CN108183484B - Decoupling semi-linear optimal power flow model based on hot start environment - Google Patents

Decoupling semi-linear optimal power flow model based on hot start environment Download PDF

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CN108183484B
CN108183484B CN201711336455.0A CN201711336455A CN108183484B CN 108183484 B CN108183484 B CN 108183484B CN 201711336455 A CN201711336455 A CN 201711336455A CN 108183484 B CN108183484 B CN 108183484B
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CN108183484A (en
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卫志农
朱梓荣
孙国强
臧海祥
张清松
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Hohai University HHU
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • H02J3/04—Circuit arrangements for ac mains or ac distribution networks for connecting networks of the same frequency but supplied from different sources
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
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Abstract

The invention discloses a decoupled semi-linear optimal power flow model based on a hot start environment. Firstly, performing polynomial fitting on a trigonometric function term in a power balance equation, decoupling a voltage amplitude value and a phase angle by using the operating characteristics of a system, then performing linearization processing on a phase angle quadratic term in the power balance equation in a Taylor series expansion mode, and finally obtaining a semi-linearization optimal power flow model with a linear voltage phase angle and a nonlinear amplitude value by using the result of a direct current optimal power flow model based on a network loss equivalent load as an operating point of the Taylor series expansion of the phase angle quadratic term. The invention better solves the problem of dependence of the hot start model on the operating environment, and simultaneously improves the calculation precision of the linear model and the adaptability to a large system.

Description

Decoupling semi-linear optimal power flow model based on hot start environment
Technical Field
The invention relates to a linear optimal power flow model of an electric power system, and belongs to the technical field of electric power systems.
Background
The calculation of the Optimal Power Flow (OPF) is firstly proposed by a French scholarer Carpentier in the 60's of the 20 th century, and is an important means for ensuring the safe and economic operation of a power system. However, an Alternating Current Optimal Power Flow (ACOPF) model has a strong nonlinear characteristic, and coupling between variables is very tight, which results in low computation efficiency of the model and fails to meet the online real-time computation requirement of a large-scale system. Therefore, it is important to find a suitable linearized OPF model. The Direct Current Optimal Power Flow (DCOPF) is the linear OPF model with the fastest solving speed at present. However, as the network loss is ignored and the influence of the voltage amplitude and the reactive power is not considered, the calculation error is larger, complete scheduling information cannot be obtained, and a certain application bottleneck exists, so that the research of a more complete and accurate linearization model has important practical significance.
The ac optimal power flow model is the most primitive model in the field of the invention, and is originally derived from the document Carpentier j]//Bull.Soc.
Figure BDA0001507478620000011
D' Electrical element 1962, 3: 431 and 447.
In the prior art, a linearization model which has higher precision and considers voltage amplitude and reactive power is mostly a linearization model based on a hot start mode. The hot start mode is an operation point of taylor series expansion using the historical data of the previous section scheduled in the power system day or the data of the type such as the current of the current section as a nonlinear term. The solving process of the OPF model under the hot start mode is always carried out around the selected operation point, so the quality of the operation point directly influences the precision of the calculation result. The simultaneous linearization processing of the voltage and the amplitude requires providing an amplitude and a phase angle operating point for the model, and the existing technology usually adopts the current power flow value as the operating point and needs to acquire the current power flow of the system, so that when the power flow of the system is unknown, the adaptability of the linearization model in the hot start mode is poor. Meanwhile, the calculation accuracy of the model cannot be effectively guaranteed by taking the load flow value as an operation point, and the situation of non-convergence may occur when the model is applied to some large systems with higher requirements on operation environments. Therefore, the research of the linearized OPF model with wider application range has important practical significance.
Disclosure of Invention
The purpose of the invention is as follows: the invention provides a voltage amplitude and phase angle decoupling semi-linear model with nonlinear voltage amplitude and linear phase angle, aiming at the problems that the calculation efficiency of an alternating current optimal power flow model is difficult to meet the requirement of operation analysis of a large power grid, a DCOPF model does not take the influence of voltage amplitude and reactive power into account, complete scheduling information cannot be obtained, a hot start linear model is high in dependence on an operation point and a method for quickly providing a good operation environment is lacked. Aiming at the phase angle operation point information required by the phase angle linearization processing of the model, the invention reasonably utilizes the existing DC optimal power flow model with higher precision and based on the network loss equivalent load, and takes the obtained result as the phase angle operation point required by the model, thereby effectively solving the problem of high dependence of the hot start model on the operation environment and expanding the application range of the model.
