Analysis sensitivity

Sensitivity analysis is a very important tool in analysing the relative importance of the model parameters and in the design of experiments for their optimal determination. In many cases, it is be found that a model may be rather insensitive to a particular parameter value in the region of main interest, and then the parameter obviously does not need to be determined very accurately. 

Model parameters are usually determined from expterimental data. In doing this, sensitivity analysis is valuable in identifying the experimental conditions that are best for the estimation of a particular model parameter. In advanced software packages for parameter estimation, such as SIMUSOLV, sensitivity analysis is provided. The resulting iterative procedure for determining model parameter values is shown in Fig. 2.39. 

Equilibrium Constants in a Reversible Esterification Reaction Using ESLand SIMUSOLV 

The objective is to demonstrate the power of modern simulation packages in the estimation of model parameters. Here the parameters are estimated using SIMUSOLV running on VAX systems and on a PC using the ESL simulation package, the main features of which can be found on the last page of this book. 

Ethanol and acetic acid react reversibly to ethyl acetate, using a catalyst, ethyl hydrogensulfate, which is prepared by reaction between sulfuric acid and ethanol. 

Sensitivity analysis can answer a number of key questions, such as the following  

A controllable source of variability is also known as a critical control point and refers to a point, step or procedure at which control can be applied and a hazard can be prevented, eliminated or reduced to an acceptable level. A critical limit is a criterion that must be met for each preventive measure associated with a controllable source of variability or critical control point. 

Sensitivity analysis can be used to identify and prioritize key sources of uncertainty or variability. Knowledge of key sources of uncertainty and their relative importance to the assessment end-point is useful in determining whether additional data collection or research would be useful in an attempt to reduce uncertainty. If uncertainty can be reduced in an important model input, then the corresponding uncertainty in the model output would also be reduced. Knowledge of key sources of controllable variability, their relative importance and critical limits is useful in developing risk management options. 

Sensitivity analysis is useful not only because it provides insight for a decision-maker, but also because it assists a model developer in identifying which assumptions and inputs matter the most to the estimate of the assessment end-point. Therefore, sensitivity analysis can be used during model development to identify priorities for data collection, as well as to determine which inputs matter little to the assessment and thus need not be a significant focus of time or other resources. Furthermore, insights obtained from sensitivity analysis can be used to determine which parts of a model might be the focus of further refinement, such as efforts to develop more detailed empirical or mechanistic relationships to better explain variability. Thus, sensitivity analysis is recommended as a tool for prioritizing model development activities. 

The sensitivity analysis of a system of chemical reactions consist of the problem of determining the effect of uncertainties in parameters and initial conditions on the solution of a set of ordinary differential equations [22, 23], Sensitivity analysis procedures may be classified as deterministic or stochastic in nature. The interpretation of system sensitivities in terms of first-order elementary sensitivity coefficients is called a local sensitivity analysis and typifies the deterministic approach to sensitivity analysis. Here, the first-order elementary sensitivity coefficient is defined as the gradient 

The set of equations described by Eq. (2.67) can be rewritten to show the functional dependence of the right-hand side of the equation as 

For a local sensitivity analysis, Eq. (2.69) may be differentiated with respect to the parameters a to yield a set of linear coupled equations in terms of the elementary sensitivity coefficients, cXM /daj. 

Since the quantities df/d(Mj) are generally required during the solution of Eq. (2.69), the sensitivity equations are conveniently solved simultaneously with the species concentration equations. The initial conditions for Eq. (2.70) result from mathematical consideration versus physical consideration as with Eq. (2.69). Here, the initial condition [c M /da, ]f=0 is the zero vector, unless a, is the initial concentration of the /th species, in which case the initial condition is a vector whose components are all zero except the y th component, which has a value of unity. Various techniques have been developed to solve Eq. (2.70) [22, 23], 

It is often convenient, for comparative analysis, to compute normalized sensitivity coefficients 

The sensitivity analysis for the right-hand side of constraints was studied, as shown in Table 2.17. The upper limits for some of the constraints are positive infinity while the lower limits vary. Lower limits for most of the constraints assumed a negative value. Constraints whose upper limits are allowed to go to positive infinity imply that they are not critical to the production process. As an illustration, the demand for heating oil can be decreased without inflicting a change in the current optimal basis. The other constraints can be analyzed in a similar manner. 

