CN112564084A - Method for rapidly determining voltage stability of power distribution network accessed by large-scale distributed photovoltaic - Google Patents

Method for rapidly determining voltage stability of power distribution network accessed by large-scale distributed photovoltaic Download PDF

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CN112564084A
CN112564084A CN202011136969.3A CN202011136969A CN112564084A CN 112564084 A CN112564084 A CN 112564084A CN 202011136969 A CN202011136969 A CN 202011136969A CN 112564084 A CN112564084 A CN 112564084A
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reactive
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CN112564084B (en
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蔡永翔
付宇
肖小兵
杨安黔
文贤馗
张锐锋
赵霜霜
刘安茳
何洪流
郑友卓
吴鹏
李前敏
张洋
王卓月
郝树青
李跃
张恒荣
苗宇
柏毅辉
李忠
安波
黄伟
郭素
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Guizhou Power Grid Co Ltd
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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
    • 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
    • H02J3/12—Circuit arrangements for ac mains or ac distribution networks for adjusting voltage in ac networks by changing a characteristic of the network load
    • H02J3/16—Circuit arrangements for ac mains or ac distribution networks for adjusting voltage in ac networks by changing a characteristic of the network load by adjustment of reactive power
    • 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
    • H02J3/38—Arrangements for parallely feeding a single network by two or more generators, converters or transformers
    • H02J3/381—Dispersed generators
    • 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
    • H02J2203/00—Indexing scheme relating to details of circuit arrangements for AC mains or AC distribution networks
    • H02J2203/10—Power transmission or distribution systems management focussing at grid-level, e.g. load flow analysis, node profile computation, meshed network optimisation, active network management or spinning reserve management
    • 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
    • H02J2203/00—Indexing scheme relating to details of circuit arrangements for AC mains or AC distribution networks
    • H02J2203/20—Simulating, e g planning, reliability check, modelling or computer assisted design [CAD]
    • 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
    • H02J2300/00—Systems for supplying or distributing electric power characterised by decentralized, dispersed, or local generation
    • H02J2300/20—The dispersed energy generation being of renewable origin
    • H02J2300/22—The renewable source being solar energy
    • H02J2300/24—The renewable source being solar energy of photovoltaic origin
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
    • Y02E10/00—Energy generation through renewable energy sources
    • Y02E10/50—Photovoltaic [PV] energy
    • Y02E10/56—Power conversion systems, e.g. maximum power point trackers
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
    • Y02E40/00—Technologies for an efficient electrical power generation, transmission or distribution
    • Y02E40/30—Reactive power compensation

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Abstract

The invention discloses a method for quickly determining voltage stability of a large-scale distributed photovoltaic accessed power distribution network, which is characterized in that a network power flow equation is utilized to express the relevance between voltage change and node reactive power change, meanwhile, a piecewise linear function equation is utilized to express the relevance between node reactive power regulating quantity and voltage of a measuring point, and two types of equations are integrated on the basis of a function transformation rule to describe the relation between a network state and a voltage measuring result of any node; in order to describe a state equation of network voltage change, the voltage stability problem of the power distribution network is further expressed as a process of describing two adjacent iterative reactive outputs, and the voltage stability problem of the power distribution network is converted into a reactive change process of an inverter and subjected to differential solution; the method solves the technical problems that in the prior art, when large-scale cluster distributed photovoltaic grid connection is faced, all nodes or most nodes in a network are connected to a photovoltaic power supply, the calculation amount and the calculation difficulty of the traditional voltage stability calculation method are increased sharply, the parameter setting is very lack of flexibility, and the like.

Description

Method for rapidly determining voltage stability of power distribution network accessed by large-scale distributed photovoltaic
Technical Field
The invention belongs to the technical field of voltage stability of a power distribution network, and particularly relates to a method for quickly determining the voltage stability of the power distribution network of large-scale distributed photovoltaic access.
Background
At present, fossil energy still occupies a pillar position in the global energy structure, and power generation still mainly depends on fossil fuels such as coal and natural gas. However, environmental problems are increasingly highlighted due to the emission of a large amount of carbon dioxide and toxic and harmful gases, global temperature rises due to the emission of greenhouse gases, glaciers at the two poles melt, and extreme meteorological problems frequently occur; toxic and harmful substances generated after fossil energy combustion enter air, gases such as nitrogen oxides and sulfur dioxide cause ecological problems such as photochemical smog, acid rain, ozone layer damage and the like, and serious danger is formed on human bodies and natural ecological systems.
At present, the proportion of the grid-connected capacity of the distribution network in distributed photovoltaic power generation is rapidly improved, and the trend of large-scale cluster grid connection is presented in the future. In the early stage, the photovoltaic power generation is mainly centralized grid connection, and a large amount of photovoltaic power is collected and boosted and then transmitted through a power transmission line to supply power for a remote load. However, the light abandonment ratio is still high due to the limit of the capacity of a power transmission corridor, and according to the national energy agency 'statistics information on the national photovoltaic power generation in 2018', the light abandonment amount of the national photovoltaic power generation is 54.9 hundred million kilowatts hours, and the average utilization hours is only 1115 hours. Compared with centralized photovoltaic power generation, plug and play can be realized in distributed photovoltaic power generation, photovoltaic electric energy can be utilized on the spot, and the problems of capacity occupation and line loss caused by large-capacity power transmission are avoided. By the end of 2018, 12384 ten thousand kilowatts of a centralized power station are increased, 2330 thousand kilowatts are increased in the last year, and the year is increased by 23 percent; distributed photovoltaic 5061 ten thousand kilowatts, 2096 ten thousand kilowatts are newly added in the last year, the year-on-year increase is 71%, and the large-scale grid-connected power generation of the distributed photovoltaic in medium and low voltage distribution networks becomes a necessary trend.
