Nonlinear Dynamic Model-Based Multiobjective Sensor Network

Article pubs.acs.org/IECR

Nonlinear Dynamic Model-Based Multiobjective Sensor Network Design Algorithm for a Plant with an Estimator-Based Control System Prokash Paul,† Debangsu Bhattacharyya,*,† Richard Turton,† and Stephen E. Zitney†,‡ †

Department of Chemical and Biomedical Engineering, West Virginia University, Morgantown, West Virginia 26506, United States National Energy Technology Laboratory, U.S. Department of Energy, Morgantown, West Virginia 26507, United States

‡

ABSTRACT: A novel sensor network design (SND) algorithm is developed for maximizing process efficiency while minimizing sensor network cost for a nonlinear dynamic process with an estimator-based control system. The multiobjective optimization problem is solved following a lexicographic approach where the process efficiency is maximized first followed by minimization of the sensor network cost. The partial net present value, which combines the capital cost due to the sensor network and the operating cost due to deviation from the optimal efficiency, is proposed as an alternative objective. The unscented Kalman filter is considered as the nonlinear estimator. The large-scale combinatorial optimization problem is solved using a genetic algorithm. The developed SND algorithm is applied to an acid gas removal (AGR) unit as part of an integrated gasification combined cycle (IGCC) power plant with CO2 capture. Due to the computational expense, a reduced order nonlinear model of the AGR process is identified and parallel computation is performed during implementation.

1. INTRODUCTION

A number of researchers have looked into synthesizing cost optimal sensor networks1−4 while considering constraints on precision, reliability, error detectability, and resilience. Cost of sensors can also be minimized considering constraints on hardware and software redundancy.5 Redundancy in measurements can improve the estimation accuracy.6 Other objective functions that have been considered in the existing literature are maximizing reliability7−9 and maximizing the value of precision.10,11 Bagajewicz12 and co-workers13−15 have investigated the economic value of precision, instrumentation upgrades, and economic value of data reconciliation. Kadu et al.16 solved an implicit multiobjective optimization problem by evaluating the impact of sampling frequencies on state estimation accuracy. However, advanced process and power plants aim at achieving higher process efficiency and therefore the loss in the efficiency due to the sensor network should be minimized for these plants. Considering an estimator-based control loop in these plants, measurements from the sensor network are used to generate estimates of the controlled variables. Since these estimated values are used by the controllers, performance of the control loop is affected by the sensor network and, therefore, plant efficiency. However, there is hardly any work in the

Optimal sensor network design (SND) is crucial for economic and efficient plant operation, monitoring, and control. In large chemical and energy plants, there are many candidate locations for sensor placement. In each of these physical locations, it would be possible to measure one or more process variables of interest. Candidate sensors can vary widely in terms of accuracy, precision, signal-to-noise ratio, reliability, and speed of response. In addition, it may be expensive, difficult, or infeasible to measure all variables of interest. These unmeasured variables should be estimated, if desired. However, it is neither economically nor physically feasible to install sensors at every candidate location. Plus such a sensor network will be highly redundant and can lead to information overload. Evaluation of all possible locations for all variables of interest for all sensor types would lead to a highly complex and computationally expensive combinatorial problem. As an alternative, usually sensors are placed using available process experience and knowledge. Such suboptimal sensor networks can lead to a number of issues. If control actions are taken based on measurements from such a network, they can lead to suboptimal operation. If process monitoring is performed using measurements obtained from a suboptimal sensor network, it can lead to unwanted product qualities, undesired process conditions, and/or violation of environmental regulations. Therefore, it is very important to find a systematic strategy for optimal SND. © XXXX American Chemical Society

Received: Revised: Accepted: Published: A

October 18, 2016 June 5, 2017 June 6, 2017 June 6, 2017 DOI: 10.1021/acs.iecr.6b04020 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

variables by trading off economic and control objectives.34 If primary controlled variables are selected with due consideration of plant economic performance and are used in an estimatorbased control system, then lower estimation accuracy of these primary controlled variables leads to lower plant efficiency. As noted earlier, the NDMSND algorithm typically leads to increased computational expense, primarily because of solution of the nonlinear dynamic model, both as a “true process” and as a process model that is used in the nonlinear estimator, until the process reaches its settling time. The computational cost of the nonlinear estimator itself is also high. These added computational expenses significantly increase because of the increased number of state variables and candidate sensor variables. In the existing literature on SND using nonlinear models, the extended Kalman filter (EKF) is considered as a state estimator.20,35 However, EKF can lead to large errors in the true posterior mean and covariance of the transformed state variables that are approximated as Gaussian random variables and can lead to suboptimal performance and divergence. Use of an unscented Kalman filter (UKF)36−43 addresses these issues by representing the state distributions by carefully chosen sample points, known as sigma points, and then propagating them through the nonlinear process.36 This results in a better approximation of the moments and also avoids numerical issues related to the linearization step of EKF. There is hardly any work on using UKF for SND. To reduce the computational expense of the NDMSND, one alternative is to use a reduced order model. Use of a reduced order model for SND is also rarely investigated in the literature. It should be noted that it is not the intent of this paper to show the superiority of one estimation technique over the other. Rather the focus of this paper is on development of an SP algorithm and our intent here is to demonstrate that the proposed SP algorithm is applicable and tractable for fully nonlinear and computationally expensive filters such as the UKF. The algorithm, of course, can be readily applied for systems where the KF or EKF is used as the state estimator. As noted earlier, typical objectives considered for SND are maximization of precision, error detectability, reliability, or minimization of sensor cost. Since these lead to a multiobjective optimization problem, the typical approach is to maximize or minimize one objective while treating others as constraints. Obviously, such an approach fails to take into account the trade-off between the competing objectives. Comprehensive reviews of multiobjective optimization for designing wireless sensor networks have been presented recently.44,45 Recently, an excellent work on multiobjective optimization has been presented by Sen et al. by considering maximization of Fisher information as well as efficiency for an IGCC plant.46 In this work, it is desired to maximize process efficiency while minimizing sensor budget, thereby requiring the solution of a multiobjective optimization problem. This optimization problem can be formulated in two stages using lexicographic optimization47,48 where the most important objective is optimized first followed by optimization of the second important objective subject to the constraint that the optimal value of the first objective obtained in the first stage optimization does not deteriorate. Solution of such lexicographic optimization is Pareto-optimal.47 To the best of our knowledge, there is no work on lexicographic optimization for SND in the open literature. Instead of solving a multiobjective optimization problem, if the sensor placement objectives can be expressed in terms of a cost, then they can be unified as a single

