# Mathematical models on the line

## Tune combustion process for goodness sake.

##### By Jeffery Williams, Frederick C. Huff, and Peter Francino

The use of mathematical models and optimization is becoming an important part of the design and operation of power plants.

Optimization techniques make plants more profitable and thus better able to meet the demands and opportunities of the deregulated electricity markets and to help them comply with environmental requirements.

There are areas of a power plant, or islands, where optimization is possible. Some of these areas include emissions monitoring, condenser/cooling tower subsystems, combustion optimization, and fleet-wide economic load dispatching.

Depending on the island of optimization, different types of models need to be developed, and different optimization techniques are required.

We will examine some areas where optimization is possible and highlight the types of models required and the optimization techniques required. We also have a general-purpose optimization program that can run in real time and communicate with the plants Distributed Control System (DCS) leveraging these optimization opportunities for consideration.

## Different mathematical models

With utility deregulation and more stringent pollution regulations in place, minimization of overall generation cost for a power company has become increasingly important.

One way to minimize the cost of producing power is to utilize optimization in the process. There are different types of power plants, and within them, various islands of optimization are possible. Just as there are different power plants, the optimization of these plants often require different mathematical models and optimization techniques.

Depending on the application, the component performance models may be linear or nonlinear.

Coal-fired plants: In a coal-fired electric generating unit, there are many islands where optimization is possible. Tuning the combustion process to minimize NOx is one. This process is highly nonlinear and requires a neural network model to predict the boiler emission.

Another island of optimization in a coal-fired plant is steam temperature. Fuzzy models have successfully controlled this subsystem. In the typical condenser/cooling tower subsystem, several cooling water circulating water pumps operate in parallel with one another while the cooling tower has several fans or groups of fans that might be able to operate at several speeds. Depending on the ambient conditions at the plant, it might be possible to change the equipment selection, which could result in lower auxiliary power costs.

Combined cycle plants: In these plants, there are two combustion turbo-generators (CTG). The hot exhaust from each CTG feeds a heat-recovery steam generator (HRSG). Gas-fired duct burners can augment this heat. The high-, medium-, and low-pressure steam produced by the HRSG feed into a steam turbo generator (STG). Depending on the plant demand, often, there are multiple equipment configurations that satisfy demand, and some configurations are more efficient than others are.

Hydroelectric: Many dams contain multiple hydroelectric turbo-generators. In this case, the goal is to satisfy the plant demand by loading the turbo-generators to minimize the amount of water consumed. This is a nonlinear optimization problem. However, it is further complicated due to vibration problems occurring at certain loads. The load ranges that cause vibration to occur become "forbidden zones." The optimization program must be aware of these ranges and prevent any machine from taking on a load that lies in one of these areas.

Cogeneration plants: The dual goal of producing both steam and power in a cogeneration plant causes complexity in building and operating these facilities. A typical plant may have six boilers that can burn gas or oil and supply steam to a common header. Power generates internally and can be purchased from the utility and sold back. Optimization of this type of plant involves dispatching the loads on the boilers and turbines so the process steam demand and power demand is satisfied at least cost. This can happen using linear programming.

Fleet-wide economic dispatch: With utility deregulation and more stringent regulations on power plant pollution, scheduling and operating power generation from the corporate level have become important. In addition to the fact that load forecast and production scheduling are more dynamic at the corporate level, each boiler unit at the plant level also has to face the challenge of reducing exhaust gas pollution. In coal-fired units, the task normally comes down to reducing NOx, SO2, and opacity levels below certain limits mandated by the Environmental Protection Agency. These limitations can normally translate to NOx or SO2 control set points during each unit's daily operation. Significant reduction in pollutant level can bring in credit for the company that directly leads to financial savings. However, pollution control inevitably incurs cost. These costs must be a part of the model and the overall objective cost function. For a power company with multiple boiler units, it is often difficult to determine the load dispatch profile and also specify the optimal pollutant control levels for all of the different units, such that the overall material and maintenance cost is minimal, while the environmental pollution constraints are also met at the same time. With the modern day control system and advances in network technology, it is possible to do a real-time fleet-wide economic dispatch that considers not only plant heat rates but also environmental costs. This optimization requires a mixture of linear and nonlinear models.

