Multiple Model Adaptive Estimation of a Hybrid Solid Oxide Fuel Cell Gas Turbine

Power Plant Simulator

Alex Tsai

1, David Tucker

2, Tooran Emami

1

1United States Coast Guard Academy, New London CT, USA

2US Department of Energy, National Energy Technology Laboratory, Morgantown WV, USA

ABSTRACT

Operating points of a 300kW Solid Oxide Fuel Cell Gas Turbine

(SOFC-GT) power plant simulator is estimated with the use of a

Multiple Model Adaptive Estimation (MMAE) algorithm, aimed at

improving the flexibility of controlling the system to changing

operating conditions. Through a set of empirical Transfer Functions

derived at two distinct operating points of a wide operating envelope,

the method demonstrates the efficacy of estimating online the

probability that the system behaves according to a predetermined

dynamic model. By identifying which model the plant is operating

under, appropriate control strategies can be switched and

implemented upon changes in critical parameters of the SOFC-GT

system - most notably the Load Bank (LB) disturbance and FC

cathode airflow parameters. The SOFC-GT simulator allows testing

of various fuel cell models under a cyber-physical configuration that

incorporates a 120kW Auxiliary Power Unit, and Balance-of-Plant

components in hardware, and a fuel cell model in software. The

adaptation technique is beneficial to plants having a wide range of

operation, as is the case for SOFC-GT systems. The practical

implementation of the adaptive methodology is presented through

simulation in the MATLAB/SIMULINK environment.

INTRODUCTION

The Department of Energy’s National Energy Technology

Laboratory (NETL) has researched Solid Oxide Fuel Cell / Gas

Turbine (SOFC-GT) power generation for over a decade [1-3]. The

Hybrid-Performance, or HyPer project, employs a cyber-physical

simulation of a hybrid SOFC-GT plant which enables the study of the

interaction between balance of plant (BoP) components and different

fuel cell arrangements under a variety of control strategies [4,5]. One

noticeable find is the effect fuel cell mass flow rate has on the

efficiency and performance of the system. In order to regulate a

synchronous turbine speed while successfully track a changing power

demand, an airflow bypass methodology was devised to effectively

control FC power robustly [6]. This approach, together with an

implementation of modern control theory is the basis for the research

undertaken at NETL.

Among the control algorithms tested in HyPer, are model-based

controllers. These controllers are dependent on either a model of the

BoP components by themselves, or a combined overall BoP/FC

model. This work focuses on the latter approach of empirically

deriving the BoP model of the plant without the FC model. After the

initial proof-of-concept stage, the FC model can be then incorporated,

to adequately complete the design. A Transfer Function (TF) matrix

serves as the model, obtained from Open Loop (OL) tests around

various nominal operating points.

This work aims to demonstrate that an adaptive estimation algorithm

can be useful in hybrid plants which are susceptible to nonlinearities.

If the operating points of a large operating envelope can be

adaptively identified, and a different control algorithm be attached to

each operating region, then the difficulties in mitigating nonlinear

interactions can be addressed one at a time. The adaptive

identification technique which facilitates such a merge is known as

Multiple Model Adaptive Estimation (MMAE) [7]. When a

particular controller is ‘attached’ to a model of a specific operating

region, the methodology is known as Multiple Model Adaptive

Control (MMAC) [8,9]. This paper is the precursor to the

implementation of the MMAC approach, intended as a follow-up

study on the HyPer SOFC/GT simulator.

To effectively test the MMAE method, the sensed FC mass flow rate

and turbine speed signals are used as inputs to the algorithm. The

corresponding actuators used for control are an airflow bypass valve

and the electrical load attached to the turbine. The system

identification technique then statistically selects a model from a bank

of models assembled offline. The output of the estimation algorithm

is thus a probability value which identifies a model of a specific

operating point to that of the real-time data. Hence real-time data is

‘matched’ to a model, further enabling the matching of unique

controllers to particular operating regions. The following sections

outline the methodology and plant description.

FACILITY DESCRIPTION

Built in 2002 for the purpose of investigating all related issues

concerning the design, control, and operation of pressurized

SOFC/GT plants, NETL developed a hybrid prototype known as the

Hybrid Performance project, or HyPer, shown in Fig.1-2.

