Robust combustor design based on flame transfer function modification

Abstract

Operation under stable conditions is an important prerequisite for gas turbine safety. While recent studies have focused primarily on acoustic design to avoid thermoacoustic instabilities, in the present study we shift the focus to improving stability margins by flame transfer function (FTF) modification. The flame transfer function of premixed flames is affected by various mechanisms such as variations in equivalence ratio, swirl fluctuations and shear layer instabilities. These mechanisms can be influenced by modifying parameters such as fuel distribution, injection location, swirl number or gas composition. Based on the Nyquist stability method we formulate criteria for how and at what frequencies the flame transfer function needs to be modified, in order to increase the stability margins of a thermoacoustic system. Gain and phase margin as well as the sensitivity function serve as measures of stability. The criteria are limited to single frequencies, which allows experimental FTF optimization with manageable effort. In the second part of this study it is shown that the Nyquist method can also be used as an efficient and compact way to determine whether the uncertainties of subsystems can affect the overall stability, without requiring eigenvalue calculations.

Keywords

Thermoacoustics passive control nyquist criterion flame transfer function stability analysis uncertainty quantification

Introduction

Thermoacoustic instability describes the phenomenon of heat release and acoustic interaction resulting in an increased oscillation amplitude of acoustic quantities and heat release rate. Commonly observed in stationary gas turbines and rocket engines, thermoacoustic instabilities can shorten engine life and even cause severe mechanical damage¹. To ensure stable operation of combustion systems, stability analysis is performed throughout the machine design process. One reliable and efficient method to investigate the thermoacoustic stability of a combustion system is network modelling. The system is subdivided into several subsystems, each with an individual acoustic input-output behaviour, that is represented as a transfer function or transfer matrix. These subsystems can either originate from theoretical considerations, numerical models or experimental measurements. By linking all subsystems together the network model is formed, which can be analyzed for thermoacoustic stability. This method is widely used in academia and industry for single-can, can-annular and annular combustors^2–4. The stability of the full thermoacoustic system is determined by the growth rates of its eigenvalues. Critical eigenvalues are characterized by a positive growth rate, corresponding to eigenmodes that grow exponentially in time and destabilize the system. In addition to the usual methods based on the direct determination of the eigenvalues of the network model, other methods have been proposed to analyze the stability of thermoacoustic systems. These are especially relevant regarding the fact that for measured subsystems generally no analytical expressions are available and thus eigenvalues cannot be calculated directly. In a recent study,⁵ Yoon investigates a one-dimensional thermoacoustic model of a dump combustor with only two acoustic impedances, just before and after the area expansion and formulates a condition for combustion stability based on the maximum gain of the flame transfer function. In a study by Polifke et al.⁶ the Nyquist criterion,⁷ which is commonly used in control applications, is applied to a lumped thermoacoustic model to investigate its global stability. In following studies, it was shown that the Nyquist curve can also be used to obtain good estimates of the eigenvalues when they are located close to the stability line^8,9. A comprehensive review of passive control of combustion dynamics in the light of the Nyquist method is given by Richards et al.¹⁰. Difficulties that arise in the context of stability analysis are model and parameter uncertainties that can cause a system predicted to be stable to become unstable nonetheless. Therefore, not only boolean stability criteria for modelled systems are needed, but also information about the robustness of the systems is of interest. Recent studies have focused on taking into account uncertainties in the modelled subsystems and their effects on the thermoacoustic eigenvalues. Guo et al. presented a method based on a Gaussian process surrogate model¹¹, that allows for a cost-effective estimation of thermoacoustic dynamics to effectively perform Monte Carlo simulations for uncertainty analysis. Other approaches for uncertainty analysis feature polynomial chaos expansion¹² and adjoint perturbation theory¹³. In this study, we propose to use the Nyquist criterion as a cost-effective method to investigate the stability of a thermoacoustic system and to extend the analysis to investigate its robustness to parameter uncertainties.

When despite all efforts an instability occurs in an engine, costly measures often have to be taken. Common practice is to focus on the acoustics and to install passive acoustic damping devices such as Helmholtz dampers or liners^14,15. For can-annular systems, adaptation of the coupling interface is also conceivable¹⁶. In the work of Bade et al.^17,¹⁸ a modification of both, combustor acoustics and flame response is analyzed. Using analytic models calibrated to measured data, an optimization method is proposed that aims at finding an optimal burner configuration based on an analysis of the eigenvalues of a network model.

In contrast to this study, we restrict our attention to design parameters that affect only the flame transfer function but have negligible effects on acoustics. The flame dynamics are influenced by multiple physical mechanisms such as swirl fluctuations^19–21, perturbations in the equivalence ratio^22,23 and shear layer instabilities^24–26. Positive and negative interactions between the effects of each mechanism can cause frequency bands of low and high amplitude response, respectively. Design parameters for FTF modification could for example be the fuel injection distribution, the swirl number or geometric measures^27,28. That even small geometric changes can be used for tailoring the flame transfer function was recently demonstrated by Æsøy et al.²⁹. By placing elements upstream of a bluff body stabilized flame, the gain and phase of the flame transfer function could be modified at targeted frequencies by convective-acoustic interference.

Optimizing the stability margins of a thermoacoustic system by means of shaping the flame transfer function has, to the knowledge of the authors, not been considered yet. This is probably due to the high costs associated with flame transfer function measurements and the lack of suitable optimization criteria.

When the Nyquist method is used for the stability analysis, the phase and gain margin and the maximum gain of the sensitivity function can be used as stability margins^30,31. The higher these stability margins are, the more robust the thermoacoustic system is against parameter uncertainties. In the present study we show that these stability margins can, for a thermoacoustic system, be directly influenced with small adaptions of the flame transfer function. We formulate criteria on how to adapt the flame transfer function at explicit frequencies to design a robustly stable thermoacoustic system with large stability margins for a given combustor. It is shown that even a small change in one of the mechanisms affecting the flame transfer function can have a large effect on the stability margins. The results of this work pave the way for designing robustly stable thermoacoustic systems based on flame transfer function modification at low experimental efforts.

