CN114529074A - Multi-main-body planning method for power distribution network driven by end-to-end decentralized transaction - Google Patents

Multi-main-body planning method for power distribution network driven by end-to-end decentralized transaction Download PDF

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CN114529074A
CN114529074A CN202210132983.9A CN202210132983A CN114529074A CN 114529074 A CN114529074 A CN 114529074A CN 202210132983 A CN202210132983 A CN 202210132983A CN 114529074 A CN114529074 A CN 114529074A
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向月
代佳琨
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Abstract

The invention discloses a power distribution network multi-main-body planning method driven by end-to-end decentralized trading, which comprises the following steps: the initialization load increases; carrying out load flow calculation on the power grid, judging whether a power distribution network line exceeds a load upper limit, and if so, entering a third step; the method comprises the steps of upgrading a power distribution network line, establishing a virtual end-to-end transaction model, carrying out virtual end-to-end transaction, automatically matching an electricity selling main body according to the electricity purchasing price and the electricity purchasing quantity of the electricity purchasing main body, automatically matching the electricity purchasing main body according to the electricity selling price and the electricity selling quantity of the electricity selling main body, enabling an objective function to be the maximum social benefit, optimizing the scheme of the best matching in a transaction set, and obtaining the scheme of the best matching in the optimized transaction set. The invention integrates virtual end-to-end transaction to establish a multi-main-body extension planning model of the power distribution network. The multi-main-body extension planning of the power distribution network combining investment planning and economic operation is realized.

Description

Multi-main-body planning method for power distribution network driven by end-to-end decentralized transaction
Technical Field
The invention relates to the field of power planning, in particular to a power distribution network multi-main-body planning method driven by end-to-end decentralized transaction.
Background
More investors will participate in the investment and operation of distributed power sources in the power distribution network. At the same time, local grid companies are responsible for line consolidation and installation of other distributed power sources. However, how to maximize the benefit of each investor while the power company obtains profits is a problem in the multi-subject expansion planning of the power distribution network.
The existing power distribution network extension planning method comprises single-main-body extension planning and multi-main-body extension planning, and the existing single-main-body extension planning cannot effectively improve the resource utilization efficiency and save the investment cost of the power distribution network extension planning on the problem of load increase in a power distribution network. For the multi-principal extension planning method, games of investment schemes of a plurality of investment principals are generally considered, and although the load increase demand can be met, an effective electric energy transaction mode is lacked to combine planning and operation, so that benefit maximization of the plurality of investment principals is difficult to realize. With the increase of distributed power sources and the further opening of an electric power market, many scholars begin to research an end-to-end trading mode of electric energy in the electric power market, but most of the existing research only considers an operation mode of a power distribution network, so that the economic operation of the power distribution network is realized, and the multi-main-body extension planning of the power distribution network is not effectively combined, so that the resource waste is caused.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a power distribution network multi-main-body planning method driven by end-to-end decentralized transaction, which comprises the following steps:
step one, initializing load increase;
step two, carrying out load flow calculation on the power grid, judging whether a power distribution network line exceeds the upper limit of load, and if so, entering step three;
step three, upgrading the power distribution network line, establishing a virtual end-to-end transaction model, performing virtual end-to-end transaction, automatically matching the electricity selling main body according to the electricity purchasing price and the electricity purchasing quantity of the electricity purchasing main body, automatically matching the electricity purchasing main body according to the electricity selling price and the electricity selling quantity of the electricity selling main body, optimizing the scheme of the best matching in the transaction set to obtain the scheme of the best matching in the optimized transaction set, wherein the objective function is the maximization of social benefits;
and step four, obtaining a power distribution network multi-main-body planning scheme according to the optimized most matched scheme in the transaction set.
Further, the end-to-end transaction model:
the end-to-end transaction model comprises the following steps:
Figure BDA0003496676630000011
Dm,min≤dm≤Dm,max
Gn,min≤gm≤Gm,max
Figure BDA0003496676630000021
Figure BDA0003496676630000022
pw≥0 w∈W
ProP2Pis a social benefit, NBIs a collection of electricity purchasing parties, NSIs a collection of electricity vendors, dmIs the active power demand of user m, gnGenerated energy of distributed power supply of power seller n, UmIs a benefit function of buyer m, CnIs a benefit function of seller n, Dm,minAnd Dm,maxIs the upper and lower limits of the active power demand of user m, Gn,minAnd Gn,maxLower and upper power generation limits, p, of the generator n, respectivelywIs the active power of the transaction W, which is the transaction set.
