining an appropriate parameter set that can be estimated reliably.13,15 A systematic framework for reliable parameter estimation is depicted as Figure 2, which includes parameter subset selection, parameter estimation, reliability evaluation, and subset refinement. Initial subset selection Parameter estimation Confidence interval evaluation NO

Reliable results?

Subset update

YES Optimal subset obtained Figure 2. Systematic parameter estimation procedure.

3.1 Initial Subset Selection Some parameters may have similar effects on the model outputs such that the outputs behave similarly in response to their local perturbations. This similarity can be investigated by measuring collinearity between output sensitivity vectors. Specifically, cosine distance between a pair of parameters is defined by 16-18 1 where and

and

cos

1

∙ ‖ ‖∙

,

(2)

are sensitivity vectors of the outputs with respect to parameters

, respectively. Parameters with cosine distance less than a given tolerance 8


Page 9 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 are considered to be indistinguishable and clustered into one group. Checking the pairwise cosine distance between the parameters, leads to grouping among parameters that cannot be estimated separately. The groups need to be clustered further until the distance between all the groups are greater than

. Here

the group distance is decided by the largest distance between their members, because parameters collected into one group are required to be pairwise indistinguishable. In the end, the parameter with the largest sensitivity magnitude is picked from each group to make up the estimable subset, while others are fixed at their nominal values. 3.2 Reliability Evaluation and Subset Refinement In order to validate the parameter selection, confidence intervals are computed to evaluate reliability of the resulting estimates. For parameter

, the confidence

interval is defined by 19

where

∗

∙

/

is the number of experimental data points;

distribution for the given confidence level covariance matrix

∗

∗

where

∗

and

(3) is the two-tails Student’s

/

and

degrees of freedom. The

is approximated by 19 ∗

∗

,

∗

∗

,

(4)

are the objective and the reduced Hessian at the solution

. We define the reliability factor

1, … ,

as the ratio of the length of

parameter confidence interval to its estimated value. Hence a large reliability factor corresponds to an unreliable estimate. For a given threshold

1, … ,

, if the ratio satisfies

, the corresponding estimates are regarded as acceptable.

Otherwise, the parameter with the largest ratio is fixed at its nominal value, and problem (1) is solved again with the reduced parameter subset. This subset refinement procedure continues until all reliability factors are within the given threshold. 3.3 Numerical Results In this numerical experiment, we demonstrate the parameter estimation and subset selection approach at 100%RFP. The output sensitivities are scaled to provide 9



uniform comparisons, i.e. , where

(5)

is the kth component of sensitivity vector

value of parameter

;

and

and the value of output variable

are the nominal

at 100%RFP, respectively.

Based on the scaled values, cosine distances between parameter pairs (2) are shown in 0.05, the parameters are firstly divided into

Table 6. For the given threshold

,

four clusters for each NSSS module, i.e.

,

,

,

and

,

,

. The smallest distance between the groups is 0.087

and

, associated with

, and the clustering result remains unchanged. Referring to the

sensitivity norms in Table 5, parameters

,

,

and

for each NSSS

module are then selected for parameter estimation. Table 6. Cosine Distance between Parameter Pairs at 100%RFP NSSS1

NSSS1

0

NSSS2

0.524

0.0001

0.506

0.726

0.747

0.812

0.751

0.891

0.751

0.886

0.890

0.908

0.956

0

0.522

0.0002

0.811

0.800

0.787

0.891

0.952

0.891

0.950

0.952

0.959

0.980

0

0.504

0.726

0.747

0.813

0.751

0.891

0.751

0.886

0.890

0.908

0.956

0

0.819

0.808

0.794

0.886

0.950

0.886

0.948

0.950

0.958

0.979

0

0.008

0.087

0.890

0.952

0.890

0.950

0.952

0.960

0.982

0

0.044

0.908

0.959

0.908

0.958

0.960

0.969

0.991

0

0.956

0.980

0.956

0.979

0.982

0.991

0.990

0

0.524

0.0001

0.506

0.726

0.747

0.812

0

0.522

0.0002

0.811

0.800

0.787

0

0.504

0.726

0.747

0.813

0

0.819

0.808

0.794

0

0.008

0.087

0

0.044

NSSS2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 36

0

For parameter estimation, the experimental data of both NSSS modules contains 0.5% white Gaussian noise. The parameter estimation results are summarized in Table 7 with a reliability threshold

20% imposed. Initially,

reliability ratio 50.5% associated with

8, while the largest

fails to satisfy the threshold

. This

indicates the necessity to refine the parameter selection. As shown in the following two 10


Page 11 of 36

columns,

and

are successively removed from the subset. This procedure 4, where estimation of

terminates after three rounds of estimation with

and

of both NSSS modules leads to reliability factors that are sufficiently small. Based on the available data, at most four of the parameters can be estimated with the desired level of reliability. Table 7. Results of Parameter Estimation and Reliability Evaluation at 100%RFP Estimation (reliability factor %) 8 6

NSSS1

Parameters

NSSS2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4

0.0999(2.3)

0.0999(1.8)

0.1001(1.5)

1357.79(6.3)

1353.51(5.7)

1356.83(2.0)

1300.35(40.0)

1338.51(38.3)

x

1331.29 (50.5)

x

x

0.1000(2.3)

0.0999(1.8)

0.0999(1.5)

1351.22(6.2)

1355.33(5.7)

1355.83(2.0)

1366.06(43.3)

1313.69(37.1)

x

1394.62(50.3)

x

x

4. Iterative Two-Step (ITS) Method for Significant Load Changes The model determined at the last section is a reasonable approximation to the plant operating around the conditions where the parameters are estimated. However, when operating conditions change significantly, we cannot expect this local model to remain applicable, which indicates that implementing significant load change of HTR-PM usually requires model calibration. A direct calibration method takes a sequence of relatively small load change steps, namely, |∆

|

,

(6)

where the right-hand side is positive and small, hopefully yielding plant status where the model still holds approximately. At each step, (i) apply the current inputs to the plant and solve a PE problem with the corresponding measurement to determine the model parameters; (ii) use the model with the parameter estimates to determine the inputs that minimize some performance index, which is chosen for the load change task to be the differences between the model predicted and design values of the key outputs at the end of this step, i.e. 11



