CN111628498A - Multi-target power distribution network reconstruction method and device considering power distribution network reliability - Google Patents
Multi-target power distribution network reconstruction method and device considering power distribution network reliability Download PDFInfo
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Abstract
The invention relates to a multi-target power distribution network reconstruction method and a device considering the reliability of a power distribution network, wherein the method comprises the following steps: a data acquisition step: acquiring a topological relation and element information of a power distribution network system; a power distribution network reconstruction model construction step: constructing a power distribution network reconstruction model, wherein the power distribution network reconstruction model comprises a power distribution network reliability objective function and a network loss objective function; acquiring a power distribution network reconstruction scheme: solving the power distribution network reconstruction model to obtain a plurality of power distribution network reconstruction schemes; determining an optimal reconstruction scheme: and (4) adopting a hierarchical analysis model, and obtaining the optimal reconstruction scheme from the plurality of power distribution network reconstruction schemes obtained in the power distribution network reconstruction scheme obtaining step through expert scoring. Compared with the prior art, the improved Monte Carlo algorithm is adopted for reliability calculation, and meanwhile, the minimum line loss and the reliability index are used as main targets, so that the calculation efficiency of the reliability of the power distribution network is improved, and the economy of the reconstruction scheme of the power distribution network is improved.
Description
Technical Field
The invention relates to the field of power distribution network reconstruction, in particular to a multi-target power distribution network reconstruction method and device considering power distribution network reliability.
Background
The optimization of the operation mode of the power distribution network can reduce the benefits brought by the network loss, but the influences of load balance, the quality of power supply voltage, the safety and the reliability of the power grid and the like must be considered during implementation, and certain constraints are imposed. Therefore, the objective function of the optimization of the operation mode of the power distribution network is not single network loss but a multi-objective decision problem.
The network reconstruction is taken as an important measure for optimizing the structure of the power grid, and is paid attention by scholars at home and abroad, and the optimization target of the network reconstruction at present is generally to reduce the line loss of the power grid, balance the load and the like. And commonly used algorithms for power distribution network reconstruction, such as an analytic algorithm, a heuristic method, a load point exchange method and the like. In recent years, with the increase in demand for electric power and the rapid increase in research on a Distribution Management System (DMS), reconfiguration of a distribution network has been noticed by more students as an important function of the DMS. The main forward direction they have explored is to find a search algorithm that can optimize the topology of the distribution network. The reconstruction of the power distribution network is a multi-target nonlinear hybrid optimization problem, most of the existing algorithms select a main target by a single target function or by adopting a dimension reduction method, and other targets are used as constraint processing.
For example, in the conventional power distribution network reconstruction, the minimum line loss is used as an objective function, and the requirements on system safety and reliability cannot be met.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a multi-target power distribution network reconstruction method and device considering the reliability of a power distribution network, which are comprehensive in consideration and high in calculation efficiency.
The purpose of the invention can be realized by the following technical scheme:
a multi-target power distribution network reconstruction method considering power distribution network reliability comprises the following steps:
a data acquisition step: acquiring a topological relation and element information of a power distribution network system;
a power distribution network reconstruction model construction step: constructing a power distribution network reconstruction model, wherein the power distribution network reconstruction model comprises a power distribution network reliability objective function and a network loss objective function;
acquiring a power distribution network reconstruction scheme: solving the power distribution network reconstruction model to obtain a plurality of power distribution network reconstruction schemes;
determining an optimal reconstruction scheme: and (4) adopting a hierarchical analysis model, and obtaining the optimal reconstruction scheme from the plurality of power distribution network reconstruction schemes obtained in the power distribution network reconstruction scheme obtaining step through expert scoring.
Further, the calculation expression of the power distribution network reliability objective function is as follows:
in the formula (f)ASAIFor the reliability of the power supply of the AISI system, fEENSFor the power supply reliability of EENS systems, NiIs the number of users of the load point i, UiMean annual outage time at load point i, PaiIs the average load of the access load point i.