How the FTF itself can be modified is beyond the scope of this study, but previous studies have shown that modification of the flame response is feasible. Besides theoretically based studies like the one from Æsøy et al.²⁹, several data-driven approaches were shown to be suitable for similar problems. Various algorithms for optimization-based pulsation reduction of a generic thermoacoustic system are numerically investigated by Reumschüssel et al.³². Paschereit et al. used a genetic algorithm to experimentally optimize a combustor with the aim of reducing pressure pulsations while maintaining low NO_X emissions²⁷. Due to the dependence on the acoustic boundary conditions, the combustor optimization by pressure pulsation reduction must be carried out in the real combustion chamber of the gas turbine and cannot be performed in a test rig. In contrast, FTF-based optimization can be performed in a test rig and additionally provides more insight into the interaction between acoustics and heat release rate.

The present study is based on the theory and results of Richards et al., who analyzed the potential of passive control of combustion dynamics by varying system time delays using the Nyquist method¹⁰. Building on this we define explicit criteria for shaping the flame transfer function with the objective of designing a robust thermoacoustic system.

Finally, it should be noted that improving system stability by modifying the flame transfer function is not a new idea in general. Rather, it is often considered indirectly in the design of combustion systems. For example, the use of a pilot burner or a change in fuel injection affects the flame response and thus the flame transfer function. The objective of this study is to present the underlying theory to provide a framework for robust combustor design based on modifications of the flame transfer function.

The remainder of this study is structured as follows: first, the Nyquist method is explained and applied to a generic combustor that features an unsteady heat source. Explicit criteria for increasing the stability margins by modifying the flame transfer function are derived in a second step. In the last section it is shown how the Nyquist criterion can also be used as a cost-effective method to take into account parameter uncertainties.

Feedback loop interpretation of thermoacoustics

Thermoacoustic dynamics result from the interaction between the flame and the surrounding acoustic elements. The overall system can be partitioned into two separate subsystems, the unsteady heat release of the flame and the acoustics in the combustion chamber¹⁰. These can be represented in the form of a flame transfer function (FTF) and an acoustic transfer function (ATF). The flame transfer function is defined as the heat release rate fluctuation of the flame, related to an acoustic velocity fluctuation upstream of the flame,

FTF : \frac{\hat{u}}{\bar{u}} \mapsto \frac{\hat{q}}{\bar{q}} .

(1)

The acoustic transfer function is then defined accordingly, relating the velocity fluctuation upstream of the flame to the heat release fluctuation,

ATF : \frac{\hat{q}}{\bar{q}} \mapsto \frac{\hat{u}}{\bar{u}} .

(2)

This representation enables a stability analysis as it is common in control theory, where a feedback cycle features an input and an output quantity. To apply the theory to a thermoacoustic system, the input can be thought of as a disturbance

d (s)

to the heat release rate and the output are velocity fluctuations in the combustor

\hat{u} (s)

, as sketched in Figure 1. For a full analogy to control theory, the feedback is subtracted from the disturbance. This can be accounted for by simply including a factor of -1 in the open loop system as sketched in Figure 1; without loss of generality as we did not make any restrictions to the disturbance. The open loop system is thus given by

L (s) = - 1 FTF (s) ATF (s),

(3)

where

s = σ + j ω

denotes the complex frequency. The transfer function of the closed loop, from a heat release disturbance

d (s)

to acoustic oscillations in the duct upstream of the flame

\hat{u} (s)

, is then given by

G (s) = \frac{L (s)}{1 + L (s)} .

(4)

The full thermoacoustic system is represented by the closed loop cycle. However, the stability analysis can be performed on the basis of the open loop system

L (s)

, as will be shown in the following.

Figure 1.

Interpretation of a thermoacoustic system as a closed loop feedback cycle with disturbance $d (s)$ as input and velocity fluctuation $\hat{u} (s)$ as output. $L (s)$ denotes the open loop system.

Open loop-based stability analysis

The stability of the entire system is determined by the poles of the closed loop system $s_{t}$ , Eq. (4), which are equal to the roots of the denominator and represent the solutions of the dispersion relation

1 + L (s) = 0.

(5)

Depending on the real part of a pole (eigenvalue), the corresponding eigenmode grows or decays exponentially in time. By use of Cauchy’s argument principle, which allows to count poles within a closed contour, the Nyquist criterion can be formulated⁷. It states that Eq. (5) has no solutions in the right half plane (i.e. the closed loop is stable), if the number of times the Nyquist curve of the open loop circles the critical point (-1,0) in the complex plane in clockwise direction is equal to the number of unstable poles of the open loop. Therein, the Nyquist curve is obtained by mapping the stability limit in the complex plane (i.e. the imaginary axis,

j ω

) through the open loop system,

L (j ω)

. Per Eq. (5), the critical point represents the location of all thermoacoustic eigenvalues (poles)

s_{t}

mapped through

L (s)

For the case of a stable open loop system, it follows that the closed loop is stable if the Nyquist curve does not encircle the critical point. The acoustic and flame subsystems in thermoacoustic setups are typically stable when considered individually and thus form a stable open loop system. As for $ω \to \infty$ the gain of most physical systems is zero, while for $ω \to 0$ it is a real value, the Nyquist criterion can be rephrased to a left-hand rule. The latter can also be derived from the preservation of handedness for conformal mappings, as it was for example explained by Polifke et al.⁶:

For a stable open-loop system, the corresponding closed loop is stable, if the critical point is on the left hand side, when following the Nyquist curve.