Fig.1 NETL HyPer Test Facility

Proceedings of the ASME 2016 14th International Conference on Fuel Cell Science, Engineering and Technology FUELCELL2016

June 26-30, 2016, Charlotte, North Carolina

FUELCELL2016-59656

1This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release. Distribution is unlimited.

The HyPer project utilizes the cyber-physical concept, where the

hybrid system is simulated partly through hardware and software.

The gas turbine and Balance of Plant components constitute the

hardware part, while the fuel cell electrochemistry and thermal

dynamics are captured in a 1-D high fidelity model [10,11]. This

model calculates the heat effluent a 300kW-700kW fuel cell would

produce under measured temperature, pressure, and airflow states

throughout the physical plant, which represents the cathode side of

the SOFC. The calculated thermal load is then imparted to the plant

by a fast acting fuel valve that burns natural gas and is controlled by

the model. Hence, the effect of a fuel cell loss or increase in load can

be studied as a disturbance to the thermal equilibrium of the hybrid

system Balance of Plant components. A description of the sensors,

actuators, and Balance of Plant equipment, is summarized below.

Fig.2 CAD Rendering of HyPer Hardware Facility

Gas Turbine

A 120kW Garrett Series 85 auxiliary power unit is used for the

turbine and compressor system, and consists of single shaft, direct-

coupled turbine operating at a nominal 40,500rpm, a two-stage radial

compressor with a gear driven synchronous (400Hz) generator. An

isolated 120kW continuously variable resistor load bank loads the

electrical generator. The compressor is designed to deliver

approximately 2.3 kg/s at a pressure ratio of about four. The

compressor discharge temperature is typically 475K for an inlet

temperature of 298K.

Fuel Cell Simulator

The thermal characteristic of the effluent exiting the post combustor

of an SOFC system is simulated in hardware using a natural gas

burner with an air-cooled diffusion flame. The fuel cell dynamics are

coupled to the system dynamics through sensor measurements fed to

the model as shown in Figure 1.

Heat Exchangers

The project facility makes use of two counter flow primary surface

recuperators with a nominal effectiveness of 89% to preheat the air

going into the pressure vessel used to simulate the fuel cell cathode

volume.

Pressure Vessels

Pressure vessels are used to provide the representative fuel cell air

manifold, cathode volume, and the post combustion volume of a solid

oxide fuel cell. The total volume of the airside components is

approximately 2,000L.

Bleed Air Bypass Valve

The bleed air bypass valve is used to bleed air from the compressor

plenum to the atmosphere through the stack as shown in Fig.2. The

bleed air valve and associated piping is a nominal 15.2cm diameter.

Cold-Air Bypass

The cold-air bypass valve, (15.2cm nominal diameter), is used to

bypass air from the compressor directly into the turbine inlet through

the post combustor volume as shown in Fig.2.

Hot-Air Bypass

The hot-air bypass valve (15.2cm nominal diameter) is used to

bypass air preheated by the recuperators into the turbine inlet through

the post combustor volume as shown in Fig.2.

Load Bank

A 120kW Avtron continuously variable load bank is used to load the

turbine electric generator. The load bank is capable of imposing up

to 95kW resistive load and up to 25kW reactive load.

Previous work includes the empirical derivation of input/output

dynamic relationships in the frequency domain, known as Transfer

Functions (TF) [12]. When an actuator is commanded a step input,

classical control theory provides a means to identify all pertinent

dynamic parameters of a First Order Plus Delay (FOPD) TF as seen

in Eq.1.

( )( )

S

ij

ij

ij es

ksg

⋅−

+⋅= θ

τ 1 (1)

Where k, �, and θ are the static gain, time constant, and delay time

between the output and input signals. By applying a step input one

actuator at a time, a TF matrix can be built around a nominal

operating point from each of the individual TF equations. Figures 3-

6 show the step response of the fuel cell mass flow rate and turbine

speed as a function of the Cold Air (CA) bypass valve, while Figs.7-8

show responses to Electrical Load (EL) changes. Note that the mass

flow rate response is obtained at open loop, using the load bank as the

actuator and those of the turbine speed response are obtained at open

loop, i.e. no control, and no fuel cell model incorporated at the time

of the steps.