Further, the reliability of the power distribution network reliability objective function is calculated by adopting an improved Monte Carlo method, wherein the improved Monte Carlo method comprises the following steps:
s101: the method comprises the steps of initializing a power distribution network system and setting simulation times according to topological relation and element information of the power distribution network system, wherein the element information comprises load point information;
s102: sampling the system state, generating a random number for each load point in the power distribution network system, respectively judging whether the random number of each load point is in a load point fault interval corresponding to the load point, and if so, judging that the load point has a fault; otherwise, the load point is in a normal state;
s103: analyzing the fault state, and calculating and recording the power failure time of each load point;
s104: repeating and sequentially executing the steps S102 to S103, and performing fault simulation until the preset simulation times are reached;
s105: and calculating the average power failure time of each load point, thereby calculating the reliability of the power distribution network system.
Further, in step S103, the failure state analysis specifically includes dividing the load points into three categories, including a load point at which the power supply is recovered after the failed component is repaired, a load point at which the power supply is recovered by switching the power supply, and a load point at which the power supply can be recovered after the section switch is operated.
Further, the power failure time T of the load point of power restoration after the fault element is repairedTTTR1The calculation expression of (a) is:
TTTTR1＝TTTR
in the formula, TTTRRepair time for failed components;
the power failure time T of the load point for recovering power supply through power supplyTTTR2The calculation expression of (a) is:
TTTTR2＝Ts
in the formula, TsFor the transfer of operating time;
the power failure time T of the load point capable of recovering power supply after the section switch is operatedTTTR3The calculation expression of (a) is:
TTTTR3＝Tcut
in the formula, TcutThe switch operation time is segmented.
Further, the calculation expression of the network loss objective function is as follows:
in the formula, PlossFor the active power loss of the system, kiIs the state of the load point i, k i1 or 0, k i1 is that the load point i is in a closed state, k i0 is the load point I in the off state, IiIs the current flowing at the load point i, riIs the resistance of the load point i and n is the total number of load points in the system.
Further, the power distribution network reconstruction model further comprises a line constraint, and the expression of the line constraint is as follows:
in the formula, pijAs active load of line ij, qijFor reactive loading of line ij, line ij being the line between load point i and load point j, pdjIs the active load of load point j, qdjIs the reactive load of the load point j, k is the number of load points, ViIs the voltage at the load point i, VjIs the voltage at the load point j, zijIs the impedance of line ij, iijIs the current of line ij, IiA current flowing at a load point Ij,maxMaximum current flowing at load point j, Sl' is the actual load of line ij, SlIs the maximum load of line ij.
Further, the power distribution network reconstruction model further comprises topology constraints, wherein the topology constraints are that the system topology structure is kept to be radiation and no island is generated in the power distribution network reconstruction process.
Further, the power distribution network reconstruction model further comprises load point voltage offset constraints, and the expression of the load point voltage offset constraints is as follows:
Vj,min≤Vj≤Vj,max
in the formula, VjIs the voltage at the load point j, Vj,minLower voltage limit of load point j, Vj,maxThe upper voltage limit at load point j.
The traditional method does not consider the problems of load rate, voltage offset and the like of a line.
Further, before the step of determining the optimal reconstruction scheme, the method further comprises:
a reconstruction scheme pre-screening step: acquiring the load rate of a power distribution network reconstruction scheme, if the load rate of part of lines in the power distribution network reconstruction scheme exceeds a preset heavy load operation threshold value, deleting the power distribution network reconstruction scheme, otherwise, reserving the power distribution network reconstruction scheme, and traversing each power distribution network reconstruction scheme; and executing the optimal reconstruction scheme determination step on the reserved power distribution network reconstruction scheme.
The invention also provides a multi-target power distribution network reconstruction device considering the reliability of the power distribution network, which comprises a memory and a processor, wherein the memory stores a computer program, and the processor calls the computer program to execute the steps of the method.
Compared with the prior art, the invention has the following advantages:
(1) the power distribution network reconstruction model takes the reliability of a power distribution system as one of the objective functions, is favorable for improving the reliability of the reconstructed system, and reduces the reliability calculation time by utilizing an improved Monte Carlo simulation method; the minimum line loss is taken as one of the objective functions, so that unnecessary loss is reduced, and the economy is improved; when the power distribution network reconstruction model is solved, firstly, feasible schemes are preliminarily screened out through constraint conditions and objective functions, and then, an optimal scheme is selected out through an analytic hierarchy process.
(2) The method can realize reconstruction quickly and reduce economic loss of people caused by power failure.
(3) The invention realizes the reconstruction of the power distribution network by taking the requirements of load rate, voltage offset and the like as constraint conditions, obtains the optimal reconstruction scheme, and is close to the actual power grid operation.