literature on SND to maximize process efficiency. Authors of this work have presented two SND algorithms for maximizing process efficiency.17,18 In the earlier work,17 a steady-state model-based SND (SSND) algorithm was proposed for maximizing process efficiency. However, due to consideration of the steady-state condition in the SSND algorithm, the algorithm could not capture the transient loss in efficiency of the estimator-based control system. Therefore, the later work18 developed a dynamic model-based SND (DMSND) algorithm for maximizing efficiency. However, both the SSND and DMSND algorithms used a linear process model with the linear Kalman filter for state estimation for computational tractability.19 However, use of linear models for highly nonlinear processes can lead to suboptimal SND. There are very few works published in the area of nonlinear dynamic model-based SND (NDMSND) using a nonlinear process model. The SND problem for a nonlinear continuousstirred tank reactor has been solved by Salahshoor et al.20 using a genetic algorithm (GA) approach. An optimal SND problem for a catalytic fixed-bed reactor has been investigated by Vande Wouwer et al.21 using a nonlinear distributed parameter model. Alonso et al.22 have studied optimal location and type of sensors in a low dimensional nonlinear convection−diffusion− reaction process through an efficient guided search algorithm that minimizes orthonormality distortion. Georges23 has used an approach based on nonlinear observability functions24 for determining sensor location. Lopez and Alvarez25 have presented a geometric approach to determine the degree of estimability for nonlinear systems. However, geometric approaches26,27 are computationally expensive for determining sensor locations. Nguyen and Bagajewicz28 have used an equation-based tree search method for the design of a sensor network for a nonlinear system. Singh and Hahn29 have performed an observability analysis of a system over an operating region for placing a single sensor. The authors later extended the analysis and considered measurement redundancy for placing multiple sensors.30 Lee and Diwekar have considered sensor placement in an integrated gasification combined cycle (IGCC) plant by using Fisher information as a metric of observation.31 Seenumani et al. have developed an algorithm for sensor placement in an IGCC gasifier model by minimizing the cost of the sensor network subject to the constraint of estimation accuracy.32 There is hardly any work in the literature that has explicitly considered a nonlinear dynamic model for sensor placement. As noted in our earlier papers,17,18 the objective functions commonly used in SND algorithms are variances of estimated variables or trace of the estimation error covariance matrix. These objective functions cannot be used as a measure of the process efficiency as the process efficiency depends on the actual estimation error. Therefore, the objective function must take into account dynamics of the process efficiency until its “settling time”, i.e. until the time the variation in the process efficiency settles within some desired “steadystate” value. Therefore, the SND problem needs to be solved using a nonlinear dynamic model and a nonlinear estimator explicitly until the process reaches its settling time for all candidate sensor sets. This is a challenging computational problem that has yet to be investigated in the open literature. First, it should be noted that not all control loops have a strong effect on the process efficiency. Selection of the primary controlled variables from the economic perspective has been investigated by Skogestad,33 while some of the authors of this paper have investigated selection of the primary controlled B

DOI: 10.1021/acs.iecr.6b04020 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 1. Flowchart of the NDMSND algorithm for maximizing efficiency.

• The NDMSND algorithm is presented for the first time for a nonlinear dynamic process with an estimator-based control system. • Use of a reduced order model for SND is investigated in order to achieve computational efficiency. • The NDMSND algorithm is developed for solving a novel multiobjective optimization problem for maximizing process

objective. However, the process efficiency would impact the operating cost while the cost of the sensor network is a capital cost. Therefore, for unifying these costs, use of net present value (NPV) is appropriate. In summary, the main contributions of this paper are as follows: C


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Figure 2. Flowchart of the NDMSND algorithm for minimizing budget for sensors.

2. NDMSND ALGORITHM DEVELOPMENT The estimator-based control system that is used to develop the NDMSND algorithm can be seen in Figure 1 of our earlier work.18 In this work, ud and uc are used to denote disturbance and control variables rather than d and u used in our previous publication.18 The estimator receives the noisy measurements, ynoisy,β, from the sensor network and estimates the controlled variables (ŷcont,est) and the variables for monitoring ŷmom process

efficiency and minimizing sensor budget in a lexicographic order. • A novel unifying objective function is proposed based on an NPV analysis by combining the cost of process efficiency with the cost of the sensor network. • The algorithm is applied to a large size problem that consists of thousands of process states and a large number of candidate sensors. D


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Industrial & Engineering Chemistry Research ⎡ tol(1) ⎤ ⎢ ⎥ TOL = ⎢⋮ ⎥ ⎢ ⎥ ⎣ tol(m)⎦

performance.18 The controller (controllers) then implements (implement) the corrective action on the process based on the estimated controlled variables. For developing the NDMSND algorithm, the set of equations corresponding to each block in the estimator-based control system is first organized. The NDMSND algorithm is developed for multiobjective optimization. Several techniques for multiobjective optimization exist in the literature.47 In this work, the lexicographic approach to multiobjective optimization is used. Several mathematical nonlinear lexicographic optimization problems have been reported in the literature.48 In this approach, the objective functions are arranged in lexicographic order where goals are optimized in order of decreasing importance. A lexicographic optimization can be written as min f1 (x), f2 (x), ..., fk (x)

Here Esterr(m)i denotes estimation error of the mth variable at the ith instant and tol(m) denotes tolerance for the mth variable. In step A1, the integer problem is solved subject to the constraints corresponding to each block of the estimator-based control system in Figure 1. The additional constraints on the budget and the estimation accuracy are represented by eqs 3 and 4, respectively, where ci denotes the cost of a sensor of type i. In step A2, the budget for the sensors is minimized subject to the same constraints as those in step A1, except that the additional constraints are on the deviation in efficiency and the estimation accuracy as denoted by eqs 6 and 7, respectively. Δηest and Esterr are defined as

(1)

s.t. g (x ) ≤ 0

Δηest = ηopt − η(xact , β)

(8)

h(x) = 0

Esterr = yma,act − yma ̂

(9)

