# Mathematical models and process data How to use process data to improve process monitoring and control

Application of mathematical models for the purposes of process monitoring is rising due to improved support of data historization and data handling. In most modern industries, process control systems are connected to systems that collect process data on a regular basis (e.g. 1 minute) and by using special data compression techniques, data can be stored for years and used for purposes of improved monitoring and control. Models developed for process monitoring enable faults recognitions, prediction of properties and improved quality and process control.

### The classification of mathematical models

There are many classifications of mathematical models. The most important one is dividing mathematical models into two categories, based on:

- Physical theory (first principles models, white-box models),
- Empirical descriptions (empirical models, black box models, data-driven models)

### First principles models

The first group of models is based on physical and chemical laws: mass and energy balances, thermodynamics, chemical reaction kinetics. They are also called **“white-box”** **models** or **first principles models**.

Those models give physical insight into the process and explain the process behavior in terms of state variables and measured variables. To build those models, common practice is to use a specialized process simulation software.

But they can also be developed using any programming language. In this category, only the system parameters are measured or known from the literature. Because they are explaining the process behavior in details, these models are applicable for engineering analysis and optimization and can be developed for any system before it is constructed.

### Empirical models

** Empirical models**, on the other hand, are attractive when a physical model cannot be developed due to limited time or resources and they are not able to describe physical phenomena of the process.

A mathematical model is developed from interactions between variables that are detected from the large set of data. This is the reason that those kinds of models require a large set of input-output data and have limited application for engineering analysis.

However, the contribution of this kind of models is significant for advanced process control and improved monitoring, for quality control, predictions, and fault detection applications.

The selection of the appropriate model requires engineering judgment in accordance to defined goals as well as some skill in recognizing how response patterns match possible algebraic functions.

### “Grey box” models

Between those two groups, there is a gray area and sometimes those two extremes are used together. It could be that much physical insight is available, but that certain information or understanding is lacking. In those cases, physical models could be combined with black box models; the resulting models are called** gray box or hybrid models**.

### Applications in process monitoring

All the groups may be applied with the purpose of improving process monitoring. However, due to simpler calculation methods and more practical approach to complex industrial units, empirical models have been applied more and are referred to as soft sensors, inferential monitoring, and data-driven sensors and similar.

Lately, more detailed first principles models are also finding their role in this operational area and SimulaterLive.com will be providing information about the way of the progress and how they are proving their competence in practice.

Soft sensors are a valuable tool in many different industrial fields of application, including refineries, chemical plants, cement kilns, power plants, pulp and paper industry, food processing, nuclear plants, urban and industrial pollution monitoring, just to give a few examples. They are used to solve a number of different problems such as measuring system back-up, what-if analysis, real-time prediction for plant control, sensor validation and fault diagnosis strategies. Development of advanced process control application would also be less efficient without those tools.

### Mathematical tools

There are many mathematical and statistical methods which find their applications as soft sensors. Some of the most applied are linear and non-linear regression, partial least squares algorithms, principal component analysis, artificial neural networks etc.

The most important factor in the selection of the proper method is the process behavior: is it linear or non-linear. Process behavior usually is non-linear. However, whether the soft sensor model should also be non-linear depends on the operating range in which the model will be used. If the process is controlled and the operating range is small, a linear process model may be an adequate approximation of reality.

Another question to select the right method is whether the model needs to be dynamic or static. For control and prediction type applications, models are usually dynamic. But, if the process has small time to steady state, static model will be good enough for the satisfactory prediction.

### Examples of applications

Potential applications of soft sensors can be applied in any industrial field. Refineries have a wide range of applications.

Soft sensors are used to predict property and/or complement on-line analyzer measurement, to estimate column composition, to monitor the whole process section, to warn for critical variables. Examples will be available on SimulateLive.com.

They can be applied in the polymerization processes to predict properties and grades of produced polymers, in the fermentation process to estimate the behavior of a critical variable.

Using neural networks application can be used to estimate paper curl measurement or predict potential corrosion in a pulp and paper mill.

As emissions are a great concern of all industries, soft sensors can also be used to predict potential emissions.

They can be applied in a nuclear plant, e.g. to estimate feed water.

Food factories are another interesting field for the application of soft sensors to improve product quality control in a food cooking. An on-line dynamic model between the influential variables and the product quality attributes can be developed together with feedback control.

Generally speaking, it should be pointed out that it can be difficult to discover all the right dependencies among the data and very often the engineering experience and engineering judgment is the most important component of successful applications. SimulateLive.com will help you and guide you through mathematical tools and software available on the market whose aim is to make that applications closer to industrial practice.