The technical scheme is as follows: a decoupled semi-linear optimal power flow model based on a hot start environment is characterized in that: the method comprises the following steps:
(1) analyzing the nonlinear characteristics of the alternating current optimal power flow model;
(2) performing polynomial fitting on a trigonometric function item in a system power balance equation in the alternating current optimal power flow model, and decoupling a voltage amplitude and a voltage phase angle by using the running characteristic of the system;
(3) carrying out linearization processing on a voltage phase angle quadratic term in the decoupled power balance equation in a Taylor series expansion mode, so as to reduce the nonlinearity degree of the equation;
(4) the result obtained by the direct current optimal power flow model based on the network loss equivalent load is used as an operation point of the phase angle quadratic term Taylor series expansion, and the calculation precision of the model is improved;
(5) and (3) verifying the accuracy and the efficiency of the model through an example test.
Further, in the step (1), nonlinear characteristics of the alternating current optimal power flow model are analyzed, and a standard form of the alternating current optimal power flow model is given firstly:
Figure BDA0001507478620000021
Figure BDA0001507478620000022
in the formula: n isgIndicates the number of generators, a2i、a1iAnd a0iConsumption of a characteristic parameter, P, for the ith generatorGi、QGiActive and reactive power, P, of the ith generator, respectivelyGk、QGkActive power of kth generator connected to node iReactive and reactive power, PDi、QDiActive and reactive loads, U, respectively, of node iiIs the voltage amplitude of node i, θij＝θi-θjIs the voltage phase angle difference of node i and node j, Gij、BijReal and imaginary parts, G, of the ith row and jth column element, respectively, of the admittance matrixii、BiiThe real and imaginary parts, n, of the self-admittance of node i, respectivelybIs the number of nodes of the system, PLi、QLiFor active and reactive power flow of the ith branch, nLThe number of branches of the system is,*、
Figure BDA0001507478620000037
respectively, the lower limit and the upper limit of each variable;
it can be seen from the ac optimal power flow model that the nonlinear characteristics are mainly reflected in the first two constraints in the constraint conditions, that is, the node power balance constraint and the line power flow constraint, and since the node power is the algebraic sum of the line power flow, the linearization of the line power flow is the key to improve the model solution efficiency.
Further, in the step (2), polynomial fitting is performed on a trigonometric function term in the system power balance equation, and the voltage amplitude and the voltage phase angle are decoupled by using the system operation characteristics, wherein the method specifically comprises the following steps:
3.1 the line tidal current expression can be deduced by the alternating current optimal tidal current model to be as follows:
Figure BDA0001507478620000031
in the formula: pij、QijActive power flow and reactive power flow, g, of line ij respectivelyij、bijRespectively, the conductance and susceptance of line ij;
3.2 the phase angle difference between two ends of the line is usually in the process of operating the power system
Figure BDA0001507478620000032
To
Figure BDA0001507478620000033
According to the characteristic, the invention utilizes MATLAB fitting tool box to fit trigonometric function items in the system power balance equation to obtain the following expression:
Figure BDA0001507478620000034
to facilitate the subsequent expression, let C1＝0.97，C2＝0.49；
3.3 since the node voltage is always maintained at about 1pu during the operation of the power system, there is a UiUj1, the following approximation can be obtained:
Figure BDA0001507478620000035
3.4 substituting the expression of 3.2-3.3 into the line power flow equation of 3.1 to obtain a line power flow expression with voltage amplitude and phase angle decoupling, wherein the expression is as follows:
Figure BDA0001507478620000041
further, step (3) is to carry out linearization processing on a voltage phase angle quadratic term in the decoupled power balance equation in a Taylor series expansion mode, so as to reduce the nonlinearity degree of the equation, and the method specifically comprises the following steps:
expressing theta in line power flow expressionijThe second order term of the first order term is expanded by Taylor series, and the truncation error is ignored, so that the following approximation can be obtained:
Figure BDA0001507478620000042
substituting the above formula into the decoupled line power flow equation to obtain:
Figure BDA0001507478620000043
in the formula: thetaij,0Is composed of
Figure BDA0001507478620000044
A reference point for performing taylor series expansion, namely an operation point;
at this point, the power balance equation can be written as:
Figure BDA0001507478620000045
and further, using the result obtained by the direct current optimal power flow model based on the network loss equivalent load as an operation point of the phase angle quadratic term Taylor series expansion.