∗

∑ where and

,

∗

∈

,

Page 12 of 36

(7)

are vectors of key outputs whose elements are listed in Table 3,

represents the weighting factor. We then repeat steps (i) and (ii) over the

sequence of load change steps until the load change task is completed. For this ITS method it is worth noting that the model determined under the current conditions is used to optimize the inputs for the next step, whereas the resulting inputs will lead the plant to a new status where the model characteristics of the plant change and the previous model becomes obsolete. When there exists structural error between the model and the plant, the inputs generated from the model will not lead to the plant optimum, because the updated parameters that compensate for structural error prevent the model from generating directions that can catch up with the first-order optimality conditions of the plant. To demonstrate the effect of structural mismatch on load change, conservative and aggressive step sequences are designed for the ITS procedure. The conservative sequence limits the maximal load change in each RTO step to 2%RFP, while the aggressive one allows larger steps of 9%RFP. In this example both reactors of HTR-PM change load from 100%RFP to 50%RFP synchronously. The conservative scheme takes 25 steps to complete the load change. Figure 3a compares the values of heat transfer coefficients (

) used by the model and estimated

with the measurement at the resulting steps, indicating good agreement between the model and the plant. As a result, there is no large difference between the model predicted outputs and measured ones, and all the key outputs satisfy their requirements during the course of load change (Figure 3b). The input sequence is presented in Figure 3c. The performance index of the plant, (7) evaluated with the real plant outputs, at the end of load change is 62.2.

12


Page 13 of 36


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1400

-2

-2

1200 1100 1000

1200 1100 1000

900 800 0

model plant

1300

NSSS2: K2(Wm /K

1300

NSSS1: K2(Wm /K)

1400

model plant

900

5

10

15

iteration, k

20

800 0

25

5

10

15

iteration, k

20

25

(3a) Heat transfer coefficients. 1

1030

model plant

1020

0.9

nr

T6(K)

0.8 0.7

1010

1000

0.6

990

0.5 0

bound design value model plant

5

10

15

iteration, k

20

980 0

25

14.2

10

15

iteration, k

850


14.4

5

20

25


848

T7(K)

P7(MPa)

846 14 13.8

844 842

13.6 840 13.4 0

5

10

15

iteration, k

25

0

10

15

iteration, k

841

13.25 13.2

20

25


842

Tso(K)

13.3

5

843


13.35

Pso(MPa)

20

840 839 838 837

13.15

836 13.1 0

5

10

15

iteration, k

20

25

0

5

10

15

iteration, k

20

25

(3b) Key outputs. 13



-3

x 10

100

2

90 80

ext

Gin(Kg/s)

1.5

1

70 60 50

0.5 0

5

10

15

iteration, k

20

40 0

25

100

1.1

90

1

70 60

10

15

20

25

5

10

15

20

25

iteration, k

0.8 0.7 0.6

50 40 0

5

0.9

80

xgv

Gpump(Kg/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 36

0.5 5

10

15

iteration, k

20

0.4 0

25

iteration, k

(3c) Input sequence. Figure 3. Load change by conservative strategy.

On the other hand, the aggressive scheme takes only 6 steps, but it demonstrates the obvious effect of the structural mismatch. As shown in Figure 4a, the model determined by the parameter values cannot represent the plant accurately; successive parameter estimation cannot help this situation, because it always one step behind the change of the plant status. Consequently, the plant outputs are much different from the model predictions, and deviate significantly more from their design values than in the conservative case. Moreover, steam temperature (

) violates the lower bound (Figure

4b). Figure 4c gives the input sequences. At the end of the load change, performance index (7) is 821.77. This experiment illustrates that updating local models conservatively often leads to safe but inefficient operations, while aggressive schemes give rise to fast but insecure ones.

14


Page 15 of 36


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1400

-2

-2

1200 1100 1000

1200 1100 1000

900 800 0

model plant

1300

NSSS2: K2(Wm /K

1300

NSSS1: K2(Wm /K)

1400

model plant

900

2

iteration, k

4

800 0

6

2

iteration, k

4

6

(4a) Heat transfer coefficients. 1

1030

model plant

1020

0.9

nr

T6(K)

0.8 0.7

1010

1000

0.6

990

0.5 0


2

iteration, k

4

980 0

6

14.2

iteration, k

4

850


14.4

2

6


848

T7(K)

P7(MPa)

846 14 13.8

844 842

13.6 840 13.4 0

2

iteration, k

4

0

iteration, k

4

841

13.25 13.2

6


842

Tso(K)

13.3

2

843


13.35

Pso(MPa)

6

840 839 838 837

13.15

836 13.1 0

2

iteration, k

4

6

0

2

iteration, k

4

6

(4b) Key outputs. 15



-3

2.2

x 10

100

2

90

1.8 80

Gin(Kg/s)

ext

1.6 1.4 1.2

0.8

50

0.6 0

70 60

1

2

iteration, k

4

40 0

6

100

1.1

90

1

70 60

iteration, k

4

6

4

6

0.8 0.7 0.6

50 40 0

2

0.9

80

xgv

Gpump(Kg/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 36

0.5 2

iteration, k

4

0.4 0

6

2

iteration, k

(4c) Input sequence. Figure 4. Load change by aggressive strategy.

