Improved monitoring and / or prediction of uncertainties in industrial processes and electrical networks

Abstract

A computer-implemented method (100) for calibrating that an amount of uncertainty in an industrial process (1a) and / or an electrical network (1b) will exceed a given threshold, the method comprising the steps of: providing (110) an initial desired value; providing (120) a predictive model (2) configured to predict future values of the quantity from the values of the quantity; a history of past values of measurements of the quantity is provided (130). Determining (140), for this history, a difference between a value predicted by the prediction model and a measured past value as a residual (3); determining (150) an estimated probability density function (4) of the residuals (3); determining (160), based at least in part on the history and / or residual (3), a confidence set to which the estimated probability density function (4) belongs, said confidence set comprising a probability density function that is equivalently reasonable given the history and / or residual (3); and determining (170) a new value based at least in part on the dimensions of the confidence set such that if less than is exceeded, it is ensured that the initial desired value is not exceeded even if the residual should be represented according to any probability density function in the confidence set instead of the estimated probability density function (4).

Description

TECHNICAL FIELD

[0001] The present invention relates to the monitoring and / or prediction of uncertain quantities in industrial processes and electrical networks. In order to ensure safe and reliable operation of industrial processes and / or electrical networks, it is desirable to have as accurate as possible knowledge of these uncertain quantities. BACKGROUND

[0002] When monitoring and / or controlling an industrial process, the development of certain quantities can be uncertain. For example, the temperature, the pressure, the power load and / or the power generation can fluctuate.

[0003] Many of these tasks are performed under the constraint that a certain quantity must not exceed a given threshold , but at the same time this quantity is to some extent uncertain. In this case, it is not feasible to guarantee that the quantity will never exceed the threshold . In most cases, it is still possible to quantify the probability that this happens . If the quantity is an uncertain quantity, the constraint can be explicitly constructed as a chance constraint, i.e. the uncertain quantity is to remain below the threshold with a probability (confidence) of at least .

[0004] There exist stochastic methods for estimating . Such stochastic methods can make use of the history of past data and thus use the indications that are evident from the history without relying on any prior knowledge. They start from the worst case, i.e. with zero knowledge about the behavior of the uncertain quantity . However, in most cases, the situation is far less bad, because at least some information is available. SUMMARY OBJECTIVE OF THE INVENTION

[0005] It is therefore an object of the present invention to improve the prediction of the probability that an uncertain quantity in an industrial process and / or electrical network will exceed a given threshold in the case where there is non-zero knowledge about the behavior of this uncertain quantity .

[0006] This object is achieved by the computer-implemented method according to the independent claims. Further advantageous embodiments are specified in the dependent claims. DISCLOSURE OF THE INVENTION

[0007] The invention provides a computer-implemented method for calibrating an uncertain quantity in an industrial process and / or an electrical network exceeds a given threshold . .

[0008] A quantity may be uncertain for any reason. For example, a quantity may behave in a technical process in a fluctuating or not completely predictable way. In another example, a quantity may only be measurable within a large error range and / or only indirectly based on other measurable quantities, so that any prediction of future values based on history of past values has a large error range. Examples of uncertain quantities in technical applications include liquid levels, pressures, temperatures, power loads, and power generation. For example, in an electrical network, the behavior of consumers and the power generation of renewable energy sources is only predictable to a limited extent.

[0009] In the course of the method, an initial expected value of the quantity is provided. Furthermore, a prediction model for the uncertain quantity is provided. This prediction model is used to predict future values of the quantity from values of the quantity . The prediction model can take any form. For example, it can be a parameterized function of past values , and the parameters of this function can have been obtained by fitting. In another example, the prediction model can be a stochastic model, where past values determine a distribution from which new samples are drawn. If only very little information is available, the prediction model can also be, for example, a simple periodic function, or even a constant function. Even a little knowledge is better than none.

[0010] A history of measured past values of the quantity is provided. For this history , the difference between the values predicted by the prediction model and the measured past values is determined as a residual. In this context, the prediction model can apply to any suitable prediction horizon. For example, the prediction model can give one predicted value at a time, or it can give multiple predicted values at a time.

[0011] ​An estimated probability density function of the residual is determined. That is, the work product of the prediction model output is directly used, while the estimated probability density function is focused on the task of completing a full and accurate reproduction of the past values in the history .

