روش تبدیل فوریه برای تجزیه و تحلیل حساسیت در سوخت زغال سنگ نیروگاه

This work proposes a Fourier transform method to determine the sensitivities associated with a real coal power plant using a Rankine cycle. Power demand determines the plant revenue and is supposed to be the most important parameter to be accurately measured, and this hypothesis is at the center of this study. The results confirm that under full design load, variables such as steam pressure, temperature and mass flow rate are closely dependent on power demand, though overall thermal efficiency is more sensitive to boiler efficiency. Partial load simulation shows that the overall thermal efficiency remains strongly dependent on the boiler parameters, but other operational variables such as steam temperature at the turbine outlet changes its sensitivity according to the load. The results from the Fourier transform method are in good agreement with those determined by classical differential and Monte Carlo methods. However, the Fourier transform method requires only a single run, providing major savings in computational time as compared to the Monte Carlo method, a major advantage for analysis of power systems whether operating under full or partial load.

Open energy markets allow energy to be sold in short intervals, ranging from 5 min segments upwards. In the case of integrated electricity grids, unexpected outages of a major power train at a power station can cause supply shocks and large spikes in spot prices. In a tight competitive energy market, power generation plants must have a capability to predict demand with high accuracy and cope with small discrepancies while minimizing standby provisions but still retain enough supply flexibility to manage (or take advantage of) unanticipated infrastructure failures. These issues are outside the control of the plant and have a high level of uncertainty. The profitability of the plant relies on the difference between the revenue generated, i.e., the power that is exported, and the costs of production, i.e., the efficiency of the plant. To evaluate plant efficiency and, consequently, gain insight on how best to control costs, measurement of the plant operation is required. All of these measurements have some degree of uncertainly, related to instrument accuracy, calibration, maintenance and so on, and they also affect the outcome with different degrees of sensitivity. It is, consequently, important for plant management to understand and know which measurements are most critical and how accurate they need to be, since this will influence maintenance and upgrading decisions and resource allocations. The techniques for both uncertainty and sensitivity analysis are quite similar, and their difference lies principally in the interpretation of results. The main objective is to calculate the uncertainties of the results due to the uncertainties of the inputs, which leads to the reliability of the system. Sensitivity analysis helps to resolve how variation in the input data affects the output of a system. Both uncertainties of measurements and variation in input data may be seen as biases of mean values, and therefore, the same techniques can be employed for different purposes. Deviations in sensitivity analysis can be taken as arbitrated rates of mean values, for example 1% of every mean value of data input. In this paper, we focus on sensitivity analysis as applied to routine operation but note that the same kinds of techniques can be applied to uncertainty analysis applied to demand variation. There are a variety of methods that have been developed to examine the issues of process sensitivity and uncertainty. Lomas and Epperl [1] suggested that the differential method is preferable for individual parameters, but the Monte Carlo Method may be better for global sensitivity identification. Hamby [2] compared fourteen sensitivity analysis techniques when applied to a common model (of dispersion of radiation pollution in the atmosphere) and concluded that most provide very similar outcomes. Macdonald and Stracham [3] also reviewed the application of these methods to predict uncertainties of thermal models, including possible sources of uncertainty in simulated models. Taking an output Y from a set of equations Y = f(x1, …, xn), where x1, …, xn are the input data with known probability distributions, the probability distribution of Y can be found by sensitivity methods. The deviation of a given input datum will be propagated into the solution of the equation set. Indices relating probability distributions of input and output data provide useful connections, and two classical methods are reviewed in this paper. The differential method (DM) with a global covariance View the MathML sourceuY2 is calculated after the product of each first order partial derivative of Y with respect to its xi parameters by the corresponding deviation ui [1] and [4] equation(1) View the MathML sourceuY2=∑i=1n∂Y∂xi2ui2. Turn MathJax on Two indices were proposed by Hamby [2]. A dimensionless number called the importance index Ii of a given data i relates to sensitivity as follows: equation(2) View the MathML sourceIi=∂Y∂xi·x¯iY¯. Turn MathJax on This index indicates a rate or proportion between deviations, and it takes into account the relative rate of variation of the input data rather than its absolute deviation. A sensitivity index Si, is given by equation(3) View the MathML sourceSi=∂Y∂xi2·ui2uY2, Turn MathJax on which has a wider meaning than Ii because it represents the relative contribution of i in respect to the total uncertainty of Y, or the rate of local to global variance. The error propagation equation (Eq. (1)) is obtained by approximating Y using a Taylor series and first order derivatives to linearize the model [5]. The Monte Carlo method (MCM) consists of performing multiple evaluations of a system, imposing random perturbations to the input data set. Every element follows a particular probability distribution [6]. The system global variance VY can be obtained by simultaneous variation of all input data, but the local variance VY∣X can also be determined for a single input or small sets of data. The sensitivity index is defined as: equation(4) View the MathML sourceSi=VY|xiVY. Turn MathJax on The MCM requires a large computational effort compared with that of the DM to obtain either total or individual uncertainties. The index depends on the variance confidence range, given by equation(5) View the MathML sourceS2·nχp′2⩽σ2⩽S2·nχp″2, Turn MathJax on where S2 is the variance of an n element sample, σ2 is the population variance, χ2 is the chi-square coefficient for a given confidence p, and p′ and p″ are the upper and lower confidence range limits, respectively. The sample size n must be found in order to satisfy the confidence interval. In the DM, the derivative of an interval View the MathML source(x¯-ux;x¯+ux) is always calculated around its mean valueView the MathML sourcex¯. By doing this, values in this interval generate Y values that follow a straight line. The response deviates in proportion to the non-linear behavior of the actual function Y. Furthermore, the deviation of Y is non-symmetrical with respect to the mean value. In the MCM, the Gaussian deviation of the input data x matches the Y axis by following the original function of the system instead of its derivative. In the case of high non-linearities, one can easily choose different statistical distributions for the independent data. Based on these general methodologies, there are several options available in the literature for addressing sensitivities in power generation. In this work, a real coal fired power plant was chosen as a case study to investigate sensitivity analysis methods for power delivered. Specific codes based on the DM and MCM were built in order to simulate the plant running at design conditions or partial loads and to perform sensitive analysis. However, these codes (in particular MCM) require major computational time, which is always intrinsically linked with time to get the correct information and opportunity costs. To address this shortcoming, an alternative method based on Fourier transforms is proposed and compared to the DM and MCM. The Fourier transform method is applicable for analysis of power systems operating either under full or partial loads, and it requires only a single run, thus providing major savings in computational time, while allowing any quick decision making process in a competitive commercial energy market.