The values of the slack or surplus variables and the dual prices in Table 2.15 provide the most economical average operating plan for a 30-day period. For instance, it indicates that the primary distillation unit is not at full capacity as the solution 

Constraints Right-hand side of constraints ranges (t/day)  

The dual prices of slacks on mass balance and product requirement rows can be interpreted more specifically. Consider a mass balance constraint  

The product stream is increased by an and the feed streams x3 and x2 must increase correspondingly. The dual price of an indicates the effect of making marginally more products without taking into account its realization (which is on x3), that is, it indicates the cost added by producing one extra item of the product, or, in other words, the marginal cost of making the product. In addition to that, consider the product requirement constraint  

Below the results of Sensitivity Runs with MADONNA are given from the BIOREACT example that is run as a batch fermenter system. This example involves Monod growth kinetics, as explained in Section 1.4. In this example, the sensitivity of biomass concentration X, substrate concentration S and product concentration to changes in the Monod kinetic parameter, Ks, was investigated. Qualitatively, it can be deduced that the sensitivity of the concentrations to Ks should increase as the concentration of S becomes low at the end of the batch. This is verified by the results in Fig. 2.30. The results in Fig. 2.31 give the sensitivity of biomass concentration X and substrate concentration S to another biological kinetic parameter, the yield coefficient Y, as defined in Section 1.4. 

The results of a sensitivity analysis are usually presented as plots of an economic criterion such as NPV or DCFROR vs. the parameter studied. Several plots are sometimes shown on the same graph using a scale from 0.5 x base value to 2 x base value as the abscissa. 

The purpose of sensitivity analysis is to identify those parameters that have a significant impact on project viability over the expected range of variation of the parameter. Typical parameters investigated and the range of variation that is usually assumed are given in Table 6.12. 

The choice of which feed and product prices to use in the sensitivity analysis depends strongly on the method of price forecasting that has been used. Typically, total raw material cost is studied rather than treating each feed separately, but if raw material costs are found to be the dominant factor, then they may be broken out into the costs of individual raw materials. 

In a simple sensitivity analysis, each parameter is varied individually, and the output is a qualitative understanding of which parameters have the most impact on project viability. In a more formal risk analysis, statistical methods are used to examine the effect of variation in all of the parameters simultaneously and hence quantitatively determine the range of variability in the economic criteria. This allows the design engineer to estimate the degree of confidence with which the chosen economic criterion can be said to exceed a given threshold. 

A simple method of statistical analysis was proposed by Piekarski (1984) and is described in Humphreys (2005). Each item in the estimate is expressed as a most likely value, ML an upper value, H and a lower value, L. The upper and lower values 

Some sources of uncertainty and variability may have so little influence on risk that they can be held constant and not treated probabilistically in the assessment. The analysis plan should state the rationale for deciding which variables and hypotheses this applies to (USEPA 1998). 

Sensitivity analysis provides a good tool for this purpose (USEPA 1997). It quantifies the change in model outputs as a function of changes in each model input and enables the influence of different inputs to be compared. 

Different methods of sensitivity analysis will produce different results, so they should be chosen carefully (Warren-Hicks and Moore 1998). A comprehensive account of alternative approaches is provided by Saltelli et al. (2000). 

As well as guiding problem formulation, sensitivity analysis can be valuable in optimizing the use of resources. By revealing which uncertainties have the most influence on the results of the assessment, sensitivity analysis can also help target additional research or monitoring and by revealing which of the controllable sources of variability have the most influence, sensitivity analysis can help identify and evaluate practical options for managing risk. 

In order to test the economic performance of the project to variations in the base case estimates for the input data, sensitivity analysis is performed. This shows how robust the project is to variations in one or more parameters, and also highlights which of the inputs the project economics is more sensitive to. These inputs can then be addressed more specifically. For example if the project economics is highly sensitive to a delay in first production, then the scheduling should be more critically reviewed. 