In eq 8, ηopt denotes the optimal efficiency, i.e., efficiency obtained under perfect measurement and η(xact,β) denotes the efficiency obtained in the estimator-based system. Equation 9 gives the estimation error in key variables, denoted by Esterr, calculated by subtracting the actual values from the estimated values of process variables. The constraint on estimation accuracy in eq 4 can be eliminated reducing the number of constraints to just 1 for each of the steps, and the objective function for STEP A1 is reformulated as follows:

x∈S where the objective functions f i(x) are optimized based on their priority in a specific problem. In problem 1, the most important objective function is f1(x) while the least important is f k(x). In the NDMSND algorithm, maximization of process efficiency (or minimization of the deviation from optimal efficiency) is considered to be the most important objective followed by minimization of cost of the sensor network. Therefore, the following lexicographic optimization problem is solved: step A1

step B1 k

minβ[λ1 ∑ (Δηest, i)2 + (SSEE)T λ 2]

k

minβ ∑ (Δηest, i)2

(2)

i=1

∑ ciβi ≤ b;

βi = 0, 1

∀ i ∈ Ns

∀i

∑ ciβi ≤ b;

βi = 0, 1

∀ i ∈ Ns

∀i

In eq 10, λ1 and λ2 are the weighting factors. Similarly, step A2 is reformulated as

(3) (4)

SSEE < TOL

step B2

step A2 minβ ∑ ciβi ;

(10)

i=1

minβ[λ1′ ∑ ciβi + (SSEE)T λ 2′]; βi = 0, 1

∀ i ∈ Ns

∀i

∀ i ∈ Ns (11)

(5) k

k

∑ (Δηest,i)2 ≤ optimal value obtained from step A1

βi = 0, 1

∀i

∑ (Δηest,i)2 ≤ k′·(optimal value obtained from step B1)

i=1

(6)

SSEE < TOL

(7)

i=1

In eq 11, λ′1 and λ′2 are the weighting factors and k′ is the goal factor. This reformulation greatly helped in avoiding issues with constraint violation while using MATLAB’s genetic algorithm toolbox for solving this two-stage optimization problem. In eq 10, λ1 and λ2 can be adjusted to weigh the deviation from optimal efficiency over the estimation error. Similarly, in eq 11, λ1′ and λ2′ can be adjusted to weight the cost of the sensor network over the estimation error. These adjustments would reflect the user perspective with due consideration of the requirements of the underlying process. For example, if the user would like to sacrifice the estimation error in one or more variables, a lower weighting factor would be preferred for those

Here ⎡ k ⎤ ⎢∑ Esterr(1)i 2 ⎥ ⎢ i=1 ⎥ ⎢ ⎥ SSEE = ⎢⋮ ⎥ ⎢ k ⎥ ⎢ 2⎥ ⎢∑ Esterr(m)i ⎥ ⎣ i=1 ⎦ E


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Industrial & Engineering Chemistry Research variables. However, if the variable of interest is used for monitoring of a critical equipment item, then there may be strict requirements in the tolerable error. If there is a strict tolerance that must be satisfied, one can consider penalty or barrier terms that penalize the objective function heavily in case of constraint violation. Readers are referred to the rich literature in this area.49,50 2.1. NDMSND Algorithm. The two-step NDMSND algorithms are shown in Figures 1 and 2. The optimization problems for both steps are solved using a GA. In the first step, process efficiency is maximized by following the algorithm shown in Figure 1. The algorithm for the next step is shown in Figure 2, where the cost of sensors is minimized. In this step, the estimation accuracy and the efficiency obtained for optimal sets of sensors for different budgets are used as constraints. The same optimal set of sensors is obtained if the solution is unique. For better understanding of the perspective underlying the NDMSND algorithm, interested readers are referred to our previous publication18 of linear dynamic model-based SND algorithm. 2.2. Estimator. In the NDMSND algorithm, a nonlinear estimator is considered to estimate states given the measurements and the nonlinear model. While the proposed algorithm can be readily applied to other linear and nonlinear filters such as the KF and extended Kalman filter (EKF), in this work we have considered only UKF that can be used for a general nonlinear system given by xk = f (xk − 1 , uk − 1) + wk − 1

W 0c =

λ + 1 − αsc 2 + βsc L+λ

Wim = W ic =

1 , 2(L + λ)

(16)

i = 1, ..., 2L

(17)

Initialization (Each Step Completed Once before Filtering). Q k = E[wkwkT],

R k = E[vkvkT]

(18)

P0 = E[(x0 − x0̂ )(x0 − x0̂ )T ]

x0̂ = E[x0],

(19)

Executing the Filter Recursively (Each Step at Every Discrete Time). Four steps are included. Step 1. Generation of the sigma points: Pk − 1 | k − 1 = chol(Pk − 1 | k − 1)

(lower Cholesky decomposition) xk − 1 | k − 1 = [xk̂ − 1 | k − 1 xk̂ − 1 | k − 1 −

xk̂ − 1 | k − 1 +

L + λ Pk − 1 | k − 1

L + λ Pk − 1 | k − 1 ]

(20)

Step 2. Prediction transformation: xk(|i)k − 1 = f (xk(−i) 1 | k − 1 , uk − 1),

i = 0, ..., 2L

(21)

2L

xk̂ | k − 1 =

∑ Wimxk(|i)k− 1

(22)

i=0

(12)

2L

yk = h(xk) + vk

Pk | k − 1 = Q k − 1 +

(13)

i=0

(23)

xk and yk are the state and measurement vectors, respectively. f(.) and h(.) are the process and measurement nonlinear vector functions. Random vectors w and v denote process and measurement noise, which are both assumed to be zero-mean white noise with known covariance and uncorrelated. The unscented transformation (UT) is the central technique of the UKF. Here, x is a random variable which is typically assumed to be normally distributed (Gaussian) with mean, x,̅ and covariance, Px. The UT uses a small set of deterministically selected points, called sigma points. The spread of these points depends on the values of the scaling parameters and the weight vectors. The scaling of the UT can be fully represented by three scaling parameters.42,43 The primary scaling parameter, αsc, determines the spread of the sigma points. The secondary scaling parameter, βsc, is used to include information about the prior distribution (for Gaussian distributions, βsc = 2 is optimal). The tertiary scaling parameter, κsc, is usually set to zero.40 Using these three scaling parameters, an additional scaling parameter, λ, and weight vectors, Wm (mean) and Wc (covariance), are calculated. Notations and steps used here have mainly followed the work of Rhudy and Gu51 as well as Wan and Merwe.43 Off-Line Calculations (Each Step Completed Once before Filtering). Define scaling parameters and weight vectors: λ = αsc 2(L + κ ) − L

