# Digital Regulation of Accelerator Power Supplies

Particle accelerators require magnet power supplies with very high performance. Important power-supply considerations include reproducibility, stability and resolution—typical values are 100 PPM, 50 PPM, and 16 bits, respectively. Digital technology advances have made it possible to implement very complicated regulation schemes using DSPs or PLDs. These advances along with increased performance demands have resulted in designers using digital instead of analog regulators. This article presents a new digital-regulation scheme based on state estimation that may replace traditional analog PID regulators. The article presents a state-space model of the proposed system, including power supply, regulator, and estimator, along with Matlab and Simulink simulations.

**Figure 1**shows a typical power-supply block diagram. The output of the power circuit may be constant, as is the case for storage-ring power supplies, or time varying, such as for booster power supplies. The supply achieves power conversion using silicon-controlled rectifiers (SCRs) or switched-mode DC choppers. The regulator, which may be digital or analog, uses the actual output current and voltage to generate the current and voltage error signals, which are then applied to the power circuit. The regulator must ensure that the power-supply output matches the reference value despite fluctuations on the incoming grid and environmental noise. The local controller provides the means to set the reference and to monitor the output current of the power supply for the control system.

**Figure 1**does not show the ADC and DAC, since their placement depends on the type of the regulator (for example, digital or analog).

**Figure 1:** Generic accelerator-magnet power supply

**Figure 2** shows the usual regulation scheme, which comprises one [outer] current loop and one [inner] voltage loop. The reference current, set by the local controller after being converted to an analog value, is compared to the actual current measured by a direct-current current transformer (DCCT). The resultant current error is then fed to the PI regulator whose output is the reference of the voltage loop. The voltage loop has a much higher bandwidth than the current loop—typical values are 1 kHz and 1 Hz, respectively).

**Figure 3**shows the simplified model of the power circuit.

**Figure 3:** Simplified model of the power circuit

The numerical values in **Figure 3** correspond to the correction magnet power supply, used in the simulations. In the figure, *L*_{2} is the inductance of the output filter, *C* is the capacitance of the output filter, *L*_{1} is the inductance of the super-conducting magnet, and *R*_{1} is the internal resistance of the cables. You can write the state space equations of the system shown in **Figure 3** as:

The three states are chosen to be the current through the inductance of the output filter, output voltage of the power supply and the output voltage of the DCCT. In the state-space equations, 1/*g* is the gain of the DCCT, *A* is the gain of the voltage amplifier, *u* is the steering signal of the power supply, and *y* is the actual current through the magnet. In the next step, we calculated the equivalent digital model of the system. The sample time, *T*, was 10^{-4} seconds in the analog-to-digital conversion.

The numerical values of the digital state matrices after substituting the model parameters will be:

**Figure 4**.

**Figure 4:** The proposed regulation scheme, based on state estimation and pole placement

In the scheme shown in **Figure 4**, the pole-placement feedback is used to place the closed-loop poles of the system in our desired locations. Use of a digital integrator eliminates the steady-state error and improves system behavior.

where we have:

In these equations, *X* is the state vector and *q* the output of the integrator. The pole-placement feedback vector was calculated for the augmented system. The numerical value of the feedback vector is written as:

*K*_{d} = (*K*_{d1} *K*_{d2}), *K*_{d1} = (1.7056 0.0820 -41.9419), *K*_{d2} = 1.7780

Where *K*_{d1} is the feedback vector and *K*_{d2} is the integrator gain.

where is the estimated states vector and *C*_{d} • (*k*) is the estimated output current at instant *k*. A block diagram of the system comprising the plant and estimator is shown in **Figure 5**.

**Figure 5:** Block diagram of the plant and state estimator

You can show through some rather straightforward calculations that the states error vector, *e*(*k*), decays with the following dynamics:

From the last equation, it is clear that the dynamics of the error are determined from the eigenvalues of the (*A*_{d} - *G* • *C*_{d}) matrix. The three poles of the estimator should then be fast so that the estimated states reach the actual states after a very short time. You can find the estimator gain vector, *G*, by calculating a pole-placement feedback that locates the poles of the (*A*_{d}, *TRANS*(*C*_{d})) system in our desired locations where *TRANS*(*C*_{d}) is the transposition of the output vector. The numerical value of the *G* vector in the simulations, is written as:

In these equations, *X* is the real states vector, is the estimates states vector, *q* is the output of the integrator, and *U*_{ref} is the reference input.

**Figures 6**and

**7**.

**Figure 6:** Simulation model of the power supply and the proposed regulator

**Figure 7:** The upper graph shows output voltage while the lower graph shows output current for step-input reference

**Figure 8:** The prototype power supply and regulation board

**Summary**

This article proposes and shows the simulation results for a new digital-regulation technique for magnet power supplies. Design of the pole-placement feedback and the estimator was done using a state-space approach. Matlab simulations show the validity of the proposed scheme.

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