Further, the power seller benefit model is as follows:
Figure BDA0003496676630000023
wherein theta is1,n、θ2,nAnd theta3,nIs an a priori non-negative parameter related to seller n.
Further, the power purchasing side benefit model comprises:
Figure BDA0003496676630000024
wherein tau is1,mAnd τ2,mIs a priori non-negative parameter related to the power supplier m.
Further, the node electricity price model is as follows:
Figure BDA0003496676630000025
Figure BDA0003496676630000026
Figure BDA0003496676630000027
Figure BDA0003496676630000028
Figure BDA0003496676630000029
Figure BDA00034966766300000210
Figure BDA0003496676630000031
Figure BDA0003496676630000032
for marginal electricity price in line area l, A1-A5Is the electricity price coefficient lambdareceiver(l)Is the basic electricity price, mu, of the line end nodeoutput(l)And mureceiver(l)The highest electricity prices at the beginning and end of the line, respectivelyThe limit is set to the limit value of the,
Figure BDA0003496676630000033
and
Figure BDA0003496676630000034
increasing or decreasing electricity prices, f, at the beginning of the line, respectivelyl pAnd fl qRespectively active power flow and reactive power flow, R of the linelAnd XlRespectively the resistance and reactance of the line l, alIs the current square of line l;
further, the net charge model is as follows:
Figure BDA0003496676630000035
Dw,n,DLMPand Dw,m,DLMPThe marginal node prices of buyer n and seller m, respectively, in transaction w.
Further, the power distribution network multi-main-body planning scheme is that under a power distribution network multi-main-body extension planning mode, the power transmission capacity of the system is improved, and the operation of the power distribution network is maintained, and the objective function is the maximum economic benefit of a power grid company:
ProPC,P2P＝ProPC,sell+ProNUC-CStr,invest
wherein Pro isPC,sellFor electricity-selling purposes, ProNUCFor the net charge gain, CStr,investTo extend the cost for the line.
Further, the electricity selling benefits are as follows:
Figure BDA0003496676630000036
Ppg,m,tis the active load of user m at t, σsell,tIs the unit price of electricity sold by the electric company at t, λbuy,tIs the unit price of electrical energy purchased by the utility from the main grid at time t.
Further, the net charge profit
Figure BDA0003496676630000037
αNUCUnit price of passing net charge generated by end-to-end transaction
Further, the line capacity expansion cost
Figure BDA0003496676630000041
NlineNumber of lines to be expanded for electric power company, Cline,lUnit cost for extension of line L, LlIs the actual length of the line l.
The invention has the beneficial effects that: the invention integrates virtual end-to-end transaction to establish a multi-main-body extension planning model of the power distribution network. The model can drive the expansion planning of multiple main bodies in the power distribution network by using end-to-end transaction, can meet the requirement of load increase in the power distribution network on power supply capacity, reduces the expansion planning cost of the power distribution network, and simultaneously adjusts a transaction result by using a node electricity price and network charge mechanism. The multi-main-body extension planning of the power distribution network combining investment planning and economic operation is realized.
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FIG. 1 is a flow chart of the automatic planning of the present invention;
FIG. 2 is a diagram of a power distribution network multi-agent extended planning architecture used in the present invention;
fig. 3 is a diagram of a virtual end-to-end transaction architecture of the present invention.
Detailed Description
The technical solutions of the present invention are further described in detail below with reference to the accompanying drawings, but the scope of the present invention is not limited to the following.
For the purpose of making the object, technical solution and advantages of the present invention more apparent, the present invention will be further described in detail with reference to the accompanying drawings and embodiments. It should be understood that the detailed description and specific examples, while indicating the preferred embodiment of the invention, are intended for purposes of illustration only and are not intended to limit the scope of the invention. The components of embodiments of the present invention generally described and illustrated in the figures herein may be arranged and designed in a wide variety of different configurations.
Thus, the following detailed description of the embodiments of the present invention, presented in the figures, is not intended to limit the scope of the invention, as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments of the present invention without making any creative effort, shall fall within the protection scope of the present invention. It is noted that relational terms such as "first" and "second," and the like, may be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions.
Also, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising an … …" does not exclude the presence of other like elements in a process, method, article, or apparatus that comprises the element.
The features and properties of the present invention are described in further detail below with reference to examples.
As shown in fig. 1, a method for planning multiple power distribution networks driven by end-to-end decentralized transactions includes the following steps:
step one, initializing load increase;
step two, carrying out load flow calculation on the power grid, judging whether a power distribution network line exceeds the upper limit of load, and if so, entering step three;
step three, upgrading the power distribution network line, establishing a virtual end-to-end transaction model, performing virtual end-to-end transaction, automatically matching the electricity selling main body according to the electricity purchasing price and the electricity purchasing quantity of the electricity purchasing main body, automatically matching the electricity purchasing main body according to the electricity selling price and the electricity selling quantity of the electricity selling main body, optimizing the scheme of the best matching in the transaction set to obtain the scheme of the best matching in the optimized transaction set, wherein the objective function is the maximization of social benefits;
and step four, obtaining a power distribution network multi-main-body planning scheme according to the optimized most matched scheme in the transaction set.
The end-to-end transaction model comprises the following steps:
the end-to-end transaction model comprises the following steps:
Figure BDA0003496676630000051
Dm,min≤dm≤Dm,max
Gn,min≤gm≤Gm,max
Figure BDA0003496676630000052
Figure BDA0003496676630000053
pw≥0 w∈W
ProP2Pis a social benefit, NBIs a collection of electricity purchasing parties, NSIs a collection of electricity vendors, dmIs the active power demand of user m, gnGenerated energy of distributed power supply of power seller n, UmIs a benefit function of buyer m, CnIs a benefit function of seller n, Dm,minAnd Dm,maxIs the upper and lower limits of the active power demand of user m, Gn,minAnd Gn,maxLower and upper power generation limits, p, of the generator n, respectivelywIs the active power of the transaction W, which is the transaction set. The optimal (social benefit maximization) set of electric quantity transaction is formed through the transaction game of the electricity purchasing party and the electricity selling party in the power distribution network, namely the distribution of trend, so that the operation of the power distribution network is influencedThe line state.
And the power distribution network multi-main-body extension planning model carries out investment planning on parts (lines and the like) with out-of-limit possibility according to the running state formed by the end-to-end transaction model so as to upgrade the power distribution network.
The power selling party benefit model comprises the following steps:
Figure BDA0003496676630000054
wherein theta is1,n、θ2,nAnd theta3,nIs an a priori non-negative parameter related to seller n. The function of the transaction model of the electricity selling party is as follows: and calculating the benefit of the electricity selling party in the end-to-end transaction. Inputting: calculating parameter theta of generated energy and benefit of power selling party distributed power supplyi,n(ii) a And (3) outputting: the total benefit of the power seller.
The power purchasing side benefit model comprises the following steps:
Figure BDA0003496676630000061
wherein tau is1,mAnd τ2,mIs a priori non-negative parameter related to the power supplier m. For calculating the benefit of the power purchasing party, inputting: calculating parameters of the active load demand and the benefit of the power purchasing party; and (3) outputting: the total benefit of the power purchasing party.
The node electricity price model is as follows:
Figure BDA0003496676630000062
Figure BDA0003496676630000063
Figure BDA0003496676630000064
Figure BDA0003496676630000065
Figure BDA0003496676630000066
Figure BDA0003496676630000067
Figure BDA0003496676630000068
for marginal electricity price in line area l, A1-A5Is the electricity valence coefficient, lambdareceiver(l)Is the basic electricity price, mu, of the line end nodeoutput(l)And mureceiver(l)The highest electricity price limit at the beginning and end of the line respectively,
Figure BDA0003496676630000071
and
Figure BDA0003496676630000072
increasing or decreasing electricity prices, f, at the beginning of the line, respectivelyl pAnd fl pRespectively active power flow and reactive power flow, R of the linelAnd XlRespectively the resistance and reactance of the line l, alIs the current square of line l; the model is used for calculating the real-time node electricity price of each line in the power distribution network, and inputs: and power flow and resistance reactance values of the power distribution network line. And (3) outputting: node electricity prices.
The net charge model is as follows:
Figure BDA0003496676630000073
Dw,n,DLMPand Dw,m,DLMPThe marginal node prices of the buyer n and the seller m in the transaction w are respectively, and the model is used for calculating the power grid company according to the node electricity price differenceThe network fee to be charged to the electricity seller is input as follows: and node electricity prices of nodes where the electricity selling party and the electricity purchasing party are located are output: the grid company requires a grid fee to be charged to the electricity seller.
The power distribution network multi-main-body planning scheme is that under a power distribution network multi-main-body extension planning mode, the power transmission capacity of a system is improved, and the operation of a power distribution network is maintained, and an objective function is the maximum economic benefit of a power grid company:
ProPC,P2P＝ProPC,sell+ProNUC-CStr,invest
wherein Pro isPC,sellFor electricity-selling purposes, ProNUCFor the net charge gain, CStr,investFor line capacity cost.
The electricity selling benefits are as follows:
Figure BDA0003496676630000074
Ppg,m,tis the active load of user m at t, σsell,tIs the unit price of electricity sold by the electric company at t, λbuy,tIs the unit price of electrical energy purchased by the utility from the main grid at time t.
The net fee income
Figure BDA0003496676630000075
αNUCUnit price of passing net charge generated by end-to-end transaction
The line capacity expansion cost
Figure BDA0003496676630000076
NlineNumber of lines to be expanded for electric power company, Cline,lUnit cost for extension of line L, LlIs the actual length of the line l.
Specifically, the power distribution network multi-main-body extension planning method integrating virtual end-to-end transaction comprises the following steps: an end-to-end transaction model modeling method and a power distribution network multi-main-body investment operation model modeling method.
An end-to-end transaction model modeling method comprises the following steps:
according to the virtual end-to-end trading model adopted by the invention, traders in the market can be producers, consumers and producers and consumers of electric energy, and a trading matching set among the producers, the consumers and the producers and consumers only needs to meet the physical constraint and the trend balance of the power distribution network. On the basis, the electricity-purchasing main body is automatically matched with the electricity-purchasing quantity according to the electricity-purchasing price of the electricity-purchasing main body in the market, and the electricity-purchasing main body is automatically matched with the electricity-selling quantity according to the electricity-selling price of the electricity-purchasing main body. The final objective function is social benefit maximization. The optimization goal is the best matching solution in the trade set.
1) End-to-end transaction model:
Figure BDA0003496676630000081
Figure BDA0003496676630000082
Figure BDA0003496676630000083
Figure BDA0003496676630000084
Figure BDA0003496676630000085
pw≥0 w∈W (6)
ProP2Pis a social benefit, NBIs a collection of electricity purchasers, NSIs a collection of power sellers, dmIs the active power demand, g, of user mnGenerated energy of distributed power supply of power seller n, UmIs the benefit of buyer mFunction, CnIs a benefit function of seller n, Dm,minAnd Dm,maxIs the upper and lower limits of the active power demand of user m, Gn,minAnd Gn,maxLower and upper power generation limits, p, of the generator n, respectivelywIs the active power of the transaction W, which is the transaction set.
2) The power seller benefit model:
Figure BDA0003496676630000086
θ1,n、θ2,nand theta3,nIs a priori non-negative parameter related only to seller n.
3) The power purchasing side benefit model:
Figure BDA0003496676630000087
τ1,mand τ2,mIs an a priori non-negative parameter related only to the electricity purchaser m.
4) Node electricity price model:
Figure BDA0003496676630000088
Figure BDA0003496676630000091
Figure BDA0003496676630000092
Figure BDA0003496676630000093
Figure BDA0003496676630000094
Figure BDA0003496676630000095
Figure BDA0003496676630000096
for marginal electricity price in line area l, A1To A5Is the electricity price coefficient; lambda [ alpha ]receiver(l)、μoutput(l)、μreceiver(l)
Figure BDA0003496676630000097
And
Figure BDA0003496676630000098
are dual variables in (10-14).