(14)

λ L+λ

(15)

W0m =

∑ Wic(xk(|i)k− 1 − xk̂ | k− 1)(xk(|i)k− 1 − xk̂ | k− 1)T

Step 3. Observation transformation: ψk(|ik)− 1 = h(xk(|i)k − 1),

i = 0, ..., 2L

(24)

2L

yk̂ | k − 1 =

∑ Wimψk(|ik)− 1

(25)

i=0 2L

Pkyy| k − 1 = R k − 1 +

∑ Wic(ψk(|ik)− 1 − yk̂ | k− 1 )(ψk(|ik)− 1 − yk̂ | k− 1 )T i=0

(26) 2L

Pkxy| k − 1 =

∑ Wic(xk(|i)k− 1 − xk̂ | k− 1)(ψk(|ik)− 1 − yk̂ | k− 1 )T i=0

(27)

Step 4. Measurement update:

Kk = Pkxy| k − 1(Pkyy| k − 1)−1

(28)

xk̂ | k = xk̂ | k − 1 + Kk(yk − yk̂ | k − 1 )

(29)

Pk | k = Pk | k − 1 − KkPkyy| k − 1Kk T

(30)

It should be noted that, in the UKF algorithm presented above, the resampling step has been omitted; i.e. the sigma points obtained from the time update have been used instead of generating new sigma points. This was done to save computational effort, albeit at the cost of the estimator performance, since the estimator algorithm is solved as part of the SP algorithm which is a large-scale combinatorial problem. Even though the resampling step is omitted here, F


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Industrial & Engineering Chemistry Research readers are strongly encouraged to investigate the effect of the resampling step on the results and computational expense of the SP algorithm. It should be noted that the SP algorithm proposed here can be used for other formulations/algorithms of nonlinear estimators as well. Interested readers are referred to an excellent reference on nonlinear estimators for more information.52 2.3. System Identification. In this work, a discrete-time linear state space model given by eq 31 is used as the process model while an additive, second order model, given by eq 32, is used as the measurement model. Obviously other forms of nonlinear process and measurement models can be readily used in this SP algorithm.

xk = Φxk − 1 + Buk − 1 r

yk(m) =

Table 1. List of Candidate Sensors in the AGR Unit 1 2 3 4 5 6 7 8 9 10 11 12

(31) r

r

13 14 15 16 17 18 19 20 21 22

∑ θ1,(im) xk(i) + ∑ ∑ θ2,(im,j)xk(i)xk(j) i=1

i=1 j=1

(32)

y(m) k

In eq 32, denotes the mth measurement at the kth instant. It can be observed that eq 32 is a linear-in-parameter (LIP) model of the general form Fy = Gxθ + ε

(33)

θ = (Gx TGx)−1Gx TFy

(34)

Gx and Fx denote the time-series data of regressors and regressand, respectively. Ordinary least-squares estimates of the parameter vector θ are given by eq 34. Here the size of the parameter vector is optimally selected by considering the Akaike information criterion (AIC) of the candidate models53 by utilizing the data from the high-fidelity process model. More information about the methodology used for model identification can be found in a paper coauthored by the corresponding author of this paper.54 2.4. Net Present Value (NPV) Optimization. Cost of the sensor network is a capital cost, while the resulting efficiency due to a sensor network affects the operating cost. Both costs can be unified through NPV calculation. As all operating and capital costs do not need to be considered in such a formulation; a partial NPV (pNPV) analysis55 as shown in eq 35 will suffice.

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

pNPV ($) = −sensor network cost ($) − OC ($/year) ⎛ (1 + i)n − 1 ⎞⎛ 1 ⎞ ⎟ ⎜ ⎟⎜ ⎝ i(1 + i)n ⎠⎝ 1 + i ⎠ (35)

In eq 35, OC, i, and n denote operating cost, internal hurdle rate, and operating life, respectively. Since pNPV will be negative, minimization of pNPV is considered to be the objective function. Obviously, minimization of pNPV also turns the multiobjective optimization problem mentioned in the beginning section 2 into a single objective optimization problem. Therefore, only step 1 of the two-step optimization is carried out and the objective function is modified to be min[λ3(pNPV) + (SSEE)T λ4]

Temperature Sensors syngas cooler inlet inlet to H2O knockout drum top outlet of H2O knockout drum clean syngas at top of CO2 absorber stripped gas cooler outlet off gas from top of H2S absorber rich solvent at H2S absorber bottom rich solvent at inlet to H2S concentrator N2 gas flow to H2S concentrator H2S concentrator vapor outlet stripped solvent at bottom of Selexol stripper stripped solvent at bottom of Selexol stripper Pressure Sensors inlet to H2O knockout drum top outlet of H2O knockout drum clean syngas at top of CO2 absorber loaded solvent at bottom of CO2 absorber rich solvent at inlet to H2S concentrator stripped solvent at bottom of Selexol stripper top outlet of acid gas knockout valve outlet at bottom of acid gas knockout drum Selexol stripper top outlet stripped solvent at bottom of Selexol stripper Flow Sensors inlet to H2O knockout drum clean syngas at top of CO2 absorber off gas from top of H2S absorber H2S concentrator vapor outlet stripped solvent at bottom of Selexol stripper top outlet of acid gas knockout CO2 Analyzers H2O knockout drum bottom outlet off gas from top of H2S absorber H2S concentrator vapor outlet top outlet of acid gas knockout Selexol stripper top outlet H2S Analyzers H2O knockout drum bottom outlet loaded solvent at bottom of CO2 absorber off gas from top of H2S absorber rich solvent at H2S absorber bottom H2S concentrator vapor outlet stripped solvent at bottom of Selexol stripper top outlet of acid gas knockout valve outlet at bottom of acid gas knockout drum Selexol stripper top outlet

with CO2 capture.17 The two-stage, Selexol solvent-based AGR unit for selectively removing hydrogen sulfide (H2S) and CO2 from syngas has been described in detail in our earlier publications.17,18 The linear process model is identified by linearizing the nonlinear Aspen Plus Dynamics model of the AGR unit around the nominal operating condition as described in our earlier publications.17,18 The nonlinear measurement model, involving 62 dominant state variables, is identified by using the measurement results obtained from the model of the AGR unit by simulating a pseudorandom binary signal of multiple inputs. Additive Gaussian white noise is used for characterizing process and measurement noises. Variations in the syngas flow rate are considered as the disturbance.19The

(36)

Clearly, the budget constraint is eliminated in this case.

