Proceedings of the Institute of Acoustics

 

 

Bayesian optimization based adaptive controller for thermoacoustics instability

 

Bayu Dharmaputra¹, ETH Zurich, Zurich, Switzerland

Alain Williner², ETH Zurich, Zurich, Switzerland

Bruno Schuermans³, ETH Zurich, Zurich, Switzerland

Nicolas Noiray⁴, ETH Zurich, Zurich, Switzerland

 

ABSTRACT

 

Thermoacoustics instability is one of the factors that limits the operational range of gas turbines. Both active and passive control strategies have been developed to suppress thermoacoustic instability. Active control strategies can adapt to the operational condition of the gas turbine and hence providing a wider effectivity compared to the passive control strategies. However, it remains a chal-lenge on how to best optimize the control parameters. In this work, we propose an adaptive control strategy by using Gaussian Process Regression with a modified Safe-Optimization (SafeOpt) algo-rithm. The main benefit of the algorithm is it can safely explore the parameter space and minimizing the cost function simultaneously. The performance of the algorithm in suppressing the thermoacoustic instability is experimentally demonstrated by stabilizing a turbulent combustor with loudspeakers actuation equipped with a gain-delay ( n -t ) controller. It is found that the proposed algorithm can safely optimize both n and t parameter in under 20 iterations. The final iteration is then fed to the ordinary Extremum Seeking Controller (ESC) to further optimize the parameters around the optimum point.

 

1. INTRODUCTION

 

Future gas turbine combustor for power generation requires high fuel flexibility to be competitive in the market [1]. However, gas turbine combustors are prone to thermoacoustics instability. There are multiple mechanisms that can lead to thermoacoustic instabilities, the readers are advised to look at [2]. To be operatable at different conditions, ideally, a combustor needs to have a reasonably wide range of thermoacoustic stability. One way to do it is to apply an active control strategy [3].

 

To be able to stabilize the system at different operating conditions, the controller needs to be adap-tive. One of the most popular adaptive control algorithms is the extremum seeking controller (ESC). A successful implementation of ESC in combustion application has been done in [4].However, ESC has a disadvantage of being trapped in a local minimum and it is not straight forward to implement safety constraint into the algorithm.

 

Recently, an adaptive control strategy based on Bayesian optimization has been developed [5]–[7]. It has the advantage of being able to implement safety constraints and at the same time optimize the control parameters.

 

In this work we focus on optimizing the control parameters of our gain-delay controller to minimize the root mean-squared (RMS) of pressure fluctuations (objective function) in the combustor and also keeping the safety constraint satisfied.

 

2. Methods

 

The safety constraint in our case is the maximum voltage applied to the loudspeaker. Note that in principle the constraint can be some other quantities, such as NOx and CO level at the exhaust. We denote the objective function and constraint function as j(a) and g(a)respectively, where a is the vector of parameters, which in our case is the gain (𝑛) and delay (𝜏) parameter. Both j(a) and g(a) are not known a priori, and hence the goal is to iteratively optimize and explore the safe parameters within a reasonable number of iterations. Both functions are approximated with a Gaussian Process Regressor, and it will be further explained in the next subsection.

 

2.1. Gaussian Process

 

A Gaussian Process (GP) can approximate unknown function based on a nonparametric model [8]. It works by assuming that the function values of j(a) (and consequently g(a)) with different a jointly gaussian. A GP is described by a prior mean function and covariance function 𝑘 described as:

 

 

Where a_i and a_j are two different points on the parameters space. The covariance or the kernel function encodes the smoothness and the rate of change of the reconstructed function which are con trolled by the value of 𝜎₁ and l. The GP framework can predict the function value 𝑓(a^*) at arbitrary a^* based on past observations by using the bayes rules. The posterior mean and covariance then become:

 

 

Where 𝑓_n is the vector n observed values, 𝜎 is the standard deviation of the observation noise, k_n (𝑎^∗) is the covariance vector between the new point 𝑎^∗ and the observed data points, and K_n is the covariance matrix of the observed data points.

 

2.2. SafeOpt Algorithm

 

The optimization scheme is based on the SafeOpt algorithm presented in [6], [7]. The GP was trained on seven initial data points. The parameter domain was divided into a 100-by-100 grid. This first iteration of the GP was used to predict the mean objective function values and their standard deviation at the parameter combinations at the grid points. The predicted mean and standard deviation define the upper and lower bounds, u(a) l(a) and shown in equation (4) and (5) respectively, of the expected objective function values. Together with the safety threshold J_max the so-called safe set was calculated. In this case, a minimum of the cost functions is searched. If u_g (a) is smaller thanJ_max , the corresponding parameter combination is part of the safe set.

 

In the next step, the minimizer set is calculated. The points in the safe set are assessed to verify whether their l_j (a) is smaller than the minimum u_j (a) of all the grid points in the domain. If this is the case, the point is part of the minimizer set, otherwise the point is further assessed for the expander set. Hence, if the remaining points have a 𝜔 = u(a) - l(a) which is larger than the maximum standard deviation found in the minimizer set, then they become a part of the expander set. The union of the minimizer and expander set defines the set of new possible experimental points.

 

 

The upper and lower bounds have the indices J and g corresponding to the cost function J(a) and constraint function g(a).

 

In some cases, the algorithm ends up with multiple possible points for the next experimental evaluation. Therefore, the next point with the highest value is chosen to increase the information gain. Then, the experiment is performed, and the GP is fitted again with the new added data.