This work presents a new method for sensitivity analysis based on the fast Fourier transform. The results obtained for the Fourier method are shown to be within the same range as those obtained from the classical differential and Monte Carlo methods. The sensitivity index was used to sort the most relevant inputs by order of impact on the calculated outputs of the system. Although the numerical values of these indices do not match exactly for every calculated variable, their ranking and quantitative importance are very similar. The significant advantage of the fast Fourier transform method is that it simultaneously provides individual and overall sensitivities, resulting in large savings in terms of computational time as compared to the Monte Carlo Method. The sensitivity index (SI) can be interpreted in the form of major design parameters, and the choices of inputs and outputs can be studied in terms of plant response. Although high thermal efficiency correlates to profit maximization, the SI shows that it is important to have a good measure of the boiler efficiency. Thus, it may be more cost effective to focus on cost control through better information regarding thermal efficiency than revenue control through ever more accurate measurement of exported power. In addition, the turbine inlet steam temperature (T1) is one of the most important considerations for plant designers. In the case that T1 is held constant and the inlet pressure (P1) is varied to follow the power demand, the latter is seen to be most sensitive to power demand. The cooling tower outlet temperature does not appear to influence any of the outputs, indicating that it is not very worthwhile expending resources to get very good measures of ambient conditions. However, ambient conditions are critically important in determining the overall thermal efficiency and one of the main concerns of cycle design, but the sensitivity index shows they become relatively unimportant once the plant design conditions have been fixed and the hardware selected. It may be noted that the sensitivity index for the turbine inlet steam temperature T1 increases when the power load is reduced. The steam at the turbine output changes from saturated to increasingly superheated as load reduces, and therefore, having an accurate measure of T1 becomes increasingly important. Overall, design condition simulation showed that operational variables such as steam pressure, temperature and mass flow rate are very sensitive to the uncertainties of the prescribed power, and the overall plant thermal efficiency is directly related to the boiler efficiency. On the other hand, partial load operation sensitivities can be easily determined, and variations were observed for the steam thermodynamic state at the turbine outlet. The Fourier transform method proposed in this paper is, therefore, a powerful method for sensitivity analysis whether a power plant is at full or partial load.