CN111178956A - Trading model and quotation strategy applicable to cross-country electricity-carbon market - Google Patents

Trading model and quotation strategy applicable to cross-country electricity-carbon market Download PDF

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CN111178956A
CN111178956A CN201911336666.3A CN201911336666A CN111178956A CN 111178956 A CN111178956 A CN 111178956A CN 201911336666 A CN201911336666 A CN 201911336666A CN 111178956 A CN111178956 A CN 111178956A
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周原冰
宁叶
黄瀚
冯骅
丁涛
秦博宇
杨青润
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Abstract

The invention relates to a trading model and a quotation strategy suitable for a cross-country electricity-carbon market, which are characterized in that: firstly, establishing a double-layer transaction model suitable for a cross-country electricity-carbon market, wherein the double-layer transaction model is divided into an upper-layer cross-country power high-low matching transaction model and a lower-layer cross-country carbon quota high-low matching transaction model which are established in series; and then, proposing a quotation strategy suitable for the cross-country electricity-carbon market double-layer trading model, predicting the reported contents of other countries in the market through neural network learning, and proposing a reported price determination method capable of maximizing the benefit of the country. The method provided by the invention can be used in a transnational electricity-carbon market with the global energy Internet as the background, provides a relevant market mechanism and a quotation strategy, and provides a reference for the development of the transnational electricity-carbon market in the future.

Description

Trading model and quotation strategy applicable to cross-country electricity-carbon market
Technical Field
The invention relates to a trading model and a quotation strategy suitable for a cross-country electricity-carbon market.
Background
In recent years, the global energy internet proposed by china first is considered to be accepted by more and more countries, and corresponding physical connection and market mechanisms are continuously built and promoted. With the development of the global energy internet, the international power trade is more and more frequent, and the improvement of the international power market is imperative. Meanwhile, the global carbon emission reduction action taking Paris protocol as a benchmark enables the future multinational carbon market to be developed vigorously. Power is closely related to carbon quota, but the current power market operates separately from the carbon market, so it is very important to come up with a reasonable cross-country electricity-carbon market solution.
In addition, each country must maximize its own benefits as much as possible while meeting its own needs in the cross-country electricity-carbon market. How to reasonably quote in the cross-country electricity-carbon market to achieve the maximization of the benefit of the country is also a question to be researched.
Disclosure of Invention
The invention aims to provide a trading model and a quotation strategy suitable for a cross-country electricity-carbon market.
In order to achieve the purpose, the invention adopts the following technical scheme:
a transaction model and quotation strategy suitable for cross-country electricity-carbon markets is characterized in that: the method comprises the following steps:
1) establishing a double-layer transaction model suitable for a cross-country electricity-carbon market;
2) and establishing a quotation strategy suitable for the cross-country electricity-carbon market double-layer trading model.
Further, the step 1) specifically includes the following steps:
1.1) establishing an upper-layer cross-country power high-low matching transaction model, completing cross-country power high-low matching transaction, reporting transaction electric quantity and transaction electricity price by each country simultaneously, and clearing the market based on a high-low matching principle;
1.2) establishing a lower-layer cross-country carbon quota high-low matching transaction model: firstly, calculating the carbon emission considering transnational power trading, and determining the carbon quota trading volume to be reported by each country; and then, in the carbon quota trading market, reporting the carbon quota trading volume and the trading price by each country at the same time, and clearing the market based on a high-low matching principle.
Further, the step 1.1) specifically includes the following steps:
1.1.1) the electric power is sent out of the country and sorted from low to high according to the transaction electricity price reported by the electric power, and the reported transaction electricity quantity is taken as the equivalent length of the country in the sequence during sorting; and when the transaction electricity prices are the same, the transaction electricity prices are ranked in parallel.
1.1.2) the power accepting countries are sorted from high to low according to the transaction electricity prices reported by the power accepting countries, and the reported transaction electricity quantity is used as the equivalent length of the power accepting countries in the sequence during sorting; and when the transaction electricity prices are the same, the transaction electricity prices are ranked in parallel.
1.1.3) the power sending country and the power receiving country form price pairs according to the above sequence, namely, the power price difference pairs are sequenced from high to low.