[0012] A confidence set to which the estimated probability density function belongs is determined, based at least in part on the history and / or on the residual. This confidence set comprises probability density functions that are equally reasonable given the history and / or the residual. In this way, it is taken into account that the history on which the estimated probability density function is based is finite: for an infinitely long history, the probability density function would converge to the true distribution of the past values according to the law of large numbers. The finiteness of the history implies that there is necessarily some error in estimating the probability density function. Given the estimated probability density function, the confidence set basically means: "given that this probability density function is apparently considered reasonable by virtue of the estimation, then this probability density function could also be any of the other functions in the set

[0013] ". A new ' is determined based at least in part on the dimension of the confidence set . This new ' means: if exceeds ' by less than , then it is guaranteed that even if the residual behaves according to any probability density function in the confidence set instead of the estimated probability density function, this would not exceed the initial expectation . In this context, the dimension can for example be understood as the number of degrees of freedom constituting the set , or as the number of arguments that need to be provided in order to characterize an element of this set .

[0014] The inventors have found that in this way, the part of the uncertainty that remains in the quantity of values after the prediction model has done its work can be adapted to, and thus also the quantity the available knowledge about the behavior of the residuals. The more knowledge available, the more accurate the prediction model, which in turn means that the residuals are smaller and behave according to an easily catchable distribution, the members of which can be characterized by a small number of independent variables (meaning that the dimensionality of the confidence set is small). Another factor influencing the dimensionality of the confidence set is the size of the available history of past measurements. As discussed before, even if given zero prior knowledge, the distribution of the residuals can be perfectly estimated if an infinite long history of past measurements is available. This means that a longer history can at least partially compensate for the lack of prior knowledge about the behavior of the uncertainty quantities .

[0015] In general, the new ' is lower than the initial expectation . This can be understood as follows: assuming that the residuals behave according to the estimated probability distribution function, providing the initial expectation means that there is indeed a risk in reality. However, the uncertainty of the estimated probability distribution function, which is reflected in the dimensionality of the confidence set means that the risk can actually be higher than the initial expectation if the true distribution of the residuals does not behave according to the estimated probability density function but according to another probability density function from the confidence set . To compensate for this and to guarantee that the new ' is at most , only the new ' is acceptable from the beginning. That is, the uncertainty of the estimated probability distribution function is translated into an additional safety margin that has to be taken into account when operating the industrial process and / or the electrical network. The better the prediction model, the richer the history of past measurements , the smaller the additional safety margin.

[0016] Thus, the end result is that the chance constraint, i.e. specifying a threshold for at least one uncertainty quantity , can be satisfied more reliably, also specifying an expectation of the number of times the constraint is violated. This greatly helps to develop countermeasures for cases in which the chance constraint is violated.

[0017] For example, in an electrical network, there can be a certain i.e. the power supply and demand deviate too much, resulting in the electrical network not being able to operate at a stable AC frequency, and measures have to be taken to bring them closer again, for example by shedding load, buying the power deficit from a neighbouring electrical network, or by storing the excess power or delivering it to a neighbouring electrical network (usually for an additional fee). The latter phenomenon manifests itself in negative electricity prices, even in day-ahead electricity auctions. This countermeasure comes at a cost, and the cost of some countermeasures is higher than that of others. For example, if there is enough pumped hydro storage available to store the excess power available at a certain time, then the excess power can be stored instead of being sold at a negative price. This means that it is more profitable to sell it later, instead of paying to dispose of it. It also means that the connection to a neighbouring electrical network, which usually has limited power capacity, does not need to be taxed.

[0018] In another particularly advantageous embodiment, the cumulative density function of the estimated probability density function is calculated. Then, the inverse function of this cumulative density function is calculated. Thereby, the new threshold value must be met to guarantee the minimum requirement is valid. The evaluation is carried out in this way. In this way, the new threshold value can be directly translated into a safety margin for the uncertain quantity . That is, in all chance constraints in which the uncertain quantity occurs, the newly calculated minimum requirement can be directly replaced as residual. For example, the uncertain quantity can be replaced by the output of the prediction model plus the minimum requirement

[0019] as residual. In a simple example, it is to be guaranteed that the level of a container does not fall below 1 meter with a probability of 80% (corresponding to a risk of 20% that this occurs). The combination of the quality of the prediction model and the availability of historical measurements of the past values can lead to the result that, due to the uncertainty in the probability density function reflected in the confidence set

[0020] , it must be planned to always keep the level above 1.1 meters in order to ensure that the risk of the level falling below 1 meter does not exceed 20%.

[0021] In another particularly advantageous embodiment, the prediction model is combined with a minimum requirement The residuals are combined to form an updated prediction model. That is, the output of the new prediction model is the output of the old prediction model plus the value based on... Determined residuals, for example, Confidence levels can be used To estimate or otherwise determine the confidence level The function, that is, Use this updated prediction model to begin a new iteration of the method. Reducing a given residual in multiple steps may be easier than reducing it all at once. In particular, the first improvement can be obtained quickly, while further calibration requires more time.