5. Trust-Region Load Change Strategy From the last section we learn that it is not easy to determine the appropriate pace to implement significant load changes. Implementation of each load change step leads the plant to a new status where the constraints for key outputs may be violated, although they hold for the predicted model. Therefore, how should proper stepsizes be determined for a load change? Are uniform steps a good choice? To answer this, we require a model that depicts the plant well enough to generate reasonable decisions. We assume the plant corresponds to an underlying mechanism , , , ̅,

0.

Here we divide the parameters into two categories.

̅∈

(8) is the vector of

parameters, such as helium leakage ratio ( ), that vary slowly with respect to load ∈

is the vector of

parameters that change with load and thus have varying values;

represents an

change and thus can be treated as constants. In contrast,

16


Page 17 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


unknown parameter mapping based on the variable u, such as the heat transfer coefficients (

) shown in Figures 3a and 4a.

5.1 Parameter Mapping is accurate at this operating

We assume the model determined at conditions point and in a certain region around

, these parameters can be predicted by ∙ ̂

where

̂

is

parameters

,

, and ∂ /

estimated at

(9)

represents the sensitivities of

with respect to the inputs. The local parameter mapping (9) requires

evaluation of plant derivatives, which is not an easy task. Moreover, not only the unknown structure of

are

but also the disturbances that the plant suffers at

reflected in (9). To evaluate the sensitivities, we apply consecutive perturbations to the plant. For the current conditions

, perturbations are chosen to be ∙

where

,

is the ith unit vector and

1…

and

, (10)

is a small constant; i in the superscript for

indicates the perturbation index, while in the subscript refers to the ith component of the vector. Perturbing in this way introduces fewer errors to the plant, in the sense that each subsequent input vector varies based on its predecessor, not necessarily based on , as commonly does. With the group of measurements, corresponding to each input,

∈

1, … ,

,

, we solve separate parameter estimation problems ,

min , . .

∑ , ,

, ̅, 0

,

̂ ,

and obtain parameter estimates

,

0

1, … ,

(11)

accordingly. Now with the

definitions ∆ ∆ sensitivities ∂ /

⋯ ̂

,

̂

̂

,

̂

,

⋯

̂

,

can be approximated by the matrix ∆

∙

∆

.

17


̂ , from the linear system (12)


Page 18 of 36

Then the local parameter mapping in (9) is computed as ̂

∙

.

(13)

5.2 Integrated Parameter Mapping and RTO We substitute

for the unknown

in mechanism (8), yielding the

composite model , ̅,

0,

(14)

where the description has been simplified and given in terms of the independent variables and the parameters. Usually, there exists large mismatch between this model and the real plant (depicted by (8)) if linearization error of

moves far away from

; when

→

, the

tends to zero and model accuracy increases. This

condition leads us to impose a trust-region constraint ‖

‖

∆ (15)

to model (14), where ∆ limits operating conditions

to a region within which we

trust the model is an adequate representation of the plant. Based on models (14) and (15) we design an iterative trust-region algorithm to implement desired load changes and maintain an accurate model. The optimization problem is given by min . .

, ̅, 0 ‖

‖ where

∗

∑

,

0

(16)

∆ ,

is constraints on the inputs. The last constraint is imposed to confine the

optimization on operations to a valid model. The algorithm is stated below, where ,target

and

denote the target load level and the measured load level of the plant,

respectively. 5.2.1 Algorithm I Step 0 (Initialization): Given the operating conditions

at the initial level of load,

and give the initial trust-region radius ∆ . Choose constants 0 0

1

, ∆

∆

∆

, and termination tolerance

Step 1 (Parameter estimation): If

,target

1, . Set

← 0.

, stop. Otherwise, apply the

strategy in Figure 2 to estimate the heat transfer coefficients to obtain ̂ . 18


Page 19 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Step 2 (Local parameter mapping): Perturb the plant with the sequence of inputs in (10) and solve the estimation problems (11) with the corresponding measurements. Calculate sensitivity matrix

according to (12), and .

approximate the parameters with (13) to obtain

Step 3 (Optimization): Solve problem (16) and denote the solution as Step 4 (Model evaluation): Apply

to the plant and compute

,

, , ̂

If

0, accept

.

,

; otherwise, restore

Step 5 (Trust-region radius adaptation): Set ∆ ,∆ min ∆ max ∆ , ∆ ∆ Let

.

←

.

if , if otherwise.

,

1. If the new iterate is accepted, go to Step 1; else go to Step 3.

Algorithm I has an essential difference from iterative parameter estimation and optimization methods. In particular, Steps 1 and 3 are not performed in sequence. The latter may be repeated with an updated trust-region radius until the solution of problem (16) is accepted at Step 4; this is guaranteed because the model built with (14) and (15) sufficiently agrees with the plant (i.e.

is close to 1) when the trust region

is small enough. We then return to Step 1 to estimate new parameters corresponding to the new status of the plant. In this sense, it is better to say that parameter mapping and RTO are integrated by the trust-region framework. Whether or not to renew the local parameter mapping (including parameter estimation) depends on whether or not the solution of the trust-region constrained optimization problem is accepted. The management of the trust-region radius is a mechanism that adjusts the region of approximating the parameters with

, based on the quality of the optimization solution. In contrast

to the uniformly bounded load change steps (6), Algorithm I gives rise to a sequence of adaptive steps, depending on how much input correction is allowed at each iteration, which is decided by the trust-region framework. 19



Page 20 of 36

There are a few comments on Algorithm I. Firstly, the local mapping is constructed at

and its perturbations

1, … ,

, which ensures that the

model coincides with the plant at these sampling points. This is important, since otherwise model accuracy will not be maintained even if the trust region tends to zero, and the unknown relation

can not be characterized around

. This also

suggests that the concept of trust-region does not apply to the fixed-model methods11,20. Secondly, the dominant cost of the algorithm lies in the sensitivity evaluation in Step 2. At least