 

2.3. Experimental Setup.

 

The schematic of the setup can be seen in Figure 1. The turbulent combustor setup consists of a plenum section, combustion chamber, one loudspeaker, one microphone, and a swirl burner. The fuel, which is a blend of natural gas and hydrogen, is delivered through the lance. The airstream is injected through the plenum, entering the duct and mixing with the fuel inside the burner block. The technically premixed mixture then burns inside the combustion chamber.

 

The microphone signal is then fed to a gain-delay controller which then supplies the voltage signal to the loudspeaker. The loudspeaker is then acoustically forcing the flame and with the appropriate tuning, stabilizes the flame.

 

3. Results

 

3.1 Single Parameter (𝒏) optimization

 

The algorithm was first tested to optimize the 𝒏 value and fixing the 𝜏 value. 13 iterations were performed, and 3 initial points were added to start the GP regression. Note that in principle the initial points can be added also from simulations. However, in our case, we chose the first three initial points to be n = -2.25,0,3. Following the steps in the previous section, the algorithm will give out the next parameter(s) to be evaluated. The suggested point will then improve the knowledge and still satisfy the constraint.

 

 

Figure 1: Experimental Setup

 

For every measurement point, the microphone and loudspeaker voltage signal are recorded for five seconds. The rms value of the pressure fluctuations is then extracted, as well as the maximum loudspeaker voltage value. In this case, the inequality constraint is the maximum loudspeaker volt age lower than 1.2 Volts. Furthermore, after 5 iterations, another inequality constraint is added to the algorithm: the rms pressure value should be lower than 350 Pa. This is done to accelerate the convergence to the optimum point.

 

The results of the algorithm can be seen in Figure 1. As can be seen at every iteration, both objective and constraint function are reconstructed. After 8 iterations the algorithm already finds to correct optimum region. From 8^th to 13^th iteration, the algorithm keeps evaluating close to the constraint limit of the voltage to improve the knowledge. Almost at all iterations the safety constraints are satisfied. This is a strong advantage of the algorithm compared to other adaptive method such as Extremum Seeking Controller (ESC).

 

The time trace of pressure signal with and without the control action can be seen in Figure 3. The density of the pressure signal can be seen in Figure 3c). It is apparent that the controller has successfully stabilized the system.

 

3.2 Two parameters (𝒏 − 𝜏) optimization.

 

Both the gain and delay parameter are optimized with the SafeOpt algorithm. In this case, seven initial points were added to start the initial GP. The algorithm is then running until 20 iterations. Note that after 8 iterations, the algorithm has already arrived at the point where the system is thermoacoustically stable, although it is not the most optimum location yet. In the subsequent iterations, the algorithm simply explores the region close to it. Hence, the combustor was operated under high pulsation for about 40 seconds. As can be seen from Figure 4, the surface shows that there are multiple minima and they all satisfy the constraint. In this case, the algorithm picks the minimum at n close to 1 and 𝜏 close to 7 milliseconds. As it turns out, this point is the global optimum of the real surface as well.

 

The 𝑛 − 𝜏 pair from the last iteration of SafeOpt algorithm is then fed into an Extremum Seeking Controller (ESC) to further optimize the parameters locally. The results can be seen in Figure 5. It is apparent that the ESC only slightly changes the mean value of both parameters after five seconds. This indicates that the last iteration of the SafeOpt is already a good one.

 

 

Figure 2: n parameter optimization. The algorithm is terminated after 13 iterations. Top: measured rms pressure signal and the GP prediction. Bottom: measured maximum voltage and the corresponding GP Prediction.

 

 

Figure 3: Time trace of pressure signal a) without control action, b) with control action, parameters are obtained from the final iteration of SafeOpt. c) Normalized probability density of pressure signal with and without control action

 

 

Figure 4: Two parameters optimization. The algorithm is terminated after 20 iterations.

 

 

Figure 5: a) The gain and delay variation by applying Extremum ESC with the last iteration of GP as the initial point. b) The reconstructed surface with GP, superimposed with the points explored by the ESC.

 

4. CONCLUSIONS

 

It has been shown that SafeOpt algorithms can be used in conjunction with Extremum Seeking Controller to adaptively control the thermoacoustic instability of a turbulent combustor. SafeOpt can be used to safely explore the parameter space and then final fine optimization can be done by Extreme Seeking Controller. The framework can be extended to include more quantities to be incorporated into the constraint function.

 

5. ACKNOWLEDGEMENTS

 

This study was supported by the European Research Council under the ERC Consolidator Grant (No: 820091) TORCH (2019-2024).

 

6. REFERENCES

 

1. Welch, M., Pym, A., Flexible Natural Gas / Intermittent Renewable Hybrid Power (2017).

2. Candel, S., Combustion dynamics and control: Progress and challenges, Proc. Combust. Inst., 29(1), 1–28 (2002).

3. Schuermans, B., Modeling and Control of Thermoacoustic (2003).

4. Moeck, J. P., Analysis, Modeling, and Control of Thermoacoustic Instabilities (2010).

5. König, C., Turchetta, M., Lygeros, J., Rupenyan, A., Krause, A., Safe and Efficient Model-free Adaptive Control via Bayesian Optimization (2021).

6. Berkenkamp, F., Krause, A., Schoellig, A. P., Bayesian Optimization with Safety Constraints (2021).

7. Berkenkamp, F., Schoellig, A. P., Krause, A., Safe Controller Optimization for Quadrotors with Gaussian Processes, IEEE ICRA, 491–496 (2016).

8. Eberhard, J., Geissbuhler, V., Gaussian Processes for Machine Learning (2006).

 

---

¹ bayud@ethz.ch

² alainw@ethz.ch

³ bschuermans@ethz.ch

⁴ noirayn@ethz.ch