(4) The method also comprises a reconstruction scheme pre-screening step before the optimal reconstruction scheme determining step is executed, and the power distribution network reconstruction scheme is screened based on the load rate, so that the workload of the expert for scoring is reduced, and the reliability of the actual operation of the power distribution network reconstruction scheme is improved.
Drawings
FIG. 1 is a flow diagram of a multi-target power distribution network reconstruction method considering power distribution network reliability according to the present invention;
FIG. 2 is a schematic diagram of an improved Monte Carlo method power system reliability evaluation process according to the present invention;
fig. 3 is a schematic diagram of an initial grid structure of a load point system according to embodiment 14 of the present invention, in which a dotted line is a communication line, which is normally open at ordinary times, and a solid line is a closed line;
FIG. 4 is a graph of normalized line profiles for various metrics, FIG. 4(a) is a graph of line loss for various metrics, FIG. 4(b) is a graph of line profile for various metrics voltage offset, FIG. 4(c) is a graph of line profile for various AISI metrics, and FIG. 4(d) is a graph of line profile for various EENS metrics;
fig. 5 is a schematic diagram of a final rack reconstruction result obtained in the embodiment of the present invention.
Detailed Description
The invention is described in detail below with reference to the figures and specific embodiments. The present embodiment is implemented on the premise of the technical solution of the present invention, and a detailed implementation manner and a specific operation process are given, but the scope of the present invention is not limited to the following embodiments.
Example 1
The embodiment provides a multi-target power distribution network reconstruction method considering the reliability of a power distribution network, which is characterized in that an improved Monte Carlo algorithm is adopted for reliability calculation, the minimum line loss and reliability indexes are used as main targets, the conditions of load rate and voltage offset are met, mutual constraint is achieved, and power distribution network reconstruction is achieved.
As shown in fig. 1, the multi-target distribution network reconstruction method includes the following steps:
data acquisition step S1: acquiring a topological relation and load data of a power distribution network system;
a power distribution network reconstruction model construction step S2: constructing a power distribution network reconstruction model, wherein the power distribution network reconstruction model comprises a power distribution network reliability objective function and a network loss objective function;
the calculation expression of the power distribution network reliability objective function is as follows:
in the formula (f)ASAIFor the reliability of the power supply of the AISI system, fEENSFor the power supply reliability of EENS systems, NiIs the number of users of the load point i, UiIs the annual average of load point iTime of power failure, PaiIs the average load of the access load point i.
The calculation expression of the network loss objective function is as follows:
in the formula, PlossFor the active power loss of the system, kiIs the state of the load point i, k i1 or 0, k i1 is that the load point i is in a closed state, k i0 is the load point I in the off state, IiIs the current flowing at the load point i, riIs the resistance of the load point i and n is the total number of load points in the system.
The constraint conditions of the power distribution network reconstruction model comprise:
1) line constraint, wherein the power distribution network line constraint conditions comprise power balance constraint, kirchhoff voltage law constraint, line limit constraint, voltage limit constraint and the like. The computational expression of the harness constraint is as follows:
in the formula, pijAs active load of line ij, qijFor reactive loading of line ij, line ij being the line between load point i and load point j, pdjIs the active load of load point j, qdjIs the reactive load of the load point j, k is the number of load points, ViIs the voltage at the load point i, VjIs the voltage at the load point j, zijIs the impedance of line ij, iijIs the current of line ij, IiA current flowing at a load point Ij,maxMaximum current flowing at load point j, Sl' is the actual load of line ij, SlIs the maximum load of line ij.
2) And (4) topological constraint, wherein the system topological structure is kept to be radiation and no island is generated all the time in the reconstruction process.
3) And load point voltage offset constraint, wherein the expression of the load point voltage offset constraint is as follows:
Vj,min≤Vj≤Vj,max
in the formula, VjIs the voltage at the load point j, Vj,minLower voltage limit of load point j, Vj,maxThe upper voltage limit at load point j.