1 steady-state operating points are needed to estimate the

sensitivities, which is expensive and introduces additional perturbations to the plant. With ISOPE the plant derivatives can be constructed by gathering active measurements for derivative evaluation.21 This method can be used to generate the sequence of inputs, with Δ

and Δ

derived from successive iterations instead of

from perturbations. However, there exists a compromise between optimizing the inputs with respect to the RTO objective and approximating the sensitivities accurately. Also, an initial phase is required to start the method, where standard ITS information sets for later sensitivity

iterations are applied to accumulate

evaluation. Therefore, this phase suffers from structural mismatch and may lead to violation of the output constraints. Due to the above reasons, we calculate the sensitivities as in Section 5.1. The trust-region subproblem is solved to convergence at Step 3, with the expectation to reduce the overall number of iterations required to solve the original problem. Finally, the success of the solution of Step 4 is determined only from the plant evaluation. 5.2.2 Numerical Results - Load Change Example Revisited I We apply Algorithm I to the load change problem in Section 4, where both reactors change load from 100%RFP to 50%RFP synchronously. Algorithm parameters are

0.5 ,

0.75 ,

0.7 ,

1.4 ,

1%RFP ; the

trust-region radius ∆ is set componentwise to 0.45 times the difference between the input values at 100%RFP and 90%RFP; ∆

and ∆

20


are set to 0.7∆ and 4∆ ,

Page 21 of 36

respectively. The load change is completed after 4 steps of Algorithm I. All the iterates are 0, and the results are presented in Figure 5; local parameter

successful, i.e.

mapping is performed at the iterations marked by ‘▲’. Figure 5a compares the values of heat transfer coefficients (

) used by the model and estimated with the

measurement at the resulting steps, indicating that the model with parameter mapping agrees with the plant very well during the course of load change. As a result, the plant outputs are almost the same as the model predictions, and all the constraints on key outputs are satisfied (Figure 5b). Figures 5c presents the input sequence. Compared with the results in Section 4, the results of Algorithm I suggest advantages of model-base optimization integrated with local parameter mapping in addressing structural mismatch. Algorithm I takes (much) fewer steps to implement load change. Different from the uniform load change steps in Section 4, the adaptive steps are {9.95%, 14.95%, 17.83%, 6.29%}. All these steps except the last are much larger than those of the aggressive strategy, while the model-plant agreement is even better than in the conservative case from Section 4. As the key outputs of the plant stay closer to their design values at the end of load change, performance index of the plant, namely, the objective of (16) evaluated with the real plant outputs, is much better, taking the value of 1.18. 1400

-2

1200 1100 1000 900 800 0▲

model plant

1300

NSSS2: K2(Wm /K

-2

1400

model plant

1300

NSSS1: K2(Wm /K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1200 1100 1000 900

1▲

2▲ iteration, k

3▲

800 0▲

4

1▲

2▲ iteration, k

3▲

4

(5a) Heat transfer coefficients.

21



1

model plant

Page 22 of 36


1020

0.9 1010

nr

T6(K)

0.8 0.7

1000

0.6 990 0.5 0▲

1▲

2▲ iteration, k

3▲

0▲

4

14.2

2▲ iteration, k

850


14.4

1▲

3▲

4


848

T7(K)

P7(MPa)

846 14 13.8

844 842

13.6 840 13.4 0▲

1▲

2▲ iteration, k

0▲

4

2▲ iteration, k

841

13.25 13.2

3▲

4


842

Tso(K)

13.3

1▲

843


13.35

Pso(MPa)

3▲

840 839 838 837

13.15

836 13.1 0▲

1▲

2▲ iteration, k

3▲

0▲

4

1▲

2▲ iteration, k

3▲

4

1▲

2▲ iteration, k

3▲

4

(5b) Key outputs. -3

x 10

100

2 90

ext

Gin(Kg/s)

1.5

1

70 60 50

0.5 0▲

80

1▲

2▲ iteration, k

3▲

0▲

4

22


100

1

90

0.9

80

0.8

xgv

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Gpump(Kg/s)

Page 23 of 36

70

0.7

60

0.6

50

0.5

40 0▲

1▲

2▲ iteration, k

3▲

0.4 0▲

4

1▲

2▲ iteration, k

3▲

4

(5c) Input sequence. Figure 5. Load change by Algorithm I (▲ iterations with local parameter mapping).

5.3 Nonlocal Parameter Mapping As we discussed in the previous section, sensitivity evaluation is expensive, especially for problems with many degrees of freedom22. Is it possible to evaluate the sensitivities less frequently? If the parameters of interest do not vary nonlinearly with operating conditions around

, parameter mapping built at this point will remain

valid in some region larger than ∆ . In this case,

can be applied to the

successive iterations without changing the structure of the parameters, that the model

, provided

is still able to produce iterates that improve performance of the

plant. If not, the reuse of the sensitivities terminates, and it is necessary to evaluate the sensitivities again. This leads to a modification of Algorithm I, namely, an adaptive model update mechanism. We describe the resulting algorithm with a nonlocal parameter mapping below, where we define the set

which contains the index of

iterations that allow a reuse of previous sensitivities. The set

is initially empty

and it is augmented whenever the current iteration is very successful (i.e.