Changing just one of the individual input parameters at a time gives a clearer indication of the impact of each parameter on NPV (the typical indicators under investigation), although in practice there will probably be a combination of changes. The combined effect of varying individual parameters is usually closely estimated by adding the individual effects on project NPV. 

Typical parameters which may be varied in the sensitivity analysis are  

If the fiscal system is negotiable, then sensitivities of the project to these inputs would be appropriate in preparation for discussions with the host government. 

When the sensitivities are performed the economic indicator which is commonly considered is the true value of the project, i.e. the NPV at the discount rate which represents the cost of capital, say 10%. 

In order to simplify the presentation, the reaction will be considered to take place in a batch reactor of constant volume, working under isothermal conditions. The mass balance equations can be written as follows  

The rate Rj is a function of the concentrations of all the species present in the reaction medium. 

The numerical integration of equations (1) gives the values of the concentrations as functions of the reaction time, t  

Chapter XIII - Analysis and reduction of reaction mechanisms 

The calculated concentrations Cj are functions of the reaction time t and of the initial 

For each distribution with a number of bars, start with the first bar of the first distribution and add successively all combinations of the bars in the second distribution, then go to the 

To generalize, let x, represent the nth ab.scis.sa of the first distribution and Pj the corresponding ordinate of a normalized distribution. Then c is formed of the ordered pairs (equation 2.7-.30). As an example, for x, 

Suppose the components are redundant, then their probabilities, if independent, are combined by multiplication (equation 2.7-34). this is generalized by analogy with the preceding (equation 2.7-35). As a numerical example, if the distributions z=(x 

When a risk or reliability analysis has been performed, it is appropriate to inquire into the sensitivity of the results to uncertainties in data. One type of sensitivity analysis is the effect on system reliability that results from a small change in a component s failure probability. A problem in doing this is determining the amount of data uncertainty that is reasonable. The amount of change 

Significance of risk contribution may be done by ordering the risk contributors from most-to-least (rank order), but because of the arbitrariness of variation of the variables, this may be meaningless A more systematic approach is to calculate the fractional change in risk or reliability for a fractional change in a variable. 

The investment criteria discussed in this section are set out in Table 6.7, which shows the main advantage and disadvantage of each criterion. 

There is no one best criterion on which to judge an investment opportunity. A company will develop its own methods of economic evaluation, using the techniques discussed in this section, and will have a target figure of what to expect for the criterion used, based on their experience with previous successful, and unsuccessful, projects. 

Criterion Abbreviation Units Main advantage Main shortcoming 

Investment — , Shows financial resources needed No indication of project performance 

Net future worth NFW , Simple. When plotted as cash-flow diagram, shows timing of investment and income Takes no account of the time value of money 

In this section we consider the parametrized vector differential equation 

These partial derivatives provide a lot of information (ref. 10). They show how parameter perturbations (e.g., uncertainties in parameter values) affect the solution. Identifying the unimportant parameters the analysis may help to simplify the model. Sensitivities are also needed by efficient parameter estimation procedures of the Gauss - Newton type. Since the solution y(t,p) is rarely available in analytic form, calculation of the coefficients Sj(t,p) is not easy. The simplest method is to perturb the parameter pj, solve the differential equation with the modified parameter set and estimate the partial derivatives by divided differences. This brute force approach is not only time consuming (i.e., one has to solve np+1 sets of ny differential equations), but may be rather unreliable due to the roundoff errors. A much better approach is solving the sensitivity equations 

The sensitivity equations (5.39) are derived by differentiating (5.37) with respect to Pj, and changing the order of differentiation on the left hand side. The initial values to (5.39) are given by 

Example 5.3 Parameter sensitivities in the microbial growth model 

In order to discuss the practical identifiability of the model studied in Examples 5.1.1 and 5.1.2, Holmberg (ref. 3) computed the sensitivities of the microorganism concentrations y and substrate concentration 2 with 

As there is considerable uncertainty in many of fhe assumptions for various technologies, HjSim is designed to allow the user to easily change them. These changes in key component costs, fuel prices, and process efficiency have varying effects on hydrogen production costs. 

