(689h) Application of Gaussian Processes to Online Approximation of Compressor Maps for Load-Sharing Optimization

Devising optimal operating strategies for a compressor station relies on the knowledge of the compressor characteristics. As the compressor characteristics change with time and use, it is necessary to provide accurate models of the characteristics that can be used in optimization of the operating strategy. We propose a new algorithm for online learning of the compressor characteristics using Gaussian Processes.

The transport of natural gas along pipelines can range over several thousand kilometres resulting in severe pressure losses due to friction. These pressure losses are compensated by compressor stations at regular intervals such that the pressure of the gas is boosted to ensure adequate transport to the destination of choice. These compressors will have performance characteristics defined in the form of compressor maps, which express the efficiency of the compressor as a function of mass flow and pressure ratio. Additionally, the performance characteristics tend to differ from one machine to the other due to design differences and various degrees of degradation. In general, the compressor operating point can be manipulated to alter the compressor efficiency in order to minimize the power consumption. However, when multiple compressors are combined as part of a compressor station, all with different performance characteristics, the minimization of the power consumption of the whole station becomes a complex optimization problem referred to as load sharing optimization (LSO) [1,2,3,4,5].

LSO works as follows: the compressor station will first get a mass flowrate target for the whole station from a dispatch centre. Load sharing optimization then attempts to determine the allocation of the total flow to each of the individual compressors which minimizes the total power consumption of the station. Whilst simple in principle, the complexity of the problem arises from the fact that an accurate model of the compressor performance characteristics is necessary to achieve an accurate solution to the optimization problem. Without an up-to-date and accurate model, there is a risk that the compressor station can be operated in a sub-optimal manner due to plant-model mismatch. This is a known issue in the real-time optimization literature and several adaptation strategies have been proposed to this end [6]. Namely, three primary approaches can be classified, (i) Model-parameter adaptation or the two-step approach whereby the model parameters are estimated and updated based on output measurements from the system before the updated model is used for optimization. (ii) Modifier adaptation where the cost function and constraints are modified before optimization is performed to ensure the model and plant share the same optimality conditions. (iii) Direct input adaptation whereby the optimization problem is reformulated into a feedback control problem.

Specifically, model-parameter adaptation [1,2,3] and modifier adaptation [4,5] have been previously used to address the issue of load sharing optimization applied to compressor stations. However, both approaches have their challenges. A major challenge for the modifier adaptation approach is the accurate estimation of the gradients of the cost function and/or constraints from data [5,6]. Additionally, modifier adaptation can be highly sensitive to noisy measurements which may make the approach less robust when the level of measurement noise is significant [6,7]. On the other hand, the robustness of model-parameter adaptation is reliant on the ability of the adaptation scheme to reduce the plant-model mismatch. When a parametric model is used to represent the system of interest, a structural mismatch may exist between the model and plant if the model structure is not sufficient to capture the real behaviour of the plant [6]. The challenge with compressor performance maps is that a structurally correct representation is seldom available as degradation of compressor performance typically occurs over the lifetime of the compressor. As a result, a parametric model of the compressor map, such as a polynomial expression as used in [1,2,3], may quickly begin to exhibit a structural mismatch after some time. However, [7] argues and demonstrates that if the model used is flexible enough to overcome any structural plant-model mismatch then model-parameter adaptation is a robust and reliable approach in comparison to modifier adaptation. This is especially the case when considering different degrees of plant-model mismatch as well as measurement noise. It is for this reason, we propose the use of a non-parametric model, namely Gaussian Processes (GP) to model the compressor performance characteristics.

The chief advantage of using a GP is that, as a non-parametric model, no underlying assumptions regarding the functional form of the system to be represented are made. Consequently, GPs are highly flexible, and the issue of structural mismatch can be avoided while the overall plant-model mismatch can be minimized. This addresses the major limitations of the model-parameter adaptation approach [8].

For the above reasons, we propose a new algorithm that uses Gaussian Processes for online learning of the characteristics of the compressors. The performance of the new approximation is demonstrated in a case study with three compressors. The case study shows that the proposed algorithm accurately captures the characteristics of the compressors even if prior knowledge is initially unavailable. The results show that the flexible nature of Gaussian Processes allow them to adapt to the data online, making them amenable for use in real-time optimization applications. This is clearly depicted in Fig. 1 which shows the evolution of the model of a compressor map (blue surface) over a 72-hour period as data is collected from the system. The collected data is represented by the black points. The green surface represents the true plant compressor map. The red surface represents a polynomial model of the compressor map which clearly exhibits a significant mismatch with the plant. On the other hand, the blue surface represents the Gaussian Process approximation over the 72-hour period. It is clear to see that, although the Gaussian Process approximation initially exhibits a mismatch with the system, after additional data is collected, the blue surface starts to better approximate the true plant compressor map (green surface). This highlights the flexibility of using Gaussian Processes in the model-parameter adaptation approach.

References

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