),

which is an indicator to the next iteration to continue using, instead of renewing, the existing parameter mapping. It is worth noting that, with nonlocal parameter mapping, the incumbent model may become inappropriate in the interval between model updates. If this is the case, reducing the trust-region radius does not necessarily produce a better step. Therefore, we need to ensure that the trust-region size is never reduced due to poor quality of the model. Instead, if there is possibility of degeneration, the local parameter mapping 23



Page 24 of 36

must be rebuilt around the current conditions and the parameter structure needs to be re-evaluated. Despite the irregular update of the model, parameter estimation after each step acceptance is still indispensable. 5.3.1 Algorithm II The following algorithm is a variant of Algorithm I that attempts to apply a nonlocal parameter mapping in Step 2. This may avoid a new evaluation of the sensitivity matrix and has the potential to find improved points in Step 3. Step 0 (Initialization): Given the operating conditions

at the initial level of load,

and the initial trust-region radius ∆ . Choose constants 0 , ∆

1

∆

∆

1, 0

, and termination tolerance

. Set

← 0, and

← . Step 1 (Parameter estimation): If

, stop. Otherwise, apply the

,target

strategy in Figure 2 to estimate the heat transfer coefficients to obtain ̂ . 1∉

Step 2 (Parameter mapping): If

, perturb the plant with the sequence of

inputs in (10), solve the estimation problems (11) with the corresponding measurements, and calculate sensitivities

according to (12). Set

,

and approximate the parameters with

denote

̂ 1∈

Else, if

∙

.

(17)

with (17) and the last , and go to Step

, then apply

3. Step 3 (Optimization) Solve problem (16) with parameter mapping denote the solution as

.

Step 4 (Model evaluation): Apply

to the plant and compute

,

, , ̂

If

0, accept

, and

.

,

; otherwise, restore

←

.

Step 5 (Trust-region radius adaptation): Set min ∆

∆ ,∆

,

if

max ∆ , ∆

if

∆ ∆

if if

and and and 24


i.e. i.e. .

is local ,

is not local ,

Page 25 of 36

If

, add

to set

1. For the case where the new iterate

. Let

is rejected, go to Step 3 if the model is local, otherwise go to Step 2. Else, if the new iterate is accepted, go to Step 1. 5.3.2 Numerical Results - Load Change Example Revisited II We apply Algorithm II to the load change problem in Section 4. The algorithm parameters are the same as in Section 5.2.2. This leads to only 4 steps to realize the 0, and a local parameter

load change. All the iterates are successful, i.e.

mapping occurs twice, at the beginning and the 3rd iterations. The results on heat transfer coefficients, key outputs, and input sequence are presented in Figures 6a-6c; local parameter mapping is performed at the iterations marked by ‘▲’. Compared with the strategies in Section 4, Algorithm II takes (much) fewer steps, and at the same time catches up well with the structures of the plant, demonstrating advantages in handling structural mismatch. Instead of taking uniform load change steps, Algorithm II results in adaptive steps, i.e. {9.95%, 14.08%, 17.11%, 7.88%}. All the key outputs of the plant satisfy their constraints and stay closer to their design values at the end of load change, where we obtain a better performance index of the plant, i.e. 3.02. 1400

-2

1200 1100 1000 900 800 0▲

model plant

1300

NSSS2: K2(Wm /K

-2

1400

model plant

1300

NSSS1: K2(Wm /K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1200 1100 1000 900

1

2

iteration, k

3▲

800 0▲

4

1

2

iteration, k

3▲

4

(6a) Heat transfer coefficients.

25



1

model plant

Page 26 of 36


1020

0.9 1010

nr

T6(K)

0.8 0.7

1000

0.6 990 0.5 0▲

1

2

iteration, k

3▲

0▲

4

14.2

2

iteration, k

850


14.4

1

3▲

4


848

T7(K)

P7(MPa)

846 14 13.8

844 842

13.6 840 13.4 0▲

1

2

iteration, k

0▲

4

2

iteration, k

841

13.25 13.2

3▲

4


842

Tso(K)

13.3

1

843


13.35

Pso(MPa)

3▲

840 839 838 837

13.15

836 13.1 0▲

1

2

iteration, k

3▲

0▲

4

1

2

3▲

4

2

3▲

4

iteration, k

(6b) Key outputs. -3

x 10

100

2 90

ext

Gin(Kg/s)

1.5

1

70 60 50

0.5 0▲

80

1

2

iteration, k

3▲

0▲

4

1

26


iteration, k

100

1

90

0.9

80

0.8

xgv

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Gpump(Kg/s)

Page 27 of 36

70

0.7

60

0.6

50

0.5

40 0▲

1

2

iteration, k

3▲

0.4 0▲

4

1

2

iteration, k

3▲

4

(6c) Inputs sequence. Figure 6. Load change by Algorithm II (▲ iterations with parameter mapping).

Further, we compare the results with those by Algorithm I. The two algorithms differ in the way they maintain parameter mapping and thereby update the model. Specifically, Algorithm I updates model at every step, while Algorithm II updates model only if necessary. As expected, the former captures the characteristics of the plant more accurately (Figures 5a and 6a) and leads to better decisions; therefore, the key outputs of the plant stay closer to their design values during the course of load change (Figures 5b and 6b). On the other hand, Algorithm I needs more CPU time because of the step-wise model update. Each update takes about 100 seconds on 2.2 GHz Intel Core i5 CPU. Furthermore, the step-wise update strategy introduces more frequent perturbations to the plant. In contrast, adaptive model update mechanism does not necessarily lead to more iterations. In this case, both algorithms take 4 iterations to implement the significant load change. As mentioned above, when the parameters of interest do not have strong nonlinear variations around the operating point, a new sensitivity update may not be necessary. From a practical perspective, Algorithm II is more desirable even if a few more iterations are needed, especially for systems with many manipulated variables, because we avoid perturbing the system at each iteration. Figure 7 illustrates the load change steps by these two algorithms, as well as the prediction error for the objective function at each step. The iterations where local parameter mapping is performed are marked by ‘▲’ for Algorithm I and by ‘△’ for Algorithm II. For Algorithm I, the prediction error remains very small in the full 27



iteration process. Accordingly, the model is allowed to work on larger local regions, and the resulting load change steps are a little larger than with Algorithm II, except at the last iteration, where a shorter step reaches the target load level. Algorithm II encounters a large prediction error at the 3rd iteration, which happens in that the model has been kept unchanged in the first three steps, while the stepsize has been increasing. After rebuilding the local model and solving problem (16) in the same trust-region, Algorithm II completes load change in the next step. Algorithm I Algorithm II

Algorithm I Algorithm II

0.8

0.15

0.6

0.1

0.4

0.05

0.2

0

▲0△

▲1

▲2

iteration, k

▲3 △

abs(k-1)

0.2

abs(Δnr)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 36

0

Figure 7. Load change steps and prediction errors on objective function (▲ parameter mapping in Algorithm I, △ parameter mapping in Algorithm II).