[0022] In another particularly advantageous embodiment, optimization is performed on at least one variable of the industrial process. Alternatively or in combination with this, at least one equation can be solved with respect to at least one variable of the industrial process and / or the electrical network and / or any other problem. Further alternatively or in combination with this, a control output for at least one variable of the industrial process and / or the electrical network is obtained from a controller model (such as a model used in model predictive control (MPC)). In each case, the above-described optimization, solution, or acquisition of the control output is performed with uncertainty... It is executed under at least one chance constraint. Where, for the use of confidence levels (1- ') Estimated threshold Minimum requirements It is used as an approximation of the residual. That is, as discussed earlier, the uncertainty... The output of the prediction model can be added (1- ') can be used instead. For example, the parameter (1- ') Calculate the inverse cumulative density function to obtain (1- Then, the algorithm itself can be used as before without modification.

[0023] In another particularly advantageous embodiment, at least one actuator that causes a physical impact on the industrial plant and / or electrical network is actuated based on the results obtained from the optimization, solution, or control output described above. In this way, the calibration of the chance constraint directly translates into an improvement in the operation of the industrial plant and / or electrical network.

[0024] In another particularly advantageous embodiment, the confidence set of the probability density function Selected to include all probability distribution functions Based on the given divergence metric, the probability distribution function The divergence between the estimated probability density function and the target probability set is less than the predetermined confidence set size. For example, the confidence set may be defined as: .

[0025] In this context, is a divergence, while is the estimated probability density function. For a one-dimensional random variable, the divergence is defined as: .

[0026] In this context, is a random variable with distribution function To take into account the uncertainty in the probability distribution estimation, the classical chance constraint

[0027] i.e. the probability that a randomly drawn value will be smaller than or equal to is This chance constraint can be reformulated as: .

[0028] In one example where the - divergence is a second order - divergence, it is defined as

[0029] For , a closed-form expression for the reduced can be derived: .

[0030] In another particularly advantageous embodiment, the probability density function is estimated using at least one kernel density estimator. For example, a Gaussian kernel, an Epanechnikov kernel or a uniform kernel can be used. For example, for a Gaussian kernel, the kernel function bandwidth is obtained according to the Scott rule, and the resulting estimated probability density function can be written as: . where is the kernel function, is the bandwidth of the kernel function, is the number of samples, while is the used sample. The advantage of the Gaussian kernel is that it has a smoother, more general distribution on small data sets compared to other kernel functions. The choice of the bandwidth has a much greater impact on the probability density function than the choice of the kernel function, and should therefore be chosen according to the number of available samples to select the bandwidth. For example, the bandwidth can be chosen according to Scott's rule is chosen to be That is, in a particularly advantageous embodiment, the at least one kernel density estimator is a Gaussian kernel whose bandwidth minimizes the integrated mean squared error.

[0031] There can be any number of kernel functions. The same kernel function can be used for all samples, a different kernel function can be used for each sample, or anything in between. For example, a parametric kernel function can be fitted to the data.

[0032] In another particularly advantageous embodiment, the confidence set size is determined based at least in part on the sample variance corresponding to the initial expected value of the statistic quantile. This is independent of the size of the histogram bin to which the given data is applied.

[0033] The general confidence interval for the estimated density of the probability distribution of X can be written as: .

[0034] Since the variance of the estimate is unknown, the sample variance is used to construct the confidence interval. The values and are the statistic quantile and quantile, respectively, of the lower statistic: .

[0035] With this statistic, the quantiles are defined as and , respectively.

[0036] In another particularly advantageous embodiment, the estimate of the probability distribution function is performed on a selected bootstrap subset of the residuals. This has two benefits. First, the computation becomes more efficient if only a subset of the data is used that preserves the essential concept of the data behavior. Second, if the confidence set size is determined based on the statistic quantile, it can be ensured that the statistic is unbiased, i.e., and .

[0037] ​In another particularly advantageous embodiment, the prediction model comprises a linear time-invariant auto-regressive model with exogenous inputs. These exogenous inputs comprise a synthetic input signal that models the non-linear behavior of the uncertain quantity within the context of the linear model. In this way, the model itself can remain linear and good prediction accuracy can be achieved even if trained on only small batches of data.