1.1.4) clearing in turn according to the reported transaction electric quantity, wherein the clearing price is the average value of each price difference pair; clearing until the valence difference pair is 0, and failing to make a trade when the valence difference pair is a negative value; and if two or more than two price difference pairs are the same, distributing the reported electric quantity equally according to the current residual reported electric quantity of each country.
Further, the step 1.2) specifically includes the following steps:
1.2.1) calculating and considering the carbon emission of transnational power transaction, and determining the carbon quota transaction amount to be reported by each country by combining the power carbon emission quota distributed in the current year;
1.2.2) in the carbon quota trading market, reporting the carbon quota trading volume and the trading price simultaneously by each country, and clearing the market based on the high-low matching principle.
Further, the step 1.2.1) specifically comprises the following steps:
1.2.1.1) predicting the monthly cross-country trade electric quantity and the load level of the country in the current year;
1.2.1.2) predicting the power generation proportion of various energy sources in the current year of the country every month;
1.2.1.3) calculating the predicted electric power carbon emission in the current year in China;
1.2.1.4) calculating the transaction amount of the carbon quota which should be reported by the country.
Further, the step 1.2.2) specifically comprises the following steps:
1.2.2.1) ordering the carbon quota selling countries according to the respective reported transaction prices from low to high, and taking the reported carbon quota transaction amount as the equivalent length of the country in the sequence during ordering; when the transaction prices are the same, the transaction prices are ranked in parallel;
1.2.2.2) ordering the carbon quota buying countries according to the transaction prices reported by the countries from high to low, and taking the reported carbon quota transaction amount as the equivalent length of the country in the sequence during ordering; when the transaction prices are the same, the transaction prices are ranked in parallel;
1.2.2.3) forming price pairs for the carbon quota selling country and the carbon quota buying country according to the ordering, namely ordering the carbon quota price difference pairs from high to low;
1.2.2.4) clearing in turn according to the reported carbon quota transaction amount, wherein the clearing price is the average value of each price difference pair; clearing until the valence difference pair is 0, and failing to make a trade when the valence difference pair is a negative value; and if two or more than two price difference pairs are the same, distributing the carbon quota according to the current residual reported carbon quota of each country.
Further, the step 2) specifically includes the following steps:
2.1) the study and prediction of the contents reported by other countries: the three-layer BP neural network is taken as a frame, and network training is completed through a gradient descent algorithm; then substituting the input parameters of other countries in the current year to obtain the next reported transaction amount and transaction price of other countries;
2.2) the method for determining the price reported by the home country: after the content reported by other countries is determined according to the method, the transaction price reported by the home country is further determined so as to obtain the maximum benefit.
Further, the step 2.1) specifically comprises the following steps:
2.1.1) constructing a three-layer neural network, comprising an input layer, a hidden layer and an output layer. The input parameters are historical information published by the researched country, and the output parameters are transaction electric quantity and transaction electricity price reported by the researched country all the year round.
2.1.2) calculating an output parameter through a neural network according to the input parameter, and obtaining a training error.
2.1.3) judging whether the training error meets the given threshold value requirement. If so, finishing training; if not, updating the weight and returning to the step 2.1.2).
2.1.4) after the network training is finished, substituting the network training into the input parameters of the country studied in the current year to obtain the transaction electric quantity and the transaction electricity price reported next time by the country.
Further, the step 2.2) specifically includes the following steps:
2.2.1) assuming that the country does not participate in the transaction, namely only other countries participate in the transaction, and determining the final clearing result according to the predicted contents reported by the other countries and the high-low matching principle.
2.2.2) if the country is a trade selling country, selecting the matching pair with the highest clearing price as a reference; and if the country is a trading buying country, selecting the matching pair with the lowest clearing price as a reference, wherein the clearing price is the price difference pair average value.
2.2.3) if the country is a trade selling country, determining the previous selling party of the matched pair selling party, and subtracting a very small value from the reported trade price to be used as the final reported trade price; if the country is a transaction buying country, determining the former buying party of the matched pair of buying parties, and adding a small value to the transaction price reported on the former buying party to be used as the final reported transaction price.