3. CASE STUDY This section illustrates the application of the NDMSND algorithm to the acid gas removal (AGR) unit as part of an integrated gasification combined cycle (IGCC) power plant G


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Figure 3. Sequential changes in disturbances: (a) inlet pressure and (b) CO2 flow rate in inlet syngas.

Figure 4. Comparison between APD model and identified model for (a) COS molar holdup in the cooler located immediately after the high pressure CO2 compressor and (b) H2O molar holdup on the first tray of the CO2 absorber.

Figure 5. Step change in (a) inlet pressure and (b) CO2 flow rate in the inlet syngas. Identified model.

disturbance in the inlet flow rate of the acid gas is simulated by changing the inlet pressure of the syngas to the AGR unit. The SND algorithm is generic. Changes in the controller set points and the multiple disturbances can be readily implemented. Equation 37 defines the efficiency of the AGR unit where the numerator represents the amount of CO2 captured and the denominator is the megawatt hours (MWh) power consumption. η(xact , β) =

For the pNPV calculation as shown in eq 35, the partial operating cost is calculated by eq 38. It should be noted that the electricity consumption in eq 37 is calculated by considering not only the direct electricity consumption in the AGR unit, but also the thermal energy consumption being converted to equivalent electric consumption. For more details, interested readers are referred to our earlier publications.17,18 OC =

FCO2,in(xact) − FCO2,out(xact) 3

aFsolvent(xact) + ∑c = 1 Pc(xact)

×

(37) H

mol of CO2 capture MWh × 1000 × mol of CO2 capture year

price ($) of electricity kWh

(38) DOI: 10.1021/acs.iecr.6b04020 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 6. Response in (a) CO2 molar holdup on the first tray of the CO2 absorber and (b) CO2 molar holdup on the fifth tray of the Selexol stripper.

In addition, for eq 35, n = 9 years and i = 0.1 per year are assumed. A total of 42 candidate sensors in the AGR unit were evaluated, and Table 1 shows the distribution of these candidate sensors in the AGR unit. Information on the accuracy and cost of commercially available sensors was obtained from Liptak56 and was also provided in our earlier publications.17,18 Jones et al.34 have identified a number of primary controlled variables for the AGR unit. These primary control loops implement the estimatorbased control action. The developed NDMSND algorithm is implemented by using the GA18 available in the Global Optimization Toolbox in MATLAB. The Parallel Computing Toolbox from MathWorks was used in GA during implementation of the NDMSND algorithm. For the first step of optimization the CPU time was approximately 4−5 h, while for the second step the CPU time was approximately 3−4 h. Finally, the CPU time for the pNPV optimization was approximately 4 h.

from the APD model. The error in the identified model is found to be acceptable. Two such comparison studies will be presented here. Figure 3 shows sequential step changes in multiple disturbances; namely the change in the inlet pressure is shown in Figure 3a, while the composition of the inlet syngas (simulated by changing the CO2 content) is shown in Figure 3b. Parts a and b of Figure 4 show that the results for the identified nonlinear model compare well with those for the rigorous APD model for carbonyl sulfide (COS) molar holdup in the cooler located immediately after the high-pressure CO2 compressor and H2O molar holdup on the first tray of the CO2 absorber, respectively. Parts a and b of Figure 5 show single step changes in the inlet pressure and composition of the inlet syngas, respectively. Parts a and b of Figure 6 show a good comparison between the APD model and the identified model for the CO2 molar holdup on the first tray of the CO2 absorber and the CO2 molar holdup on the fifth tray of the Selexol stripper, respectively. AIC values for models of the variables presented in Figures 4 and 6 are listed in Table 2. Overall, the maximum relative error was less than 10%. It is also important to note that the reduced model is much more computationally efficient than the rigorous process model. With the reduced model the CPU time for evaluation of one candidate sensor was less than 1 s while the rigorous process model took more than 300 s. The sampling interval was 0.02 h. 4.2. NDMSND Results. Table 3 shows the results of the case studies for different budgets while maximizing (CO2 capture) efficiency, i.e., in step 1 of the two-step lexicographic optimization approach. It should be noted that the values of the efficiency for all results presented below are in terms of deviation from the maximum efficiency obtained without any estimation error, while the ΔpNPV values are given in terms of the deviation from the ΔpNPV obtained without placing any sensors. The case studies in Table 3 show that, as the budget increases, the number of sensors changes, which, in turn, increases efficiency.

4. RESULTS AND DISCUSSION 4.1. Nonlinear AGR Model. The identified nonlinear AGR model is tested by simulating a number of disturbances in the Table 2. AIC Values for the Models of the Variables in Figures 4 and 6 Figure

variable

AIC

4a

COS molar holdup in the cooler located immediately after the high pressure CO2 compressor H2O molar holdup on the first tray of the CO2 absorber CO2 molar holdup on the first tray of the CO2 absorber CO2 molar holdup on the fifth tray of the Selexol stripper

−2288.8

4b 6a 6b

−507.5 −98.79 −552.68

rigorous AGR process model implemented in Aspen Plus Dynamics (APD). Transient responses of a large number of variables from the identified model are compared with those

Table 3. Number of Sensors, Value of Objective Functions, and Integral Deviation from Optimal Efficiency for Different Cost of Sensors While Maximizing Efficiency in Step 1 case study

total no. of sensors

cost of sensors

∑Δηest (kmol of CO2/MWh)

ΔpNPV ($ million)

computation time

1a 2a 3a

24 21 22

$321,600 $65,400 $47,200

2.57 14.79 21.05

−2.074 −1.245 −0.701

2 h 24 min 3 h 10 min 3 h 55 min

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Table 4. Number of Sensors, Value of Objective Function, and Integral Deviation from Optimal Efficiency While Minimizing Budget case study

total no. of sensors

cost of sensors

∑Δηest (kmol of CO2/MWh)

ΔpNPV ($ million)

computation time

1b 2b 3b

24 22 20

$151,200 $63,400 $40,400

2.65 15.12 23.07

−2.237 −1.217 −0.525

2 h 40 min 3 h 4 min 4 h 8 min

Table 5. Analysis of Sensor Sets in Tables 3 and 4 solution sets from Table 3 (maximizing efficiency)

solution sets from Table 4 (minimizing budget)

sensor type

cost per sensor

case 1a ($321,600)

case 2a ($65,400)

case 3a ($47,200)

case 1b ($151,200)

case 2b ($63,400)

case 3b ($40,400)

temperature pressure flow H2S analyzer CO2 analyzer

$1,000 $2,200 $4,000 $70,000 $10,000

8 8 4 4 −

8 7 3 − 3

10 6 6 − −

8 6 5 1 4

8 7 5 − 2

9 7 4 − −

Figure 9. Comparison of transient efficiency profiles obtained using the sensor networks synthesized by the DMSND and NDMSND algorithms.