The TF matrices of Eqs.2 and 3 show two distinct operating points

created by the nonlinearity inherent when the CA bypass valve is

opened beyond a certain range. In both matrices, y1 and y2 represent

the turbine speed and FC mass flow rate respectively, while u1, u2 are

the electric load and CA bypass valve. Eq.4 depicts one such

operating point in TF matrix form. This matrix can be built in the

Simulink/Matlab environment shown in Fig.9, where the electrical

load (EL) and bypass valve (CA) affect the FC mass flow rate and

turbine speed denoted by outputs 1 and 2 respectively.

In Simulink, the set of TF equations characterizing the data of Figs.3-

6, do not include delay terms. At the expense of generating a small

error, these delays are omitted, given the significantly low magnitude

as compared to the time constants of the response. Equations 5-9

describe the State Space representations of the TF matrix, where ∆t is

the sampling time, and ‘I’ the identity matrix. Eqs.10-11 denote the

equivalent -1discrete State Space model.

2

Fig.3 FC �� Response: CA 20%-40% Step

Fig.4 Open Loop (OL) Turbine Speed Response: CA 20%-40% Step

Fig.5 FC �� Response: CA 40%-80% Step

Fig.6 OL Turbine Speed Response: CA 40%-80% Step

Fig.7 OL Turbine Speed Response: EL 50kW-30kW Step

Fig.8 FC �� Response: EL 50kW-30kW Step

3

������ = ��.��∙���.�∙�(���.���) ��.���∙���.�∙�

(���.���)��.��∙���.��∙�(���.��) ��.��∙���.�∙�

(���.��) ∙ �!�!�� (2)

������ = ��.��∙���.�∙�(���.���) ��.�"∙���.#$∙�

(���.��)��.��∙���.��∙�(���.��) ��∙���.%%∙�

(���."�) ∙ �!�!�� (3)

������ = &'�� '��'�� '��( ∙ �!�!�� (4)

Fig.9 TF Matrix Model: 1st Operating Point, Eq.(4)

1111 uAxax ⋅+⋅−=& 2322 uAxcx ⋅+⋅−=& (5)

1233 uAxbx ⋅+⋅−=& 2444 uAxdx ⋅+⋅−=& (6)

uBxAxrrr

& ⋅+⋅= uDxCyrrr

⋅+⋅= (7)

−

−

−

−

=

d

b

c

a

A

000

000

000

000

=

4

2

3

1

0

0

0

0

A

A

A

A

B (8)

=

1100

0011C

=

00

00D (9)

tAIAd ∆⋅+= tBB d ∆⋅= (10)

)()()1( kuBkxAkx dd +=+ )()()( kuDkxCky ⋅+=

(11)

NONLINEAR SYSTEM RESPONSE

Among the various nonlinear interactions in the HyPyer plant, none

is more notable than the effect the CA valve has on turbine speed as a

function of electrical load, as seen in Fig.10. When the CA operates

between 20-35%, the speed increases to a maximum, while from 35%

onwards it decreases at approximately the same rate.

Fig.10 Operating Envelope for ) = f(CA, LB), 50kW Load

By opening the CA at its lower range, the turbine inlet pressure

increases at a higher rate than the rate at which the turbine inlet

temperature (TIT) decreases, ramping up the speed. At the higher

CA operating range, the TIT decreases at a faster rate than the rate at

which the pressure increases, decreasing the turbine speed. The

dynamic phenomena observed during the operation of the CA are

described in detailed by Pezzini and Tucker [13].

MULTIPLE-MODEL ADAPTIVE ESTIMATION

In order to adequately implement a control law which performs well

at a particular operating point under a constantly changing system, it

is beneficial to first identify the operating point itself. Multiple-

Model Adaptive Estimation (MMAE) is a system identification

methodology which implements a bank of Kalman Filters, or optimal

estimators, for the purpose of matching a system model to the true

system at hand. It estimates the true system states under the presence

of system and measurement noise in a probabilistic manner, by

minimizing the covariance of the error signal [14-16]. When the

estimated output is compared to the true measured output, a residual

error is computed. If the residual is close to zero, a model has

‘matched’ the true system, and a corresponding control algorithm can

be paired to the chosen system model. The pairing of plant estimator

with control law is known as Multiple Model Adaptive Control

(MMAC) [17]. This scheme will be studied in depth on subsequent

work.