Has the advantages that: compared with the prior art, the invention has the advantages that: according to the decoupling semi-linear optimal power flow model based on the hot start environment, the phase angle quadratic term in the decoupling power balance equation is subjected to linearization processing through decoupling processing of the voltage amplitude and the phase angle, the existing direct current optimal power flow model result based on the network loss equivalent load with higher precision is used as the phase angle operation point information required by the model, the solving efficiency of the OPF model is effectively improved, the scheduling information provided by the model is enriched, the problem that the hot start model has higher dependence on the operation environment is solved to a certain extent, and the application range of the model is expanded.
Drawings
Fig. 1 is a schematic diagram of a network loss equivalent load model.
Detailed Description
The invention will be further elucidated with reference to the following specific examples.
The method comprises the steps of starting from an alternating current optimal power flow model, analyzing nonlinear characteristics in the alternating current optimal power flow model, carrying out approximate equivalence on trigonometric function terms in a system power balance equation in a polynomial fitting mode, decoupling a voltage amplitude value and a phase angle by using the operating characteristics of a system, carrying out linearization processing on a phase angle quadratic term in the decoupled power balance equation in a Taylor series expansion mode, and using the result of the existing direct current optimal power flow model with higher precision and based on the network loss equivalent load as phase angle operating point information required by the method, so that the problem of dependency of a hot start type model on an operating environment is effectively solved.
The alternating current optimal power flow is a typical nonlinear programming problem, and the standard form of the alternating current optimal power flow comprises three parts of an objective function, an equality constraint and an inequality constraint. The invention selects the common generating cost as the target function of OPF:
Figure BDA0001507478620000051
in the formula: n isgRepresenting the number of generators; a is2i、a1iAnd a0iConsuming characteristic parameters for the ith generator; pGiThe active power output of the ith generator is obtained.
The equality constraints mainly include the power balance equations of the nodes:
Figure BDA0001507478620000052
in the formula: pDi、QDiRespectively an active load and a reactive load of a node i; u shapeiIs the voltage amplitude of node i; thetaij＝θi-θjIs the voltage phase angle difference of node i and node j; gij、BijRespectively a real part and an imaginary part of the ith row and jth column element of the admittance matrix; n isbThe number of nodes of the system.
Meanwhile, the inequality constraints mainly include:
Figure BDA0001507478620000053
in the formula: qGiThe reactive output of the ith generator is obtained; pLi、QLiThe active power flow and the reactive power flow of the ith branch; n isLThe number of branches of the system;*、
Figure BDA0001507478620000054
lower and upper limits for each variable, respectively.
As can be seen from the standard model of the alternating current optimal power flow, the nonlinear characteristics of the alternating current optimal power flow are mainly reflected in the node power balance constraint and the line power flow constraint, so that the linear processing of the line power flow is the key for improving the model solving efficiency. If i and j are serial numbers of nodes at two ends of the line L, the line flow can be written as follows:
Figure BDA0001507478620000061
in the formula: gij、bijRespectively the conductance and susceptance of the line.