Figure 7. Optimal profile vs performance of sensor network obtained by NDMSND.

Figure 8. Comparison of efficiency profiles obtained using the sensor network synthesized by the SSND and NDMSND algorithms.

Table 4 shows the results from the step 2 optimization. The minor difference in the respective optimal efficiency between step 1 and step 2 is due to the goal factor. It is observed that, in case 1b, the sensor network cost is reduced by more than 50% in comparison to case 1a with a minor change in the efficiency. There is a minor difference in costs of sensors between cases 2a and 2b, while there is about 14% reduction in the cost of sensors between cases 3a and 3b. Case 1b has the best ΔpNPV value out of all the cases presented in Tables 3 and 4. Table 5 shows the analysis of sensor sets obtained from the multiobjective optimization. It is observed that, while maximizing efficiency, GA selects more expensive sensors for achieving desired estimation accuracy in comparison to the

Figure 10. (a) ΔpNPV vs budget; (b) deviation in efficiency vs budget.

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$151,200 for NDMSND compared to $180,400 for the DMSND algorithm.18 4.3. NPV Optimization. For the NPV calculation, the following economic parameters were used: plant life = 10 years, period of operation (n) = 9 years, and i = 0.1 per year. Figure 10a shows the change in ΔpNPV ($ million) with the increase in budget ($ thousand). ΔpNPV values other than the optimal value are also shown in Figure 10a for comparison with the multiobjective optimization approach. Figure 10b shows the deviation in efficiency with an increase in the budget. It is observed that the ΔpNPV optimization yields efficiency and sensors cost that are closer to that obtained in case 1b. The final optimal senor set is the one that is obtained from the ΔpNPV optimization study at a sensor cost of $124,000. This sensor set is shown in Table 6.

Table 6. Optimal Set of Sensors at $124,000 1 2 3 4 5 1 2 3 4 5 1 2 1 1 2 3

Temperature inlet to H2O knockout drum clean syngas at top of CO2 absorber stripped gas cooler outlet H2S concentrator vapor outlet stripped solvent at bottom of Selexol stripper Pressure clean syngas at top of CO2 absorber loaded solvent at bottom of CO2 absorber stripped solvent at bottom of Selexol stripper top outlet of acid gas knockout stripped solvent at bottom of Selexol stripper Flow clean syngas at top of CO2 absorber top outlet of acid gas knockout H2S Analyzer off gas from top of H2S absorber CO2 Analyzer H2S concentrator vapor outlet top outlet of acid gas knockout Selexol stripper top outlet

5. CONCLUSIONS In this paper, a novel NDMSND algorithm for efficiency maximization of an estimator-based control system is presented. The nonlinear UKF is used for estimation. The nonlinear model used in this sensor placement study is a polynomial model. The NDMSND algorithm follows a lexicographic approach to multiobjective optimization where, in this study, the most important objective (process efficiency) is optimized first followed by optimization of the least important objective (the sensor network cost). Optimization of the pNPV that includes the cost of the sensor network as a capital cost and the cost of the deviation from the optimal efficiency as an operating cost is proposed as an alternative to the multiobjective optimization problem. The algorithm is applied to a large-scale problem comprising a large number of state variables and candidate sensor locations. The polynomial model with cross-terms is found to be satisfactory for this process. It is observed that the lexicographic optimization helps to achieve almost the same efficiency even at a lower sensor cost for each case study. It is observed that the SSND can lead to loss in efficiency during transients. While the sensor network synthesized by the DMSND algorithm can yield a similar transient efficiency profile as the sensor network synthesized by the NDMSND algorithm, the corresponding sensor network cost is lower for the NDMSND algorithm. The ΔpNPV optimization helps to identify the final optimal set of sensors by evaluating the trade-off between the cost of sensors and process efficiency. It is observed that the efficiency and sensor cost obtained from the ΔpNPV optimization is similar to the step 2 solution from the multiobjective optimization for higher budget.

sensors that are selected while minimizing the sensor cost. In case 1a, four H2S analyzers are selected that account for more than half of the cost, but the corresponding step 2 solution, i.e., case 1b, selects only one H2S analyzer. Comparison of the sensors between cases 2a and 2b shows that even though there is insignificant difference in the sensor cost between these two cases, case 2b has more flow sensors while case 2a has more CO2 analyzers. It can be observed that the total number of sensors for case study 2b in Table 4 is higher than the total number of sensors for case study 2a in Table 3. This is because case 2a selects three flow sensors ($4,000 each) while case 2b selects five. In contrast, case 2a selects three CO2 analyzers ($10,000 each) while case 2b selects two CO2 analyzers. Figure 7 shows the comparison between the optimal efficiency profile, i.e., efficiency obtained in the absence of any estimation error and the efficiency profile obtained using the sensor network from the NDMSND algorithm. There is hardly any difference between the two profiles. It should be noted that, during 1.5−3.5 h, the optimal profile is slightly exceeded by the optimal sensor network mainly because of the minor mismatch between the Aspen Plus Dynamics model that is used to generate the optimal profile and the identified model. Figure 8 shows the comparison of the efficiency profiles obtained using sensor networks synthesized by the SSND17 and the NDMSND algorithms. The sensor network obtained from the SSND algorithm results in an efficiency profile that approaches the optimal efficiency at steady state, but the profile is inferior to that obtained using the NDMSND algorithm during transient operation. Figure 9 shows the comparison of the efficiency profiles obtained using sensor networks synthesized by the DMSND and NDMSND algorithms. The efficiency obtained using the sensor network from the NDMSND algorithm matches with that obtained using the sensor network from the DMSND algorithm. Also, they compare well during transient response. However, due to use of the nonlinear model and UKF and due to the lexicographic optimization, the desired estimation accuracy is obtained at a lower sensor network cost of

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AUTHOR INFORMATION

Corresponding Author

*Tel.:304-293-9335. Fax: 304-293-4139. E-mail: Debangsu. [email protected]. ORCID

Debangsu Bhattacharyya: 0000-0001-9957-7528 Notes

This project was funded by the Department of Energy, National Energy Technology Laboratory, an agency of the United States Government, through a support contract with URS Energy & Construction, Inc. Neither the United States Government nor any agency thereof, nor any of their employees, nor URS Energy & Construction, Inc., nor any of their employees, makes any warranty, expressed or implied, or assumes any legal K


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Industrial & Engineering Chemistry Research v = measurement noise vector W = weighting factor w = process noise vector X = observation matrix x = vector of states y = vector of measurements ŷ = vector of estimated variables z = noisy measurements, composition

liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. The authors declare no competing financial interest.