In order to successfully implement the MMAE technique, it is

necessary to have knowledge of all the possible system models the

true plant could be operating as. If the operating envelope is

partitioned evenly, such that each region represents one unique

operating point with equal probability of occurrence, and known

mathematical model, a probability can be calculated and assigned to

this model. The highest probability among all available models

corresponds to the true system for that particular range of operation.

4

In order to estimate the states for this partition of ‘N’ models with the

operating envelope, a bank of ‘N’ Kalman Filters is pre-designed

offline. Thus, the true system parameters of A, B, C, D are unknown,

but are described by one of the following pre-established models.

State and output estimates are then used in the MMAE algorithm to

compute the needed probabilities.

The objective is to evaluate at each time step, which of the N models

best characterizes the true system in a statistical sense. Each model is

assigned a conditional probability (PB) based on Bayes rule as given

in Eq.12, where H is a random variable which denotes the event that

model “i” is the most exact characterization of the true system, and

‘z’ a vector containing all available input and output data, Eq.(13).

( ) ( )( )kzHHPBkPB ii == (12)

{ })()...2(),1(),1()...2(),1()( kyyykuuukz −= (13)

Figure 11 shows the operating envelope of the fuel cell mass flow

rate as a function of CA and LB inputs. The operating area has been

partitioned into 5 distinct regions, each having its own TF matrix.

Thus, during operation, the true system lies in one of the 5 sections

whose mathematical model is known.

Fig.11 Operating Envelope for �� = f(CA, LB)

The maximum airflow rate is given at 41,500rpm and 0kW, whereas

the minimum flow is at 39,500rpm and 50kW. The inflection point is

somewhere around 30%-40%, as observed in Fig.9, where there is a

nonlinearity in the efficiency. From these plots it is evident that a

separate control strategy is required below and above the inflection

point of approximately 40%. The operational limits of the CA valve

are between 0-20% and 60-80%. The linear range is approximately

30%-50%, or between regions 2 and 3.

Once the operating envelope is partitioned into ‘N’ system models for

the ‘N’ separate regions, ‘N’ Kalman Filters (KF) are designed

offline for the discrete State Space model of Eqs.14, 15. The matrix

L reflects the noise propagation matrix, and is chosen as the first 2

columns of the Ad matrix, since the system disturbance uncertainty is

assumed to be larger for the turbine speed than for the FC mass flow

rate.

The w(k) and v(k) vectors are the system and measurement sensor

noise respectively. These coefficients are uncorrelated, independent,

and follow a Gaussian probability density function with zero mean

and known variance.

)()()()1( kLwkuBkxAkx dd ++=+ (14)

)()()( kvkxCky d += (15)

Since the KF minimizes the variance of the error, or the expected

value of the covariance matrix P in Eq.16, it is practical to group the

variances of the measurement and system noise vectors into

covariance matrices Q and R, as shown in Eq.18. The diagonal

elements of these matrices comprise the elements of the w(k) and

v(k) vectors used in the MMAE algorithm, whereas the off-diagonals

express the covariance, or to what degree the noise streams are

related.

The expectation operator ‘E’ is minimized according to error defined

as the difference between the true state and the estimated state in

Eq.17.

)]1()1([)1( +⋅+=+ kekeEkPT (16)

)1(ˆ)1()1( +−+=+ kxkxke (17)

=

2

2

2

1

0

0

w

wQσ

σ

=

2

2

2

1

0

0

v

vRσ

σ (18)

The best possible estimate of a state, given all available

measurements, is produced by the Predictive Type KF, shown in

Fig.12. The figure shows the discrete State Space model followed by

the discrete KF, highlighted in yellow. Note that the outputs of the

filter are the estimated states *+(,) which can then fed as the input to

the system u(k) through a controller gain Kc(k) – not shown. The

residuals, or difference between y(k) and �+(,) is the feedback

mechanism later used to assign probabilities to the stored models.

Fig.12 Prediction Type Kalman Filter (PTKF)

In order to test the robustness of the MMAE algorithm, normal

random variable noise vectors were generated using the Box-Muller

method [18], shown in Eqs.19, 20. The k1,2,3,4 constants are

independent random numbers and -wi is the plant’s noise covariance.