This method allows the stability to be evaluated without calculating eigenvalues and furthermore enables an analysis of the stability margins of the system, as will be explained in the following section. The Nyquist curve can also be used to estimate eigenvalues⁸. In particular, the corresponding method allows an effective determination of the critical eigenvalues, which are located close to the stability line and thus represent the eigenvalues relevant for engineering systems. A more detailed description of the Nyquist criterion applied to combustion systems can be found in the study of Richards et al.⁶ and references therein.

Stability analysis – generic combustor example

Before discussing stability margins, the applicability of the Nyquist method is demonstrated using a generic thermoacoustic system. Figure 2 shows a sketch of the generic combustor with a total length of 1.3 m. The acoustic model is based on planar waves with a discontinuity in temperature at the location of the flame, $x = 0$ . The system is thus separated into two regions. Upstream of the flame the temperature is set to $T_{1} = 300$ K and downstream of the flame a hot exhaust gas region is modelled by setting the temperature to $T_{2} = 2 T_{1}$ . In the following we first consider the governing acoustic equations and subsequently present a generic unsteady heat release model.

Figure 2.

Sketch of a generic combustor with heat source located at x = 0.

We assume an ideal gas in the two regions at a pressure of $1$ bar, with a gas constant of $R_{g a s} = 287$ J/kgK and a ratio of specific heats of $γ = 1.4$ . The corresponding speed of sound in each of the two regions can be computed as $c = \sqrt{γ R_{g a s} T}$ . The tube diameter is set to $0.4$ m. By assuming only plane acoustic waves, the acoustic velocity and pressure in the two regions read³³:

p^{'} (x, t) = \hat{p} (x) e^{s t} = (f e^{- \frac{x s}{c}} + g e^{+ \frac{x s}{c}}) e^{s t},

(6)

u^{'} (x, t) = \hat{u} (x) e^{s t} = \frac{1}{ρ c} (f e^{- \frac{x s}{c}} - g e^{+ \frac{x s}{c}}) e^{s t} .

(7)

The reflection coefficients at the upstream end at

x = - l_{1} = - 0.25

m, is set to

R_{1} = - 0.75

and the coefficient at the downstream end

R_{2}

is set to

1

. Upstream and downstream reflection coefficients of

- 1

and

1

are commonly used to model the large compressor plenum and the choked turbine inlet of a gas turbine combustor, respectively. To additionally account for acoustic damping in the model, we use

R_{1} = - 0.75

in this study. The Rankine–Hugoniot jump conditions are applied at

x = 0

, to relate the acoustic fluctuations across the flame³³:

{\hat{p}}_{2} - {\hat{p}}_{1} = 0,

(8a)

{\hat{u}}_{2} - {\hat{u}}_{1} = \frac{γ - 1}{ρ_{1} c_{1}^{2} A_{c}} \hat{q},

(8b)

where

A_{c}

denotes the cross-sectional tube area. These two equations together with the up- and downstream reflection coefficients yield the following system of equations:

\begin{aligned} \underset{M (s)}{\underset{⏟}{[\begin{matrix} - R_{1} e^{- τ_{1} s} & 1 & 0 & 0 \\ 1 & 1 & - 1 & - 1 \\ 1 & - 1 & - \frac{ρ_{1} c_{1}}{ρ_{2} c_{2}} & \frac{ρ_{1} c_{1}}{ρ_{2} c_{2}} \\ 0 & 0 & 1 & - R_{2} e^{- τ_{2} s} \end{matrix}]}} \underset{g}{\underset{⏟}{[\begin{matrix} g_{1} \\ f_{1} \\ g_{2} \\ f_{2} \end{matrix}]}} = \underset{q}{\underset{⏟}{[\begin{matrix} 0 \\ 0 \\ \frac{(γ - 1) \hat{q}}{c_{1} A_{c}} \\ 0 \end{matrix}]}}, \end{aligned}

(9)

where

τ_{1} \equiv 2 l_{1} / c_{1}

and

τ_{2} \equiv 2 l_{2} / c_{2}

denote acoustic time lags. For

\hat{q} = 0

, Eq. (9) can be solved for pure acoustic eigenvalues. By solving

g = M^{- 1} (s) q

(10)

and substituting the solution into Eq. (7), the acoustic transfer function as defined in the theory section can be determined.

While in practice FTFs are usually determined experimentally or numerically, here we relate the heat release fluctuations to acoustic fluctuations by means of a generic flame transfer function

FTF (s) \equiv \frac{\hat{q}}{\bar{q}} \frac{\bar{u}}{\hat{u}} = {FTF}_{a} (s) + {FTF}_{b} (s),

(11)

where the two transfer functions

{FTF}_{a, b}

represent the flame response caused by two individual effects, for example resulting from shear layer instabilities, equivalence ratio perturbations or swirl fluctuations. Both transfer functions are assumed to have the same generic form

{FTF}_{a, b} (s) = \frac{K_{a, b}}{C s + 1} e^{- τ_{a, b} s}

(12)

with different frequency dependent gain and phase, characterized by gain parameter

K_{a, b}

and characteristic time lag

τ_{a, b}

. The parameters of the two transfer functions are chosen differently, as they are intended to represent different effects. Without intending to represent a specific configuration or specific effects, the individual parameters are set to

K_{a} = 0.8

K_{b} = 0.2

C = 1 / 700

τ_{a} = 2 \times 10^{- 4}

and

τ_{b} = 5 \times 10^{- 3}

. The ratio of mean heat release to mean velocity is determined by conservation of energy. The black line in Figure 3 shows the absolute value and phase of the total flame transfer function for the specified parameter set. It can be observed that the two mechanisms interact in a constructive and destructive manner, depending on the frequency, as it is commonly observed for flame transfer functions²⁰.

Figure 3.