The invention has the beneficial effects that:
1. the invention provides a trading model and a quotation strategy suitable for a cross-country electricity-carbon market, which establish a double-layer trading model suitable for the cross-country electricity-carbon market, wherein the double-layer trading model is divided into an upper cross-country electricity high-low matching trading model and a lower cross-country carbon quota high-low matching trading model, so that the defect that the current electricity market and the carbon market are operated separately is effectively overcome;
2. the invention provides a quotation strategy suitable for the cross-country electricity-carbon market double-layer trading model, so that the benefit of each country is maximized as far as possible on the premise that the needs of each country in the cross-country electricity-carbon market are met;
3. the method provided by the invention can be used in a transnational electricity-carbon market with the global energy Internet as the background, provides a relevant market mechanism and a quotation strategy, and provides a reference for the development of the transnational electricity-carbon market in the future.
Drawings
FIG. 1 is a flow chart of a two-tier trading model applicable to a cross-country electricity-carbon market in accordance with the present invention;
FIG. 2 is a flow chart of the quotation strategy applicable to the proposed cross-national electricity-carbon market double-tier trading model.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention more apparent, the technical solutions of the embodiments of the present invention will be described clearly and completely with reference to the accompanying drawings of the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all embodiments of the present invention. All other embodiments, which can be obtained by a person skilled in the art without any inventive step based on the embodiments of the present invention, are within the scope of the present invention. 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.
The trading model and the quotation strategy suitable for the cross-country electricity-carbon market are suitable for all trading time scales, and cross-country annual trading is taken as an example for explanation.
1) The double-layer transaction model suitable for the cross-country electricity-carbon market can be established in series by two parts, namely an upper cross-country electricity high-low matching transaction model and a lower cross-country carbon quota high-low matching transaction model, and specifically comprises the following steps:
1.1) transaction model for high and low matching of upper-layer cross-country power
The method comprises the steps of firstly carrying out cross-country power high-low matching transaction, reporting transaction power and transaction electricity price by each country at the same time, and carrying out market clearing based on a high-low matching principle. And assuming that the construction of the power grid across countries is mature, the power trading of any two countries can be completed. The method comprises the following specific steps:
1.1.1) the electric power is sent out of the country and sorted from low to high according to the transaction electricity price reported by the electric power, and the reported transaction electricity quantity is taken as the equivalent length of the country in the sequence during sorting; and when the transaction electricity prices are the same, the transaction electricity prices are ranked in parallel.
1.1.2) the power accepting countries are sorted from high to low according to the transaction electricity prices reported by the power accepting countries, and the reported transaction electricity quantity is used as the equivalent length of the power accepting countries in the sequence during sorting; and when the transaction electricity prices are the same, the transaction electricity prices are ranked in parallel.
1.1.3) the power sending-out country and the power receiving country form price pairs according to the above sequence, namely, the power price difference pairs (receiving price-sending-out price) are sequenced from high to low.
1.1.4) clearing in turn according to the reported transaction electric quantity, wherein the clearing price is the average value of each price difference pair; clearing until the valence difference pair is 0, and failing to make a trade when the valence difference pair is a negative value; and if two or more than two price difference pairs are the same, distributing the reported electric quantity equally according to the current residual reported electric quantity of each country.
1.2) lower-layer cross-country carbon quota high-low matching transaction model
1.2.1) carbon emissions and carbon quota trading volume calculation considering transnational power trading
After the cross-country power transaction is finished, the transaction electric quantity among the countries is determined accordingly. Due to the existence of transnational power trading, the carbon emissions of the power industries of various countries cannot calculate only the portion corresponding to the load of the country, but are increased (decreased) according to the specific power delivery (admission) situation. Taking a certain sending country as an example, a specific method for calculating carbon emission considering transnational power trading comprises the following steps:
1.2.1.1) the monthly cross-country trade electricity and domestic load level of the current year of the country are predicted. The monthly transnational transaction electric quantity can be obtained by decomposing annual transaction contracts, the monthly load level can be obtained by historical data prediction, and a mature load prediction method is available at present.
1.2.1.2) the power generation proportion of various types of energy sources in the current year of the country is estimated. Different energy power generation has different carbon emission coefficients, for example, the carbon emission coefficients of coal and gas power generation are higher, and the carbon emission coefficients of wind power and photovoltaic power generation are almost 0. In addition, the time distribution of the same power generation energy also has differences, for example, the photovoltaic power generation is obviously higher in summer than in winter. The specific monthly energy power generation proportion value can be obtained by historical data statistics.
