In this section, the results are applied to the case of a synchronous generator
con-nected to a CPL. This system can be modeled by^{2}

˙
wherep >0 is the angular momentum, M > 0 is the total moment of inertia of the
turbine and generator rotor,ω∈R^{+} is the rotor shaft velocity,ω^{∗} >0 is the angular
velocity associated with the nominal frequency of 50 Hz, D_{m} > 0 is the damping
coefficient of the mechanical losses,D_{d}>0 is the damping-torque coefficient of the
damper windings, τ_{m} > 0 is the constant mechanical torque (physical input), and
P_{e} is the constant power load.

Let

∆ := (D_{d}ω^{∗}+τ_{m})^{2}−4(D_{d}+D_{m})P_{e} ,
and assume that ∆>0,i.e.,

P_{e} < (D_{d}ω^{∗}+τ_{m})^{2}
4(D_{d}+D_{m}) .

Then, the dynamics (5.28), (5.29) has the following two equilibria

¯

2This model is called the improved swing equation in [130, 76]. An inverter with a capacitive inertia can be modeled by similar dynamics; see [77].

Table 5.2: Simulation Parameters of the Synchronous Generator (p.u.)
M D_{m} D_{d} P_{e} τ_{m}

0.2 10^{−}^{6} 10^{−}^{4} 3 0.0027

0 10 20 30 40 50 60 70

0 20 40 60 80

Figure 5.3: Solutions of the system (5.28),(5.29) with different initial conditions.

Moreover,

R+Z(ω) =D_{d}+D_{m}− P_{e}

¯
ω_{s}ω .
The equilibrium point ω= ¯ω_{s} is asymptotically stable since

R+Z(¯ω_{s}) = τ_{m}+D_{d}ω^{∗}

¯

ω_{s} >0.

Through straightforward computations, it can be shown that the set Ω_{p} in (5.11)
can be written as

Ω_{p} ={ω∈R^{+} : ω >ω¯_{u}}. (5.30)
In this set, the shifted HamiltonianH= ^{1}_{2}M(ω−ω¯_{s})^{2} is strictly decreasing.
There-fore the solutions tend to the equilibrium ¯ω_{s} and move away from the point ¯ω_{u}
when time increases. Consequently the set Ω_{p} in (5.30) is forward invariant, and
represents an estimate of the ROA.

Figure 5.3 shows the trajectories of a number of solutions of the system (5.28)-(5.29), with the parameters given by Table 5.2, and with different initial conditions.

It is clear that the proposed method successfully identifies a very precise estimate of the ROA (blue), as all the solutions starting from outside the ROA estimate (black) diverge from the equilibrium. The result, which is derived from the systematic approach proposed in the chapter, shows an improvement over the results of [76], as both derived ROAs have the same lower bound, while here, the estimate has no upper bound.

Interestingly, it can be shown that any solution that does not start in Ω_{p}, diverges
from the stable equilibriumω_{s}. Therefore, here, the set Ω_{p} is not only an estimate,
but is the exact region of attraction.

### 5.7 Summary

This chapter has explored a class of pH systems in which the control input (or
disturbance) acts on the power of the system. These systems have been referred here
as Power-controlled Hamiltonian (P_{w}H) systems. First, a model for such systems
was proposed, and second, a condition for shifted passivity was derived. Using these
results, the stability of equilibria was investigated. Furthermore, an estimate of the
region of attraction was derived for P_{w}H systems with quadratic Hamiltonian.

The proposed modeling procedure and stability analysis were illustrated with two cases of practical interest: A DC circuit and a synchronous generator, both connected to constant power loads. The validity and utility of the proposed method was confirmed by numerical simulations of these case studies.

### STABILIZATION: AN ADAPTIVE CONTROL

### APPROACH

### Voltage Control of a Buck-Boost Converter

Synopsis This chapter addresses the problem of regulating the output voltage of a DC-DC buck-boost converter that supplies electric energy to a constant power load. The model of the network is a nonlinear, second order dynamical system that is shown to be non-minimum phase with respect to both states. Moreover, to design a high-performance controller, the knowledge of the extracted load power, which is difficult to measure in industrial applications, is required. In this chapter, an adaptive interconnection and damping assignment passivity-based control—that incorporates the immersion and invariance parameter estimator for the load power—

is proposed to stabilize the system around a desired equilibrium. Some detailed simulations are provided to validate the transient behavior of the proposed controller and compare it with the performance of a classical proportional-derivative scheme.