The actual variance and statistical properties of the measurement

noise were calculated, while the variance of the system noise was

estimated. System noise is harder to derive and distinguish from

sensor noise. Altogether, these signals follow a normal Gaussian

distribution.

( ) ( ) wii kkw σπ ⋅⋅⋅⋅−= 12 2coslog2 (19)

5

( ) ( ) vii kkv σπ ⋅⋅⋅⋅−= 43 2coslog2 (20)

Eqs.21-24 outline the KF algorithm. Having known the system

matrices of Eqs.14-15, and those of Eq.18, an arbitrary state vector *+(1) and covariance matrix P(1) is assigned at time k = 1. The

influence these random numbers have is on the convergence time of

the algorithm, not on the convergence itself. Once these initial

estimates are chosen, the filter gain Ke in Eq.21 is computed,

followed by the updated covariance matrix P(k+1) in Eq.22. The

estimated state *+(, + 1) is then generated in Eq.23, and a residual

calculated. The residual is utilized in the conditional probability

equation, comprised of Eqs.25-27. Since these equations are

computed recursively, the updated covariance matrix P feeds into the

KF gain Ke at the next time step. Thus, the output of the algorithm is

essentially Ke, P and the estimate*+(,).

( )4434421

H

TT

de CPCRPCAK1−

+= (21)

T

d

T

d

T

dd

T

i HCPAPCAPAALQLP −+= (22)

( )321

residual

einddp yyKuBxAx ˆˆˆ −++= (23)

xCy ˆˆ ⋅= (24)

RCPCST

iiii += (25)

i

Ni

S22

1

π

β = (26)

( )( ) ( )

( )( ) ( )

( )∑=

⋅⋅⋅−

⋅⋅⋅−

−⋅⋅

−⋅⋅=

N

j

j

krSkr

j

i

krSkr

i

i

kPBe

kPBekPB

jjT

j

iiT

i

1

2

1

2

1

1

1

β

β (27)

In Eq.25, Si is the covariance matrix of the residual error. Si-1 is

equivalent to the standard deviation -x2. In Eq.26, 0I is a scalar

weight and N is the size of the measurements or number of outputs.

The denominator of Eq.27 is used to normalize the response, so that

the probability lies between 0 and 1.

To adequately track and identify the model that best describes the

actual system in a noisy environment, an input signal ‘rich’ in

frequency content is required. Figure 13 shows the CA and LB

sinusoids used in the excitation of the system. These are the only

inputs used to evaluate the performance of the MMAE. Note that the

magnitude of the excitation is within the actuator’s linear range, zero

being the actuator’s nominal value.

In order to test the MMAE algorithm in Matlab/Simulink, the inputs

of Fig.13 were fed to 2 State Space models representing 2 different

operating points: Eqs.2-3. The output of these models is shown in

Fig.14 for a predetermined sequence of events. The times in which

each operating point occurs is chosen randomly. A turbine speed

sensor failure is also simulated at various times. Thus, a sequence of

events is created with noisy data and serves as the input to the

MMAE algorithm. From these simulated “online” events, the

MMAE algorithm then decides which data best matches one of

various stored offline State Space models. The outcome of the

algorithm is an assigned probability to each offline model,

corresponding to the true operating point. Once implemented, the

input to the MMAE routine becomes real data from the FC mass flow

rate sensor and the speed sensor. All operating points must be

mapped to the data if the operating point is to be properly identified.

Fig.13 CA and LB ‘Rich’ Input Signals: 0 = Nominal Value

Fig.14 MMAE Input Signals: 0 = Nominal Value

RESULTS

The results of a sequence of system changes are plotted in Figs.15-

22. Fig.15 displays the probabilities assigned to 3 different system

models, i.e. the 2 TF matrices of Eqs.2, 3, and one condition

exhibiting sensor failure. For the measurement breakdown scenario,

the C matrix is changed from its nominal value. C is the observation

matrix, which when different from the identity matrix I, implies an

inability to access or measure all the system states, hence an

indication that a sensor has malfunctioned. This is considered to be a

disturbance operating point.

6

To evaluate the methodology, a sequence of events was established to

mimic a switch between system models at specified time intervals.