Since a large number of trigonometric function terms exist in the formula (4), the coupling of the voltage amplitude and the phase angle is tight, and the model is not beneficial to the linearization processing, so that equivalent replacement is required. The phase angle difference between two ends of the system line is usually in
Figure BDA0001507478620000062
To
Figure BDA0001507478620000063
The invention utilizes MATLAB fitting tool box to fit the same, thereby obtaining the following equivalent relation:
Figure BDA0001507478620000064
to facilitate the subsequent expression, let C1＝0.97，C2＝0.49。
At the same time, due to UiUj1, the following approximation can therefore be obtained:
Figure BDA0001507478620000065
the substitution of formulae (5) to (6) into formula (4) can give:
Figure BDA0001507478620000066
the voltage amplitude and the phase angle in the formula (7) are completely decoupled, but secondary terms of all variables are still included, and the solving efficiency of the model is influenced to a certain extent by the existence of the secondary terms, so that the reduction or removal of the secondary terms in the model is the key for improving the solving efficiency of the model. Currently, the linear processing of the quadratic term is usually performed by means of taylor series expansion, but this needs to be performed at a certain ideal operating point, so that a good operating environment needs to be provided for this method. The DCOPF only expresses a system balance equation through a phase angle, has the advantage of high solving speed, and can provide a good phase angle operating point as long as the calculation precision is improved. Therefore, the invention carries out linearization processing on the quadratic term of the phase angle in the formula (7) so as to reduce the nonlinearity degree of the quadratic term. To this end, θ isijThe second order term of the first order term is expanded by Taylor series, and the truncation error is ignored, so that the following approximation can be obtained:
Figure BDA0001507478620000067
when formula (8) is substituted into formula (7), it is possible to obtain:
Figure BDA0001507478620000068
in the formula: thetaij,0Is composed of
Figure BDA0001507478620000071
And (5) performing Taylor series expansion operation points.
In this case, equation (2) can be converted into:
Figure BDA0001507478620000072
the calculation accuracy of the above model is requiredThe method utilizes the result of the existing DC optimal power flow model with higher precision and based on the network loss equivalent load as the phase angle operating point of the model of the invention to depend on a good phase angle operating environment, and the main idea of the model is to averagely distribute the line loss into the loss of the equivalent ground resistance at two ends of the line by considering the line loss factor on the basis of DCOPF, wherein the equivalent model is shown as figure 1. In the figure: r isequ,ijEquivalent ground resistance at two ends of the access branch; ploss,ijActive loss is generated for the branch.
Since in DCOPF, U is takeni≈Uj1, so when requ,ij＝2/Ploss,ijThe active power consumed by each equivalent ground resistor is
Figure BDA0001507478620000073
At this time, P is satisfiedij＝Pji+Ploss,ijAccording to the actual power flow condition of the line, for Ploss,ijDocument 1 (where rain, Weishinong, Sun Guo, etc. improved DC optimal power flow algorithm based on network loss equivalent load model [ J]Power system automation, 2016, 40 (6): 58-64), the present invention will not be described in detail. Therefore, the dc model based on the grid loss equivalent load can be expressed in the following form:
Figure BDA0001507478620000074
Figure BDA0001507478620000075
in the formula:
Figure BDA0001507478620000076
the network loss equivalent load of the node i is obtained;
Figure BDA0001507478620000077
are the nodal admittance matrix elements established as the inverse of the branch reactance.
In order to verify the advantages of the model of the invention compared with the existing technology, the calculation precision and the calculation efficiency are verified, and the model of file 2(Yang Z, Zhong H, Xia Q, et al. optimal power flow based on scientific linear adaptation of power flow equations [ J ]. IET Generation Transmission & Distribution, 2016 (14): 3654-3662.) is selected for comparison. The model described in the document 2 belongs to a hot-start linear model, in order to more intuitively explain the problem of dependence of the hot-start model on an operating environment, the present invention respectively selects the current tidal current value and the commonly used flat start value (namely, the voltage amplitude is set to 1pu, and the voltage phase angle is set to 0) of the system as the operating point of the model, and for convenience of later description, an alternating current optimal tidal current model is defined as AC, the model described in the present invention is M _1, the model 2 of the document in the tidal current operating environment is M _2, and the model 2 of the document in the flat start operating environment is M _ 3.
In the invention, a primary-dual interior point method (PDIPM) is adopted to solve each model, and algorithm programming is realized on an MATLAB 2014a platform. The example test is carried out on an IEEE 300 node system, a Polish 2383 node system, a Polish 2736 node system and a 8304 node large system. In order to ensure the uniformity of the test environment, the same sparse technology and convergence precision are adopted in the process of solving the OPF problem of each system, so that the inaccuracy of the test result caused by the difference in the algorithm is avoided.