[0038] In another advantageous embodiment, the prediction model is selected that has been trained with an objective function comprising a regularization term. This regularization term depends on a norm of the parameter set characterizing the behavior of the model . In this way, overfitting of the model can be avoided in the process of training the prediction model for rolling horizon prediction on small training data sets. For example, with this regularization, multi-step prediction of the uncertain quantity can be performed in each control step of stochastic model predictive control (SMPC) instead of only a single day-ahead prediction.

[0039] In another particularly advantageous embodiment, the history of measured past values comprises at most half of the past values used for training the prediction model. This allows the prediction model to be able to clearly distinguish between long-term behavior captured by the prediction model (on the one hand) and unexpected short-term behavior that does not conform to the model’s expectations and has to be inferred from the history (on the other hand).

[0040] As the present method is computer-implemented, it can be embodied in the form of software. Therefore, the present invention also relates to a computer program having machine-readable instructions which, when executed by one or more computers and / or computing instances, cause the one or more computers and / or computing instances to perform the method described above. Examples of computing instances include virtual machines, containers, or serverless execution environments in the cloud. The present invention also relates to a machine-readable data carrier and / or a download product via which the computer program is available. A download product is a digital product having a computer program, for example, which can be sold in an online store for immediate delivery and download onto one or more computers. The present invention also relates to one or more computing instances having the computer program and / or having the machine-readable data carrier and / or the download product.

[0041] The present disclosure proposes a novel method to construct stochastic optimization and / or stochastic model predictive control problems based on a data-driven non-parametric self-tuning chance constraint method, whose applications include but are not limited to energy management problems, asset planning problems, scheduling problems, and control problems. The proposed method does not require any prior assumption on the probability distribution of the uncertain quantities, but instead uses an improved method to estimate the probability density function of the residual of the uncertain variables predicted based on the past data, which uses an automatic and self-tuning approach. In addition, the proposed method can construct the stochastic part of the optimization and control problems as algebraic (linear) constraints, thus not negatively impacting the computational effort required to solve such stochastic problems.

[0042] Optimization-based solutions to problems like energy management, production scheduling, plant control, etc. typically operate in environments where certain quantities (e.g., process variables, weather conditions, parameters, etc.) are uncertain and / or subject to unknown disturbances. A common strategy is to use prediction models to predict the future behavior of the uncertain quantities and to use this prediction as fact when planning the future behavior of the system under consideration. Based on the construction of optimization problems on the nominal "trajectories" of the uncertain quantities obtained from the prediction models, this approach lacks robustness when the actual behavior of the uncertain quantities differs from the predicted behavior (which is very common). This can lead to performance degradation and compliance issues, as the solution of the "nominal" optimization problem can be suboptimal or infeasible once implemented.

[0043] Robust and stochastic optimization methods have been proposed over the years to address this problem, relying on different philosophies (e.g., deterministic vs. stochastic) and different constructions. However, these solutions still suffer from several issues. Deterministic (or worst-case) methods tend to provide overly conservative results. Stochastic methods also face some challenges: they either require prior knowledge of the exact potential probability distribution of the uncertainty (which is often difficult to obtain), or they result in constructions that are difficult to handle, often too computationally expensive to implement in real industrial applications. Available optimization-based industrial solutions still cannot include the stochastic contribution of the uncertain quantities without requiring unrealistic prior assumptions, resulting in constructions that are easy to handle and computationally reasonable.

[0044] For the problem described, the proposed solution: relies on a database non-parametric method that is used to estimate the probability density function (PDF) of the amount of uncertainty from the available data; relies on a bootstrap method to automatically improve the size of the confidence set of the estimated PDF; and relies on a linear (algebraic) formulation of the chance constraint that can be easily added to any type of optimization problem without increasing its complexity and computational burden. Moreover, the proposed solution is intended to be implemented in combination with any kind of prediction / estimation model, since it is applied to the estimation residuals (i.e. the estimation errors), rather than directly to the uncertain variables. This allows more flexibility in the predictions and improves the adaptability of the estimations, since they can be built from input signals such as weather data, process dependencies or synthetic inputs that improve the overall performance (e.g. weekday / weekend indicators). The adjustable confidence level is provided as a decision variable to the operator / user to select the desired level of robustness (and conservatism) of the optimization results in the face of uncertainty.

[0045] This solution allows to build a stochastic optimization problem to solve a variety of tasks without any kind of a priori assumption, without negatively affecting the overall amount of computation required, and it allows to create an automated process where the desired probability density function (PDF) is estimated from data almost without engineering effort and human intervention, and the confidence level is self-tuned based on the information content of the available data.

