## Efficient automation – modelling and control technology

In industrial production, centrifugal pump systems are usually characterized by their H-Q characteristic curves in order to determine the corresponding operating point (see **Fig. 1**). However, these same characteristic curves only apply to the stationary case, i.e. the settled resting state of the system. Nevertheless, even when calculating the stationary points in the real pump system, there can be considerable deviations from the characteristic curves supplied by the manufacturer for each pump. On the one hand, this is because the pumps are measured at the manufacturer under defined ideal conditions and, according to the standard DIN EN ISO 9906, may deviate by up to 10% from this “ideal line” (see **Fig. 2**). On the other hand, the course of the system characteristic curve is not known with sufficient accuracy and can also vary considerably due to external influences and internal conditions affecting the centrifugal pump system. Both the pump characteristic curve and the system characteristic curve are therefore not completely quantifiable in reality.

These circumstances motivate dynamic modelling according to system theory. For this purpose, a mathematical model is to be designed with the aid of differential equations to calculate the operating point between the centrifugal pump and the system in the stationary state of the system. For the test bench in **Fig. 3**, consisting of two centrifugal pumps with a connected piping system, such a system-theoretical model is being developed – initially only for one centrifugal pump with connected piping. Both centrifugal pumps can be characterized in the stationary state by means of a quadratic characteristic curve (cf. **Fig. 2**).

From a system theoretical point of view, electrical, mechanical, pneumatic, thermal and hydraulic systems can always be modelled using the characterizing physical variables of resistance, inductance and capacity. These variables can be found in almost every field of physics and are known as concentrated parameters in reduced, combined models. In the hydraulic field, such a model can be usefully designed using electrohydraulic analogies, i.e. pressure *p* corresponds to electrical voltage *u,* and volume flow *Q* corresponds analogously to electric current *i*. The concentrated parameters can therefore be described as hydraulic resistance, hydraulic inductance and hydraulic capacity. In general, pressure and voltage are known as potential variables, while volume flow and current are correspondingly known as flow variables.

The necessary technology diagram of the test bench in** Fig. 3** for the model development is shown in the following **Fig. 4**. If only the pressure side of the centrifugal pump is considered, possible measured output variables are pressure *p _{2}* downstream of the centrifugal pump and system pressure

*p*somewhere in the piping system. Since, in practice, often only the pressure is available as a measured variable and not the volume flow rate, this will be handled in the same way here. The input variable is defined as the angular frequency

_{A}*ω*of the motor or centrifugal pump, which is determined metrologically either by a measuring shaft or by the frequency converter.

The mathematical model of the centrifugal pump system from **Fig. 4 **is shown as a block diagram in **Fig. 5**. In this diagram, the dark blue blocks model the dynamics of the centrifugal pump, while all other blocks denote the piping system. This system-theoretical structure can be clearly explained as follows.

The “centrifugal pump” system is delimited by its input variable *ω* and its output variable *p*. The same pressure p is used as input for the “pipeline” system, whereby the pipeline system itself consists of two subsystems. The first output variable is system pressure *p _{A}*, which completes the first part. This forms the input variable of the second system part, which represents the pipeline section that transports the water into the upper tank after the system pressure sensor. Here, in turn, atmospheric pressure

*prevails, to which all pressure measurements are related. Thus, the model works with relative pressures.*

_{pa}The block diagram clearly shows the concentrated parameters R, L and C of the respective subsystems. All three subsystems each contain a quadratic non-linear component, which is known as the non-linear resistance component. Within the centrifugal pump system, there is also a bilinear component (multiplication of input variable ω with state variable Q), which introduces an additional resistance component.

Based on the electrohydraulic analogy already mentioned, the block diagram can also be clearly represented as an electrical equivalent circuit diagram with non-linear resistances (see **Fig. 6**).

For hydraulic inductivity, the following relationship can be represented by formula: *L* = *m/A²*

Accordingly, hydraulic capacities can be described as follows: *C* = *V*⋅*β*.

In the two equations above, m stands for the fluid mass in the system part under consideration, *A* for a calculated cross-sectional area, *V* correspondingly for the volume within this area, and *β* characterizes the compressibility of the fluid used. The differential equation system on which the block diagram is based results in:

It is assumed that only the non-linear structure of the differential equation system is known, whereas the concentrated parameters cannot be quantified. A one-time identification would only be satisfactory for a certain working range, since the parameters are strongly dependent on temperature, the medium itself, and on changes in the operating point. To make matters worse, the volume flow could only be measured within an initialization measurement. During ongoing plant operation, on the other hand, it is assumed to be unknown.

A first approach to obtain a simple transfer behaviour, in the form of a Laplace transfer function, from angular frequency *ω* to system pressure *p _{A}*, can be realized by means of an operating point linearization. This is clearly illustrated in

**Fig. 7**. It should be noted that, within this working range, a certain error is made in the linearization.

As a simple, reduced Laplace transfer function, based on empirical relationships and derived from the underlying system of differential equations, the following transfer structure between *ω* and *p _{A}* can be given:

In theory, the unknown parameters *a _{i}* and b

*represent functions of the concentrated parameters*

_{i}*R*,

*L*and

*C*.

The modelling of simple centrifugal pump systems, which is presented here in compact form, has several intended objectives:

- Estimation of varying operating points in centrifugal pump systems
- Estimation of varying characteristic curves
- Estimation of varying parameters or parameter changes
- Design of robust controller structures for pressure, volume and height control
- Use for condition monitoring
- Use for soft sensor technology

## Control technology

The preceding modelling of centrifugal pump systems serves as a basis for a corresponding controller design for system pressure control. PID controllers are used for such purposes as standard in industrial production. However, PID controllers have the disadvantage that they only function well in non-linear systems around the operating point under consideration – assuming that parameter drifts change this point significantly. An adaptive and robust control method would be better suited for this.

Based on the structure of the linearized transfer function, adaptive or robust controllers can be designed in such a way that disturbing influences are compensated as far as possible and a large working range of the non-linear system can be covered despite linearization.