Gain and phase of flame transfer function for the generic combustor example for different parameter values. Black, solid: $τ_{b} = 5.0 \times 10^{- 3}$ . Red, dashed: $τ_{b} = 5.5 \times 10^{- 3}$ . All other parameter values are kept constant. Vertical lines highlight the frequencies $ω_{a}^{*} / (2 π) = 257.3$ Hz and $ω_{ϕ}^{*} / (2 π) = 83.9$ Hz.

The thermoacoustic eigenvalues of the combustor $s_{t}$ can be determined by first rearranging Eq. (9) into the form

M_{f} (s) g = 0

(13)

and then by solving

det (M_{f}) = 0

. Another possibility is to determine the acoustic transfer function as stated above and solve the dispersion relation, Eq. (5), or to investigate the Nyquist curve in the complex plane.

Figure 4 (left) shows the thermoacoustic eigenvalues $s_{t}$ between $0$ and $1000$ Hz of the thermoacoustic system, identified with the determinant method explained above. It can be observed that all thermoacoustic eigenvalues have a negative growth rate and thus the closed loop thermoacoustic system is stable. Two eigenvalues found at frequencies of $85.7$ Hz and $257$ Hz, respectively, are located relatively close to the stability line compared to the other eigenvalues.

Figure 4.

Left: Thermoacoustic eigenvalues in the complex plane. The coloured line shows the stability line. Right: Nyquist curve (mapping of the stability line through the open loop transfer function $L (j ω)$ ) in the complex plane. The colour indicates the frequency. The dash-dotted line indicates the unit circle. The frequencies of the two thermoacoustic eigenvalues closest to instability are marked with crosses on the stability line (Nyquist curve).

Figure 4 (right) shows the corresponding Nyquist curve in the complex plane. It can be observed that the critical point lies on the left hand side of the Nyquist curve. Since the critical point represents the location of all thermoacoustic eigenvalues mapped through the open loop system $L (s_{t})$ and the Nyquist curve represents the mapped stability line $L (j ω)$ , this criterion yields, as expected, the identical result: the thermoacoustic system is stable. The two frequencies with thermoacoustic eigenvalues closest to the stability line, compare Figure 4 (left), are highlighted with crosses on the Nyquist curve (stability line). It can equally be seen in Figure 4 (left) and Figure 4 (right) that at these two distinct frequencies, the thermoacoustic eigenvalues (the critical point) are close to the stability line (the Nyquist curve).

Gain and phase margin

The Nyquist curve allows to define stability margins describing the system’s proximity to instability³¹. The gain margin $a_{r}$ is defined as the factor which multiplied with the open loop system causes the Nyquist curve to cross the critical point. The phase margin $ϕ_{r}$ is the angle that can be added to (or subtracted from) the Nyquist curve until the critical point is exceeded. For a given open loop system gain and phase margin can easily be determined from the crossings of the Nyquist curve with the negative real axis and the unit circle, respectively. Mathematically, they are defined as

a_{r} = \frac{1}{| min (L (j ω_{a, i})) |}, with ℑ (L (j ω_{a, i})) = 0

(14)

ϕ_{r} = min (| ∠ (L (j ω_{ϕ, k})) - π |), with | (L (j ω_{ϕ, k})) | = 1,

(15)

where

ω_{a, i}

are all frequencies at which the Nyquist curve crosses the real axis and

ω_{ϕ, k}

are all frequencies at which the Nyquist curve crosses the unit circle. The frequencies out of these, where the Nyquist curve is additionally closest to the critical point determine the stability margins. In the following, they are denoted as

ω_{a}^{*}

and

ω_{ϕ}^{*}

, respectively.

A gain margin of $2$ , for example, allows the gain to be uncertain by $100 %$ without loosing stability, while a gain margin of $1$ corresponds to a marginal stable system. The phase margin directly translates into additive uncertainty, a phase margin of $θ$ for example means that the open loop system could be uncertain in phase by $θ$ without crossing the critical point. A system with large phase and gain margins is considered robustly stable, as the system can feature uncertainties in phase and gain without undergoing instability. Figure 5 visualizes the stability margins for the generic combustor example. The gain margin is $a_{r} = 1.13$ and is found at $f_{a}^{*} = ω_{a}^{*} / (2 π) = 257.3$ Hz. The phase margin is $ϕ_{r} = {18.2}^{\circ}$ and is found at $f_{ϕ}^{*} = ω_{ϕ}^{*} / (2 π) = 83.9$ Hz.

Figure 5.

Nyquist curve $L (j ω)$ in the complex plane with gain and phase margin indicated.

Sensitivity function

Generally it is possible that the gain (phase) margin is very sensitive to changes in phase (gain) and therefore systems with initially high gain and phase margins can still become unstable if the uncertainties affect phase and gain simultaneously. A more robust measure for quantifying the robustness to uncertainties is the distance of the Nyquist curve from the critical point $| L (j ω) - (- 1) |$ . The minimum distance between the Nyquist curve and the critical point is usually determined by examining the gain of the sensitivity function $S (j ω)$ , which is the inverse of the distance. The critical minimum distance at frequency $ω_{μ}^{*}$ is located at the point of maximum gain of the sensitivity function³¹:

μ = max | \underset{S (j ω)}{\underset{⏟}{\frac{1}{1 + L (j ω)}}} | .

(16)

The black line in Figure 6 shows the gain of the sensitivity function for the generic combustor example. The sensitivity function shows narrow frequency bands of high and low gain, respectively. The frequencies at the peaks,

ω_{μ, l}

are critical as the Nyquist curve gets closest to the critical point for these frequencies. The vertical dashed lines highlight the frequencies at which the gain and phase margins were found. It can be observed that for this example the frequencies of gain and phase margin are very close to the frequencies of high sensitivity gain. We expect that this is common in thermoacoustic processes, since they are dominated by time delays and therefore should normally not exhibit complex-shaped Nyquist curves. However, this needs not generally to be the case.