1.2.1.3) calculating the predicted electric power carbon emission in the current year of the country
Figure BDA0002331115100000061
Wherein E is the predicted electric power carbon emission in the current year of the country; t is the number of months; m is a number of the type of the power generation energy; m is the total number of the types of the power generation energy; qtThe transnational transaction electric quantity sent out for the native country in the tth month; dtThe load capacity of the country in the t month; p is a radical ofm,tThe proportion of the domestic power generation energy type m in the t th month is 1 in total every month; e.g. of the typemThe carbon emission coefficient of unit electric quantity of the power generation energy type m in China.
1.2.1.4) calculating the transaction amount of the carbon quota which should be reported by the country
After the carbon emission considering the transnational power transaction is calculated, the carbon emission quota distributed in the current year is combined, and the transaction amount of the carbon quota which should be reported by the country can be determined to be
Etrade=E-E0(2)
Wherein E istradeThe carbon quota transaction amount to be reported by the country is bought in positive number and sold in negative number; e0The carbon emission quota of the power distributed for the country in the current year.
1.2.2) Cross-country carbon quota high-low matching transaction model
In the carbon quota trading market, the carbon quota trading volume and the trading price are reported by each country at the same time, and the market clearing is carried out based on the high-low matching principle. The method comprises the following specific steps:
1.2.2.1) ordering the carbon quota selling countries according to the respective reported transaction prices from low to high, and taking the reported carbon quota transaction amount as the equivalent length of the country in the sequence during ordering; and when the transaction prices are the same, the transaction prices are ranked in parallel.
1.2.2.2) ordering the carbon quota buying countries according to the transaction prices reported by the countries from high to low, and taking the reported carbon quota transaction amount as the equivalent length of the country in the sequence during ordering; and when the transaction prices are the same, the transaction prices are ranked in parallel.
1.2.2.3) the carbon quota sell country and the carbon quota buy country form price pairs according to the above sequence, namely, the carbon quota price difference pair (buy price-sell price) is sequenced from high to low.
1.2.2.4) clearing in turn according to the reported carbon quota transaction amount, wherein the clearing price is the average value of each price difference pair; clearing until the valence difference pair is 0, and failing to make a trade when the valence difference pair is a negative value; and if two or more than two price difference pairs are the same, distributing the carbon quota according to the current residual reported carbon quota of each country.
2) The method is suitable for the quotation strategy of the cross-country electricity-carbon market double-layer trading model. In the present method, the proposed pricing strategy is considered from a national perspective standing in the cross-country electricity-carbon market, with the goal of maximizing the benefit of itself in the bid game with other countries as much as possible. The concrete steps can be divided into two parts of the method for learning and predicting the contents reported by other countries and determining the price reported by the home country, which are established in series:
2.1) study prediction of contents reported by other countries
In the cross-country electricity-carbon market, the transaction amount and the transaction price reported by each country are independent, namely, the reported contents of other countries cannot be known when a certain country reports. If the reported contents of other countries can be reasonably predicted by a certain method, the effective information mastered by the country can be more comprehensive, so that the self reported price can be guided. The method takes a BP neural network as a framework and finishes network training through a gradient descent algorithm. Taking the case that one country learns to predict the transaction electricity quantity and the transaction electricity price reported by another country as an example, the following description is given:
2.1.1) constructing a three-layer neural network, comprising an input layer, a hidden layer and an output layer. The input parameters are historical information published by the researched country, including power information such as power load, power generation type and quantity, domestic power market price and the like of each year, and non-power information such as total GDP amount and composition condition of each year, and are marked as xiD in total; the output parameters are the transaction electric quantity and the transaction electricity price reported all the year round by the researched state and are marked as yjL in total; each element of the hidden layer is marked as bhAnd q in total. It should be noted that, in the training process, the input parameters and the output parameters need to be staggered by one bit in time, for example, the input parameters of the nth year correspond to the output parameters of the (N + 1) th year, that is, the goal is to predict the reported content of the (N + 1) th year through the information of the nth year. The weight from input layer to hidden layer is denoted vjqThe weight from the hidden layer to the output layer is denoted as wqjInitial weight average assignment. The activation function is a sigmod function, and the expression is as follows:
Figure BDA0002331115100000081
2.1.2) calculating the output parameter by the neural network according to the input parameter to obtain the error of
Figure BDA0002331115100000082
Wherein epsilon is a training error, y'jIs the output parameter value calculated by the neural network.
