multiple model adaptive estimation of a hybrid solid oxide
TRANSCRIPT
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
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(���.���)��.��∙���.��∙�(���.��) ��.��∙���.�∙�
(���.��) ∙ �!�!�� (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.
9
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