The embodiment adopts an improved Monte Carlo method to evaluate the reliability of the power system, and comprises the following steps:
s101: the method comprises the steps of initializing a power distribution network system and setting simulation times according to topological relation and element information of the power distribution network system, wherein the element information comprises load point information;
s102: sampling the system state, generating a random number for each load point in the power distribution network system, respectively judging whether the random number of each load point is in a load point fault interval corresponding to the load point, and if so, judging that the load point has a fault; otherwise, the load point is in a normal state;
s103: analyzing the fault state, and calculating and recording the power failure time of each load point;
specifically, the load points are classified into three categories through fault state analysis, including a load point for recovering power supply after a fault element is repaired, a load point for recovering power supply through power supply transfer and a load point capable of recovering power supply after a section switch is operated;
power failure time T of load point for recovering power supply after fault element repairTTTR1The calculation expression of (a) is:
TTTTR1＝TTTR
in the formula, TTTRRepair time for failed components;
power failure time T of load point for recovering power supply through power supply transferTTTR2The calculation expression of (a) is:
TTTTR2＝Ts
in the formula, TsFor the transfer of operating time;
power failure time T of load point capable of recovering power supply after section switch operationTTTR3The calculation expression of (a) is:
TTTTR3＝Tcut
in the formula, TcutIs the section switch operation time;
s104: repeating and sequentially executing the steps S102 to S103, and performing fault simulation until the preset simulation times are reached;
s105: and calculating the average power failure time of each load point, thereby calculating the reliability of the power distribution network system by adopting a calculation expression of a power distribution network reliability objective function.
The specific description is as follows:
the Monte Carlo method is a method commonly used for evaluating the reliability of a power system at present, and mainly comprises three parts, namely system state sampling, system state analysis and system evaluation index statistics, in the application process, and the evaluation main steps are shown in FIG. 1.
As can be seen from fig. 2, when applying the monte carlo method, it is necessary to start with the random state sampling of the system, and the state of the system can be randomly sampled without regard to the duration and transition rate of the state. Higher accuracy is required in the evaluation process, so that the sampling algorithm can be improved in the sampling process, the contrast is reduced, and the calculation precision is improved.
After the system state sampling is completed, the state needs to be analyzed and a test function value is calculated, and the calculation method of the test function corresponds to the reliability evaluation index. The state analysis of the power generation and transmission system mainly starts from two aspects: on one hand, the power flow analysis is carried out on the state of the power transmission system to judge whether a fault exists in the system; on the other hand, on the basis of trend analysis, the actual state of the system is optimized, the fault state existing in the system is eliminated, and the safe operation of the system is ensured.
The implementation manner of this step in this embodiment is:
the improved Monte Carlo method comprises the following steps:
s201: and inputting the network topology and parameter information, including all element information in the system, and initializing the system. The simulation times NY are set.
S202: sampling the system state; randomly generating [0,1 ] for each element j in the system]Distributed random number RjJudgment ofCutting off the random number RjWhether it falls in the element failure region [0, λj) If n indicates the failure of the element j1 is ═ 1; otherwise, the element is in normal state nj＝0。
S203: and analyzing the fault state, classifying the load points to determine the type of the influence of the fault on the load, and classifying the load points into the following three types.
1) And restoring the power supply load point after the fault element is repaired.
2) And (4) restoring the power supply through the power supply.
3) The load point of the power supply can be recovered after the section switch is operated.
For the load point of the type 1), the total power failure time is the element repair time TTTTR＝TTTTR+TTTRWherein T isTTRTime is repaired for the failed component. For class 2) load point, total time of power failure TTTTR＝TTTTR+TsWherein T issFor transferring operating time. For class 3) load point, total time of power failure TTTTR＝TTTTR+TcutWherein T iscutThe switch operation time is segmented. Accumulated power off time TTTTRAnd recorded.
S204: updating the reliability index of the load point;
s205: and judging whether the simulation times reach currently, if not, returning to the step S2 to continue the simulation, wherein the simulation times NY is NY + 1. If yes, returning to the step S6;
s206: total power off time T for all load pointsTTTRTaking an average value, namely: u shapej＝TTTTRand/NY. Reliability indexes of each load point can be obtained;
s207: calculating a system reliability index according to the load point reliability index; and a calculation expression of a power distribution network reliability objective function can also be adopted to directly calculate the reliability index of the power distribution network system.
A power distribution network reconstruction scheme acquisition step S3: solving the power distribution network reconstruction model to obtain a plurality of power distribution network reconstruction schemes;
optimal reconstruction scheme determination step S4: firstly, carrying out primary screening by calculating the load rate of each power distribution network reconstruction scheme; and then, a hierarchical analysis model is adopted, and an optimal reconstruction scheme is obtained from the plurality of power distribution network reconstruction schemes obtained in the power distribution network reconstruction scheme obtaining step through expert scoring.