Figure 6.

Absolute value of sensitivity function over frequency for $τ_{b} = 5 \times 10^{- 3}$ (black) and $τ_{b} = 5.5 \times 10^{- 3}$ (red). Vertical lines highlight the frequencies $ω_{a}^{*} / (2 π) = 257.3$ Hz and $ω_{ϕ}^{*} / (2 π) = 83.9$ Hz.

Robust combustor design with FTF modification

With the Nyquist method and stability margins at hand, we now address the question of how to modify a stable thermoacoustic system in order to increase its robustness. Generally, the most stable system is designed by maximizing the stability margins, based on either gain and phase margin or the gain of the sensitivity function. For the sake of clarity, the concept is first presented based on the gain and phase margin. For completeness, the approach based on the sensitivity function is indicated in the second part of this section.

In accordance to the definition of the open loop transfer function, Eq. (3), the Nyquist curve $L (j ω)$ consists of two elements, the acoustic transfer function and the flame transfer function. By geometric combustor modification or installation of passive damping devices such as Helmholtz dampers or liners, the acoustics can be modified in order to achieve stable operation³⁴. However, geometric variations are costly and dampers typically require cooling air, which can affect engine efficiency. In this study we want to shift the focus to modifications of the flame transfer function to increase stability margins instead.

In the following, we will consider the theoretical basis of designing a robust system using FTF modifications and not consider how the FTF itself can be modified. It is assumed that there exist tuning parameters $α$ that allow for sufficient modification of the FTF. Such parameters could for example be the location of fuel injection, the fuel and equivalence ratio distribution across several injection locations, the swirl number, the pilot fuel ratio or the gas mixture. As mentioned above, corresponding studies can be found in the literature. It is further assumed that the tuning parameters can be varied by an optimization scheme in an experimental setup, which also allows to measure the FTF. In the following, the $α$ -index indicates that the corresponding transfer function can be modified with the set of tuning parameters $α$ .

The Nyquist curve can be interpreted as a complex function with a frequency dependent gain and phase

L_{α} (j ω) = | L_{α} (j ω) | e^{j ∠ (L_{α} (j ω))},

(17)

each of them with an acoustic and a flame contribution

| L_{α} (j ω) | = | {FTF}_{α} (j ω) | | ATF (j ω) |

(18)

∠ (L_{α} (j ω)) = π + ∠ ({FTF}_{α} (j ω)) + ∠ (ATF (j ω))

(19)

By considering gain and phase of the Nyquist curve separately, Eqs. (18) and (19), it becomes obvious how gain and phase can directly be effected by a modification of the flame transfer function. Accordingly, the gain margin and the phase margin can be tuned by modifying the FTF at

ω_{a}^{*}

and

ω_{ϕ}^{*}

, respectively. The gain margin can be increased by the factor that the FTF gain is decreased at the frequency

ω_{a}^{*}

and a change of the FTF phase by

θ

ω_{ϕ}^{*}

can increase the phase margin by

θ

. This means that maximizing the stability margins is equivalent to minimizing the FTF gain at

ω_{a}^{*}

and in-/decreasing the FTF phase at

ω_{ϕ}^{*}

. Accordingly, a cost function of an optimization scheme aimed at increasing stability margins should be formulated based on these criteria.

The criteria form the basis for designing robustly stable combustors based on FTF modification. We emphasize that the optimization routine only needs to evaluate the FTF at the specific frequencies, $ω_{a}^{*}$ and $ω_{ϕ}^{*}$ , and is therefore limited in experimental effort.

As the FTF is measured at the frequency of the phase margin $ω_{ϕ}^{*}$ , the gain at this frequency $| {FTF}_{α} (j ω_{ϕ}^{*}) |$ can also be added to the cost function. In the same way, $∠ ({FTF}_{α} (j ω_{a}^{*}))$ can be included in the cost function. This extended criterion can be useful since, usually, frequencies at both stability margins are critical for stability. This can also be inferred by the high amplitudes of the sensitivity function at $ω_{ϕ}^{*}$ and $ω_{a}^{*}$ in Figure 6. The cost function for robust combustor design than reads

\begin{aligned} J (α) = & c_{1} | {FTF}_{α} (j ω_{a}^{*}) | + c_{2} | {FTF}_{α} (j ω_{ϕ}^{*}) | \\ \pm c_{3} ∠ ({FTF}_{α} (j ω_{a}^{*})) \pm c_{4} ∠ ({FTF}_{α} (j ω_{ϕ}^{*})), \end{aligned}

(20)

where

c_{i}

are arbitrary weights. In the following we will set

c_{1} = c_{2} = 1

and

c_{3} = c_{4} = 1 / (2 π)

to normalize the cost function. Depending on the initial situation, other weights may be appropriate.

However, it is unlikely to find a tuning parameter combination $α^{*}$ that effects the FTF’s phase and gain individually and furthermore at these distinct frequencies only. This in turn means that the overall optimization procedure needs to be performed iteratively, with updated $ω_{a}^{*}$ and $ω_{ϕ}^{*}$ each time the optimizer has improved the stability margins. Additional care must be taken to not optimize the FTF for certain frequencies at the cost of moving the Nyquist curve closer to the critical point for other frequencies¹⁰. These difficulties require to measure the FTF for a broader frequency band each time the optimizer has improved the stability margins.

In summary, these results allow the approach visualized in Figure 7 to optimize a given combustor based on FTF modification. First, the acoustic and flame transfer function need to be determined. In a second step the Nyquist method is used to compute the stability margins and the frequencies $ω_{a}^{*}$ and $ω_{ϕ}^{*}$ . With this information the optimization routine can start with the aim to find a tuning parameter combination that minimizes $J (α)$ , Eq. (20). This is the critical step of the routine, as it can be very costly depending on the size of the parameter space. Therefore, limiting the cost function to certain frequencies drastically decreases the costs for an FTF modification based optimization. If the optimizer finds a solution with a sufficient decrease in $J (α)$ , a new FTF needs to be determined to update the frequencies $ω_{a}^{*}$ and $ω_{ϕ}^{*}$ . Moreover, this step ensures that the optimizer has not improved the robustness at the frequencies $ω_{a}^{*}$ and $ω_{ϕ}^{*}$ at the cost of decreasing the overall stability margins for other frequencies. The routine ends when sufficient stability margins are reached or further improvements are not possible within the given tuning parameter spaces.

Figure 7.

FTF modification routine based on the gain and phase margin criterion.

Alternatively, robustness can be increased by modifying the FTF based on the maximum sensitivity gain criterion. Decreasing the FTF gain and in-/decreasing the FTF phase at $ω_{μ}^{*}$ increases the minimum distance between the Nyquist curve and the critical point. If the gain of the sensitivity function shows multiple high peaks, as for example shown in Figure 6, multiple $ω_{μ, l}$ can be considered. Consequently, the cost function for the FTF modification based on the sensitivity function reads

J_{μ} (α) = \sum_{l = 1}^{N} \frac{b_{l}}{| 1 - FTF (j ω_{μ, l}) ATF (j ω_{μ, l}) |},

(21)

where

b_{l}

are weights and

N

is the number of considered peaks in the gain of the sensitivity function. In contrast to the cost function based on gain and phase margin, Eq. (20), this cost function requires the evaluation of the acoustic transfer function as well. The optimization routine is then the same as depicted in Figure 7, with frequencies

ω_{μ, l}

instead of

ω_{a}^{*}

and

ω_{ϕ}^{*}

, cost function

J_{μ} (α)

instead of

J (α)

and stability margin

μ

instead of

a_{r}

and

ϕ_{r}

. As presented in the generic combustor example, see Figure 6, the criterion based on gain and phase margin and the criterion based on the sensitivity function will lead to very similar critical frequencies

ω^{*}

for many cases. However, the criterion based on the sensitivity function can generally be considered more robust, since it is based on the distance between the Nyquist curve and the critical point and therefore does not consider only separate changes of phase or gain.

Robust design – generic combustor example

Without performing any optimization, the generic combustor example is considered again in order to demonstrate the leverage of FTF based combustor optimization. Again, we will first focus on gain and phase margin, which for this combustor are $a_{r} = 1.13$ and $ϕ_{r} = {18.2}^{\circ}$ at frequencies of $f_{a}^{*} = ω_{a}^{*} / (2 π) = 257.3$ Hz and $f_{ϕ}^{*} = ω_{ϕ}^{*} / (2 π) = 83.9$ Hz, see Figure 5.

It is assumed that through the tuning parameters we can influence one of the two mechanisms that effect the overall flame transfer function, for example the ${FTF}_{b}$ via its time lag, $α = τ_{b}$ . The red curve in Figure 3 shows the FTF with a $10 %$ increase in $τ_{b}$ . Overall, the modified FTF (red) does not vary much from the original FTF (black). The average gain over the considered frequency range remains approximately constant. At the frequencies of interest, $ω_{a}^{*}$ and $ω_{ϕ}^{*}$ , it can be observed that the gain has decreased. The phase remains constant at $ω_{a}^{*}$ and increases by a negligible amount at $ω_{ϕ}^{*}$ . Overall, the cost function $J (α)$ is reduced by $8.5 %$ with this FTF modification. Figure 8 shows that with this FTF adaption the stability margins could be increased to $a_{r} = 1.38$ and $ϕ_{r} = {25.67}^{\circ}$ , meaning that the system could contain uncertainties of $38 %$ in gain or ${25.67}^{\circ}$ in phase without undergoing instability.

Figure 8.

Nyquist curve $L_{α} (j ω)$ for modified FTF in the complex plane with increased gain and phase margin indicated.

The improvement in stability margins can equally be seen in the sensitivity function plot, Figure 6. The gain of the sensitivity function based on the modified FTF is shown in red. The change in $τ_{b}$ decreases the maximum gain $μ$ to almost one third. For this example, optimization based on both stability criteria would lead to similar results because the critical frequencies are similar.

The example reveals two findings. First, small modifications of the FTF at the correct frequencies can largely increase the stability margins, which makes robust combustor design based on FTF modification a promising tool. As the example presented shows, such a modification can already be achieved by small changes of a single mechanism that affects the flame dynamics. And second, the strong sensitivity of stability margins to parameter changes indicates that quantifying the effects of uncertainty on stability is very important, and is therefore considered next.

Uncertainty quantification

The subsystems in a thermoacoustic network model originate either from theoretical considerations, from measurements or from simulations^33,35,36. In any case, the subsystems differ from the real physical systems and uncertainties have to be taken into account to make reliable predictions about the stability of the actual system. How uncertainties propagate to system stability is known as uncertainty quantification¹³. The Nyquist criterion enables a compact and cost-effective tool to analyze the stability of the full thermoacoustic system when subsystems have uncertainties. Computing thermoacoustic eigenvalues is generally expensive as the underlying equations are highly nonlinear. Recent studies have thus focused on reducing the degrees of freedom by exploiting symmetries^3,37 or using non-iterative methods³⁸ to solve for thermoacoustic eigenvalues. In comparison to eigenvalue calculations, computing Nyquist curves is cost-effective. We therefore propose to use the Nyquist method to effectively perform Monte Carlo simulations for uncertainty analysis. If for each uncertain open loop system $L_{u} (j ω)$ the critical point is on the left hand side, when following the Nyquist curve, no thermoacoustic eigenvalue has a positive growth rate and thus, the thermoacoustic system is stable for each considered uncertainty.