[0046] The precondition is that: a nominal optimization problem is built to solve a specific task where one or more variables are uncertain or affected by unknown disturbances; a prediction model is available to estimate the future behavior of the uncertain variables (the model can have any kind of structure or formulation); and a dataset with historical values of the uncertain variables and corresponding estimations is available. The solution is based on the following steps: a) the user / operator / system sets the desired prediction horizon length and the desired level of confidence; b) the residuals (i.e. the errors between the estimated and the actual values) of all the uncertain variables on the available historical set are computed; c) the probability density function of the residuals of all the uncertain variables from the available dataset is estimated using, for example, a kernel density estimation with a Gaussian kernel; d) the bandwidth of the employed kernel function is computed using Scott's rule; e) the sample variance and the confidence level corresponding to the one set in a) are used to compute the confidence set of the estimated PDF; f) the confidence set is used to compute the chance constraint; g) the nominal optimization problem is solved with the chance constraint; h) the solution is provided to the user / operator. - the statistical quantiles are used to construct the confidence intervals of the estimated probability density function; f) the data bootstrapping technique is used to calibrate the obtained density function, so that the confidence intervals are built in an unbiased way; g) the confidence interval size is computed and it is used to calculate a reduced risk level from the confidence level defined in a), so that the risk level is adapted to the information content of the available data; h) the cumulative density function of the considered residuals is computed from the estimated density function obtained with the bootstrapped kernel function, and the corresponding quantile function is evaluated at the reduced risk level in g), to obtain the algebraic (linear) chance constraints that can be included in the relaxed optimization problem construction.

[0047] The advantages of this solution include: • it provides an automated process to obtain from the data the algebraic chance constraints for the construction of the stochastic optimization problem, including a self-tuning functionality to adapt the risk level to the information content of the available data, without the need of prior assumptions. • it is used in combination with any type of prediction model, and it is built on the estimated residuals, instead of being directly built on the uncertain variables, so that additional inputs can be used to improve the estimation accuracy. • it uses the data bootstrapping technique to calibrate the construction of the kernel density estimator, while the state of the art uses pointwise error intervals, so that the size of the final confidence set is improved. • it is a flexible and "problem-agnostic" method, since it is applicable to all kinds of prediction / estimation models, and it can be easily adapted to various problems with uncertain quantities, such as microgrid optimal energy scheduling, building / industrial energy management, production or inventory scheduling, etc. • there is no need of prior assumptions on the probability function. • the performance is improved thanks to better estimation performance, since applying the described stochastic method to the prediction residuals instead of directly applying it to the uncertain variables allows better estimation accuracy, since it permits the use of additional input signals. • the size of the density function confidence set is improved using the data bootstrapping method, which improves the reduced risk level. • the self-tuning of the reduced risk level reduces the conservatism of the results, since the confidence level is automatically adapted to the information content of the available data set. • the obtained algebraic chance constraints do not have a significant impact on the computational effort required to solve the resulting optimization problem, and they do not have an impact on the optimization problem category (e.g. linear, quadratic, non-linear, etc.). This allows a computationally lean construction of the stochastic optimization problem, and allows its use for real industrial applications.

[0048] The solution comprises: • A computational unit, with access to relevant history (e.g. past values of uncertain variables), with a pre-trained prediction / estimation model (model structure and learning / training method are out of the scope of the invention), which is used to estimate the future behavior of uncertain variables. • A (possible) user / operator / system interface, where the desired level of robustness is specified via a confidence level parameter. • A (possible) user / operator / system / factory interface, where the results of the optimization can be visualized and / or implemented on the system or factory. • An intelligent and self-tuning algorithm that estimates the probability density function of the estimation residuals of all uncertain variables, self-tunes the kernel function bandwidth, makes the risk level automatically adaptive to the information content of the data set, calibrates the density function estimates with data bootstrapping, evaluates the corresponding quantile functions for the reduced risk level, and outputs the algebraic chance constraints that can be included in any kind of optimization problem.

[0049] The method and its overall workflow / process allow for the automatic derivation of stochastic chance constraints for optimization problem construction from data, without the need for prior assumptions on the probability distributions.

[0050] This data-driven non-parametric method is based on kernel density estimation, combined with data bootstrapping calibration, automatic kernel bandwidth computation, self-tuning of the (reduced) risk level, estimation of the probability density function, and subsequent evaluation of the corresponding quantile functions, to construct algebraic chance constraints.

[0051] The concept involves a data-driven non-parametric self-tuning chance constraint method to construct tractable and computationally reasonable stochastic optimization problems, which can be used to solve various tasks such as energy management, production scheduling, etc.