The calculation results of each model are given in table 1, where the relative error refers to the relative error between the model and the AC model, and it can be seen from the results that M _2 maintains higher accuracy for most systems, but the calculation error of M _3 exceeds 1.5%, and the calculation error of Polish 2383 node system even reaches 4.9%, which indicates that the calculation accuracy of the hot start model described in document 2 is closely related to the quality of the operating point. For a 8304 node large system, no matter the current trend or the flat start value is taken as an operation point, a good enough operation environment cannot be provided for the file 2 model, so that the system OPF problem cannot be solved effectively, which further shows that the hot start model has strong dependence on the quality of the operation environment, and therefore, the practical application of the model has certain limitation.
TABLE 1 comparison of computational accuracies of different models
Figure BDA0001507478620000081
M _1 reserves a voltage amplitude quadratic term in the power balance equation, so that the relation between the system power balance equation and the voltage amplitude is well fitted, amplitude operation point information is not required to be provided, and the model has the advantages of being good in phase angle operation environment provided by a direct current optimal power flow model based on network loss equivalent load, and controlling the calculation error within five thousandths when solving the OPF problem of each system. For a large 8304 node system with high requirement on an operation point, the model can still effectively converge and keep high precision. Therefore, the model M _1 meets the requirement of practical engineering application on the calculation precision by combining the direct current optimal power flow model based on the network loss equivalent load, and has higher practical value.
Although the objective function may reflect the accuracy of the model to some extent, the accuracy of the system reactive scheduling information cannot be effectively characterized. And the reactive power dispatching information needs to be determined by the voltage amplitude and the phase angle at the same time, so the reactive power output of the generator can indirectly reflect the accuracy of the state variable of the system. Thus table 2 gives the error between the reactive power output solved by each model and the results obtained by the AC model. The result shows that the dependency of the hot start model in the document 2 on the operation environment is high, the deviation of the reactive scheduling information under different operation environments is large, and because the model of the invention reserves the amplitude quadratic term and the existing direct current optimal power flow model based on the network loss equivalent load can provide a good phase angle operation environment for the invention, the reactive scheduling information of the model of the invention is more accurate than that of the model in the document 2, and the result is basically consistent with that of the AC model. In conclusion, the model has higher calculation precision, the problem of high dependence of the model on the operating environment is effectively solved, the robustness of the hot start model is improved, and the application range of the model is expanded.
TABLE 2 comparison of different model scheduling information (reactive power output of generator)
Figure BDA0001507478620000091
TABLE 3 comparison of the required computation times for the different models
Figure BDA0001507478620000092
In addition to the calculation accuracy, the calculation efficiency is also one of the important indexes for evaluating the quality of the linearized model. Table 3 thus gives the computation time and the number of iterations required for each model in solving the different systems. It should be noted that, since M _1 needs to provide phase angle operating point information for it based on the dc optimal power flow model of the grid loss equivalent load, the calculation time of M _1 in table 3 includes the time required to obtain the phase angle operating point, and the time in parentheses is the time required to obtain the phase angle operating point. It can be seen that, thanks to the powerful computing power of MATLAB and the application of the sparse technology, the AC model can effectively converge within 7s and 50 iterations when solving the OPF problem of most systems. However, for a 8304 node large system tested by the invention, the AC model needs to iterate 796 times, and the convergence time is about 165s, which is far beyond the requirement of online application on the calculation efficiency, so the invention has important practical significance for the exploration of the linearized OPF model. Although the solving efficiency of the model of the invention is lower than that of the model of the document 2 when an IEEE 300 node system, a Polish 2383 node system and a Polish 2736 node system are processed, for a 8304 node large system, the calculation time of the model of the invention is shortened to about 12s, which is 93% shorter than that of an AC optimal power flow model, and the model of the document 2 cannot be effectively converged, so that the model of the invention has more practical engineering application value than that of the model of the invention.
In conclusion, the following conclusion can be obtained from the comparison between the model calculation accuracy and the solution efficiency, and the decoupled semi-linear optimal power flow model based on the hot start environment has higher calculation accuracy and solution efficiency, has stronger adaptability to a practical large system, better solves the problem of high dependence of the hot start model on the operation environment, and has higher practical application value.