The grid connection of a large-scale distributed power supply can cause the voltage out-of-limit and voltage fluctuation risks of a power distribution network, and the distributed photovoltaic unit which is newly connected to the grid at present generally needs to have certain voltage-reactive power regulation capacity so as to achieve the aims of inhibiting the voltage fluctuation and voltage out-of-limit and reducing network loss. However, the distributed photovoltaic grid connection is difficult to be uniformly planned by a Distribution System Operator (DSO), and therefore, the grid connection position and capacity have certain randomness. The distributed photovoltaic grid connection related standards (such as GB/T50865 design specifications for accessing the 2013 photovoltaic power generation to the power distribution network, DL/T1773 guide rules for the voltage and reactive power technology of the 2017 power system, NB/T32015 guide rules for accessing the 2013 distributed power generation to the power distribution network, and the like) only require that the grid connection power factor of the distributed power supply meets the related requirements, and do not restrict the shape and slope of a voltage-reactive droop control curve, which causes great risk to the voltage stability of the power distribution network. According to a power flow equation (Newton Raphson method or forward-backward substitution method) of the power distribution network, the reactive change of a single node can cause the voltage linkage change of all nodes, the voltage change can cause the reactive adjustment of the nodes, the change of the network node voltage presents an iterative fluctuation process, and if the process is iteratively converged, the network voltage presents a stable characteristic; if the iteration of the process does not converge, the network voltage has instability problem, even the malfunction of the protection device is caused.
Therefore, the method has very important significance for maintaining safe and stable operation of the power distribution network in the aspects of grid connection of a large number of distributed photovoltaic power supplies, rapid judgment of network voltage stability and measurement and calculation of stable boundaries. According to the traditional method for evaluating the voltage stability of the power distribution network, when the grid-connected scale of a distributed power supply of a network is low, the power distribution network can be equivalent to a voltage source or a synchronous machine, the influence of reactive power regulation on the system after single photovoltaic grid connection is researched, and the voltage-reactive droop slope is adjusted through a disturbance-observation method; if a small amount of photovoltaic grid connection can be achieved through CPU parallel core division calculation, network voltage stability can be evaluated, and voltage-reactive parameters can be determined through multi-core combined debugging. However, in the face of grid connection of large-scale cluster distributed photovoltaic (all or most of nodes in a network are connected to a photovoltaic power supply), the calculation amount and the calculation difficulty of the traditional voltage stability calculation method will rise sharply, and the parameter setting is very lack of flexibility.
Disclosure of Invention
The technical problems to be solved by the invention are as follows: the method for rapidly determining the voltage stability of the power distribution network accessed by the large-scale distributed photovoltaic is provided, and the technical problems that in the prior art, when the grid connection of the large-scale cluster distributed photovoltaic is faced, all nodes or most nodes in the network are accessed to a photovoltaic power supply, the calculation amount and the calculation difficulty of the traditional voltage stability calculation method are increased sharply, the parameter setting is very lack of flexibility and the like are solved.
The technical scheme of the invention is as follows:
a method for quickly determining voltage stability of a power distribution network accessed by large-scale distributed photovoltaic comprises the steps of expressing the correlation between voltage change and node reactive power change by using a network tide equation, expressing the correlation between node reactive power regulating quantity and voltage of a measuring point by using a piecewise linear function equation, and integrating two types of equations based on a function transformation rule to describe the relation between a network state and a voltage measuring result of any node; in order to describe a state equation of network voltage change, the voltage stability problem of the power distribution network is further expressed as a process of describing two adjacent iterations of reactive power output, and the voltage stability problem of the power distribution network is converted into a reactive power change process of an inverter and subjected to differential solution.
The method for converting the voltage stability problem of the power distribution network into the reactive change process of the inverter and performing differential solution comprises the following steps: simplifying the voltage stability problem of the power distribution network into linear operation of a voltage-reactive sensitivity matrix and an inverter voltage-reactive droop control parameter matrix, and judging the voltage stability of the power distribution network based on a spectrum radius upper limit theory; calculating a voltage-reactive inverter parameter adjustable range set of different photovoltaic grid connection points based on the provided power distribution network voltage stability analysis solving algorithm; if all photovoltaic grid-connected unit voltage-reactive inverter parameters in the network are set within an adjustable range, even if the position capacity of grid-connected photovoltaic is always in a dynamic change process, the voltage stability of the power distribution network can be ensured.
The method for rapidly determining the voltage stability of the power distribution network accessed by the large-scale distributed photovoltaic system specifically comprises the following steps:
step 1, establishing a voltage sensitivity solving model based on a Newton-Raynaud algorithm;
step 2, establishing a voltage-reactive power control network description model based on the voltage sensitivity matrix;
step 3, establishing a power distribution network voltage stability judgment model based on a voltage sensitivity matrix and a spectrum radius upper limit theory;
and 4, establishing a network voltage stability critical parameter set based on the power distribution network voltage stability judgment model.