[0052] Let us consider the optimal energy management problem of a small grid-connected micro-electric network, consisting of a photovoltaic generator, uncontrollable uncertain electricity demand load, and a battery energy storage system. The goal is to manage the battery energy storage system to maximize the use of renewable energy and minimize the cost of electricity purchase from the electric network, while satisfying the electricity demand load. Then, a potential constraint of the considered optimization problem is the energy balance constraint, i.e., at any time instant k the sum of electricity purchase from the electric network, battery discharge, and photovoltaic system generation should be equal to zero:

[0053] Since both photovoltaic generation and load demand are unknown and uncertain, prediction models are used to estimate their values, denoted as and respectively. The resulting nominal (non-stochastic) constraint is: .

[0054] This nominal constraint does not consider the randomness of uncertain variables, thus exhibiting suboptimal and potential infeasibility issues. The corresponding algebraic chance constraint obtained using the data-driven nonparametric method proposed in this paper is: Attached Figure Description

[0055] The invention is illustrated in the accompanying drawings below, but is not intended to limit the scope of the invention. The drawings show:

[0056] Figure 1: Uncertainties used for calibration in industrial processes and / or electrical networks Will exceed the given threshold of An exemplary embodiment of method 100;

[0057] Figure 2: Comparison of predicted and actual electrical network load values ​​( Figure 2a Comparison between predicted and actual photovoltaic power generation values ​​(); Figure 2b );

[0058] Figure 3 : Possesses confidence level Examples of probability density functions for the corresponding confidence interval estimates;

[0059] Figure 4 Threshold-based Minimum requirements The estimated value of residual 3 is used to calibrate the predicted value of the load;

[0060] Figure 5 The state of charge of batteries in an electrical network and the state of battery depletion The relationship; a comparison with the nominal MPC. Detailed Implementation

[0061] Figure 1 is a schematic flowchart of an exemplary embodiment of method 100, which is used to calibrate uncertainties in industrial processes and / or electrical networks. Will exceed the given threshold of Method 100 begins when an uncertainty needs to be evaluated in an industrial process 1a and / or electrical network 1b. Will exceed the given threshold of .

[0062] In step 110, the following is provided initial expected value .

[0063] In step 120, prediction model 2 is provided. This prediction model 2 is configured to follow the quantity value To predict this quantity Future value .

[0064] According to box 121, this prediction model 2 may include a linear time-invariant autoregressive model with exogenous inputs. According to box 122, the exogenous inputs may include a synthetic input signal that, within the context of the linear model, addresses uncertainties. The nonlinear behavior is modeled.

[0065] According to box 123, an objective function can be selected to train prediction model 2, which includes a regularization term that depends on the set of parameters representing the behavior of model 2. norm .

[0066] In step 130, a quantity is provided. Past values ​​of measurement History .

[0067] According to box 131, this history At most, it includes past values. Half of it has been used to train prediction model 2.

[0068] In step 140, regarding this history Determine the value predicted by the prediction model. Compared with past values ​​of measurement The difference between them is taken as residual 3.

[0069] In step 150, the probability density function 4 of the estimated residuals 3 is determined.

[0070] According to box 151, the probability density function 4 can be estimated using at least one kernel density estimator.

[0071] According to box 151a, this at least one kernel density estimator can be selected as a Gaussian kernel with bandwidth that minimizes the integral mean square error.

[0072] According to box 152, the estimation of probability distribution function 4 can be performed on the selected bootstrap subset of residual 3.

[0073] In step 160, at least in part based on history And / or, based on residual 3, determine the confidence set to which the estimated probability density function 4 belongs. This confidence set Included in a given history and / or residual 3.

[0074] According to block 161, this confidence set of probability density functions 4 may be chosen to include all probability distribution functions that, according to a given divergence measure, have a divergence from the estimated probability density function 4 smaller than a predetermined confidence set size .

[0075] According to block 161a, the confidence set size is determined (161a) based at least partly on a sample variance and / or - a statistical quantile corresponding to the initial expectation value of the residual . .

[0076] In step 170, a new is determined based at least partly on the dimension of the confidence set . The meaning of this new is that if exceeds the initial expectation value by more than , then it is guaranteed that even if the residual behaves according to any probability density function in the confidence set instead of the estimated probability density function 4, this will not exceed the initial expectation value . .

[0077] In the example shown in Fig. 1, in step 180, the cumulative density function 5 of the estimated probability density function 4 is computed; in step 190, the inverse function 6 of the cumulative density function 5 is computed. Then, in step 200, the threshold value must satisfy in order for the guarantee about to be valid is evaluated based on the inverse function 6 of the cumulative density function 5 and the new . .

[0078] In step 210, the prediction model 2 can be combined with the residual determined based on the minimum requirement to form an updated prediction model 2'. In step 220, a new iteration of the method 100 can be started with this updated prediction model 2'.

[0079] In step 230, the following steps are performed under at least one chance constraint on the uncertainty • at least one variable of the industrial process 1a and / or the electrical network 1b is optimized; and / or • at least one variable of the industrial process 1a and / or the electrical network 1b is optimized; and / or • at least one variable of the industrial process 1a and / or the electrical network 1b is optimized; and / or• solving at least one equation regarding at least one variable of the industrial process la and / or the electrical network lb and / or any other problem; and / or • obtaining a control output of at least one variable of the industrial process la and / or the electrical network lb from the controller model. for a threshold value using a confidence level the determined minimum requirement is used as an approximation of the residual 3.

[0080] In step 240, at least one actuator that has a physical influence on the industrial plant la and / or the electrical network lb is actuated in accordance with the results of the above-described optimization, solving, or obtaining of a control output.

[0081] Figure 2a A power load in an electrical network lb that is powered by photovoltaic electricity generation, includes battery energy storage, and is connected to an external electrical network is shown . The power load is shown over time . Curve a shows the predicted value, and curve b shows the actual measured value. In the example shown in , the overall trend of the predicted value coincides with the measured value, but the predicted value misses some (literally) unforeseen details. Figure 2a

[0082] The photovoltaic electricity generation in the same electrical network lb is shown Figure 2b . Similar to , the photovoltaic electricity generation Figure 2a is shown over time . Curve a shows the predicted value, and curve b shows the actual measured value. In the example shown in , the predicted result coincides with the overall trend of the measured value. However, the predicted value underestimates the rising amplitude of the photovoltaic electricity generation Figure 2b when it reaches its maximum.

[0083] An example of a probability density function 4 of an estimate of a residual is shown, whose confidence interval corresponds to a confidence Figure 3 . The probability is plotted against the value of the residual 3. The confidence interval defines a confidence set of reasonable probability density functions. In the one-dimensional case shown in , this confidence set Figure 3 manifests itself as a region that the probability density function can cross.

[0084] Figure 4 is shown how a threshold-based​ Minimum requirements The residual 3 estimate is used for calibration. Figure 2a Electrical loads in electrical network 1b shown The prediction. Therefore, Figure 4 Corresponding to Figure 2a And added calibrated power loads. The prediction curve c. Specifically, the calibrated prediction value is always greater than the electrical load. The actual measured value. Therefore, this prediction ensures that it does not underestimate the electrical load. This reduces the risk of violating constraints.

[0085] Figure 5 The diagram illustrates the risk level. How the value of affects how the process operates. The figure plots the state of charge (SoC) of the battery in the electrical network 1b studied in Figure 2 over time. The curve showing the change in charge state. The opportunity constraint is that the charge state must not fall below a threshold. Compared to the results of traditional model predictive control (MPC), the acceptable level of risk... The lower the value, the higher the safety margin of the energy stored during the battery charging phase. List of reference numerals in the attached diagram: 1a Industrial plant 1b Electrical Network 2. For uncertainties Prediction model 2' Updated prediction model 2 3. Uncertainties The residual between the predicted value and the past value 4. Probability density function of residual 3 5. Cumulative density function of probability density function 4 6. The inverse function of the cumulative density function 100 calibration method 110 provides initial expected value 120 provides prediction model 2 121 Choosing a linear model with exogenous inputs 122. Using the synthesized input signal as the exogenous input. 123 Selecting a prediction model trained using regularization 2 130 provides an uncertainty. History 131. Select those with a shorter history. Settings 140 From history determine residual 3 150 Estimate probability density function 4 151 Estimate probability density function 4 with kernel density estimator 151a Select Gaussian kernel density estimator 152 Estimate probability density function based on bootstrap subsets 160 Determine confidence set 161 Select confidence set according to divergence 161a Select specific method for determining confidence set 170 Determine new 180 Compute cumulative density function 5 190 Compute inverse of cumulative density function 5 200 Evaluate minimum requirements on threshold 210 Combine prediction model 2 with estimate of residual 3 to obtain updated model 2' 220 Start new iteration of method 100 with updated model 2' 230 Use estimate of residual 3 in chance constraint 240 Actuate at least one actuator Uncertain quantity Risk of exceeding threshold Initial expected value of New risk in view of available data Confidence set of probability density function (S)MPC Stochastic Model Predictive Control P Probability Photovoltaic power production in 1b in electrical network Load 1b in electrical network X Uncertain quantity Uncertain quantity Threshold of uncertain quantity Past values of uncertain quantity History of​​​​ threshold value minimum requirements