The Nyquist curve realisations $L_{u} (j ω)$ , that consist of an FTF and an ATF contribution, can especially be computed at low costs if only the FTF features uncertainties. In this case, the ATF only needs to be computed once and evaluating the FTF based on an analytical model is generally cost-effective. If the acoustics are also subject to uncertainties, the ATF must be computed for each realization of uncertainty, which in turn means that matrix inversions must be performed for each frequency and each uncertainty realization, see Eq. (10). However, compared to common iterative eigenvalue methods, this can still be considered as cost-effective.

Further costs can be reduced by computing the Nyquist curves only in certain frequency bands, for example at those where the nominal Nyquist curve without uncertainties is close to the critical point. A disadvantage of this method is that it does not provide information about the location of the thermoacoustic eigenvalues, only whether the system is stable. However, if a Nyquist curve for a given uncertainty crosses the stability line, an additional eigenvalue determination for this specific uncertainty could be performed in order to gain information about the growth rate and frequency of the unstable eigenmode.

Uncertainty quantification – generic combustor example

Uncertainty quantification based on the Nyquist method is demonstrated in this last section. As nominal system serves the robust generic combustor from the previous section with large stability margins, $τ_{b} = 5.5 \times 10^{- 3}$ . The temperature $T_{1}$ is now varied in the range of $\pm 10 %$ and the flame parameters $K_{a}$ , $K_{b}$ and $τ_{b}$ in a range of $\pm 5 %$ . Note that the uncertainty in temperature $T_{1}$ effects many other parameters such as the downstream temperature $T_{2}$ , the mean heat release, the speed of sounds as well as the gas densities according to the ideal gas law. Figure 9 shows $1250$ realisations of Nyquist curves that feature parametric uncertainties. Due to the uncertainties, the array of Nyquist curves covers large areas of the complex plane. However, the robust design of the generic combustor ensures that for each uncertainty combination the corresponding Nyquist curve realization does not encircle the critical point, and thus the system remains stable. Between $200$ Hz and $300$ Hz the Nyquist curves get close to the critical point, but never pass it. Figure 10 shows the corresponding $1250$ eigenvalues determined with the significantly more expensive determinant method, Eq. (13). It can equally be seen in Figure 9 and Figure 10 that the system is likely to loose its stability feature at approximately $240$ Hz for higher uncertainties.

Figure 9.

Nyquist curve realisations $L_{u} (j ω)$ for $1250$ uncertain parameter combinations. At about 250 Hz, the system is close to the critical point due to the uncertainties.

Figure 10.

Thermoacoustic eigenvalues in the range between $220$ Hz and $280$ Hz for $1250$ combinations of uncertain parameters (blue) and for the nominal parameters (red).

Conclusion

In this study, it was proposed to increase the robustness of thermoacoustic systems by modifying the flame transfer function. Using the theory around the Nyquist method, expressions for stability margins, namely gain margin, phase margin and sensitivity function, were presented. These quantify the tendency of a system to become unstable and can be used to identify critical frequencies. Based on the stability margins, cost functions were formulated that can be minimized to maximize the robustness of the thermoacoustic system to model and parameter uncertainties. Since the criteria are limited to single frequencies, the cost of evaluation is very limited. This opens up new possibilities for methods of combustor design and passive control through experimental FTF shaping and iterative optimization routines. The proposed method takes into account the interaction between acoustics and flame and thus the origin of thermoacoustics, but at the same time allows for an optimization solely based on flame transfer function modification. Although no specific FTF modification measures are described in this study, reference is made to relevant work in the literature where the flame response could be modified by making small adjustments to the burner. At this point, however, we would also like to point out that gas turbine engineers must balance a variety of design considerations, of which thermoacoustics is only one. Small changes to the system can have large effects on other design parameters, such as emissions, hydrodynamic stability or thermal stresses. Potential effects should therefore always be assessed, even if only minor changes are made to the system. When this is taken into account, the presented theory enables the effective design of robust thermoacoustic systems.

In addition, this study proposed to use the Nyquist method for uncertainty quantification when a thermoacoustic system is subject to parametric uncertainties. Calculating Nyquist curves instead of repeated eigenvalue calculations offers a significant cost reduction.

Footnotes

Acknowledgements

The authors acknowledge the German Ministry of Economy and Technology and MAN Energy Solutions SE for funding within the scope of the AG Turbo project ROBOFLEX (grant nr. 03EE5013E). J. G. R. von Saldern and J. M. Reumschüssel would like to thank Prof. Rudibert King for his excellent teaching of control engineering.

ORCID iDs

Jakob G. R. von Saldern

Kilian Oberleithner

References

Lieuwen

. Unsteady combustor physics. Cambridge: Cambridge University Press, 2012.

Dowling

Stow

. Acoustic analysis of gas turbine combustors. J Propuls Power 2003; 19(5): 751–764.

von Saldern

JGR

Orchini

Moeck

. Analysis of thermoacoustic modes in can-annular combustors using effective bloch-type boundary conditions. J Eng Gas Turbines Power 2020; 143(7): 1374–1389.

Bauerheim

Parmentier

Salas

Nicoud

Poinsot

. An analytical model for azimuthal thermoacoustic modes in an annular chamber fed by an annular plenum. Combust Flame 2014; 161(5): 1374–1389.

Yoon

. Combustion instability analysis from the perspective of acoustic impedance. J Sound Vib 2020; 483: 115500.

Polifke

Paschereit

Sattelmayer

. A universally applicable stability criterion for complex thermo-acoustic systems. VDI Berichte 1997; 1313(1): 455–460.

Nyquist

. Regeneration theory. Bell Syst Tech J 1932; 11(1): 126–147.

Sattelmayer

Polifke

. Assessment of methods for the computation of the linear stability of combustors. Combust Sci Technol 2003; 175(3): 453–476.