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Subscripts

ACKNOWLEDGMENTS

As part of the National Energy Technology Laboratory’s Regional University Alliance (NETL-RUA), a collaborative initiative of the NETL, this technical effort was performed under RES Contract No. DE-FE0004000.

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NOMENCLATURE a = power consumption coefficient due to solvent regeneration αsc = primary scaling parameter b = scalar budget ($) βsc = secondary scaling parameter ci = cost of individual sensor ($) ψ = predicted measurement chol(.) = lower Cholesky decomposition E(.) = expected value ε(t) = deviation of the controlled variable from set point Esterr = estimation error η = efficiency F = flow rate (mol/h) f(.) = process polynomial Gen = number of generations in GA θ = parameter vector h(.) = measurement polynomial i = internal hurdle rate K = steady state Kalman gain matrix k = time instant k′ = goal factor κ = tertiary scaling parameter L = length of state vector λ = scaling parameter Ns = total number of candidate sensor n = number of states, number of years in NPV analysis OC = operating cost P = pressure P, P0 = state error variance−covariance matrix Pc = consumed power (MWh) Pyy = measurement error variance−covariance matrix Pxy = state-measurement error variance−covariance matrix pNPV = partial net present value Q = process noise variance−covariance matrix R = measurement noise variance−covariance matrix s = polynomial T = temperature tss = time to reach steady state TOL = tolerance vector u = inputs uc = vector of control inputs ud = vector of disturbance

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act = actual variable β = decision variable vector of binary numbers (0 and 1) CO2 = carbon dioxide c = covariance, control action d = disturbance est = estimated/estimator-based H2S = hydrogen sulfide in = inlet or, input ma = process monitoring variables and active constraints mom = monitoring variables noisy = noisy measurements opt = optimal out = outlet or output s = set of measured variables set = desired set point solvent = Selexol solvent sc = scale

REFERENCES

(1) Bagajewicz, M.; Cabrera, E. New MILP formulation for instrumentation network design and upgrade. AIChE J. 2002, 48, 2271−2282. (2) Bagajewicz, M. Design and retrofit of sensor networks in process plants. AIChE J. 1997, 43 (9), 2300−2306. (3) Bagajewicz, M.; Sanchez, M. Cost-optimal design of reliable sensor networks. Comput. Chem. Eng. 2000, 23 (11−12), 1757−1762. (4) Chmielewski, D. J.; Palmer, T.; Manousiouthakis, V. On the theory of optimal sensor placement. AIChE J. 2002, 48, 1001−1012. (5) Kelly, J. D.; Zyngier, D. A new and improved MILP formulation to optimize observability, redundancy and precision for sensor network problems. AIChE J. 2008, 54, 1282−1291. (6) Kretsovalis, A.; Mah, R. S. Effect of redundancy on estimation accuracy in process data reconciliation. Chem. Eng. Sci. 1987, 42, 2115−2121. (7) Ali, Y. Sensor Network Design for Maximizing Reliability of Chemical Processes. Ph.D. Thesis, Indian Institute of Technology Kanpur, 1993. (8) Ali, Y.; Narasimhan, S. Sensor network design for maximizing reliability of linear processes. AIChE J. 1993, 39, 820−828. (9) Ali, Y.; Narasimhan, S. Redundant Sensor Network Design for Linear Processes. AIChE J. 1995, 41 (10), 2237−2249. (10) Bagajewicz, M.; Markowski, M. Instrumentation design and upgrade using an unconstrained method with pure economical objectives. Presented at the conference on Foundations of Computer Aided Process Operations, Coral Springs, FL, USA, 2003. (11) Bagajewicz, M.; Markowski, M.; Budek, A. Economic value of precision in the monitoring of linear systems. AIChE J. 2005, 51 (4), 1304−1309. (12) Bagajewicz, M. On the definition of software accuracy in redundant measurement systems. AIChE J. 2005, 51 (4), 1201−1206. (13) Bagajewicz, M. On a new definition of a stochastic-based accuracy concept of data reconciliation-based estimators. Proceedings of the 15th European Symposium on Computer-Aided Process Engineering; Elsevier: 2005. (14) Bagajewicz, M. Value of accuracy in linear systems. AIChE J. 2006, 52 (2), 638−650. L