5.4 Related Work In practice the model available for optimization is usually far from being complete, which is typically a local and/or partial approximation to the real system. As a consequence, mismatch between the model and the system should be taken into account when trying to promote system performance, and to guarantee convergence to the correct optimum. ISOPE methods enforce first-order convergence through tracking the Karush-Kuhn-Tucker (KKT) conditions of the real system. However, it is undetermined how much input correction should be implemented from one iteration to the next. Besides the commonly used relaxation methods, first-order filter methods were put forward to cope with this problem carefully to ensure stable convergence.11,23 A recently developed modifier strategy is different from the previous implementations in that it emphasizes accuracy of the underlying model,24 which is in 28


Page 29 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the same spirit as this work. However, the modifier method imposes a uniform truncation error constraint to control the output prediction error, which in fact allows the parameters to vary within a certain region such that the data-fitting and the optimization objectives can be reconciled. The bound on truncation error is crucial to performance of the modifier method but its selection is case dependent. On the other hand, in this work model accuracy is controlled through the trust-region framework; moreover, this leads to an adaptive evaluation and adjustment mechanism, rather than a uniform bound. Finally, local parameter mapping approximates the plant characteristics corresponding to the unknown/complex mechanism, which relates this work to reduced model (RM) based optimization. RMs can be constructed through model order reduction25,26 or data-driven model reduction27-30. With the trust region as a globalization strategy between the RM and the original problem to ensure convergence to the correct optimum, several methods manage the local models to simplify their construction and try to avoid frequent recourse to the original detailed model.31-34 In this work, the plant derivative evaluation corresponds to the first-order approximation, and the related cost is reduced by the adaptive model update mechanism. 6. Conclusions and Future Work Iterative strategies are often used to deal with plant-model mismatch to guarantee convergence of model-based decision-making to the plant optimum. In this paper, an integrated parameter mapping and real-time optimization strategy is proposed to address structural mismatch in the context of significant load change of HTR-PM. This trust-region-based strategy has the benefit that it forces the model itself to be adequate for decision-making and thereby leads to correct convergence. Here the basic model is determined by parameter estimation at each iteration, where the parameters that can be reliably estimated are identified by a systematic approach for parameter selection. Then the reduced model is extended to be more accurate by the trust-region framework. This is designed to integrate parameter 29



Page 30 of 36

mapping and optimization, where the parameter sensitivities are included into the model, together with the trust-region constraint, to compensate structural mismatch. The resulting method (i.e. Algorithm I) has the feature that the input correction allowed at each iteration is decided adaptively through rigorous trust-region-based optimization. Furthermore, a model update mechanism is developed to reduce the cost on sensitivity evaluation. The resulting method (i.e. Algorithm II) has an additional feature that model update is performed adaptively to maintain a valid approximation to the plant without introducing frequent perturbations. Performance and effectiveness of the proposed methods are demonstrated by the load change problem of HTR-PM, where both reactors change load from 100%RFP to 50%RFP. In future work, more general objectives will be considered for the RTO problem, with in-depth analysis on convergence properties of trust-region methods with parameter mapping. For now, we only state briefly that convergence holds because the local model must be applied whenever an improved point is not found. As a result, for problems of the form (16), as the objective tends to zero, the (reduced) gradient is also zero. Additionally, the proposed method will be extended and investigated further for dynamic applications35, as well. For set-point optimization under output constraints, a multilayer control structure was developed for ISOPE to minimize safety zones introduced to ensure feasibility.12,36 Similar measures, which work with our proposed method to secure feasibility of output constraints, will also be explored. Finally, we are also interested in comparing this work with multi-model methods. As the name suggests, the latter involve multiple predefined models. They use greatly simplified mechanism model (if there is any) or polynomials directly to describe the system within levels of input uncertainties, so as to improve control efficiency37-39. Or they include models for predefined level of uncertainties to derive robust but conservative decisions in design and operation40,41. On the other hand, although there exist unknown relations between the process characteristics (i.e. parameters) and operating conditions, we do not assume their structures a priori. Instead, we compensate this model deficiency, as well as disturbance, through process feedback. It 30


Page 31 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


can be instructive to compare performance of the operations and efficiency in generating operations according to these two kinds of methods. Notations Reactor Model: area of heat transfer specific heat at constant pressure mass flow rate relative neutron density pressure rated power temperature heat transfer coefficient (

,

,

); reactivity feedback coefficient (

,

,

)

helium leakage ratio inside lower plenum , , ) density ( , ); reactivity ( pressure loss coefficient Subscripts (Reactor) c reactor core cr from core to reflector ext external f fuel fb feedback in input m moderator r reflector 0 steady-state 1 lower plenum 2 lower header 3 riser 4 upper header 5 downcomer 6 outlet header Steam Generator Model: area of the flow cross section specific heat at constant pressure fluid mass velocity friction factor of pressure gravitational acceleration mass flow rate specific enthalpy overall heat transfer coefficient length 31