2.1.3) judging whether the training error meets the given threshold value requirement. If so, finishing training; if not, the weights are updated by the following formula and the procedure returns to step 2.1.2).
Δwhj=-η(y′j-yj)×y′j×(1-y′j)×bh(5)
Figure BDA0002331115100000083
2.1.4) after the network training is finished, substituting the network training into the input parameters of the country studied in the current year to obtain the transaction electric quantity and the transaction electricity price reported next time by the country.
2.2) method for determining price reported by home country
After the content reported by other countries is determined according to the method, the transaction price reported by the home country can be further determined to obtain the maximum benefit. The specific steps are described below:
2.2.1) assuming that the country does not participate in the transaction, namely only other countries participate in the transaction, and determining the final clearing result according to the predicted contents reported by the other countries and the high-low matching principle.
2.2.2) if the country is a trade selling country, selecting the matching pair with the highest clearing price (price difference to average value) as a reference; if the country is a trading buying country, the matching pair with the lowest clearing price (price difference to average value) is selected as a reference.
2.2.3) if the country is a trade selling country, determining the previous selling party of the matched pair selling party, and subtracting a small value (such as 1 percent) from the reported trade price to be used as the final reported trade price; if the country is a transaction country, determining the previous buyer of the matched pair of buyers, and adding a small value (such as 1 percent of the ratio) to the transaction price reported by the previous buyer to be used as the final reported transaction price.
The transaction price reported by the home country determined by the steps can ensure that the transaction sold country has the largest profit and the transaction purchased country has the smallest cost.
The above description is only an embodiment of the present invention, and not intended to limit the scope of the present invention, and all equivalent structures or equivalent flow transformations made by using the contents of the specification and the drawings, or applied directly or indirectly to other related systems, are included in the scope of the present invention.

Claims (9)

1. A transaction model and quotation strategy suitable for a cross-country electricity-carbon market is characterized in that: the method comprises the following steps:
1) establishing a double-layer transaction model suitable for a cross-country electricity-carbon market;
2) and establishing a quotation strategy suitable for the cross-country electricity-carbon market double-layer trading model.
2. The trading model and quotation strategy applicable to the cross-country electricity-carbon market according to claim 1, wherein the step 1) comprises the following steps:
1.1) establishing an upper-layer cross-country power high-low matching transaction model, completing cross-country power high-low matching transaction, reporting transaction electric quantity and transaction electricity price by each country simultaneously, and clearing the market based on a high-low matching principle;
1.2) establishing a lower-layer cross-country carbon quota high-low matching transaction model: firstly, calculating the carbon emission considering transnational power trading, and determining the carbon quota trading volume to be reported by each country; and then, in the carbon quota trading market, reporting the carbon quota trading volume and the trading price by each country at the same time, and clearing the market based on a high-low matching principle.
3. The trading model and pricing strategy for the cross-country electricity-carbon market according to claim 2, wherein the step 1.1) comprises the following steps:
1.1.1) the electric power is sent out of the country and sorted from low to high according to the transaction electricity price reported by the electric power, and the reported transaction electricity quantity is taken as the equivalent length of the country in the sequence during sorting; when the transaction electricity prices are the same, the transaction electricity prices are ranked in parallel;
1.1.2) the power accepting countries are sorted from high to low according to the transaction electricity prices reported by the power accepting countries, and the reported transaction electricity quantity is used as the equivalent length of the power accepting countries in the sequence during sorting; when the transaction electricity prices are the same, the transaction electricity prices are ranked in parallel;
1.1.3) forming price pairs by the power transmitting country and the power receiving country according to the sequence, namely sequencing the power price difference pairs from high to low;
1.1.4) clearing in turn according to the reported transaction electric quantity, wherein the clearing price is the average value of each price difference pair; clearing until the valence difference pair is 0, and failing to make a trade when the valence difference pair is a negative value; and if two or more than two price difference pairs are the same, distributing the reported electric quantity equally according to the current residual reported electric quantity of each country.