Sattelmayer

Polifke

. A novel method for the computation of the linear stability of combustors. Combust Sci Technol 2003; 175(3): 477–497.

10.

Richards

Straub

Robey

. Passive control of combustion dynamics in stationary gas turbines. J Propuls Power 2003; 19(5): 795–810.

11.

Guo

Silva

Yong

Polifke

. A gaussian-process-based framework for high-dimensional uncertainty quantification analysis in thermoacoustic instability predictions. Proc Combus Inst 2020; 38(4): 6251–6259.

12.

Avdonin

Jaensch

Silva

Češnovar

Polifke

. Uncertainty quantification and sensitivity analysis of thermoacoustic stability with non-intrusive polynomial chaos expansion. Combust Flame 2018; 189: 300–310.

13.

Mensah

Magri

Moeck

. Methods for the calculation of thermoacoustic stability boundaries and monte carlo-free uncertainty quantification. J Eng Gas Turbines Power 2018; 140(6): 061501.

14.

Noiray

Schuermans

. Theoretical and experimental investigations on damper performance for suppression of thermoacoustic oscillations. J Sound Vib 2012; 331(12): 2753–2763.

15.

Zhao

Morgans

Dowling

. Tuned passive control of acoustic damping of perforated liners. AIAA J 2011; 49(4): 725–734.

16.

von Saldern

JGR

Orchini

Moeck

. A non-compact effective impedance model for can-to-can acoustic communication: analysis and optimization of damping mechanisms. J Eng Gas Turbines Power 2021; 143(12): 121024.

17.

Bade

Wagner

Hirsch

Sattelmayer

Schuermans

. Design for thermo-acoustic stability: modeling of burner and flame dynamics. J Eng Gas Turbines Power 2013; 135(11): 111502.

18.

Bade

Wagner

Hirsch

Sattelmayer

Schuermans

. Design for thermo-acoustic stability: procedure and database. J Eng Gas Turbines Power 2013; 135(12): 121507.

19.

Komarek

Polifke

. Impact of swirl fluctuations on the flame response of a perfectly premixed swirl burner. J Eng Gas Turbines Power 2010; 132(6): 061503.

20.

Palies

Durox

Schuller

Candel

. The combined dynamics of swirler and turbulent premixed swirling flames. Combust Flame 2010; 157(9): 1698–1717.

21.

Müller

Lückoff

Kaiser

Paschereit

Oberleithner

. Modal decomposition and linear modeling of swirl fluctuations in the mixing section of a model combustor based on particle image velocimetry data. J Eng Gas Turbines Power 2022; 144(1): 011021.

22.

Lieuwen

Zinn

. The role of equivalence ratio oscillations in driving combustion instabilities in low NO_X gas turbines. Symposium (International) on Combustion 1998; 27(2): 1809–1816.

23.

Bluemner

Paschereit

Oberleithner

. Generation and transport of equivalence ratio fluctuations in an acoustically forced swirl burner. Combust Flame 2019; 209: 99–116.

24.

Poinsot

Trouve

Veynante

Candel

Esposito

. Vortex-driven acoustically coupled combustion instabilities. J Fluid Mech 1987; 177: 265–292.

25.

Schadow

Gutmark

. Combustion instability related to vortex shedding in dump combustors and their passive control. Prog Energy Combust Sci 1992; 18(2): 117–132.

26.

Oberleithner

Schimek

Paschereit

. Shear flow instabilities in swirl-stabilized combustors and their impact on the amplitude dependent flame response: a linear stability analysis. Combust Flame 2015; 162(1): 86–99.

27.

Paschereit

Schuermans

Büche

. Combustion process optimization using evolutionary algorithm. In: Turbo Expo: Power for Land, Sea, and Air, volume 2: Turbo Expo 2003. 2003. pp. 281–291.

28.

Durox

Moeck

Bourgouin

Morenton

Viallon

Schuller

Candel

. Flame dynamics of a variable swirl number system and instability control. Combust Flame 2013; 160(9): 1729–1742.

29.

Æsøy

Nygård

Worth

Dawson

. Tailoring the gain and phase of the flame transfer function through targeted convective-acoustic interference. Combust Flame 2022; 236: 111813.

30.

Zhou

Doyle

. Essentials of robust control. Upper Saddle River, NJ: Prentice Hall, 1998.

31.

Skogestad

Postlethwaite

. Multivariable feedback control: analysis and design. vol. 2. New York: Wiley, 2005.

32.

Reumschüssel

von Saldern

JGR

Paschereit

Orchini

. Gradient-free optimization in thermoacoustics: application to a low-order mode. J Eng Gas Turbines Power 2022; 144(5): 051004.

33.

Dowling

. The calculation of thermoacoustic oscillations. J Sound Vib 1995; 180(4): 557–581.

34.

Dupére

IDJ

Dowling

. The use of helmholtz resonators in a practical combustor. J Eng Gas Turbines Power 2005; 127(2): 268–275.

35.

Paschereit

Schuermans

Polifke

Mattson

. Measurement of transfer matrices and source terms of premixed flames J Eng Gas Turbines Power 2002; 124(2): 239–247.

36.

Merk

Gaudron

Silva

Gatti

Mirat

Schuller

Polifke

. Prediction of combustion noise of an enclosed flame by simultaneous identification of noise source and flame dynamics.. Proc Combus Inst 2019; 37(4): 5263–5270.

37.

Mensah

Campa

Moeck

. Efficient computation of thermoacoustic modes in industrial annular combustion chambers based on bloch-wave theory. J Eng Gas Turbines Power 2016; 138(8): 081502.

38.

Buschmann

Mensah

Nicoud

Moeck

. Solution of thermoacoustic eigenvalue problems with a noniterative method. J Eng Gas Turbines Power 2020; 142(3): 031022.