5) Net-passing fee model
Figure BDA0003496676630000099
Dw,n,DLMPAnd Dw,m,DLMPThe marginal node prices of buyer n and seller m in transaction w, respectively;
(1) the power distribution network multi-main-body investment operation model is as follows:
under the multi-main-body expansion planning mode of the power distribution network, an electric power company only needs to improve the power transmission capacity of the system to maintain the operation of the power distribution network, and the smooth end-to-end transaction is guaranteed. At this time, the objective function is the maximum economic efficiency of the electric power company, and the economic efficiency of the consumers and the users needs to be considered.
1) Selling electricity benefits
Figure BDA00034966766300000910
Ppg,m,tIs the active load of user m at t, σsell,tIs the unit price of electricity sold by the electric company at t, λbuy,tIs the unit price of electrical energy purchased by the utility from the main grid at time t.
2) Net charge profit
Figure BDA0003496676630000101
αNUCIs the unit price of the net charge generated by the end-to-end transaction.
3) Line capacity expansion cost
Figure BDA0003496676630000102
NlineNumber of lines to be expanded for electric power company, Cline,lUnit cost for extension of line L, LlIs the actual length of the line l.
4) Grid company objective function
ProPC,P2P＝ProPC,sell+ProNUC-CStr,invest (25)
The foregoing is illustrative of the preferred embodiments of this invention, and it is to be understood that the invention is not limited to the precise form disclosed herein and that various other combinations, modifications, and environments may be resorted to, falling within the scope of the concept as disclosed herein, either as described above or as apparent to those skilled in the relevant art. Meanwhile, the model and/or size of each component of the present invention should be understood as being matched and matched with each other, which belongs to the implementation mode known by those skilled in the art, and the matching tightness of each component of the present invention, which is known by those skilled in the art, belongs to the part that those skilled in the art can implement as required by adopting the mode known by those skilled in the art, and specifically, on the basis of the scheme of the present invention, those skilled in the art can make up some deficiencies in the details of the present invention through logical analysis and/or reasoning, so that the technical scheme of the present invention is more perfect and optimized.
And, if the present invention is concerned with a problem not addressed by the present invention, any means known to those of ordinary skill in the art may be used to solve the problem, and all the means known to those of ordinary skill in the art are included in the embodiments known to those of ordinary skill in the art. For example, the present invention is also intended to protect the principle rather than the specific data, and also in view of the fact that the size, the data, etc. are not the object protected by the patent of the invention, and therefore are not described in detail, and those skilled in the art can understand any implementation way known to those skilled in the art based on the present invention and combined with the prior art, and the present invention may have the drawbacks mentioned in the background, and this may be the original and unprecedented one of the inventors, but in the development of a technology, it is difficult to develop a technology without any drawbacks, the solution to this drawback can be understood as the prior art, and the people using the present invention are limited to the users who need the environment mentioned in the background and can continue to choose to use the environment mentioned in the background and neglect this drawback. Based on this, what is claimed in the present application is only a part of the technical solution corresponding to the technical problems in the background art, and other parts, for example, how to implement the technical solution of the present application more optimally on the basis of the technical solution of the present application, obviously, the technical solution of the present application with clear description can not be completely realized through one development and one patent application, so the technical solution of the present application is only limited to the technical problems to be solved in the background art, other parts can be completed in other development and patent applications, and other technical personnel in the field can continuously improve the technical solution of the present application, and apply patent for achieving the optimal use effect, all belong to the existing known ways in the field, and the modification and the change which are performed by the personnel in the field do not depart from the spirit and the scope of the present invention, and all shall be within the protection scope of the appended claims.

Claims (10)

1. A multi-main-body planning method for a power distribution network driven by end-to-end decentralized transaction is characterized by comprising the following steps:
step one, initializing load increase;
step two, carrying out load flow calculation on the power grid, judging whether a power distribution network line exceeds the upper limit of load, and if so, entering step three;
step three, upgrading the power distribution network line, establishing a virtual end-to-end transaction model, performing virtual end-to-end transaction, automatically matching the electricity selling main body according to the electricity purchasing price and the electricity purchasing quantity of the electricity purchasing main body, automatically matching the electricity purchasing main body according to the electricity selling price and the electricity selling quantity of the electricity selling main body, optimizing the scheme of the best matching in the transaction set to obtain the scheme of the best matching in the optimized transaction set, wherein the objective function is the maximization of social benefits;
and step four, obtaining a power distribution network multi-main-body planning scheme according to the optimized most matched scheme in the transaction set.