Fl2Sim was built using Powersim Studio 2005, a dynamic simulation package. Powersim solves the model equations by integrating them with respect to some variable, normally time. HjSim uses the capital cost as the dynamic variable, rather than time. The advantage of this choice is that the integration automatically produces a parameter study with respect to capital cost. 

Capital Steam methane Coal Electrolysis Th Chem Th Chem NPO 

Ideally, future work on the model would investigate the connection between capital cost and hydrogen production rate. For example, the capital cost of steam methane reforming may be linked to the production rate as in Equation (2), which will provide a more complicated trade-off. This type of analysis is possible with H2Sim, but requires the user to adjust capital costs to reflect the unit size. 

The base case results are sensitive to fuel prices and the overall efficiency of the process. Tables 6.5-6.9 show the relative sensitivity of SMR, coal gasification, electrolysis, NPO, and thermochemical nuclear to fuel costs and thermal efficiency. 

Self-organization is possible without cooperativity, but cooperativity may greatly increase the effectiveness of self-organization. The hemoglobin molecule may serve as an example of cooperativity in intermolecular interactions, where its interaction with the first oxygen molecule makes its interaction with the second easier despite a considerable separation of the two binding events in space. 

The results in an LCA study can be affected by many sources of uncertainty. The sources of uncertainty can be found in the choices used for assumptions, scope, boundaries, impact assessment methods, and the quality of the available data. In addition, assumptions made for the inclusion of end-of-life, transportation, and pollution can significantly affect the results in an LCA. 

Key to the relevance of any LCA study is the quality of data. This can be measured with sensitivity analysis (Cellura et al. 2011). 

LCA studies should include a section on sensitivity analysis and identify areas in the LCA that may be unreliable or inaccurate. The uncertainty in the LCA study should require the data to be calculated with critically reviewed methods. LCA results should include a range of results that incorporate variations in the input data. 

Three procedures of analysis can be used to estimate the uncertainty in an LCA study (May and Brennan 2003)  

As an example, sensitivity analysis was used to determine the uncertainty in an LCA on Italian roof tiles (Cellura et al. 2011). In the study, uncertainty was found in several sources of data in the LCA. The results revealed that in some cases significant differences in energy usage and 

While in slurry-fed systems heat loss plays a minor role compared to heat for water vaporization, in dry-fed appUcations gasifier heat loss is the key parameter, offering a cold gas efficiency potential of -1-3.4%-pts. while switching from a 4% 

Gasifier heat loss - o %(LHV) Coal/Ng preheat-Ash content -Oxygen preheat Coal moisture - 

Because the reference case has an oxygen purity of 95vol%, the thermodynamic impact of increasing purity to 100% is nearly negligible. Taking 80vol% as the 

To summarize the sensitivity of the investigated parameters to the varied boundary conditions, the following results can be listed  

1 Baumer, D. (1989) Gase-Handbuch, Messer Griesheim GmbH. 

In addition to changing parameters to determine the influence of different phenomena, the influence of the different terms in a model can be estimated by putting a weighting factor in front of each term in the model, e.g. the accumulation term or the axial dispersion term in a chemical reactor model, and comparing the simulations. In this way, we can easily estimate which factors dominate the model, and perhaps remove the parts that have negligible influence. We can also judge whether the result supports our expectations or whether the model must be reformulated. 