The method for establishing the voltage sensitivity solving model based on the Newton-Raynaud flow algorithm comprises the following steps:
setting the initial solution of the nonlinear equation f (x) to 0 as x0，x0The difference between the real solution x and the actual solution x is delta x(0)The essence of the solution is to iterate so that Δ x(k)→ 0, where k represents the number of iterations, the true solution to the equation is finally found;
assuming that N nodes are total, the network node set is N ═ 1,2 … N, and according to kirchhoff's law, the relationship between the voltage and the current of the nodes is as follows:
Figure RE-GDA0002941673290000041
in the formula: y is a nodal admittance matrix, Z is a nodal impedance matrix,
Figure RE-GDA0002941673290000042
is the node voltageThe vector of the vector is then calculated,
Figure RE-GDA0002941673290000043
is the injected current vector of the node; after the formula (1) is developed, the following components are provided:
Figure RE-GDA0002941673290000044
respectively recording the injected active power and the injected reactive power of the node i as PiAnd QiAnd the voltage conjugate value of the node i is denoted as Vi *(ii) a The injection current at node i can be further represented as:
Figure RE-GDA0002941673290000045
substituting equation (3) into equation (2) is:
Figure RE-GDA0002941673290000046
voltage of node i
Figure RE-GDA0002941673290000047
In polar coordinate form of
Figure RE-GDA0002941673290000048
Wherein ViIs the voltage amplitude of node i, δiIs the voltage phase angle of node i; in addition Yij＝Gij+jBijThe polar form of the power equation is:
Figure RE-GDA0002941673290000049
assuming that a system has n nodes, wherein one node is a balanced node, the n node is a balanced node, m PQ nodes are provided, 1-m are PQ nodes, n-m-1 PV nodes, and m + 1-n-1 are PV nodes, the power balance equation is as follows:
Figure RE-GDA00029416732900000410
the corresponding correction equation is as follows:
Figure RE-GDA0002941673290000051
in the formula: the off-diagonal elements i ≠ j of the submatrix as in equation (8), and the diagonal elements i ≠ j as in equation (9)
Figure RE-GDA0002941673290000052
Figure RE-GDA0002941673290000053
The iterative form of equation (7) is shown in equation (10), and the convergence criterion is
Figure RE-GDA0002941673290000057
Figure RE-GDA0002941673290000054
In the formula: Δ V'(k)Is shown in (11)
Figure RE-GDA0002941673290000055
Equation (8) correction vector after each iteration
Figure RE-GDA0002941673290000058
And Vi (k+1)Comprises the following steps:
Figure RE-GDA0002941673290000056
if it is assumed that there is no PV node in the network, only a PQ node and one balance node, the size of the jacobian matrix in equation (7) becomes 2 × (n-1), and the iterative equation (10) is modified as:
Figure RE-GDA0002941673290000061
the result after the iterative convergence is recorded as
Figure RE-GDA0002941673290000062
Gaussian elimination of formula (13), assuming H1、N1、M1And L1All exist inverse matrix
Figure RE-GDA0002941673290000063
The relation between the node voltage change delta V and the node active change delta P and the node reactive change is delta Q
ΔV＝SV-PΔP+SV-QΔQ (16)
In the formula: sV-P＝(M1H1 -1N1-L1)-1M1H1 -1In the form of a voltage-active sensitivity matrix,
SV-Q＝(L1-M1H1 -1N1)-1is a voltage-reactive sensitivity matrix;
after the formula (16) is unfolded
Figure RE-GDA0002941673290000064
For any node i (i ∈ N), its voltage change Δ ViThe relation with the active and reactive changes of the other nodes is
Figure RE-GDA0002941673290000065
In the formula:
Figure RE-GDA0002941673290000067
and
Figure RE-GDA0002941673290000068
the voltage-active and voltage-reactive sensitivities are respectively, i, j belongs to N, namely the voltage change of the node i caused by the active and reactive changes of the node j; delta PiAnd Δ QiThe active and reactive changes of node j, respectively.
The method for establishing the voltage-reactive control network description model based on the voltage sensitivity matrix comprises the following steps: expressing active and reactive power of nodes as correlation equation with network voltage and phase angle
Figure RE-GDA0002941673290000066
In the formula: f. ofPF_PThe method is an associated equation of node active power, network voltage and phase angle; f. ofPF_QIs an associated equation of node reactive power, network voltage and phase angle; pload、QloadRespectively are the active and reactive of the node load; pPV、QPVRespectively are the active and reactive power of the node photovoltaic; v, δ are the voltage and phase angle of the node, respectively.
According to the voltage reactive power droop control curve, the photovoltaic inverter absorbs reactive power when the voltage is higher, injects reactive power when the voltage is lower, and keeps the reactive power to be zero when the voltage is close to a target value; qPVmax,iAnd QPVmin,iRespectively representing the maximum injection power and the maximum absorption power of the photovoltaic inverter when the node i is connected to the grid; vi,maxAnd Vi,minRespectively corresponding to the voltage values when absorbing the maximum reactive power and injecting the maximum reactive power; vref,iIs the target voltage; the distance between the target voltage and the initial reactive injection voltage is d/2, and the distance between the target voltage and the initial reactive absorption voltage is d/2; m isiPhotovoltaic inverter when node i is connected to gridM of the droop control parameteri≥0。
The equation of the voltage reactive output of the photovoltaic grid-connected node and the node voltage can be expressed as
Figure RE-GDA0002941673290000071
Figure RE-GDA0002941673290000072
The equation for reactive power regulation may be further expressed as
QPV,i,t+1＝f(Vi,t)＝f(u(QPV,i,t))＝ψ(QPV,i,t) (22)
In the formula: vi,tIs the voltage amplitude at time t of node i; u (-) is an equation of the influence of the reactive node reactive power change on the node voltage change; f (u (-) is represented by ψ (-) and represents the relationship between the reactive output at time t +1 and the reactive output at time t.
Step 3, establishing the voltage stability of the power distribution network based on the voltage sensitivity matrix and the spectrum radius upper limit theory
The method for judging the model comprises the following steps:
to describe the correlation of the reactive power changes at two adjacent times, the partial derivative may be determined, i.e.
Figure RE-GDA0002941673290000073
Figure RE-GDA0002941673290000074
In the above formula
Figure RE-GDA0002941673290000081
Is the solved voltage-reactive sensitivity matrix parameter; it can be seen that
Figure RE-GDA0002941673290000082
Namely electricity
The voltage-reactive power control slope parameter is expressed by a diagonal matrix as follows:
Figure RE-GDA0002941673290000083
wherein M is a slope matrix of the cluster distributed photovoltaic power supply;
the stability determination condition is
ρ(MSV-Q)＜1 (25)
Based on equation (25), once the distribution network voltage-reactive sensitivity matrix and the photovoltaic slope matrix are constructed, the stability for each node can be expressed in the form of analytic expression
Figure RE-GDA0002941673290000084
In the formula: n is a radical of-Is the set of network nodes except the head node, i.e., nodes {2,3, … n };
according to the formula (26), it can be seen that the problem of voltage stability of the power distribution network including the cluster distributed photovoltaic units is mainly influenced by the positions of the photovoltaic grid-connected nodes and the voltage-reactive droop slope parameter, and the positions of the photovoltaic grid-connected points mainly influence the voltage stability of the power distribution network in the formula (26)
Figure RE-GDA0002941673290000085
The parameters of (2) are related to the line resistance and reactance parameters of the network, the line length and the topological structure of the line, and are unrelated to the reactive adjustable capacity of the photovoltaic inverter; voltage-reactive droop slope, i.e. miParameter, the larger the value is, the larger is ρ (MS)V -Q) The more easily the voltage is close to 1 or exceeds 1, the situation that the voltage fluctuation is not converged in the voltage regulation process can be caused; the condition that the voltage of the distribution network is stable is that any node in the network should satisfy the judgment condition of the formula (26).
