Particle accelerators require magnet power supplies with very high performance. Important power-supply considerations include reproducibility, stability and resolutiontypical 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.
A Generic Magnet Power Supply
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 looptypical values are 1 kHz and 1 Hz, respectively).
Power Supply State-Space Model
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, L2 is the inductance of the output filter, C is the capacitance of the output filter, L1 is the inductance of the super-conducting magnet, and R1 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:
Digital Regulation Scheme
The proposed digital regulation scheme is shown in 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.
Design of the Pole-Placement Feedback
The state-space model of the augmented system (system and integrator) is given as:
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:
Kd = (Kd1 Kd2), Kd1 = (1.7056 0.0820 -41.9419), Kd2 = 1.7780
Where Kd1 is the feedback vector and Kd2 is the integrator gain.
The output current is the only state that is measured directly. The other two states are estimated by the estimator to be used for the pole-placement feedback. The input and output signals of the power supply are fed into the estimator at each sample time and the three states of the system are estimated at the next sample time through some matrix calculations. The state-space equation of the estimator can be written as:
where is the estimated states vector and Cd • (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 (Ad - G • Cd) 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 (Ad, TRANS(Cd)) system in our desired locations where TRANS(Cd) is the transposition of the output vector. The numerical value of the G vector in the simulations, is written as:
State Space Equations of the Whole System
The digital state-space equations for the whole system, including the power supply model, the estimator, and the integrator are written as:
In these equations, X is the real states vector, is the estimates states vector, q is the output of the integrator, and Uref is the reference input.
We simulated the proposed regulation scheme using Matlab and Simulink software from The Mathworks. The simulation model uses a Gaussian random-noise generator to account for the effects of the input-voltage disturbances. In the simulations, the time constant of the closed-loop system and the damping factor were chosen to be 0.25 sec and 0.9, respectively, and the estimator poles were assumed to be two times faster than the closed loop system poles. The simulation model and simulation results for a step input are shown in Figures 6
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
An Altera PLD was chosen by the DESY (Deutsches Elektronen Synchrotron) Power Supplies group generating the steering pulses of the MOSFETs and for interfacing the regulator to the supply. Since the main purpose of this chip is to do the logical operations, some software modules had to be written to do the more complicated operations, including multiplication, addition, and delay. In the first phase of the project, a digital PID-P regulation scheme for the current-voltage loops of a 10 Volt / 25 Amp prototype power supply was implemented on the Altera chip with satisfactory results. In the next phase, we will implement the proposed regulation scheme on the chip.
Figure 8: The prototype power supply and regulation board
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.