### 6.1 Introduction

The DC-DC buck-boost power converter is increasingly utilized in power distri-bution systems since it can step up or down the voltage between the source and load, providing flexibility in choosing the voltage rating of the DC source [125, 116, 124]. Although the control of these converters in the face of classical loads is well-understood, this is not the case when constant power loads (CPLs) are considered.

This scenario significantly differs from the classical one and poses a new challenge to control theorist, see [9, 71, 38, 53, 80, 128] for further discussion on the topic and [110] for a recent literature review. It should be underscored that the typical application of this device requires large variations of the operating point—therefore, the dynamic description of its behavior cannot be captured by a linearized model, requiring instead a nonlinear one.

To address the voltage regulation of a buck-boost converter with a CPL, several techniques have been proposed in the power electronics literature, but without a nonlinear stability analysis. In [88], the active-damping approach is utilized to address the negative impedance instability problem raised by the CPL. The main idea of this method is that a virtual resistance is considered in the original circuit to increase the system damping. However, the stability result is obtained by applying small-signal analysis, which is valid only in a small neighborhood of the operating point. A new nonlinear feedback controller, which is called “Loop Cancellation”, has

been proposed to stabilize the buck-boost converter by “canceling the destabilizing behavior caused by CPL” [89]. The control problem turns into the design of a controller for the linear system by using loop cancellation method. However, the construction is based on feedback linearization [47] that, as is well-known, is highly non-robust. A sliding mode controller is designed in [111] for this problem. However, for the considered nonlinear system, the stability result is obtained by adopting the linear system theory. In addition, as it is widely acknowledged, the drawbacks of this method are that the proposed control law suffers from chattering and its relay action injects a high switching gain. The deleterious effect of these factors is illustrated in experiments shown in [111], which exhibit a very poor performance.

Aware of the need to deal with the intrinsic nonlinearities of power converters some authors of this community have applied passivity-based controllers (PBCs), which is a natural candidate in these applications. Unfortunately, in many of these reports the theoretical requirements of the PBC methodology are not rigorously respected. For instance, in [60], the well-known standard PBC [83] is used for the buck-boost with a CPL. Unfortunately, the given result is theoretically incorrect due to the fact that the authors fail to validate the stability of the zero dynamics of the system with respect to the controlled output that, as explained in [83], is an essential step for the stability analysis and, as shown in this chapter, it turns out to be violated.

An additional drawback of the existing results is that all of them require the knowledge of the power extracted by the CPL, which is difficult to measure in industrial applications. Designing an estimator for the power is a hard task because the original system is nonlinear and the only available measurements are the inductor current and the output voltage.

In this chapter, the well-known interconnection and damping assignment (IDA) PBC, first reported in [82] and reviewed in [81], is applied to stabilize the buck-boost converter with a CPL. The main contributions are enlisted as follows.

1) Derivation of an IDA-PBC that ensures the desired operating point is a (lo-cally) asymptotically stable equilibrium with a guaranteed region of attraction.

2) Design of an estimator of the load’s power, which is based on the Immersion and Invariance (I&I) technique[8] and has guaranteed stability properties, to make the IDA-PBC adaptive.

3) Proof that the zero dynamics of the system, with respect to both states, is unstable—limiting the achievable performance of classical PD controllers to

“low-gain tunings”.

The rest of the chapter is organized as follows. Section 6.2 contains the model of the system and the analysis of its zero dynamics with respect to its two states.

Section 6.3 proposes the IDA-PBC assuming that the power extraction of the load is known. To make the latter scheme adaptive, in Section 6.4 an on-line power estimator is designed and some simulations, carried out in MATLAB, are provided in Section 6.5. The chapter is wrapped-up with some concluding remarks in Section 6.6. To enhance readability, the derivation of the IDA-PBC, that is conceptually simple but computationally involved, is given in a Technical Appendix at the end of the chapter.

Figure 6.1: Circuit representation of the DC-DC buck-boost converter with a CPL