Page 32 of 36

pressure heat flux temperature heat transfer coefficient angle of steam generator tube density Subscripts (Steam Generator) fw feedwater m metal tube p primary side pump feedwater pump s secondary side 1 feedwater entry point 2 average point of economizer 3 saturate water point 4 average point of evaporator 5 saturate steam point 6 average point of superheater 7 superheater outlet point Inputs and Outputs: refer to Reactor notation refer to Steam Generator notation steam mass flow rate just before the turbine refer to Reactor notation steam pressure just before the turbine refer to Steam Generator notation steam temperature just before the turbine refer to Reactor notation refer to Steam Generator notation opening of the valve prior to the turbine refer to Reactor notation Acronyms GT-MHR GTHTR300 HTGR HTR-PM HTTR ISOPE ITS KKT MHTGR MWth

gas turbine-modular helium reactor gas turbine high temperature reactor of 300 megawatt electric nominal capacity high-temperature gas-cooled reactor high temperature gas-cooled reactor pebble bed module high temperature engineering test reactor integrated system optimization and parameter estimation iterative two-step Karush-Kuhn-Tucker modular high temperature gas-cooled reactor megawatt thermal (unit symbol for thermal performance) 32


Page 33 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


NSSS PBMR PE RFP RM RTO

nuclear steam supply system pebble bed modular reactor parameter estimation reactor full power reduced model real-time optimization

Corresponding author Zhijiang Shao, College of Control Science and Engineering, Zhejiang University, Hangzhou 310027, China. Email address: [email protected] Acknowledgements This work is supported by the National Key R&D Program of China (2017YFB0603703), the National Science and Technology (S&T) Major Project of China (ZX06902 and ZX06906), the National Science Foundation Program (61773341), the Fundamental Research Funds for the Central Universities (2018QNA5013), and the State Key Laboratory Program (ICT1804). Conflicts of interest: The authors declare no competing financial interest. References (1) Vujić, J.; Bergmann, R.M.; Škoda, R.; Miletić, M. Small modular reactors: simpler, safer, cheaper? Energy, 2012, 45, 288-295. (2) Zhang, Z.; Wu, Z.; Wang, D.; Xu, Y.; Sun, Y.; Li, F. Current status and technical description of Chinese 2×250MWth HTR-PM demonstration plant. Nuclear Engineering and Design. 2009, 239, 1212-1219. (3) Zhang, Z.; Dong, Y.; Li, F.; Zhang, Z.; Wang, H.; Huang, X. The Shandong Shidao Bay 200MWe High-temperature gas-cooled reactor pebble-bed module (HTR-PM) demonstration power plant: An engineering and technological innovation. Engineering. 2016, 2, 112-118. (4) Kravaris, C.; Hahn, J.; Chu, Y. Advances and selected recent developments in state and parameter estimation. Computers and Chemical Engineering. 2013, 51, 111-123. (5) Machado, V.C.; Tapia, G.; Gabriel, D.; Lafuente, J.; Baeza, J.A. Systematic identifiability study based on the Fisher Information Matrix for reducing the number of parameters calibration of an activated sludge model. Environmental Modelling and Software. 2009, 24, 1274-1284. (6) Deng, Z.; Deng, H.; Yang, L.; Cai, Y.; Zhao, X. Implementation of reduced-order physics-based model and multi-parameters identification strategy for lithium-ion battery. Energy. 2017, 138, 509-519. (7) Yip, W.S.; Marlin, T.E. The effect of model fidelity on real-time optimization performance. Computers and Chemical Engineering. 2004, 28, 267-280. (8) Chen. C.; Joseph, B. On-line optimization using a two-phase approach: An application study. Industrial and Engineering Chemistry Research. 1987, 26, 1924-1930. (9) Marlin, T.E.; Hrymak, A.N. Real-time operations optimization of continuous processes. AIChE Symposium Series - CPC V. 1997, 93, 156-164. (10) Xenos D.P.; Cicciotti, M.; Kopanos, G.M.; Bouaswaig, A.E.F.; Kahrs, O.; Martinez-Botas, R.; Thornhill, N.F. Optimization of a network of compressors in parallel: Real Time Optimization 33



Page 34 of 36

(RTO) of compressors in chemical plants - An industrial case study. Applied Energy. 2015, 144, 51-63. (11) Marchetti, A.; Chachuat, B.; Bonvin, D. Modifier-adaptation methodology for real-time optimization. Industrial and Engineering Chemistry Research. 2009, 48, 6022-6033. (12) Tatjewski, P.; Brdys, M.A.; Duda, J. Optimizing control of uncertain plants with constrained feedback controlled outputs. International Journal of Control. 2001, 74, 1510-1526. (13) Yang, C.; Wang, K.; Shao, Z.; Huang, X. Parameter estimation with reliability evaluation and selection refinement for steady-state model of HTR-PM. The 27th Chinese Process Control Conference. 2016, July 30 - August 1, Lanzhou, China. (14) Li, H.; Huang, X.; Zhang, L. Operation and control simulation of a modular high temperature gas cooled reactor nuclear power plant. IEEE Transactions on Nuclear Science. 2008, 55, 2357-2365. (15) Lin, W.; Biegler, L.T.; Jacobson, A.M. Modeling and optimization of a seeded suspension polymerization process. Chemical Engineering Science, 2010, 65, 4350-4362. (16) Vajda, S.; Rabitz, H.; Walter, E.; Lecourtier, Y. Qualitative and quantitative identifiability analysis of nonlinear chemical kinetic models. Chemical Engineering Communications. 1989, 83, 191-219. (17) Laan, M.J.V.D.; Pollard, K.S. A new algorithm for hybrid hierarchical clustering with visualization and the bootstrap. Journal of Statistical Planning and Inference. 2003, 117, 275-303. (18) Chu, Y.; Hahn, J. Parameter set selection via clustering of parameters into pairwise indistinguishable groups of parameters. Industrial and Engineering Chemistry Research. 2008, 48, 6000-6009. (19) Marsili-Libelli, S.; Guerrizio, S.; Checchi, N. Confidence regions of estimated parameters for ecological systems. Ecological Modelling. 2003, 165, 127-146. (20) Forbes, J.F.; Marlin, T.E. Model accuracy for economic optimizing controllers: The bias update case. Industrial and Engineering Chemistry Research, 1994, 33, 1919-1929. (21) Brdyś, M.A.; Tatjewski, P. An algorithm for steady-state optimising dual control of uncertain plants. In Proceedings of the 1st IFAC Workshop on New Trends in Design of Control Systems, Smolenice, Slovakia, 1994, 249-254. (22) Kong, X.; Yang, Y.; Chen, X.; Shao, Z.; Gao, F. Quality control via model-free optimization for a type of batch process with a short cycle time and low operational cost. Industrial and Engineering Chemistry Research, 2011, 50, 2994-3003. (23) Francois, G.; Srinivasan, B.; Bonvin, D. Comparison of six implicit real-time optimization schemes. Journal Europeen des Systemes Automatises. 2012, 46, 291-305. (24) Mandur, J.S.; Budman, H.M. Simultaneous model identification and optimization in presence of model-plant mismatch. Chemical Engineering Science. 2015, 129, 106-115. (25) Antoulas, A.C.; Sorensen, D.C. Approximation of large-scale dynamical systems: An overview. International Journal of Applied Mathematics and Computer Science. 2001, 11, 1093-1121. (26) Marquardt, W. Nonlinear model reduction for optimization and control of transient chemical processes. In Rawlings, J.B., Ogunnaike, B.A., Eaton, J.W. (eds.), Chemical Process Control VI. 2002, 12-42. (27) Burnham, A.J.; Viveros, R.; Macgregor, J.F. Frameworks for latent variable multivariate regression. Journal of Chemometrics. 1996, 10, 31-45. 34


