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turbocharger speed is significantly higher. The compressor and the turbine needs to operate at much higher pressure ratios.

Two constraints are now investigated: the maximum exhaust manifold pres-sure of 4 bar and the maximum turbocharger speed of 210000 RPM.

Both variables are states of the MVEM and the required limits can be readily added to the MPC controller. Soft constraints were used in this setup. This allowed, but heavily penalised, the exhaust pressure and the turbocharger speed exceeding their limits, but ensured the feasibility of optimisations. Exceeding the limits is often unavoidable, because of the mismatch between the prediction model and the actual plant.

The results of such simulations are shown in Figure 3.13. It is observed that the demanded pressure of 2.4 bar was achieved significantly later than it was in the case when nominal conditions were investigated. Meeting the boost demand required accelerating the turbocharger to higher speeds. Before this was achieved, the pressure in the exhaust manifold had become very high and the MPC con-troller had generated a demand for opening the VGT. Maintaining a constant boost pressure required progressive opening of the VGT which resulted in a drop in the exhaust pressure. Soon after that, the turbocharger speed achieved its maximum permitted value. The MPC controller sacrificed the reference tracking to keep the turbocharger speed at the specified limit.

0 2 4 6 8 10 0.5

1 1.5 2 2.5

Time (s)

Boost Pressure (bar)

Reference Feedback

(a) Boost pressure

0 2 4 6 8 10

0 0.2 0.4 0.6 0.8 1

Time (s)

VGT Position (−)

(b) VGT rack position

0 2 4 6 8 10

1 1.5 2 2.5 3 3.5 4 4.5

Time (s)

Exh Man Pressure (bar)

Constraint

(c) Exhaust manifold pressure

0 2 4 6 8 10

0 0.5 1 1.5 2 2.5x 10⁵

Time (s)

Turbocharger Speed (RPM)

Constraint

(d) Turbocharger speed

Figure 3.13: Simulating 1D operation at high altitudes - MPC satisfies the ex-haust manifold pressure and turbocharger speed limits.

the surge line.

In this work, the definition of the surge limit as a pressure distance is used, because it can be relatively simply accommodated in the investigated controller.

The surge limit is stored as a look-up table between the mass flow parameter and the pressure ratio. The WAVE model includes sensors for compressor flow, pre-compressor pressure, temperature and post-compressor pressure. This allows the instantaneous compressor flow parameter and pressure ratio to be calculated, thus the surge limit to be expressed as a post-compressor pressure. In the MVEM used by the controller, the compressor surge limit can be calculated in a similar fashion.

Two methods for avoiding surge were investigated:

(a) Surge limit defitions (b) Control induced oscillations

Figure 3.14: Definition and challenges with compressor surge limit.

1. Setting the boost reference as the minimum of the demanded pressure and the surge limit.

2. The difference between the surge limit and the boost pressure was defined as one of the system outputs. The evolution of this difference within the prediction horizon of MPC and its dependency on inputs were in this way included in the actual optimisation procedure. A constraint of 0.15 bar was imposed on this output.

A particular difficulty with compressor surge avoidance is highlighted in Figure 3.14b. This is valid for both approaches described above. Boost reference or constraint is specified with a certain margin in pressure from the surge line.

Considering a transient trajectory starting at the bottom left corner, initial turbo-lag pushes the operating points further from the surge line. Once the turbocharger is spinning faster, the actual boost pressure overshoots the reference or closes the gap to the constraint. As a result VGT is opened and the turbocharger slows down. This leads to lower compressor flow parameter and as a result the boost reference/constraint has to drop as well. The actual pressure needs to fall below this value, before it may continue rising to the demanded level.

Such behaviour was observed during the simulations of the 1D model with the

ambient pressure of 0.8 bar. The tests shown in the previous sections (see Figure 3.13) were conducted without surge limit avoidance control and resulted in the operation shown in Figure 3.15a. In fact, the surge limit was briefly exceeded between 8 and 9 seconds.

To avoid such a situation, the two methods described above were implemented.

The first one, relying on limiting the boost reference to the surge limit, was not effective and unreliable due to the previously described difficulties.

Better results were obtained when the difference between the boost pressure and the surge limit was treated as a system output (Surge Ctrl A in Fig. 3.15).

The problem with the dependency of the constraint on the state and the input was still visible (see Figure 3.15b). The relatively low penalty on VGT actuation from the original MPC calibration, resulted in an abrupt VGT rack position change (as shown in 3.15d). This led to a characteristic drop in the value of the constraint, which was, however, not exceeded. The boost pressure had to be then recovered from value below 2 bar.

MPC was then recalibrated with an increased weight on the VGT rate of change (from 20 to 100). The resulting response is shown as Surge Ctrl B in Figure 3.15. This time the VGT was not actuated as abruptly as previously and the drop in constraint and boost pressure was avoided.

This confirms that in this idealised setup (ideal sensors and actuators) an effective surge avoidance control can be realised for 1D models using a MPC controller. The necessity of finding a suitable trade-off in VGT actuation penalty was explained. On one hand a low penalty is preferred for fast boost tracking.

On the other, too aggressive actuation results in controller induced oscillations in the vicinity of the surge limit.