4. The trading model and pricing strategy for the cross-country electricity-carbon market according to claim 3, wherein the step 1.2) comprises the following steps:
1.2.1) calculating and considering the carbon emission of transnational power transaction, and determining the carbon quota transaction amount to be reported by each country by combining the power carbon emission quota distributed in the current year;
1.2.2) in the carbon quota trading market, reporting the carbon quota trading volume and the trading price simultaneously by each country, and clearing the market based on the high-low matching principle.
5. The trading model and pricing strategy for the cross-country electricity-carbon market according to claim 4, wherein the step 1.2.1) comprises the following steps:
1.2.1.1) predicting the monthly cross-country trade electric quantity and the load level of the country in the current year;
1.2.1.2) predicting the power generation proportion of various energy sources in the current year of the country every month;
1.2.1.3) calculating the predicted electric power carbon emission in the current year in China;
1.2.1.4) calculating the transaction amount of the carbon quota which should be reported by the country.
6. The trading model and pricing strategy for the cross-country electricity-carbon market according to claim 5, wherein the step 1.2.2) comprises the following steps:
1.2.2.1) ordering the carbon quota selling countries according to the respective reported transaction prices from low to high, and taking the reported carbon quota transaction amount as the equivalent length of the country in the sequence during ordering; when the transaction prices are the same, the transaction prices are ranked in parallel;
1.2.2.2) ordering the carbon quota buying countries according to the transaction prices reported by the countries from high to low, and taking the reported carbon quota transaction amount as the equivalent length of the country in the sequence during ordering; when the transaction prices are the same, the transaction prices are ranked in parallel;
1.2.2.3) forming price pairs for the carbon quota selling country and the carbon quota buying country according to the ordering, namely ordering the carbon quota price difference pairs from high to low;
1.2.2.4) clearing in turn according to the reported carbon quota transaction amount, wherein the clearing price is the average value of each price difference pair; clearing until the valence difference pair is 0, and failing to make a trade when the valence difference pair is a negative value; and if two or more than two price difference pairs are the same, distributing the carbon quota according to the current residual reported carbon quota of each country.
7. The trading model and pricing strategy for the cross-country electricity-carbon market according to any of claims 1-6, wherein the step 2) comprises the following steps:
2.1) the study and prediction of the contents reported by other countries: the three-layer BP neural network is taken as a frame, and network training is completed through a gradient descent algorithm; then substituting the input parameters of other countries in the current year to obtain the next reported transaction amount and transaction price of other countries;
2.2) the method for determining the price reported by the home country: after the content reported by other countries is determined according to the method, the transaction price reported by the home country is further determined so as to obtain the maximum benefit.
8. The trading model and pricing strategy for the cross-country electricity-carbon market according to claim 7, wherein the step 2.1) comprises the following steps:
2.1.1) constructing a three-layer neural network, which comprises an input layer, a hidden layer and an output layer; the input parameters are historical information published by the researched country, and the output parameters are transaction electric quantity and transaction electricity price reported by the researched country all the year round;
2.1.2) calculating an output parameter through a neural network by using the input parameter, and obtaining a training error;
2.1.3) judging whether the training error meets the given threshold requirement, if so, finishing the training; if not, updating the weight and returning to the step 2.1.2);
2.1.4) after the network training is finished, substituting the network training into the input parameters of the country studied in the current year to obtain the transaction electric quantity and the transaction electricity price reported next time by the country.
9. The trading model and pricing strategy for the cross-country electricity-carbon market according to claim 8, wherein the step 2.2) comprises the following steps:
2.2.1) assuming that the country does not participate in the transaction, namely only other countries participate in the transaction, and determining a final clearing result according to the predicted contents reported by the other countries directly according to a high-low matching principle;
2.2.2) if the country is a trade selling country, selecting the matching pair with the highest clearing price as a reference; if the country is a trading buying country, selecting a matching pair with the lowest clearing price as a reference, wherein the clearing price is a price difference pair average value;
2.2.3) if the country is a trade selling country, determining the previous selling party of the matched pair selling party, and subtracting a very small value from the reported trade price to be used as the final reported trade price; if the country is a transaction buying country, determining the former buying party of the matched pair of buying parties, and adding a small value to the transaction price reported on the former buying party to be used as the final reported transaction price.
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