Page 35 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(28) Morris, A.; Montague, G.; Willis, M. Artificial neural networks: Studies in process modeling and control. Transactions of the Institution of Chemical Engineers. 1994, 72, 3-19. (29) Goel, T.; Hafkta, R.T.; Shyy, W. Comparing error estimation measures for polynomial and kriging approximation of noise-free functions. Structural and Multidisciplinary Optimization. 2009, 38, 429-442. (30) Simpson, T.W.; Poplinski, J.D.; Koch, P.N.; Allen, J.K. Metamodels for computer-based engineering design: Survey and recommendations. Engineering with Computers. 2001, 17, 129-150. (31) Biegler, L.T.; Lang, Y.D.; Lin, W. Multi-scale optimization for process systems engineering. Computers and Chemical Engineering. 2014, 60, 17-30. (32) Alexandrov, N.M.; Lewis, R.M.; Gumbert, C.R.; Green, L.L.; Newman, P.A. Approximation and model management in aerodynamic optimization with variable-fidelity models. Journal of Aircraft. 2001, 38, 1093-1101. (33) Eldred, M.S.; Giunta, A.A.; Collis, S.S. Second-order corrections for surrogate-based optimization with model hierarchies. Proceedings of the 10th AIAA/ISSMO multidisciplinary analysis and optimization conference. 2004, Aug 30-Sept 1, Albany, NY. (34) Agarwal, A.; Biegler, L.T. A trust-region framework for constrained optimization using reduced order modeling. Optimization and Engineering. 2013, 14, 3-35. (35) Huang, S.; Wang, K.; Chen, W.; You, J.; Chen, X.; Qian, J.; Shao, Z.; Biegler, L.T. Simplified MFE with power-change adaption strategy for the dynamic optimization of HTR-PM. 11th International Symposium on Process Systems Engineering, Singapore, July 15-19, 2012, 31, 1301-1305. (36) Brdyś, M.A.; Tatjewski, P. Iterative Algorithms for Multilayer Optimizing Control. Imperial College Press, London, 2005. (37) Song, C.; Wu, B.; Zhao, J.; Li, P. An integrated state space partition and optimal control method of multi-model for nonlinear systems based on hybrid systems. Journal of Process Control. 2015, 25, 59-69. (38) Coelho, L.D.S.; Almeida, O.D.M.; Sumar, R.R.; Coelho, A.A.R. Predictive control of a nonlinear process using multiple models optimization based on fast evolutionary programming. In Soft Computing and Industry. Springer: London, 2002, 179-190. (39) Galán, O.; Romagnoli, J.A.; Palazoglu, A.; Arkun, Y. Gap metric concept and implications for multilinear model-based controller design. Industrial and Engineering Chemistry Research. 2003, 42, 2189-2197. (40) Puschke, J.; Zubov, A.; Kosek, J.; Mitsos, A. Multi-model approach based on parametric sensitivities - A heuristic approximation for dynamic optimization of semi-batch processes with parametric uncertainties. Computers and Chemical Engineering. 2017, 98, 161-179. (41) Puschke, J.; Mitsos, A. Robust dynamic optimization of a semi-batch emulsion polymerization process with parametric uncertainties - A heuristic approach. 11th IFAC Symposium on Dynamics and Control of Process Systems including Biosystems, Trondheim, Norway, June 6-8, 2016, 49, 907-912.

35



Page 36 of 36

TABLE OF CONTENTS

Significant load change of HTR-PM Difficulties  Plant operation over wide conditions without a model applicable to the full operating region

Concept

Integrated parameter mapping and real-time optimization Strategies

 structural mismatch due to parameters varying with operating conditions

 Determine basic model: Parameter estimation on a refined subset

 Constraints on key outputs of the plant

 Enhance model’s ability on prediction: Parameter mapping

Transformation

 Determine input corrections: Trust-region framework

A sequence of load change subproblems

 Reduce cost on operation and computation: Adaptive model update mechanism

Features/Advantages

 The model is more accurate in prediction despite the structural mismatch  The input correction allowed at each iteration is decided adaptively by rigorous optimization  The model is updated adaptively to maintain a valid approximation to the plant without introducing frequent perturbations  Significant load change is completed in a safer and more efficient way

36


Integrated Parameter Mapping and Real-Time Optimization for Load

Recommend Documents