The hydraulic plant described in this section is composed by two centrifugal pumps that feed a hydraulic turbine. The hydraulic plant should be seen by the turbine as a water head. The static and dynamic data used in this brief were measured from this plant, composed by two centrifu- gal pumps coupled to induction motors of 7.5 kW and variable speed drive systems. The pumps can be operated alone, in parallel or in  a  series  configuration,  always at the same speed. In this work, the pumps were set in a parallel configuration working at the same instantaneous speed with a Francis turbine as load   [6].

The modelling data presented in this work were collected from a data acquisition system. The piezo-resistive pres- sure transmitter error is ±0.175 mlc (meter of liquid col- umn).

1600

1500

1400

1300

1200

1100

1000

900

800

20

18

16

14

12

10

8

6

4

0 5 10 15 20 25 30 35 40 45 50

Time (sec)

2.1 Dynamical data

One important task that has to be developed during the identification process is the input signal selection as it can influence not only parameter estimation, but also structure selection in the case of nonlinear systems  [7].

Since the presence of a ”variable time-constant” in the pumping system dynamics was verified in an earlier work [6], the input signal was chosen to excite the system at different operating points using different step sizes. The sampling time Ts = 50ms was selected according to the criterion defined in [8]. Examples of input-output data are shown in Fig. 1. In this work N = 3200 data points from the dynamical data set were used for model identification and N = 800 were used for validation.

3. PARAMETRIC  MODEL STRUCTURES

Parametric models describe systems in terms of differential equations and transfer functions. This provides  insight into the system physics and a compact model structure. Generally, you can describe a system using an equation, which is known as the general-linear polynomial model or the general-linear model [17]. The linear model structure provides flexibility for both the system dynamics and stochastic dynamics. However, a nonlinear optimization method computes the estimation of the general-linear model. This method requires intensive computation with no guarantee of global convergence.

0 5 10 15 20 25 30 35 40 45 50

Time (sec)

Fig.  1.  Dynamical  Data:  (a)  pumps  speed  reference and

(b) system output pressure.

Simpler models that are a subset of the general linear model structure shown are possible. By setting one or more of A(q), B(q), C(q) or D(q) polynomials equal to 1 you can create these simpler models such as AR, ARX, ARMAX, and  Box-Jenkins structures.

3.1 ARX modeling

The essential characteristic of the linear regression model is that a residual component e is defined which is a linear function of the unknown model coefficients. In the SISO (single input single output) situation we can write: