IDENTIFICATION OF HYDRAULIC RESISTANCE PARAMETERS IN HYDRAULIC NETWORK MODEL

The main purpose of the work is to develop and analyze new identi ﬁ cation algorithm for multivariate multi-connected implicit system. Computational experiments and nonparametric statistics are implemented. The result of the work is a qualitatively new identi ﬁ cation and output forecasting algorithm for multivariate systems. Based on the proposed algorithm a new class of generalized multivariate implicit nonparametric models is suggested. The signi ﬁ cance of the research is subject to the fact that the majority of complex technical objects is usually described by large-scale nonlinear systems of equations up to input and output variables. Even if random effects in such models could be neglected a procedure of numerical solving the equation systems were either unstable or it would last too long to be implemented in forecasting of continuous process in the controlled object. Any delay in model calculation can also impact negatively on real-time control procedure. Using the proposed approach one can substitute solving the equation by estimation of the solution based on sample including measurements of input and output variables. Moreover, compared to counterparts, using the proposed approach it is possible to develop a single model for a range of input effects as it is presented in the research. Modelling and identi ﬁ cation of multivariate systems is one of the most urgent tasks. Nowadays the development modelling methods does not keep up with the continuous and unlimited increase in the volume of information. To solve this problem, the authors proposed a new approach to constructing models of multidimensional systems. The scienti ﬁ c novelty of the article is as follows. A new modi ﬁ cation of the nonparametric algorithm for identi ﬁ cation of multidimensional systems has been proposed and tested. It makes possible to improve the accuracy and speed of computations.


INTRODUCTION
At present there are a great number of identifi cation methods and algorithms applicable to complex system modelling.Applicability of the methods depends on type of the system, intended objective and problem statement, limitations, operating conditions, information on system's structure, effects and responses etc.These facts are refl ected in preliminary (a priori) and current information of the system.The most general identifi cation problem statement is as follows.Let a system be infl uenced by both unobservable random effects having zero mathematical expectations and limited variance, and a set of observable and controllable effects.Responses of the system are synchronically measured together with the observable effects.Measurements are performed incrementally with a certain time interval.We assume additive nature of random effects in the measurements.An operator of the system establishing relationships between observable inputs and outputs should be estimated.The operator should satisfy certain quality criterion.In addition, one can assume the case, when some output variables depend not only on input effects, but also on the rest of outputs.This is an equivalent for an implicit functional relation between corresponding variables.Such case is frequently encountered in practice and it is common for continuous production systems with complex branching structure.For example, it is common with technological processes where outputs of some production steps are at the same time inputs for the others.To identify this kind of multi-connected systems it is necessary to use special implicit modelling methods.This fact confi rms relevance of the research.Here we deal with an additional requirement: in order to develop a universal model one involves measured data for the whole range of input values.It means one-time training process should cover the whole range of values and provide good quality of predictive modelling.The proposed algorithm is tested and approved for the problem of hydraulic network identifi cation.Hydraulic network is a prominent example of multi-connected system of the above mentioned type.It usually consists of a number of interconnected pipes together with pumps and other equipment.The distinctive features of the system are large-scale nature, nonlinearity, multiconnectiveness, complex topology, infl uence of diverse random factors.Control and technological parameters forecasting problems are essential for hydraulic networks and require explicit mathematical modelling.Developing and implementing effective modelling procedures can ensure more reliable control process over water and oil pipelines functioning.It could be an additional reason why the results of the work are relevant from the practical point of view.

LITERATURE REVIEW
There are three basic approaches to identifi cation.The fi rst one is parametric approach [01].The substance of the approach is that preliminarily some assumptions about the object structure based on the available information research should be made.The assumptions determine structure of the model up to parameters.This step is followed by parameters tuning within the accepted structure [02].Parametric models are convenient and, in most cases, easy to tune.Reasonable disadvantage of the approach is complexity of model structure defi ning, which cannot be explicitly formalized.The second approach is nonparametric.It involves usage of kernel estimation [03] of functional from random values.A plenty of multivariate regression estimates have been already developed [04].Nonparametric approach does not require any assumptions about the model structure.Modelling in this case is entirely based on information contained in data sets.One disadvantage of the approach is signifi cant reliance on amount of data in training samples.Lack of data can cause imprecise and biased estimation, and, as a result, limited quality of the corresponding models.Nevertheless the nonparametric algorithms are capable to deal with various problems in many spheres, for example economics [05], medicine [06], biology [07] etc.The third approach is called hybrid or combined, and it incorporates modelling by using combinations of both parametric and nonparametric methods [08, 09].This approach inherits advantages and disadvantages of the approaches mentioned above.The ratio between pros and contras depends of contribution of each component, namely, parametric and nonparametric.Hybrid models are rather hard to be constructed.Nevertheless, they represent a fl exible solution exceeding capabilities of the rest two methods in terms of modelling accuracy.Nowadays a pressing problem is identifi cation of multivariate systems, displaying situation when input and output variables are interconnected functionally or stochastically.This case is very likely to be encountered in practice [10][11].Model in the case of interconnected variables can be represented by an implicit system of equations, which is often appearing to be undefi ned.The processes in such systems have been recently named T-processes [12,13].Modelling and control in systems of the described nature are non-trivial problems, and in order to be solved they require properly designed methods and algorithms.Construction of parametric models for the T-processes is a task with great computational diffi culties: to identify output for each new input it is necessary to solve a system of implicit equations.This imposes a signifi cant constrains on their practical application in decision-making and control tasks.Control systems for processes with implicit functional and stochastic relations between variables could only be Model-Based Control (MBC) systems providing model to transfers the control effects to the process itself when the necessary output is attained [02].In this regard, the T-process models are research subject to high demands, both accuracy and speed of computation terms.In this situation, nonparametric methods for modelling of multivariate systems have become widespread and progressive [14][15][16].One well-developed method of the interconnected systems modelling is the response prognosis algorithm [13] making possible identifi cation of multivariate systems represented by the set of implicit operators under combined a priori information.Further development of the nonparametric approach to the identifi cation of the T-processes is related to the identifi cation problem under nonparametric uncertainty, when information about the operator structure is unknown [17].Except nonparametric methods, in many cases suffi ciently qualitative models are constructed by ensemble methods [18] and decision-making trees [19] in connection with their adaptive and fl exible quantities.Hydraulic network is a remarkable example of multivariate multi-connected systems.Scientifi c and specialized literature proposes various models of hydraulic networks.Some of the sources are devoted to methodology of developing dynamic models, for example [20,21].In most cases models are within the concept of the Computational Fluid Dynamics (CFD).Equations expressed in the form of partial derivatives, partial volumes or fi nite elements are widely used for distributed systems description [22][23][24].The CFD approach requires quite complete information on the physical characteristics of the pumped liquid, nature of its fl ow, internal profi le and geometric confi guration of pipeline, and explicit information on functioning of pumps.In most cases, such models are computationally complex and they require unknown, often immeasurable or unpredictable in time and space variables.For processes in complex hydraulic networks the CFD models are rather inapplicable.In this regard, steady state stochastic multivariate models with improved features are still of interest when dealing with hydraulic network modelling.

Denote by
the vector of input effects of the system, -the vector of output responses, -the data set of statistically independent observations of system's state at discrete time moments , where s is sample size; -the implicit equation system describing topology of the multivariate multi-connected object under research, where are functions known up to their parameters ; -the vectors of the reduced size with components of appearing in j-th equation; -the vector of random uncontrollable effects infl uencing respectively input and output variables of the system.
Let identifi cation problem statement be as follows.A multivariate inertia-free system is infl uenced by observable input effects x and unobservable random effects having zero mathematical expectation and limited variance .Observations of input and output variables are assumed to be available.Functional dependence of input and output variables has an implicit form which is common for description of multi-connected system, and it is described by the equation system up to parameters .Based on the available data set with observations of effects and responses and the implicit model with the known structure one should develop model of the system in the form where A is an operator describing relations between inputs and outputs in an explicit way.Structure of the operator A is unknown.This operator should estimate output response y for an arbitrary input effect x, i.e. should represent a tool for forecasting of system's response.

ALGORITHM OF MULTIVARIATE SYSTEM IDENTIFICATION
The proposed algorithm starts with a preliminary step, at which parameters of the system are found using the training data set .To adjust the parameters one can apply any suitable procedure of parametric identifi cation, for instance where is a coeffi cient satisfying Robbins-Monroe requirements [19].Further, we assume three main steps.At the fi rst step residual matrix is calculated: At the second step for each t-th observation of residual matrix solution of the system is estimated: where bandwidth parameters are optimized according to the cross-validation procedure using residuals sample : 1) 2) 3) The third step involves nonparametric regression estimation using the data set : where bandwidth parameters are optimized according to the cross-validation procedure using the sample : For the all estimates above we assume that kernels Ф(•) and bandwidths and satisfy convergence criteria [04].It must be pointed out that in estimates (3)-( 6) denote sets of selected components of the residual vector ε and the input vector x, associated with ε j and x j .Estimation of each j-th component of the estimate ys as well as the output vector y is performed by taking into account only those dimensions of ε and x that have infl uence on the mentioned component.The suggested algorithm has some unique features if compared to counterparts [07, 10].Firstly, an improved procedure of residuals calculation is introduced.While calculating residuals, one creates matrix, where residuals are placed according to corresponding input value leading to response change.Such matrix is able to be used when forecasting system's outputs with every fi xed input confi guration.In most cases, to detect how response depends on any conceivable input pattern is impossible.Therefore, estimation of such relation is essential, especially for systems, which cannot be explored by arranging experiments.Secondly, new way of model construction is proposed.Estimation of output variables forecasting involves only those inputs that directly infl uence the particular output 4) 5)

6)
component.This leads to signifi cant reducing of dimensionality of corresponding nonparametric estimates and removal of irrelevant inputs having no contribution to the result but producing extra noise and offsets.Thus, the measure results in considerable improvement of modelling accuracy and calculation speed.Thirdly, the suggested algorithm is universal, and it can easily accept further modifi cations of estimates confi guration and ways of residuals calculation.One can implement the approach for any comprehensiveness of available information on functional relations between variables, namely, for parametric, nonparametric and combined cases.In particular, this will affect residuals calculation according to the formula (2).Right part of the equality can, therefore, differ from the parametric form represented in (2).

EXPERIMENTAL RESEARCH
Simulation and computational approval of the suggested algorithm is performed while modelling fl ow distribution problem [25] in hydraulic network.The three-loop network designed for computational experiment is depicted in Figure 1.In Figure 1, arcs denote single pipeline segments; block H -an active head being a model of a pump.The amount of arcs n = 10, and of knots k = 8, the dimension of the corresponding system of equations l = 10, input vector dimension m = 7.At some knots outfl ows are implemented.Every segment of the network has its hydraulic resistance .The arrows in contours correspond to selected directions of traversal.In accordance with Kirchhoff's law for pipeline networks the equation system is as follows: In the expression (7) we denote input effects, namely outfl ows in knots of the network, and the active hydraulic head .Outputs are fl ows in the corresponding arcs.Hydraulic resistances are represented by the parameters of the equation system (7).The system of equations ( 7) has a single solution.where x is an initial value, p -variation from the initial approximation, -random sequence evenly distributed within the interval , q -shift parameter.Each value in ( 8) should be subject to random noise infl uence.Random noise is added with the procedure similar to ( 8): where ξ is random noise amplitude, -random sequence evenly distributed within the interval , q -shift parameter.Output values for given inputs are taken as numerical solutions of the system (4) according to the Newton's method.Assume an initial training sample of the state vector of the hydraulic network.Let an identifi cation quality criterion be mean squared error (MSE): where y is forecast of the output, y* is actual output.Computational research of the identifi cation algorithm was performed by repeated starting of the process for the following parameters: • Results of computational research of the proposed algorithm compared to the basic multivariate identifi cation algorithm [07] are represented in Table 1 showing criteria for the comparison are accuracy MSE and average operation time T for different noise levels ξ and amounts of measurements s in the sample.Averaging was performed over N computational experiments.Further, the authors illustrate how the proposed identification algorithm reacts on the varying input vector.Apply following vector as an initial one: where sample size s = 200 and is the function in Figure 2. In accordance with (9), impose noise ξ = 10% on the vector at q = 0.5.For the accepted input vector , generate the output vector in the course of solving the system (7) by the Newton's method.The resulting sample is taken as training sample for the next experiment.According to the procedure (1), results of parameter estimation are represented in Figure 3.
Figure 4 depicts results of the output forecasting compared to the actual output of the system (7).One experiment for sample size s = 200, additive noise level ξ = 10%, and for the fi rst parameter α1 and the fi rst component of the output vector y1 is illustrated here.

DISCUSSION
Results of computational experiments confi rm convergence of the proposed algorithm.As expected, accuracy of modelling improves for greater sizes of training sample.Noise causes negative effect on identifi cation process.The proposed algorithm provides more accurate results if compared to the basic algorithm.Moreover, it is at least two times faster than its predecessor.The improvement of basic features of the suggested algorithm if compared to the basic one is due to its selectivity.The algorithm involves in estimation of system solution as well as in response estimation only those components of input effect that have direct infl uence on it.In other words, a priori information is used more efficiently in the algorithm.Usage of the approach to system's response estimation provides reducing of dimensionality of the estimates, which is additionally resulted in the increased calculation rate.

CONCLUSION
In the paper a new approach to identifi cation and forecasting of multivariate systems response was proposed.This approach was submitted as an improved algorithm able to distinguish better accuracy and faster calculation speed due to optimal design of nonparametric response estimation procedure.The main idea was to reduce complexity by selecting relevant input components directly infl uencing the corresponding output response.This approach can be considered as substantial step in developing novice methods of multivariate identifi cation.The proposed algorithm makes it possible to forecast the system output for arbitrary input effects by one-time training.Compared to counterpart algorithms, there is no more need to solve system of equation numerically or to train the model for each new input.Prediction of output variables is performed by imputation of an input to the nonparametric estimates within the modelling algorithm.An advantage of the approach is simplifi ed training of the model.Training procedure can be performed at once for all the possible inputs.A new way of residual calculation was implemented in the algorithm.Residuals were referenced to certain combination of input values.That means residuals serve as intermediate source of information about system's behavior enabling fast and accurate forecasting of output variables.
One fundamental problem when identifying multivariate system is amount of available information of the system under research.Relationships between variables are often pure qualitative, i.e. one can only determine some subsets of interconnected variables without any parametric structure within.In this case, the proposed algorithm is applicable as well.Residuals calculation step (2) remains the same with one remark that the right side of the expression will be used to calculate pre-trained nonparametric regression up to all variables in the j-th mixture. 272 Vladimir Bukhtoyarov -Identifi cation of hydraulic resistance parameters in hydraulic network model To sum up, we can claim that the proposed approach and corresponding computational algorithm is universal and effi cient instrument of solving multivariate identifi cation problem for diverse statements and different combinations of available information of the system under research.
One of the most demanding problems of the modern theory of identifi cation and modelling is identifi cation of multivariate dynamic systems [25].The direction of future research is related to the development of algorithms discussed in the article for dynamic case.At present, there exist reliable algorithms for linear dynamical systems only [26].The development of algorithms dealing with dynamic processes is associated with noticable diffi culties such as lack of suitable mathematical methods and extremely large computational resources requirements.
In order to overcome these diffi culties new methods are required.Development of such methods is the subject of future research.

Figure 2 :
Figure 2: Function for sample size s = 200

Figure 3 :Figure 4 :
Figure 3: An example of the parameter estimation α1[t] compared to the actual value for sample size s = 200, and noise level ξ = 10%

Table 1 :
Results of Computational Experiments on the Proposed Algorithm Compared To the Basic One