The method for determining the stability determination condition comprises the following steps: the voltage stability determination condition is increased along with the increase of the iteration number, and the voltage change degree is gradually increasedDecrease and eventually tend to stabilize; the process is equivalent to the fact that the degree of reactive power change is gradually reduced and finally tends to be stable along with the rise of the iteration times of the photovoltaic reactive power output; setting rho (·) to represent the spectrum radius of any matrix, namely the maximum value of the absolute value of the matrix eigenvalue; for the equation described in equation (24), ρ (MS) according to the theorem on the spectral radiusV-Q) < 1 represents the reactive absolute value output of the most unstable node t +1 in all photovoltaic power supplies in the network at the time t and the time t
Figure RE-GDA0002941673290000086
Smaller, the photovoltaic reactive change tends to zero after infinite iterations, and a stability judgment condition rho (MS) is obtainedV-Q)＜1。
The method for establishing the network voltage stability critical parameter set based on the power distribution network voltage stability judgment model comprises the following steps:
from the result of the derivation of equation (25), if ρ (MS)V-Q) → 1, i.e. in formula (23)
Figure RE-GDA0002941673290000093
Representing that the degree of reactive power change is not gradually reduced along with the continuous rise of the control iteration times, the iteration times of the voltage-reactive power control system are close to infinite times, the stability condition of the control system cannot be met, and rho (MS) is metV-Q) The set of slopes of the drooping curve at time → 1 is noted
Figure RE-GDA0002941673290000094
Wherein
Figure RE-GDA0002941673290000095
Is the upper bound of the droop slope of each node, i.e., miThe range of variation is
Figure RE-GDA0002941673290000096
Figure RE-GDA0002941673290000097
And M is:
Figure RE-GDA0002941673290000091
Figure RE-GDA0002941673290000092
the invention has the beneficial effects that:
according to the invention, based on the voltage sensitivity matrix and the spectrum radius upper limit theory, the algorithm for analyzing and solving the voltage stability of the power distribution network and the network voltage stability boundary determination technology greatly reduce the dependence of the traditional method for solving the voltage stability of the power distribution network on the computing capability of a CPU (central processing unit), and can calculate the voltage stability condition of a large-scale distributed photovoltaic grid-connected network (thousands of nodes) after being connected to the power distribution network at an excessive speed; in addition, the voltage stability boundary of the power distribution network is provided, the voltage-reactive slope of each node is conveniently restrained, all photovoltaic grid-connected nodes in the network only need to meet the requirement of the voltage-reactive slope change range in the setting of the control strategy, even if the position capacity of the grid-connected photovoltaic is always in the dynamic change process, the voltage stability of the power distribution network can be ensured, and a user can have better flexibility in the establishment of the control strategy.
The method solves the technical problems that in the prior art, when large-scale cluster distributed photovoltaic grid connection is faced, all nodes or most nodes in a network are connected to a photovoltaic power supply, the calculation amount and the calculation difficulty of the traditional voltage stability calculation method are increased sharply, the parameter setting is very lack of flexibility, and the like.
Description of the drawings:
fig. 1 is a schematic diagram of a voltage-reactive droop control curve (type 1) in an embodiment;
fig. 2 is a schematic diagram of a voltage-reactive droop control curve (type 2) in an embodiment.
Detailed Description
The invention provides a network-level mathematical model considering a voltage-reactive droop curve and a power flow equation simultaneously; on one hand, the method utilizes a network power flow equation to express the relevance between voltage change and node reactive power change, simultaneously utilizes a piecewise linear function to express the relevance between node reactive power regulating quantity and voltage of a measuring point, and integrates two types of equations based on a function transformation rule to describe the relation between a network state and a voltage measuring result of any node; on the other hand, in order to describe the state equation of the network voltage change, the distribution network voltage stability problem is further expressed as a process of describing the adjacent two-time iteration reactive power output, and the distribution network voltage stability problem is converted into a differential solution of the reactive power change process of the inverter.
A power distribution network voltage stability analysis solving model and a voltage-reactive critical parameter calculating method based on voltage sensitivity and spectral density upper limit theory are disclosed. And transforming a differential equation of the reactive power change process of the inverter, simplifying the voltage stability problem of the power distribution network into linear operation of a voltage-reactive sensitivity matrix and an inverter voltage-reactive droop control parameter matrix, and judging the voltage stability of the power distribution network based on a spectrum radius upper limit theory. Based on the provided analysis and solving algorithm for the voltage stability of the power distribution network, a method for calculating the parameter adjustable range set of the voltage-reactive inverter for different photovoltaic grid-connected points is further provided. If all photovoltaic grid-connected unit voltage-reactive inverter parameters in the network are set within an adjustable range, even if the position capacity of grid-connected photovoltaic is always in a dynamic change process, the voltage stability of the power distribution network can be ensured.
The invention specifically comprises the following contents:
(1) a voltage sensitivity solving model based on a bull-pull power flow algorithm;
(2) a voltage-reactive control network description model based on a voltage sensitivity matrix;
(3) a power distribution network voltage stability judgment model based on a voltage sensitivity matrix and a spectrum radius upper limit theory;
(4) and (4) determining a network voltage stability critical parameter set of the model based on the voltage stability of the power distribution network.