Claims (2)

1. A decoupled semi-linear optimal power flow model based on a hot start environment is characterized in that: the method comprises the following steps:
(1) analyzing the nonlinear characteristics of the alternating current optimal power flow model;
(2) performing polynomial fitting on a trigonometric function item in a system power balance equation in the alternating current optimal power flow model, and decoupling a voltage amplitude and a voltage phase angle by using the running characteristic of the system;
(3) carrying out linearization processing on a voltage phase angle quadratic term in the decoupled power balance equation in a Taylor series expansion mode, so as to reduce the nonlinearity degree of the equation;
(4) the result obtained by the direct current optimal power flow model based on the network loss equivalent load is used as an operation point of the phase angle quadratic term Taylor series expansion, and the calculation precision of the model is improved;
(5) the accuracy and the efficiency of the model are verified through example tests;
analyzing the nonlinear characteristics of the alternating current optimal power flow model in the step (1), and firstly providing a standard form of the alternating current optimal power flow model:
Figure FDA0002801555440000011
Figure FDA0002801555440000012
in the formula: n isgIndicates the number of generators, a2i、a1iAnd a0iConsumption of a characteristic parameter, P, for the ith generatorGi、QGiActive and reactive power, P, of the ith generator, respectivelyGk、QGkRespectively generating power for kth station connected to node iActive and reactive power of the machine, PDi、QDiActive and reactive loads, U, respectively, of node iiIs the voltage amplitude of node i, θij＝θi-θjIs the voltage phase angle difference of node i and node j, Gij、BijReal and imaginary parts, G, of the ith row and jth column element, respectively, of the admittance matrixii、BiiThe real and imaginary parts, n, of the self-admittance of node i, respectivelybIs the number of nodes of the system, PLi、QLiFor active and reactive power flow of the ith branch, nLThe number of branches of the system is,*、
Figure FDA0002801555440000013
respectively, the lower limit and the upper limit of each variable;
in the step (2), polynomial fitting is carried out on a trigonometric function term in a system power balance equation, and a voltage amplitude and a voltage phase angle are decoupled by utilizing the running characteristic of the system, and the method specifically comprises the following steps:
3.1 the line tidal current expression can be deduced by the alternating current optimal tidal current model to be as follows:
Figure FDA0002801555440000021
in the formula: pij、QijActive power flow and reactive power flow, g, of line ij respectivelyij、bijRespectively, the conductance and susceptance of line ij;
3.2 the phase angle difference between two ends of the line is usually in the process of operating the power system
Figure FDA0002801555440000022
To
Figure FDA0002801555440000023
According to the characteristic, fitting a trigonometric function item in a system power balance equation by utilizing an MATLAB fitting tool box to obtain the following expression:
Figure FDA0002801555440000024
3.3 since the node voltage is always maintained at about 1pu during the operation of the power system, there is a UiUj1, the following approximation can be obtained:
Figure FDA0002801555440000025
3.4 substituting the expression of 3.2-3.3 into the line power flow equation of 3.1 to obtain a line power flow expression with voltage amplitude and phase angle decoupling, wherein the expression is as follows:
Figure FDA0002801555440000026
wherein, C1 is 0.97, C2 is 0.49;
step (3) carrying out linearization processing on a voltage phase angle quadratic term in the decoupled power balance equation in a Taylor series expansion mode to reduce the nonlinearity degree of the equation, and the method specifically comprises the following steps:
expressing theta in line power flow expressionijThe second order term of the first order term is expanded by Taylor series, and the truncation error is ignored, so that the following approximation can be obtained:
Figure FDA0002801555440000027
substituting the above formula into the decoupled line power flow equation to obtain:
Figure FDA0002801555440000031
in the formula: thetaij,0Is composed of
Figure FDA0002801555440000032
A reference point for performing taylor series expansion, namely an operation point;
at this point, the power balance equation can be written as:
Figure FDA0002801555440000033
2. the decoupled, semi-linearized optimal power flow model based on a warm start environment of claim 1, characterized by: and using the result obtained by the direct current optimal power flow model based on the network loss equivalent load as an operation point of the phase angle quadratic term Taylor series expansion.
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