For example, in Fig.15, between (0 < k < 2000) the true system

deployed in the simulation is model 2. A successful identification of

the true model by the MMAE algorithm would select model 2 as the

matching model for the existing system. According to the plot, the

model with the highest probability for that time interval is indeed

model 2. Note how the probabilities for model 1 and 3 remain at

zero. It can be seen from Fig.14 that the residuals corresponding to

model 2 between 0 < k < 2000 is minimal. There are two residuals

per model, since there are two measured outputs of interest, the

turbine speed, and the fuel cell cathode airflow. The remaining

sequence is as follows:

1. (Model 2 = True System) for (0 < k < 2000) and (5000 < k

< 7000)

2. (Model 1 = True System) for (2000 < k < 3000), (4000 < k

< 5000) and k > 7000

3. (Model 3 = True System) for (3000 < k < 4000)

The graphs demonstrate that indeed the MMAE algorithm converges

statistically to the true system’s output, with infrequent disturbances

along the time steps ‘k’. Fig.17 shows a different sequence of events

than those of Fig.15. As before, the probabilities match the assigned

true system output. A closer look at Figs.15, 17 shows that the

probability converges quite rapidly, with plummeting spikes only at

separated intervals. The magnitudes of these spike intervals never

reach the zero probability value for any of the models, when the true

value matches the model value. Given the probabilistic nature of the

methodology, there will always be oscillations of this sort, especially

when random noise is present in the system and measurement signals.

Figs.16, 18 show the residuals of the probability plots. As expected,

when the model output matches statistically the true output signal, the

residuals are less in magnitude than when the true output is outside

the convergence interval. In Fig.18 however, when the time interval

‘k’ is below 2000 simulation times, system models 1 and 3 both have

close to zero residual for one output, but not for the second. The

same observation occurs between 3000 and 5000 simulation times.

Hence the residual of both outputs must approach zero for the system

model to be appropriately matched to the true system output.

Similar phenomena are seen for all the plots as well as the residuals.

All these observations lead to conclude that the MMAE algorithm

converges statistically to the closest system model, given the location

of the true system output.

Figures 19-22 demonstrate the effect of a higher noise disturbance in

both the system and measurement signals. In the scenario of Fig.19-

20, the covariance matrix of the system noise Q1 of Eq.28 is

amplified by 5 times, resulting in Eq.30, while the covariance matrix

of the measurement signal R1 in Eq.29 remains constant. Likewise,

in Figs.21-22 the opposite is true: Q1 nominal remains constant and

R1 changes by a factor of 5, as shown in Eq.31. By modifying the

noise magnitude of the variances, creating model discrepancies, the

robustness of the estimation approach can be ascertained and

quantified.

When the magnitude of the system variance increases 5 times from

the original nominal values, the MMAE algorithm exhibits far more

oscillations between time steps, even though the model converges

and accurately selects the correct system, as reflected in Figs.19-20.

Q1 = diag[0.05(rpm)2, 0.02123� 4�] (28)

R1 = diag[0.03(rpm)2, 0.07123� 4�] (29)

Q∆ = diag[0.25(rpm)2, 0.1123� 4�] (30)

R∆ = diag[0.15(rpm)2, 0.35123� 4�] (31)

Fig.15 Model Probabilities for Q1, R1: 1

st Sequence

Fig. 16 Model Residuals for Q1, R1: 1

st Sequence

In contrast, when the measurement covariance matrix elements in R1

are increased 5 times the nominal value, the convergence oscillates

heavily between model sequences, as seen in Figs.21, 22.

7

Fig.17 Model Probabilities for Q1, R1: 2nd Sequence

Fig.18 Model Residuals for Q1, R1: 2

nd Sequence

A close look at Fig.22 reveals that the magnitude of the residual

around the zero value for simulation times corresponding to correct

model matching is larger than that seen in Figs.18 and 20. A subtle

difference in the residual magnitudes, accounts for a messier

probability plot. When the model does not match the true system, the

residual magnitude is comparable to the case where the system

covariance matrix Q1 is amplified.

It is clear that the more variance the measurement noise has, the less

accurate the statistical probability becomes. This is not the case with

the system noise variance having the same degree of noise variance.

Assuming the system covariance matrix uncertainty was modeled

correctly, these results indicate that a good measurement system is

worth investing on. Even with the sensor failure scenario, i.e. model

3, the MMAE algorithm is able to converge with little difficulty.