Voltage sensitivity solving model based on cattle-pull tide algorithm
The Newton-Raphson algorithm is a common nonlinear equation solving method and can be used for solving the power flow of a power grid. Solving thought of Newton-Raphson algorithmThe way is: setting the initial solution of the nonlinear equation f (x) to 0 as x0，x0The difference between the real solution x and the actual solution x is delta x(0)The essence of the solution is to iterate so that Δ x(k)→ 0, where k represents the number of iterations, a true solution to the equation is finally found, so the core problem of the newton-raphson algorithm is the correction and solution of the equation.
The total number of the nodes is N, and the network node set is N ═ 1,2 … N }. According to kirchhoff's law, the voltage and current of a node are related as follows:
Figure RE-GDA0002941673290000116
in the formula: y is a nodal admittance matrix, Z is a nodal impedance matrix,
Figure RE-GDA0002941673290000117
is a vector of the voltage at the node,
Figure RE-GDA0002941673290000118
is the injected current vector at the node. After the formula (1) is developed, the following components are provided:
Figure RE-GDA0002941673290000111
respectively recording the injected active power and the injected reactive power of the node i as PiAnd QiAnd the voltage conjugate value of the node i is denoted as Vi *. The injection current at node i can be further represented as:
Figure RE-GDA0002941673290000112
substituting equation (3) into equation (2) is:
Figure RE-GDA0002941673290000113
voltage of node i
Figure RE-GDA0002941673290000119
In polar coordinate form of
Figure RE-GDA00029416732900001110
Wherein ViIs the voltage amplitude of node i, δiIs the voltage phase angle of node i. In addition Yij＝Gij+jBijIn polar form, the power equation is
Figure RE-GDA0002941673290000114
Now, assume that there are n nodes in the system, wherein there are one balancing node (n node is a balancing node), m PQ nodes (1-m are PQ nodes), and n-m-1 PV nodes (m + 1-n-1 are PV nodes). The power balance equation is:
Figure RE-GDA0002941673290000115
the corresponding correction equation is as follows:
Figure RE-GDA0002941673290000121
in the formula: the off-diagonal elements (i ≠ j) of the submatrix are as in equation (8), and the diagonal elements (i ≠ j) are as in equation (9)
Figure RE-GDA0002941673290000122
Figure RE-GDA0002941673290000123
The iterative form of equation (7) is shown in equation (10), and the convergence criterion is
Figure RE-GDA0002941673290000127
Figure RE-GDA0002941673290000124
In the formula: Δ V'(k)Is shown in (11)
Figure RE-GDA0002941673290000125
Equation (8) correction vector after each iteration
Figure RE-GDA0002941673290000128
And Vi (k+1)Comprises the following steps:
Figure RE-GDA0002941673290000126
if it is assumed that there is no PV node in the network, and there are only a PQ node and one balance node, the size of the jacobian matrix in equation (7) becomes 2 × (n-1). The iterative equation (10) is modified to:
Figure RE-GDA0002941673290000131
the result after the iterative convergence is recorded as
Figure RE-GDA0002941673290000132
Gaussian elimination of formula (13), assuming H1、N1、M1And L1All exist inverse matrix
Figure RE-GDA0002941673290000133
The relation between the node voltage change delta V and the node active change delta P and the node reactive change is delta Q
ΔV＝SV-PΔP+SV-QΔQ (16)
In the formula: sV-P＝(M1H1 -1N1-L1)-1M1H1 -1In the form of a voltage-active sensitivity matrix,
SV-Q＝(L1-M1H1 -1N1)-1is a voltage-reactive sensitivity matrix.
After the formula (16) is unfolded
Figure RE-GDA0002941673290000134
For any node i (i ∈ N), its voltage change Δ ViThe relation with the active and reactive changes of the other nodes is
Figure RE-GDA0002941673290000135
In the formula:
Figure RE-GDA0002941673290000136
and
Figure RE-GDA0002941673290000137
voltage-active and voltage-reactive sensitivities (i, j ∈ N), respectively, i.e. the voltage change at node i caused by the active (reactive) change at node j; delta PiAnd Δ QiThe active and reactive changes of node j, respectively.
Network description model based on voltage-reactive power control system
Expressing active and reactive power of nodes as correlation equation with network voltage and phase angle
Figure RE-GDA0002941673290000141
In the formula: f. ofPF_PIs a node active andcorrelation equations of network voltage and phase angle; f. ofPF_QIs an associated equation of node reactive power, network voltage and phase angle; pload、QloadRespectively are the active and reactive of the node load; pPV、QPVRespectively are the active and reactive power of the node photovoltaic; v, δ are the voltage and phase angle of the node, respectively.
Fig. 1 is a relatively common voltage reactive droop control curve, in which a photovoltaic inverter absorbs reactive power when the voltage is higher, injects reactive power when the voltage is lower, and keeps the reactive power at zero when the voltage approaches a target value. QPVmax,iAnd QPVmin,iRespectively representing the maximum injection power and the maximum absorption power of the photovoltaic inverter when the node i is connected to the grid; vi,maxAnd Vi,minRespectively corresponding to the voltage values when absorbing the maximum reactive power and injecting the maximum reactive power; vref,iIs the target voltage; the distance between the target voltage and the initial reactive injection voltage is d/2, and the distance between the target voltage and the initial reactive absorption voltage is d/2; m isiIs the droop control parameter (m) of the photovoltaic inverter when the node i is connected to the gridiNot less than 0). Fig. 2 is another more common voltage reactive droop control curve, i.e. a voltage reactive droop control curve with d/2 being zero.