Claims

1. A computer-implemented method (100) for calibrating an uncertain quantity in an industrial process (la) and / or an electrical network (lb) exceeding a given threshold , the method comprising the steps of:​ • providing (110) the initial desired value ; • providing (120) a prediction model (2) configured to predict future values of the quantity based on values of the quantity ; and ; and ​ • providing (130) the quantity of measured past values of history ; • against the history , determining (140) a difference between the value predicted by the prediction model and the measured past value as a residual (3); • determining (150) a probability density function (4) of the estimate of the residual (3); • based at least partly on said history and / or said residual (3), determine (160) a confidence set to which said estimated probability density function (4) belongs , said confidence set comprising probability density functions that are equally plausible given said history and / or said residual (3); and • determining (170) a new based at least partly on the dimension of the confidence set such that if exceeds the is less than then it is guaranteed that even if the residual should behave according to any of the probability density functions in the confidence set instead of the estimated probability density function (4), the will not exceed the initial expectation value .

2. The method (100) of claim 1, further comprising: • computing (180) a cumulative density function (5) of the estimated probability density function (4); • computing (190) an inverse function (6) of the cumulative density function (5); and • the inverse function (6) of the cumulative density function (5) and the new , the threshold value is evaluated (200) The minimum requirements that must be fulfilled in order for the guarantees about the to be valid.

3. The method (100) of claim 2, further comprising: • Combine the prediction model (2) with the minimum requirement. The residuals are then combined (210) to form an updated prediction model (2'). and • starting (220) a new iteration of the method (100) using the updated prediction model (2').

4. The method (100) of any one of claims 2 to 3, further comprising one or more of: • optimizing at least one variable of the industrial process (la) and / or electrical network (lb); • solving at least one equation with respect to at least one variable of the industrial process (la) and / or electrical network (lb) and / or with respect to any other problem; and • obtaining a control output for at least one variable of the industrial process (la) and / or electrical network (lb) from a controller model, The optimization, the solution, or the obtaining of the control output mentioned above are performed with respect to the uncertainty. It is executed under at least one chance constraint, and wherein the threshold is applied. Use confidence level The determined minimum requirements It is used as an approximation of the residual (3) mentioned in (230).

5. The method (100) of claim 4, further comprising: in accordance with a result of the optimization, the solving, or the obtaining the control output, actuating (240) at least one actuator, the actuator causing a physical influence on the industrial plant (la) and / or electrical network (lb).

6. The method (100) according to any one of claims 1 to 5, wherein the confidence set of probability density functions (4) is selected (161) to include all probability distribution functions according to a given divergence measure, the divergence of which from the estimated probability density function is less than a predetermined confidence set size . ​ 7. The method (100) of claim 6, wherein the confidence set size is determined (161a) based at least in part on the initial expected value of the sample variance and / or statistical quantile.

8. The method (100) of any one of claims 1 to 7, wherein the probability density function (4) is estimated (151) using at least one kernel density estimator.

9. The method (100) of claim 8, wherein at least one kernel density estimator is selected (151a) to be a Gaussian kernel with a bandwidth that minimizes the integrated mean squared error.

10. The method (100) of any one of claims 1 to 9, wherein the estimate of the probability distribution function (4) is performed (152) on a selected bootstrap subset of the residual (3).

11. The method (100) of any one of claims 1 to 10, wherein: • the prediction model (2) comprises (121) a linear time-invariant auto-regressive model with exogenous inputs; and • The exogenous input includes (122) a synthetic input signal, which, within the context of the linear model, addresses the uncertainty. The nonlinear behavior is modeled.

12. The method (100) of claim 11, wherein the prediction model (2) has been trained (123) with an objective function comprising a regularization term that depends on a norm of a parameter set characterizing the behavior of the model (2) . .

13. The method (100) according to any one of claims 1 to 12, wherein the amount Past values ​​of the measurement The history mentioned At most (131) past values ​​used to train the prediction model (2) Half of it.

14. A computer program comprising machine-readable instructions which, when executed on one or more computers and / or computing instances, cause the one or more computers and / or computing instances to perform the method (100) of any one of claims 1 to 13.

15. A non-transitory machine-readable data carrier and / or a download product having the computer program of claim 14.

16. One or more computers and / or computing instances having the computer program of claim 14 and / or having the non-transitory machine-readable data carrier and / or the download product of claim 15.

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