Fig.19 Model Probabilities for Q∆, R1

Fig.20 Model Residuals for Q∆, R1

The problem arises when measurement noise exceeds a threshold.

For the data analyzed in Figs.3-6, the Signal to Noise Ratio (SNR) for

the turbine speed was approximately 9, whereas for the fuel cell mass

flow rate, the SNR is greater than 20. An ideal SNR for the MMAE

methodology to work well is in the vicinity of 30:1. This means that

a larger CA actuation was required to cause a 1500rpm change in

speed response, thus a 30:1 SNR. The standard deviation of the

speed noise signal is measured at 50rpm.

8

Fig.21 Model Probabilities for R∆, Q1

Fig.22 Model Residuals for R∆, Q1

DISCUSSION

The results of the MMAE algorithm illustrate the advantages and

limitations the methodology has. Although it is readily seen that

convergence occurs in selecting the appropriate model match for a

particular sequence of operation, it is also apparent that operational

limitations hinge on whether the SNR is high enough for the residual

computation to converge properly. There is also no established

convergence when the number of models is large. This study

proposed the identification of 3 system models as a means of

conveying the concept in a simple, clear manner. The HyPer system

has inherently a larger number of models for various operating

conditions. Literature suggests that models clustered within a small

region of the operating envelope will cause convergence issues. The

residuals might look similar given the fluctuation of the noise.

Therefore, a signal to noise ratio SNR of 30:1 is frequently desired.

To avoid having some residuals be similar, the models should be

equally spaced within the operating envelope as well.

When the system and measurement covariance matrix noise levels

change by 5 times, the outcome of the algorithm differs. Fortunately,

the algorithm does show a certain degree of robustness in the

presence of incorrectly modeled noise. Figs.19 and 21 show that

when the true system has a measurement covariance matrix R1 much

smaller than the R∆ value, the probabilities are very much defined,

with few or almost no perturbations along the convergence intervals.

It is close to being a perfect model match for the true system output.

In contrast, when the true system covariance matrix R is increased by

5 fold, the probability plots converge with greater difficulty and

generate far greater and frequent oscillating ‘spikes’ than any of the

tested discrepancies. In fact, some probabilities are close to not

converging, but maintain the same high level of uncertainty

throughout the entire simulation time. This should make sense, since

the higher noise in the measurements would imply greater uncertainty

of the true output signal, and thus difficulty in tracking this signal. In

summary, it seems best to have natural occurring system noise for the

convergence of probabilities in the MMAE algorithm than

measurement noise which comes from sensors and related equipment.

With regards to a practical implementation of the algorithm suitable

for online estimation, the longest computational time occurs during

the matrix inversion calculation of Eq.21. The burden can be

lessened by building the ‘N’ Kalman Filters for the ‘N’ system

models offline, as well as the covariance matrix P. Although a

computational timing study was not included in this work, the

eventual inclusion of a bank of controllers attached to the MMAE

algorithm as follow up work must examine this. It is of key

importance to adaptively track signals with minimal delay.

One benefit to consider when examining this methodology is that it

allows for any controller structure to be implemented jointly. Unlike

the adaptive gain scheduling approach, where PID gains are assigned

based on the operating range, the MMAE technique merges a

controller of any kind. Subsequent work will demonstrate how good,

robust performance is achievable by modifying the MMAE code.

This methodology is known as Multiple-Model Adaptive Control

(MMAC).

CONCLUSIONS

This paper demonstrated the ability of the MMAE methodology to

successfully identify various operating points within an operating

envelope for the HyPer simulator. The primary goal is to properly

control the system during all its operating cycles, at an optimal

performance. To attain this, various controller structures may be

required, which may differ from the commonly used PID family. By

optimally estimating a system’s true operating plant with the use of a

bank of Kalman Filters, the goal of matching a controller becomes a

feasible and practical one.

ACKNOWLEDGEMENTS

This work was completed through collaboration between the U.S.

Department of Energy Crosscutting Research program, administered

through the National Energy Technology Laboratory and the U.S.

Coast Guard Academy.