The equation of the voltage reactive output and the node voltage of the photovoltaic grid-connected node can be expressed as
Figure RE-GDA0002941673290000142
Figure RE-GDA0002941673290000143
The equation for reactive power regulation may be further expressed as
QPV,i,t+1＝f(Vi,t)＝f(u(QPV,i,t))＝ψ(QPV,i,t) (22)
In the formula: vi,tIs the voltage amplitude at time t of node i; u (-) is an equation of the influence of the reactive node reactive power change on the node voltage change; f (u (-) is denoted by ψ (-) and represents the reactive output at time t +1 and time tAnd (5) the relation of reactive power output is carved.
Power distribution network voltage stability determination model based on voltage sensitivity matrix and spectrum radius upper limit theory
To describe the correlation of the reactive power changes at two adjacent times, the partial derivative may be determined, i.e.
Figure RE-GDA0002941673290000145
Figure RE-GDA0002941673290000144
From the above formula
Figure RE-GDA0002941673290000153
Is the solved voltage-reactive sensitivity matrix parameter; it can be seen that
Figure RE-GDA0002941673290000154
I.e. the voltage-reactive control slope parameter, can be expressed by a diagonal matrix (generally, the first node is defined as a substation exit node, and is regarded as a balance node in load flow calculation, and the node has no voltage stability problem, so the description starts from node No. 2):
Figure RE-GDA0002941673290000151
in the formula: m is a slope matrix of the clustered distributed photovoltaic power supply.
The voltage stability judgment condition is that the voltage change degree gradually decreases and finally tends to be stable along with the rise of the iteration times; the process is equivalent to the fact that the degree of reactive change gradually decreases and finally tends to be stable along with the rise of the iteration times of the photovoltaic reactive output. Let ρ (·) denote the spectral radius of an arbitrary matrix, i.e., the maximum of the absolute values of the matrix eigenvalues. For the equation described in equation (24), ρ (MS) according to the theorem on the spectral radiusV-Q) < 1 represents the reactive absolute value output of the most unstable node t +1 in all photovoltaic power supplies in the network at the time t and the time t
Figure RE-GDA0002941673290000155
And the photovoltaic reactive change tends to zero after infinite iterations. The stability determination condition is
ρ(MSV-Q)＜1 (25)
Based on equation (25), once the distribution network voltage-reactive sensitivity matrix and the photovoltaic slope matrix are constructed, the stability for each node can be expressed in the form of analytic expression
Figure RE-GDA0002941673290000152
In the formula: n is a radical of-Is the set of network nodes with the head node removed, i.e., nodes 2,3, … n.
According to the formula (26), the problem of voltage stability of the power distribution network containing the cluster distributed photovoltaic units is mainly influenced by the positions of the photovoltaic grid-connected nodes and the voltage-reactive droop slope parameters. The position of the photovoltaic grid-connection point mainly influences the position in the formula (26)
Figure RE-GDA0002941673290000156
The parameters of (2) are related to the line resistance and reactance parameters of the network, the line length and the topological structure of the line, and are unrelated to the reactive adjustable capacity of the photovoltaic inverter; voltage-reactive droop slope, i.e. miParameter, the larger the value is, the larger is ρ (MS)V -Q) The easier it is to approach 1 or exceed 1, the situation that the voltage fluctuation is not converged in the voltage regulation process will be caused. Further, it is to be noted that the condition for the distribution network voltage to be stable is that any one node in the network should satisfy the determination condition of the equation (26).
Network voltage stability critical parameter set based on power distribution network voltage stability judgment model
According to the method for rapidly judging the voltage stability of the power distribution network, the condition of the voltage stability of the network can be rapidly analyzed through an analytical expression. However, as the photovoltaic grid is continuously connected, the voltage stability of the power distribution network is in a changing state, and the analysis and judgment result usually lags behind the setting of the photovoltaic inverter voltage-reactive control law. If a set of voltage-reactive upper limit values is given in advance, all photovoltaic grid-connected nodes in the network can ensure the voltage stability of the power distribution network even if the position capacity of grid-connected photovoltaic is always in a dynamic change process as long as the requirements of the voltage-reactive upper limit values are met in the setting of the control strategy, and users can have better flexibility in the establishment of the control strategy.
From the result of the derivation of equation (25), if ρ (MS)V-Q) → 1, i.e. in formula (23)
Figure RE-GDA0002941673290000163
The reactive change degree cannot be gradually reduced along with the continuous rise of the control iteration number, and the voltage-reactive control system iteration number approaches to infinite number, so that the stability condition of the control system cannot be met. Will satisfy ρ (MS)V-Q) The set of slopes of the drooping curve at time → 1 is noted
Figure RE-GDA0002941673290000164
Wherein
Figure RE-GDA0002941673290000165
Is the upper bound of the droop slope of each node, i.e., miThe range of variation is
Figure RE-GDA0002941673290000166
Figure RE-GDA0002941673290000167
And M is:
Figure RE-GDA0002941673290000161
Figure RE-GDA0002941673290000162
according to the invention, the voltage stability analysis and solving algorithm of the power distribution network and the network voltage stability boundary determination technology based on the voltage sensitivity matrix and the spectrum radius upper limit theory are adopted, the algorithm greatly reduces the dependence on the CPU computing capacity, the voltage stability condition of the large-scale distributed photovoltaic grid connection (thousands of nodes) is rapidly calculated, the voltage stability boundary of the network is given, and the voltage-reactive slope of each node is conveniently restrained.

Claims (8)

1. A method for quickly determining the voltage stability of a power distribution network accessed by large-scale distributed photovoltaic is characterized by comprising the following steps: the method comprises the steps of expressing the relevance between voltage change and node reactive power change by using a network power flow equation, expressing the relevance between node reactive power regulating quantity and voltage of a measuring point by using a piecewise linear function equation, and integrating two types of equations based on a function transformation rule to describe the relation between a network state and a voltage measuring result of any node; in order to describe a state equation of network voltage change, the voltage stability problem of the power distribution network is further expressed as a process of describing two adjacent iterations of reactive power output, and the voltage stability problem of the power distribution network is converted into a reactive power change process of an inverter and subjected to differential solution.