2. The method of claim 1, wherein the method comprises the steps of,
the end-to-end transaction model comprises the following steps:
Figure FDA0003496676620000011
Dm,min≤dm≤Dm,max
Gn,min≤gm≤Gm,max
Figure FDA0003496676620000012
Figure FDA0003496676620000013
pw≥0 w∈W
ProP2Pis a social benefit, NBIs a collection of electricity purchasing parties, NSIs a collection of electricity vendors, dmIs the active power demand of user m, gnGenerated energy of distributed power supply of power seller n, UmIs a benefit function of buyer m, CnIs a benefit function of seller n, Dm,minAnd Dm,maxIs the upper and lower limits of the active power demand of user m, Gn,minAnd Gn,maxLower and upper power generation limits, p, of the generator n, respectivelywIs the active power of the transaction W, which is the transaction set.
3. The method according to claim 1, wherein the power seller benefit model comprises:
Figure FDA0003496676620000014
wherein theta is1,n、θ2,nAnd theta3,nIs an a priori non-negative parameter related to seller n.
4. The method according to claim 1, wherein the power purchasing party benefit model comprises:
Figure FDA0003496676620000021
wherein τ is1,mAnd τ2,mIs a priori non-negative parameter related to the power supplier m.
5. The method according to claim 1, wherein the node electricity price model is:
Figure FDA0003496676620000022
Figure FDA0003496676620000023
Figure FDA0003496676620000024
Figure FDA0003496676620000025
Figure FDA0003496676620000026
Figure FDA0003496676620000027
Dl,DLMPfor marginal electricity price in line area l, A1-A5Is the electricity price coefficient; lambda [ alpha ]receiver(l)Is the basic electricity price, mu, of the line end nodeoutput(l)And mureceiver(l)The highest electricity price limit at the beginning and end of the line respectively,
Figure FDA0003496676620000028
and
Figure FDA0003496676620000029
increasing or decreasing electricity prices, f, at the beginning of the line, respectivelyl pAnd fl qRespectively, active and reactive power flows, R, of the linelAnd XlRespectively the resistance and reactance of the line l, alIs the current square of line i.
6. The method according to claim 1, wherein the network charge model is:
Figure FDA0003496676620000031
Dw,n,DLMPand Dw,m,DLMPThe marginal node prices of buyer n and seller m, respectively, in transaction w.
7. The method according to claim 1, wherein the power distribution network multi-subject planning scheme is that in the power distribution network multi-subject extension planning mode, the power transmission capability of the system is improved to maintain operation of the power distribution network, and the objective function is the maximum economic benefit of a power grid company:
ProPC,P2P＝ProPC,sell+ProNUC-CStr,invest
wherein Pro isPC,sellFor selling electric power, ProNUCFor the net charge gain, CStr,investTo extend the cost for the line.
8. The method according to claim 7, wherein the electric power selling benefits are as follows:
Figure FDA0003496676620000032
Ppg,m,tis the active load of user m at t, σsell,tIs the unit price of electricity sold by the electric company at t, λbuy,tIs the unit price of electrical energy purchased by the utility from the main grid at time t.
9. The method as claimed in claim 7, wherein the network cost and profit are calculated by the network planning method
Figure FDA0003496676620000033
αNUCIs the unit price of the net charge generated by the end-to-end transaction.
10. The method of claim 7, wherein the cost of line capacity expansion is reduced by a method of planning multiple entities in an end-to-end decentralized transaction driven distribution network
Figure FDA0003496676620000034
NlineNumber of lines to be expanded for electric power company, Cline,lUnit cost for extension of line L, LlIs the actual length of the line l.
CN202210132983.9A 2022-02-07 2022-02-07 Multi-main-body planning method for power distribution network driven by end-to-end decentralized transaction Pending CN114529074A (en)

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CN116091102B (en) * 2023-03-06 2023-09-19 四川大学 Multi-main-body planning method for power distribution system considering multiple uncertainties

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