Article

Industrial & Engineering Chemistry Research (15) Bagajewicz, M.; Nguyen, D. Stochastic-based accuracy of data reconciliation estimators for linear systems. Comput. Chem. Eng. 2008, 32 (6), 1257−1269. (16) Kadu, S. C.; Bhushan, M.; Gudi, R. Optimal sensor network design for multirate systems. J. Process Control 2008, 18 (6), 594−604. (17) Paul, P.; Bhattacharyya, D.; Turton, R.; Zitney, S. E. Sensor network design for maximizing process efficiency: an algorithm and its application. AIChE J. 2015, 61 (2), 464−476. (18) Paul, P.; Bhattacharyya, D.; Turton, R.; Zitney, S. E. Dynamic Model-Based Sensor Network Design Algorithm for System Efficiency Maximization. Comput. Chem. Eng. 2016, 89, 27. (19) Paul, P.; Bhattacharyya, D.; Turton, R.; Zitney, S. E. Adaptive Kalman filter for estimation of environmental performance variables in an acid gas removal process. Proceedings of the 2013 American Control Conference 2013, 2717−2721. (20) Salahshoor, K.; Bayat, M. R.; Mosallaei, M. Design of Instrumentation Sensor Networks for Non-Linear Dynamic Processes Using Extended Kalman Filter. Iranian J. Chem. Eng. 2008, 27 (3), 11− 23. (21) Vande Wouwer, A.; Point, N.; Porteman, S.; Remy, M. An approach to the selection of optimal sensor locations in distributed parameter systems. J. Process Control 2000, 10, 291. (22) Alonso, A. A.; Kevrekidis, I. G.; Banga, J. R.; Frouzakis, C. E. Optimal sensor location and reduced order observer design for distributed process system. Comput. Chem. Eng. 2004, 28, 27. (23) Georges, D. The use of observability and controllability gramians or functions for optimal sensor and actuator location in finite-dimensional systems. Proceedings of the 34th Conference on Decision and Control, New Orleans, LA, USA; IEEE: 1995; p 3319. (24) Scherpen, J. M. A. Balancing of nonlinear systems. Syst. Control Lett. 1993, 21, 143. (25) Lopez, T.; Alvarez, J. On the effect of the estimation structure in the functioning of a nonlinear copolymer reactor estimator. J. Process Control 2004, 14, 99. (26) Isidori, A. Nonlinear Control Systems, 3rd ed.; Springer-Verlag: New York, 1995. (27) Hermann, R.; Krener, A. J. Nonlinear controllability and observability. IEEE Trans. Autom. Control 1977, 22, 728. (28) Nguyen, D.; Bagajewicz, M. Design of nonlinear sensor networks for process plants. Ind. Eng. Chem. Res. 2008, 47, 5529− 5542. (29) Singh, A. K.; Hahn, J. Determining optimal sensor locations for state and parameter estimation for stable nonlinear system. Ind. Eng. Chem. Res. 2005, 44, 5645−5659. (30) Singh, A. K.; Hahn, J. Sensor location for stable nonlinear dynamic system: Multiple sensors case. Ind. Eng. Chem. Res. 2006, 45, 3615−3623. (31) Lee, A.; Diwekar, U. Optimal sensor placement in integrated gasification combined cycle power systems. Appl. Energy 2012, 99, 255−264. (32) Seenumani, G.; Dai, D.; Lopez-Negrete, R.; Kumar, A.; Dokucu, M.; Kumar, R. An outer-approximation based algorithm for solving integer non-linear programming problems for optimal sensor placement. 51st IEEE Conference on Decision and Control 2012, 4455−61. (33) Skogestad, S. Control structure design for complete chemical plants. Comput. Chem. Eng. 2004, 28, 219−234. (34) Jones, D.; Bhattacharyya, D.; Turton, R.; Zitney, S. E. PlantWide Control System Design: Primary Controlled Variable Selection. Comput. Chem. Eng. 2014, 71, 220−234. (35) Anderson, B. D. O.; Moore, J. B. Optimal Filtering; PrenticeHall; Englewood Cliffs, NJ, 1979. (36) Julier, S.; Uhlmann, J. Unscented filtering and nonlinear estimation. Proc. IEEE 2004, 92, 401−422. (37) Sorenson, H.; Alspach, D. Recursive Bayesian estimation using Gaussian sums. Automatica. 1971, 7, 465−479. (38) Soderstrom, T. Discrete Time Stochastic Systems; Springer: New York, 2002.

(39) Straka, O.; Dunik, J.; Simandl, M. Gaussian sum unscented Kalman Filter with adaptive scaling parameters. Proceedings of the 14th International Conference on Information Fusion; IEEE: 2011; pp 1−8. (40) Simandl, J.; Dunik, J. Sigma point Gaussian Sum Filter design using square root unscented filters. IFAC Proceedings Volumes 2005, 38, 1000−1005. (41) Kottakki, K. K.; Bhartiya, S.; Bhushan, M. State estimation of nonlinear dynamical systems using nonlinear update based Unscented Gaussian Sum Filter. J. Process Control 2014, 24, 1425−1443. (42) Julier, S.; Uhlmann, J. A New Extension of the Kalman Filter to Nonlinear System. Proc. SPIE 1997, 3068, 182−193. (43) Wan, E.; Merwe, V. D. The Unscented Kalman Filter. Kalman Filtering and Neural Networks; Wiley: New York, 2002; Chapter 7, pp 221−282. (44) Iqbal, M.; Naeem, M.; Anpalagan, A.; Ahmed, A.; Azam, M. Wireless sensor network optimization: multi-objective paradigm. Sensors 2015, 15 (7), 17572−17620. (45) Fei, Z.; Li, B.; Yang, S.; Xing, C.; Chen, H.; Hanzo, L. A survey of multi-objective optimization in wireless sensor networks: Metrics, algorithms, and open problems. IEEE Commun. Surv. Tutorials 2017, 19, 550. (46) Sen, P.; Sen, K.; Diwekar, U. A Multi-objective optimization approach to optimal sensor location problem in IGCC power plants. Appl. Energy 2016, 181, 527−539. (47) Miettinen, K. M. Nonlinear Multi-Objective Optimization; Kluwer Academic Publishers: 2002. (48) Behringer, F. A. Lexicographic quasiconcave multi-objective programming. Zeitschrift fuer operation research. 1977, 21, 103−116. (49) Nocedal, J.; Wright, S. J. Numerical Optimization, 2nd ed.; Springer: New York, NY, 2006. (50) Fletcher, R. Practical Methods of Optimization, 2nd ed.; Wiley: 2000. (51) Rhudy, M.; Gu, Y. Understanding nonlinear Kalman filters, Part II: An implementation guide. Interactive Robotics Letters; West Virginia University: June 2013. http://www2.statler.wvu.edu/~irl/page13.html. (52) Simon, D. Optimal State Estimation; Wiley: Hoboken, NJ, 2006. (53) Akaike, H. A new look at the statistical model identification. IEEE Trans. Autom. Control 1974, 19, 716−723. (54) Bhattacharyya, D.; Rengaswamy, R. System identification and nonlinear model predictive control of a solid oxide fuel cell. Ind. Eng. Chem. Res. 2010, 49, 4800−4808. (55) Turton, R.; Bailie, R. C.; Whiting, W. B.; Shaeiwitz, J. A.; Bhattacharyya, D. Analysis, Synthesis, and Design of Chemical Processes, 4th ed.; Prentice Hall: Upper Saddle River, NJ, 2012. (56) Liptak, B. G. Process Measurement and Analysis, 4th ed.; CRC Press: Boca Raton, FL, 2003.

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