Note that the fit and response sensitivities are interrelated, as can be demonstrated in the case of the weighted least squares, when 

These sensitivities provide numerical measures for the contribution of the th reaction when 9j is associated with the parameter of the corresponding rate expression, namely, with the rate constant kj for the isothermal case or with the preexponential factor Aj for a nonisothermal model. It is more convenient to deal with nondimensional, or logarithmic, measures of sensitivities (Bukhman et al, 1969 Frank, 1978 Gardiner, 1977). Thus, the rth logarithmic response sensitivity of rate constant kj is given by 

The evaluation of sensitivities can be approached in a number of ways. The simplest, so-called brute force method, consists of the computation of the response at distinct values of the parameter considered while keeping the others constant. Assuming that 

The set of logarithmic response sensitivities for all rate constants is referred to as a sensitivity spectrum. The highest absolute values of the sensitivities in the spectrum pinpoint the most important reactions in the kinetic scheme for determining the chosen set of responses. The lowest absolute values of sensitivities, however, do not necessarily identify unimportant reactions. Let 

Both reactions proceed with practically identical rates, much higher than the other reactions, and thus determine the instant of ignition. Hence, reaction (R3) can hardly be considered unimportant. Why then is the sensitivity to this reaction so low  

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A Monte Carlo simulation is fast to perform on a computer, and the presentation of the results is attractive. However, one cannot guarantee that the outcome of a Monte Carlo simulation run twice with the same input variables will yield exactly the same output, making the result less auditable. The more simulation runs performed, the less of a problem this becomes. The simulation as described does not indicate which of the input variables the result is most sensitive to, but one of the routines in Crystal Ball and Risk does allow a sensitivity analysis to be performed as the simulation is run.This is done by calculating the correlation coefficient of each input variable with the outcome (for example between area and UR). The higher the coefficient, the stronger the dependence between the input variable and the outcome. 

The uncertainty may be addressed by constructing a base case which represents the most probable outcome, and then performing sensitivities around this case to determine which of the inputs the project is most vulnerable to. The most influential parameters may then be studied more carefully. Typical sensitivities are considered in Section 13.7, Sensitivity Analysis . 

So far, the economics of developing discovered fields has been discussed, and the sensitivity analysis introduced was concerned with variations in parameters such as reserves, capex, opex, oil price, and project timing. In these cases the risk of there being no hydrocarbon reserves was not mentioned, since it was assumed that a discovery had been made, and that there was at least some minimum amount of recoverable reserves (called proven reserves). This section will briefly consider how exploration prospects are economically evaluated. 

The computational process of analysis is hidden from the user, and visually the analysis is conducted in terms of M-02-91 or R6 [6] assessment procedure On the basis of data of stress state and defect configuration the necessary assessment parameters (limit load, stress intensity factor variation along the crack-like defect edge) are determined. Special attention is devoted to realization of sensitivity analysis. Effect of variations in calculated stress distribution and defect configuration are estimated by built-in way. 

Dougherty E P and Rabitz H 1980 Computational kinetics and sensitivity analysis of hydrogen-oxygen combustion J. Chem. Phys. 72 6571... 

A major advantage of this hydride approach lies in the separation of the remaining elements of the analyte solution from the element to be determined. Because the volatile hydrides are swept out of the analyte solution, the latter can be simply diverted to waste and not sent through the plasma flame Itself. Consequently potential interference from. sample-preparation constituents and by-products is reduced to very low levels. For example, a major interference for arsenic analysis arises from ions ArCE having m/z 75,77, which have the same integral m/z value as that of As+ ions themselves. Thus, any chlorides in the analyte solution (for example, from sea water) could produce serious interference in the accurate analysis of arsenic. The option of diverting the used analyte solution away from the plasma flame facilitates accurate, sensitive analysis of isotope concentrations. Inlet systems for generation of volatile hydrides can operate continuously or batchwise. 

Sokolowski J., Zolesio J.-P. (1992) Introduction to shape optimization. Shape sensitivity analysis. Springer-Verlag. 

Fig. 7. Sensitivity analysis, (a) Changes in discount rate as indicated by a sensitivity diagram. At center axis the discount rate i = 10% and NRR = 10.74% ...

Statistical Criteria. Sensitivity analysis does not consider the probabiUty of various levels of uncertainty or the risk involved (28). In order to treat probabiUty, statistical measures are employed to characterize the probabiUty distributions. Because most distributions in profitabiUty analysis are not accurately known, the common assumption is that normal distributions are adequate. The distribution of a quantity then can be characterized by two parameters, the expected value and the variance. These usually have to be estimated from meager data. 