## Overall plant model

As seen from the examples, a variety of model types is required in doing optimization. Some models are linear such as heat vs. power on a CTG. Some are simple parabolas such as load versus efficiency curves on boilers. In the case of combustion optimization, the process is highly nonlinear requiring a neural network model to be developed.

Many of these models contain intercepts or constant terms that must become zero when the device is out of service. Other functions, such as the heat rate versus load curve for a combined cycle plant, are nonlinear and discontinuous depending on the equipment configuration. Depending on the optimization problem, combinations of these models must combine as one entity to form an overall plant model. Therefore, a program must exist that can solve a mix of model types and be able to handle equipment out of service and different equipment configurations.

In the past, classical optimization techniques developed, some for linear models and some for nonlinear. Linear programming is a classical proven optimization technique used to solve problems in real time. However, to use linear programming every individual equipment model that is part of the overall plant model must be linear. This restriction is limiting since many real world models do not conform to this linear simplification requirement. The equipment models of much of the equipment found in a power plant are nonlinear, making it difficult to use linear programming as an optimization technique.

Nonlinear optimization techniques, such as Evolutionary Operation (EVOP) and its derivative Simplex Self-Directing Evolutionary Operation Technique (SSDEVOP), worked for the economic dispatch of boilers and steam turbo-generators.

Other nonlinear techniques such as steepest-ascent also exist, but the problem remains that some of these methods only apply to nonlinear models not a mix of linear and nonlinear. In addition, in the case of EVOP the number of variables must be limited for convergence to occur.

An optimization method that can handle a mixture of linear and nonlinear equations is still necessary. This optimizer must also be able to handle discontinuity that occurs due to equipment being out of service.

## Hypothetical plant designs

The modern day microprocessor based DCS contains the computing power to solve complex iterative mathematical procedures in real time. Therefore, the optimization techniques used no longer have to be limited. Modern day optimization solvers are commercially available. These solvers typically contain the following types of algorithms:

LP/Quadratic: This solver is ideal when all of the equations that comprise the plant model are linear and the objective function is linear or quadratic.

Gradient nonlinear: This solver is ideal when some or all of the equations are nonlinear and the functions are smooth or continuous.

Evolutionary: This solver is required when the equations are nonlinear and non-smooth.

All of the solvers should be able to handle integer variables so equipment out-of-service problems are easy to handle.

Therefore, the ideal optimization package should be a general-purpose solver for mixed integer linear/nonlinear optimization problems raised from power plant operations. The software should provide the user with abilities to find a solution of x—a vector of independent decision variables—in the feasible regions, which results from a set of equality/inequality constraints, such that the local/global minimum or maximum value of the objective function is obtained.

The ideal optimization package should have an offline and online mode. The offline package should contain a graphical user interface (GUI) that builds the optimization problem that is a plant model because it is comprised of the relationships or equations of the equipment in the plant. The offline module should allow the creation of any number of optimization problems. They can be models that represent the actual plant, or they can be hypothetical plant configurations. Besides providing the capability to develop plant models, this module provides the user with the ability to perform "what if" scenarios. It provides the user with a tool that lets him study the interactions of the plant equipment and determine the equipment configurations that provide the lowest \$/hour operating cost by letting him adjust costs, demands, equipment availability, and the like.

The GUI of the offline program should be equipped with a Web-based user interface and should allow multiple users to perform "what if" operations while not interfering with the running online optimization. The users should be able to create new models and modify existing models. It should also provide the ability to configure the online optimization process from the user interface.

In addition to having an offline mode of operation, the optimizer should also contain an online mode. In the online mode, a plant model that built up using the offline package can solve in real time. Unlike the offline mode, the demands and costs should update from the plant DCS system, and the results of the program should be available to the DCS where they can become supervisory set points or used to advise the plant operators.

A state of the art optimizer integrated with the DCS can benefit power generation facilities at the multi-unit level as well as the fleet management level. The plant can leverage the new optimization opportunities that arise from examining the operational aspects of known and observed economic factors. Plant equipment and unit operation characteristics can easily receive evaluation for optimum economic benefit. IC

## Behind the byline

Jeffery Williams (Jeffery.Williams@emersonprocess.com), Frederick C. Huff (Frederick.Huff@emersonprocess.com), and Peter Francino (Peter.Francino@emersonprocess.com) work at Emerson Process Management. This article comes from their presentation at the 15th Annual Joint ISA POWID/EPRI Controls and Instrumentation Conference, June 2005.