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REFERENCES

1. Tucker, D.; Manivannan, A.; Shelton, M. S. “The Role of Solid

Oxide Fuel Cells in Advanced Hybrid Power Systems of the

Future,” Interface 18 (3) 2009

2. Winkler, W; Nehter, P.; Tucker, D.; Williams, M.; Gemmen, R.

“General Fuel Cell Hybrid Synergies and Hybrid System

Testing Status,” Journal of Power Sources, Vol.159, Issue 1

3. Williams, M.C., Strakey, J. and Surdoval, W. “U.S. DOE Fossil

Energy Fuel Cells Program”, Journal of Power Sources,

Vol.159, Issue 2, 2006

4. Tucker, D., R. Gemmen, L. Lawson “Characterization of Air

Flow Management and Control in a Fuel Cell Turbine Hybrid

Power System Using Hardware Simulation” ASME Power

Conference, April 5-7 2005, Chicago IL

5. Tsai, A., L. Banta, D. Tucker, R. Gemmen. “Multivariable

Robust Control of a Simulated Hybrid Solid Oxide Fuel Cell

Gas Turbine Plant” Journal of Fuel Cell Science and

Technology, Vol.7 Issue 4, August 2010

6. Tsai, A., D. Tucker, C. Groves. “Improved Controller

Performance of Selected Hybrid SOFC-GT Plant Signals Based

on Practical Control Schemes” Journal of Engineering for Gas

Turbines and Power, Accepted for Publication – April 19, 2010.

Paper No. GTP-10-1104.

7. Chang, C., Athans, M. “State Estimation for Discrete Systems

with Switching Parameters” IEEE Transactions on Aerospace

and Electronic Systems, AES-14, 4, 418-425, May 1978

8. K. S. Narendra and C. Xiang, “Adaptive control of discrete-time

systems using multiple models,” IEEE Trans. Autom. Control,

vol. 45, no. 9, pp. 1669–1686, Sep. 2000.

9. Athans, M. “The Stochastic Control of the F-8C Aircraft Using a

Multiple Model Adaptive Control (MMAC) Method – Part I:

Equilibrium Flight” IEEE Transactions on Automatic Control,

Vol. AC-22, No.5, October 1977

10. Haynes, C., D. Tucker, D. Hughes, W. Wepfer, K. Davies, C.

Ford “A Real-Time Spatial SOFC Model for Hardware-Based

Simulation of Hybrid Systems” Proceedings of ASME 9th Fuel

Cell Science, Engineering, and Technology Conference.

Washington, DC Aug.7-10, 2011

11. Hughes, D., D. Tucker, A. Tsai, C. Haynes, Y., Rivera

“Transient Behavior of a Fuel Cell/Gas Turbine Hybrid Using

Hardware-Based Simulation with a 1-D Distributed Fuel Cell

Model” Proceedings of ICEPAG2010, Feb.9-11, 2010, Costa

Mesa, CA

12. Pezzini, P., Banta, L, Traverso, A., Tucker, D. “Decentralized

Control Strategy for Fuel Cell Turbine Hybrid Systems”

Proceedings of the 57th Annual ISA Power Industry Division

Symposium, June 1-6, 2014, Scottsdale, AZ

13. Pezzini P., Celestin S., Tucker D., “Control Impacts of Cold-Air

Bypass on Pressurized Fuel Cell Turbine Hybrids,” Journal of

Fuel Cell Science and Technology, 12(1):011006 (2015); FC-14-

1102; doi: 10.1115/1.4029083.

14. Applied Optimal Control and Estimation, Digital Design and

Implementation, Frank Lewis. Prentice Hall 1992

15. Digital Control of Dynamic Systems 3rd Ed, Franklin, Powell,

Workman, Addison Wesley 1998

16. Napolitano, M., Window, D., Casanova, J., Innocenti, M.

“Kalman Filters and Neural-Network Schemes for Sensor

Validation in Flight Control Systems” IEEE Transactions on

Control Systems Technology, Vol.6 No.5, September 1998

17. Athans, M., Castanon, D., Dunn K., Greene, C., Lee W.,

Sandell, N. Willsky, A. “The Stochastic Control of the F-8C

Aircraft Using a Multiple Model Adaptive Control (MMAC)

Method – Part I: Equilibrium Flight” IEEE Transactions on

Automatic Control, Vol. AC-22, No.5, October 1977

18. Box G.E.P., Muller M.E. “A Note on the Generation of Random

Normal Deviates” Annals of Mathematical Statistics, Vol. 29;

610-611, 1958

10