2. The method for rapidly determining the voltage stability of the power distribution network accessed by the large-scale distributed photovoltaic system according to claim 1, wherein the method comprises the following steps: the method for converting the voltage stability problem of the power distribution network into the reactive change process of the inverter and performing differential solution comprises the following steps: simplifying the voltage stability problem of the power distribution network into linear operation of a voltage-reactive sensitivity matrix and an inverter voltage-reactive droop control parameter matrix, and judging the voltage stability of the power distribution network based on a spectrum radius upper limit theory; calculating a voltage-reactive inverter parameter adjustable range set of different photovoltaic grid connection points based on the provided power distribution network voltage stability analysis solving algorithm; if all photovoltaic grid-connected unit voltage-reactive inverter parameters in the network are set within an adjustable range, even if the position capacity of grid-connected photovoltaic is always in a dynamic change process, the voltage stability of the power distribution network can be ensured.
3. The method for rapidly determining the voltage stability of the power distribution network accessed by the large-scale distributed photovoltaic system according to claim 1, wherein the method comprises the following steps: the determination method specifically comprises the following steps:
step 1, establishing a voltage sensitivity solving model based on a Newton-Raynaud algorithm;
step 2, establishing a voltage-reactive power control network description model based on the voltage sensitivity matrix;
step 3, establishing a power distribution network voltage stability judgment model based on a voltage sensitivity matrix and a spectrum radius upper limit theory;
and 4, establishing a network voltage stability critical parameter set based on the power distribution network voltage stability judgment model.
4. The method for rapidly determining the voltage stability of the power distribution network accessed by the large-scale distributed photovoltaic system according to claim 1, wherein the method comprises the following steps: the method for establishing the voltage sensitivity solving model based on the Newton-Raynaud flow algorithm comprises the following steps:
setting the initial solution of the nonlinear equation f (x) to 0 as x0，x0The difference between the real solution x and the actual solution x is delta x(0)The essence of the solution is to iterate so that Δ x(k)→ 0, where k represents the number of iterations, the true solution to the equation is finally found;
assuming that N nodes are total, the network node set is N ═ 1,2 … N, and according to kirchhoff's law, the relationship between the voltage and the current of the nodes is as follows:
Figure RE-FDA0002941673280000021
in the formula: y is a nodal admittance matrix, Z is a nodal impedance matrix,
Figure RE-FDA0002941673280000022
is a vector of the voltage at the node,
Figure RE-FDA0002941673280000023
is the injected current vector of the node;after the formula (1) is developed, the following components are provided:
Figure RE-FDA0002941673280000024
respectively recording the injected active power and the injected reactive power of the node i as PiAnd QiAnd the voltage conjugate value of the node i is denoted as Vi *(ii) a The injection current at node i can be further represented as:
Figure RE-FDA0002941673280000025
substituting equation (3) into equation (2) is:
Figure RE-FDA0002941673280000026
voltage of node i
Figure RE-FDA0002941673280000027
In polar coordinate form of
Figure RE-FDA0002941673280000028
Wherein ViIs the voltage amplitude of node i, δiIs the voltage phase angle of node i; in addition Yij＝Gij+jBijThe polar form of the power equation is:
Figure RE-FDA0002941673280000029
assuming that a system has n nodes, wherein one node is a balanced node, the n node is a balanced node, m PQ nodes are provided, 1-m are PQ nodes, n-m-1 PV nodes, and m + 1-n-1 are PV nodes, the power balance equation is as follows:
Figure RE-FDA00029416732800000210
the corresponding correction equation is as follows:
Figure RE-FDA0002941673280000031
in the formula: the off-diagonal elements i ≠ j of the submatrix as in equation (8), and the diagonal elements i ≠ j as in equation (9)
Figure RE-FDA0002941673280000032
Figure RE-FDA0002941673280000033
The iterative form of equation (7) is shown in equation (10), and the convergence criterion is
Figure RE-FDA0002941673280000034
Figure RE-FDA0002941673280000035
In the formula: delta V'(k)Is shown in (11)
Figure RE-FDA0002941673280000036
Correction vector delta after each iteration of equation (8)i (k+1)And Vi (k+1)Comprises the following steps:
Figure RE-FDA0002941673280000037
if it is assumed that there is no PV node in the network, only a PQ node and one balance node, the size of the jacobian matrix in equation (7) becomes 2 × (n-1), and the iterative equation (10) is modified as:
Figure RE-FDA0002941673280000041
the result after the iterative convergence is recorded as
Figure RE-FDA0002941673280000042
Gaussian elimination of formula (13), assuming H1、N1、M1And L1All exist inverse matrix
Figure RE-FDA0002941673280000043
The relation between the node voltage change delta V and the node active change delta P and the node reactive change is delta Q
ΔV＝SV-PΔP+SV-QΔQ (16)
In the formula: sV-P＝(M1H1 -1N1-L1)-1M1H1 -1In the form of a voltage-active sensitivity matrix,
SV-Q＝(L1-M1H1 -1N1)-1is a voltage-reactive sensitivity matrix;
after the formula (16) is unfolded
Figure RE-FDA0002941673280000044
For any node i (i ∈ N), its voltage change Δ ViThe relation with the active and reactive changes of the other nodes is
Figure RE-FDA0002941673280000045
In the formula:
Figure RE-FDA0002941673280000046
and
Figure RE-FDA0002941673280000047
the voltage-active and voltage-reactive sensitivities are respectively, i, j belongs to N, namely the voltage change of the node i caused by the active and reactive changes of the node j; delta PiAnd Δ QiThe active and reactive changes of node j, respectively.