Sensitivity Analysis When solving differential equations, it is frequently necessary to know the solution as well as the sensitivity of the solution to the value of a parameter. Such information is useful when doing parameter estimation (to find the best set of parameters for a model) and for deciding if a parameter needs to be measured accurately. See Ref. 105. 

Sensitivity Analysis An economic study should pinpoint the areas most susceptible to change. It is easier to predict expenses than either sales or profits. Fairly accurate estimates of capital costs and processing costs can be made. However, for the most part, errors in these estimates have a correspondingly smaller effecl than changes in sales price, sales volume, and the costs of raw materials and distribution. 

It is worthwhile to make tables or plot cuives that show the effect of variations in costs and prices on profitabihty. This procedure is called sensitivity analysis. Its purpose is to determine to which factors the profitabihty of a project is most sensitive. Sensitivity analysis should always be carried out to obseive the effect of departures from expec ted values. 

Example 3 Sensitivity Analysis The following data describe a project. Revenue from annual sales and total annual expense over a 10-year period are given in the first three columns of Table 9-5. The fixed-capital investment Cfc is 1 million. Plant items have a zero salvage value. Working capital C c is 90,000, and the cost of land Ci is 10,000. There are no tax allowances other than depreciation i.e., is zero. The fractional tax rate t is 0.50. For this project, the net present value for a 10 percent discount factor and straight-line depreciation was shown to be 276,210 and the discoiinted-cash-flow rate of return to be 16.4 percent per year. 

Sensitivity analysis to analyze quantitatively the effect of changes to operating conditions and feed variables on product quality... 

Once the model parameters have been estimated, analysts should perform a sensitivity analysis to establish the uniqueness of the parameters and the model. Figure 30-9 presents a procedure for performing this sensitivity analysis. If the model will ultimately be used for exploration of other operating conditions, analysts should use the results of the sensitivity analysis to estabhsh the error in extrapolation that will result from database/model interactions, database uncertainties, plant fluctuations, and alternative models. These sensitivity analyses and subsequent extrapolations will assist analysts in determining whether the results of the unit test will lead to results suitable for the intended purpose. 

A further detail of this study included sensitivity analysis for the area of heat exchangers, discount rate, and fuel cost. The results are listed in Table 3-8. Option 2, the turboexpander scheme, was selected in terms of energy and maintenance savings, as well as enhanced reliability, availability, and safety. 

If optimization is aehieved using the proeesses ehosen, then the standard deviation estimated for eaeh eomponent toleranee ean be eompared to the required assembly standard deviation to see if overall eapability on the assembly toleranee has been aehieved. If exeessive variability is estimated at this stage on one or two eharaeteristies, then redesign will need to be performed. Guidanee for redesign ean be simplified by using sensitivity analysis, used to estimate the pereentage eontribution of the varianee of eaeh eomponent toleranee to the assembly varianee, where the varianee equals cr. It follows that from equation 3.3 ... 

Part of the design information provided by the software is the standard deviation multiplier, z, for eaeh eomponent toleranee shown in Figure 3.14 in Pareto ehart form. Additionally, sensitivity analysis is used to provide a pereentage eontribution of eaeh toleranee varianee to the final assembly toleranee varianee as shown in Figure 3.15. 

The varianee equation provides a valuable tool with whieh to draw sensitivity inferenees to give the eontribution of eaeh variable to the overall variability of the problem. Through its use, probabilistie methods provide a more effeetive way to determine key design parameters for an optimal solution (Comer and Kjerengtroen, 1996). From this and other information in Pareto Chart form, the designer ean quiekly foeus on the dominant variables. See Appendix XI for a worked example of sensitivity analysis in determining the varianee eontribution of eaeh of the design variables in a stress analysis problem. 

Figure 4.45 Sensitivity analysis of the stress on first assembly (for f = 10 000 N)...

The eontribution of eaeh variable to the final stress distribution in the ease of stress rupture ean be examined using sensitivity analysis. From the varianee equation ... 

Figure 4.56 Sensitivity analysis for the holding torque variables...

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