Optimal emission control and identification of an unknown pollution source

The advection-diffusion-reaction equation is used for describing the dispersion of a quasi-passive contaminant from industrial point sources in a limited area. The conditions established on the open boundary ensure that the problem is correct in the sense of Hadamard, that is, its solution exists, is unique, and is stable to initial perturbations. The Lagrange identity is used to construct the adjoint operator and formulate an adjoint problem. Equivalent direct and adjoint estimates are derived to assess the concentration of the pollutant at monitoring sites of the area. Formulas obtained on the basis of adjoint estimates are useful in analysing the sensitivity of the model to both variations in the intensity of pollution sources and variations in the initial distribution of the pollutant concentration in the area. New optimal emission control strategies based on using the adjoint estimates are developed in order to prevent violations of existing sanitary standards by timely reduction of emission rates of operating sources. Optimal control here lies in minimizing these reductions. In addition, this control is primarily aimed at reducing the intensity of emissions from sources that most pollute the monitoring site. Also, new methods are proposed for identifying the main parameters of an unknown point source that arose as a result of a dangerous incident (accident, explosion, etc.). These methods allow determining the location and intensity of a constant or non-stationary point source, as well as the moment of emission of a pollutant in the case of an instantaneous point source. This helps to quickly assess the scale of the incident and its consequences. Numerical results show the effectiveness of the methods.


Introduction
During the last century, anthropogenic activities emit into the atmosphere at short intervals, such large volumes of substances that the mechanisms of assimilation do not have time to recycle the excess chemicals and to clean the atmosphere. A pollutant, depending on its concentration and toxicity, causes various health problems, from respiratory discomfort in healthy people to the increase in mortality among vulnerable populations. Consequently, it is important to design methods for controlling emissions and reducing the concentration of hazardous substances to acceptable health standards.
In this work, simple mathematical models of pollutant dispersion as well as their adjoint models are used to assess pollution levels at the monitoring sites, develop optimal strategies to control emission rates of the point sources and determine the location and intensity of unknown sources of pollution. The emission control strategies are optimal in the sense that they set minimal limits on the rate of emissions from sources to avoid exceeding sanitary standards. Whenever the dispersion model predicts a sanitation violation at the current emission levels, a control (i.e., a restriction of intensity of emissions) can be applied. Thus, mathematical control methods are an effective complement to the use of "green" technologies.
The determination of the intensity and location of unknown sources using the observed data is an inverse problem that, in the presence of observational errors, requires the use of regularization methods. In the present work, we suggest methods for assessing the main parameters of an unknown point source with stationary, instantaneous and nonstationary emission rate.
is the wind velocity vector (known from observations or some dynamic model), characterizes the rate of exponential decay of ( , ) t I r due to various physical and chemical processes, ( , ) 0 t P ! r is the turbulent diffusion coefficient, is the emission rate of the i-th source, and ( ) i G r r is the Dirac function. It is assumed that Normally the pollution flux through the open boundary S of limited area D is unknown. It is therefore important to put such boundary conditions, under which the problem will be posed correctly in a limited area, both physically and mathematically.
For this purpose, we denote by n U U n the projection of velocity U on the unit external normal n to the boundary S of domain D , and divide S into the "inflow" part S (where 0 n U , and the pollution flux is directed inside D ) and "outflow" part S (where U n t 0 , and the pollution flux is directed outside D )   The solution of problem (1)-(5) satisfies two integral equations Thus, the total mass . However, if one need to answer the question "To what extent each source is responsible for the contamination of a particular zone?" then this method requires solving N problems N . Therefore, we now describe another approach that makes it easier to answer this question, especially if the number of sources N is large.
Using the Lagrange identity [2] it is possible to define the operator adjoint to the operator of model (1)-(5) and formulate the adjoint problem in the domain D and time interval (0, ) T : We will now obtain two adjoint estimates of air pollution which are equivalent to the direct estimates (12) and ( , ) If we put in (9) then (13) leads to the first adjoint estimate (9) we obtain the second adjoint estimate where it was taken into account that ( , ) 0 The advantage of adjoint estimates is that they explicitly depend on the emission rates ) (t Q i of the sources and initial distribution 0 ( ) r I in D , whose contribution is determined by the values of weight functions ( , ) i g t r and ( ,0) g r , respectively. Note that the last integrals in estimates (15) (15) and (16). Sometimes, the adjoint estimates help to quickly solve non-trivial problems. In addition, these estimates are important for controlling the emission rate of pollution sources and identifying the parameters of an unknown source of pollution.
Here are two formulas useful in studying the sensitivity of estimate (15) (or the sensitivity of the solution to the linear problem (1)-(5)): W , and 0 J is the admissible sanitary norm. Then a control can be applied to establish such reduced emission rates * ( ) T that the re-forecast with the model M and new rates * ( ) t Q in time interval (0, ) T will give the satisfactory result: Consider a simple air pollution control defined as the optimization problem Theorem 2. If 0 D ! then the optimal control problem (23) has a unique solution * 4 If there is only one source in D, then Theorem 2 can be made more precise: is the only solution to the problem of optimal control (23) provided that it is a non-negative function in the interval [0, ] T .
In connection with Theorem 2, the set of potential solutions (23) is reduced to 1 0 This set is much smaller than (23) and therefore it is preferable in calculations. Typically, problem (25) is solved using the iterative optimization method with sequential estimation of the dynamic model M [6], and therefore if the model M is complex, this process is computationally intensive. However, the numerical solution of the optimal control problem can also be obtained using a highly efficient numerical algorithm of sequential orthogonal projections [5].

Example. Control of lead particle emissions
We now apply Theorem 3 for controlling lead particle emissions. The dispersion model (1)-(5) and adjoint model (9) [7]. We will consider four different emission rates of the source:   ). It means that the optimal strategy (25) does not require radical changes in the operation of the industrial source.

Identification of main parameters of unknown point sources
Dispersion models are commonly used to estimate the impact of pollutant emissions on air quality under various meteorological conditions. In addition, the previous sections have shown the importance of using the dispersion models and the adjoint approach in developing optimal emission control strategies. We will now show that the same methods can be used to evaluate the main parameters of unknown constant, instantaneous and nonstationary point pollution source.
Since problem (1)-(5) is linear, we can, without loss of generality, exclude from consideration all known sources in the domain D and the pollutant concentration created by them. Therefore, we assume that only of the concentrations ( , ) j t I R contain errors j GI whose mean value is equal to zero: However, it should be noted the methods described below with minimal changes are also applicable in the general case when condition (28) is not satisfied.
In what follows, we will consider only useful monitoring sites. Proof. By applying equation (  for different monitoring sites R and 1 R . It should be noted that in all cases, the zero of ( ) x ) correctly determines the site of emission source. Besides, Theorem 4 allows us to correctly calculate the intensity 100 c Q of the source. for different monitoring sites R and 1 R . It is seen that in all instances, the zeros of localization function correctly determine the site and the emission moment. Besides, Theorem 5 allows us to correctly calculate the intensity 120 e Q of the source.   9. Variational method for determining the intensity of nonstationary sources Dispersion models are a fundamental tool for determining the main parameters of pollution sources using the measurements. Due to the presence of errors in the measurements, these inverse problems are usually illposed, since they may have several solutions or not have solutions. Moreover, solutions may be unstable with respect to variations in external and internal parameters of the problem. Therefore, to suppress instability and choose an appropriate solution to the inverse problem, special regularization methods should be used.
We now describe a variational method to determine the unknown nonstationary emission rate ( ) Q t of a point source if its location 0 r in domain D is known. Suppose that a time series ( ) ( , ) j j j t I I GI