5. The method for rapidly determining the voltage stability of the power distribution network accessed by the large-scale distributed photovoltaic system according to claim 4, wherein the method comprises the following steps: the method for establishing the voltage-reactive control network description model based on the voltage sensitivity matrix comprises the following steps:
expressing active and reactive power of nodes as correlation equation with network voltage and phase angle
Figure RE-FDA0002941673280000051
In the formula: f. ofPF_PThe method is an associated equation of node active power, network voltage and phase angle; f. ofPF_QIs an associated equation of node reactive power, network voltage and phase angle; pload、QloadRespectively are the active and reactive of the node load; pPV、QPVRespectively are the active and reactive power of the node photovoltaic; v and delta are the voltage and phase angle of the node respectively;
according to the voltage reactive power droop control curve, the photovoltaic inverter absorbs reactive power when the voltage is higher, injects reactive power when the voltage is lower, and keeps the reactive power to be zero when the voltage is close to a target value; qPVmax,iAnd QPVmin,iRespectively representing the maximum injection power and the maximum absorption power of the photovoltaic inverter when the node i is connected to the grid; vi,maxAnd Vi,minRespectively corresponding to the voltage values when absorbing the maximum reactive power and injecting the maximum reactive power; vref,iIs the target voltage; the distance between the target voltage and the initial reactive injection voltage is d/2, and the distance between the target voltage and the initial reactive absorption voltage is d/2; m isiIs a droop control parameter m of the photovoltaic inverter when the node i is connected to the gridi≥0；
The equation of the voltage reactive output of the photovoltaic grid-connected node and the node voltage can be expressed as
Figure RE-FDA0002941673280000052
Figure RE-FDA0002941673280000053
The equation for reactive power regulation may be further expressed as
QPV,i,t+1＝f(Vi,t)＝f(u(QPV,i,t))＝ψ(QPV,i,t) (22)
In the formula: vi,tIs the voltage amplitude at time t of node i; u (-) is an equation of the influence of the reactive node reactive power change on the node voltage change; f (u (-) is represented by ψ (-) and represents the relationship between the reactive output at time t +1 and the reactive output at time t.
6. The method for rapidly determining the voltage stability of the power distribution network accessed by the large-scale distributed photovoltaic system according to claim 5, wherein the method comprises the following steps: step 3, the method for establishing the power distribution network voltage stability judgment model based on the voltage sensitivity matrix and the spectrum radius upper limit theory comprises the following steps:
to describe the correlation of the reactive changes at two adjacent times, the partial derivative is calculated over psi (·), i.e.
Figure RE-FDA0002941673280000061
Figure RE-FDA0002941673280000062
In the above formula
Figure RE-FDA0002941673280000063
Is the solved voltage-reactive sensitivity matrix parameter; it can be seen that
Figure RE-FDA0002941673280000064
Namely, the voltage-reactive power control slope parameter is expressed by a diagonal matrix as follows:
Figure RE-FDA0002941673280000065
wherein M is a slope matrix of the cluster distributed photovoltaic power supply;
the stability determination condition is
ρ(MSV-Q)＜1 (25)
Based on equation (25), once the distribution network voltage-reactive sensitivity matrix and the photovoltaic slope matrix are constructed, the stability for each node can be expressed in the form of analytic expression
Figure RE-FDA0002941673280000066
In the formula: n is a radical of-Is the set of network nodes except the head node, i.e., nodes {2,3, … n };
according to the formula (26), it can be seen that the problem of voltage stability of the power distribution network including the cluster distributed photovoltaic units is mainly influenced by the positions of the photovoltaic grid-connected nodes and the voltage-reactive droop slope parameter, and the positions of the photovoltaic grid-connected points mainly influence the voltage stability of the power distribution network in the formula (26)
Figure RE-FDA0002941673280000067
With respect to the line resistance, reactance parameters, line length and topology of the line of the network, withThe photovoltaic inverter has nothing to do with the reactive adjustable capacity; voltage-reactive droop slope, i.e. miParameter, the larger the value is, the larger is ρ (MS)V-Q) The more easily the voltage is close to 1 or exceeds 1, the situation that the voltage fluctuation is not converged in the voltage regulation process can be caused; the condition that the voltage of the distribution network is stable is that any node in the network should satisfy the judgment condition of the formula (26).
7. The method for rapidly determining the voltage stability of the power distribution network accessed by the large-scale distributed photovoltaic system according to claim 6, wherein the method comprises the following steps: the method for determining the stability determination condition comprises the following steps: the voltage stability judgment condition is that the voltage change degree gradually decreases and finally tends to be stable along with the rise of the iteration times; the process is equivalent to the fact that the degree of reactive power change is gradually reduced and finally tends to be stable along with the rise of the iteration times of the photovoltaic reactive power output; setting rho (·) to represent the spectrum radius of any matrix, namely the maximum value of the absolute value of the matrix eigenvalue; for the equation described in equation (24), ρ (MS) according to the theorem on the spectral radiusV-Q) < 1 represents the reactive absolute value output of the most unstable node t +1 in all photovoltaic power supplies in the network at the time t and the time t
Figure RE-FDA0002941673280000078
Smaller, the photovoltaic reactive change tends to zero after infinite iterations, and a stability judgment condition rho (MS) is obtainedV-Q)＜1。
8. The method for rapidly determining the voltage stability of the power distribution network accessed by the large-scale distributed photovoltaic system according to claim 6, wherein the method comprises the following steps: the method for establishing the network voltage stability critical parameter set based on the power distribution network voltage stability judgment model comprises the following steps:
from the result of the derivation of equation (25), if ρ (MS)V-Q) → 1, i.e. in formula (23)
Figure RE-FDA0002941673280000071
Representing that the degree of reactive power change does not gradually decrease with increasing number of control iterations, and voltage-noneThe iteration times of the control system of work will approach to infinite times, the stability condition of the control system cannot be met, and rho (MS) will be metV-Q) The set of slopes of the drooping curve at time → 1 is noted
Figure RE-FDA0002941673280000072
Wherein
Figure RE-FDA0002941673280000073
Is the upper bound of the droop slope of each node, i.e., miThe range of variation is
Figure RE-FDA0002941673280000074
Figure RE-FDA0002941673280000075
And M is:
Figure RE-FDA0002941673280000076
Figure RE-FDA0002941673280000077
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