The action object of the embodiment is a 14-load point system, the failure rate of the overhead line in the system is 0.00276 times/km, and the repair time is 12.6 h/time; the fault rate of the transformer is 0.0024 times per unit, and the repair time is 26.3068h per unit; the operation time of the isolating switch is 1h, the operation time of the standby power supply is 1h, and the topological structure of the grid is shown in figure 3.
The method comprises the steps of carrying out reconstruction calculation on an example under the condition that various constraint conditions are met, generating 9 reconstruction schemes to meet initial requirements, calculating the reliability indexes of the 9 reconstruction schemes by adopting the Monte Carlo method based on equally dispersed sampling, wherein the result is shown in table 1, and the table 1 is the load rate of each line in the 9 schemes.
TABLE 1 post-system reconstruction scheme
Through calculation, the load rate of part of the lines in the schemes 1, 2, 3, 4, 6 and 8 reaches more than 70%, so that the lines are in heavy-load operation and cannot meet the requirements of safety and economy. Therefore, schemes 5, 7 and 9 were selected as the final screening targets.
The calculation results of the load factor in this example are shown in table 2.
In table 2, the maximum load per line is taken as 20MW, and F is taken as the exclusion scheme.
An analytic hierarchy process model is adopted, expert scoring is introduced, the network loss, the voltage offset, the AISI scheme and the EENS scheme are scored, the weights of all indexes in various schemes are comprehensively compared, the optimal reconstruction scheme is obtained, and the result is shown in table 3. An analytic hierarchy process model is adopted, and experts are introduced to score the model into the prior art, so that the detailed description is omitted.
As shown in fig. 4, it is a normalized line graph of each index under different schemes.
The final rack reconstruction results are shown in fig. 5.
As can be seen from table 3, in the conventional power distribution network reconstruction, the minimum line loss is taken as an objective function, and the problems of load rate, voltage offset and the like of a line are not considered, so that the requirements on system safety and reliability cannot be met. The optimal reconstruction scheme provided by the method comprehensively considers the system safety problem and the system reliability, and has practical feasibility.
The embodiment also provides a multi-target power distribution network reconstruction device considering the reliability of the power distribution network, which comprises a memory and a processor, wherein the memory stores a computer program, and the processor calls the computer program to execute the steps of the multi-target power distribution network reconstruction method considering the reliability of the power distribution network.
TABLE 2 line load Rate for each case
TABLE 3 comprehensive consideration of optimality
The foregoing detailed description of the preferred embodiments of the invention has been presented. It should be understood that numerous modifications and variations could be devised by those skilled in the art in light of the present teachings without departing from the inventive concepts. Therefore, the technical solutions available to those skilled in the art through logic analysis, reasoning and limited experiments based on the prior art according to the concept of the present invention should be within the scope of protection defined by the claims.
Claims (10)
1. A multi-target power distribution network reconstruction method considering power distribution network reliability is characterized by comprising the following steps:
a data acquisition step: acquiring a topological relation and element information of a power distribution network system;
a power distribution network reconstruction model construction step: constructing a power distribution network reconstruction model, wherein the power distribution network reconstruction model comprises a power distribution network reliability objective function and a network loss objective function;
acquiring a power distribution network reconstruction scheme: solving the power distribution network reconstruction model to obtain a plurality of power distribution network reconstruction schemes;
determining an optimal reconstruction scheme: and (4) adopting a hierarchical analysis model, and obtaining the optimal reconstruction scheme from the plurality of power distribution network reconstruction schemes obtained in the power distribution network reconstruction scheme obtaining step through expert scoring.
2. The multi-target power distribution network reconstruction method considering power distribution network reliability according to claim 1, wherein the calculation expression of the power distribution network reliability objective function is as follows:
in the formula (f)ASAIFor the reliability of the power supply of the AISI system, fEENSFor the power supply reliability of EENS systems, NiIs the number of users of the load point i, UiMean annual outage time at load point i, PaiIs the average load of the access load point i.
3. The method for reconstructing the multi-target power distribution network considering the reliability of the power distribution network according to claim 1, wherein the reliability of the reliability objective function of the power distribution network is calculated by using an improved Monte Carlo method, and the improved Monte Carlo method comprises the following steps:
s101: the method comprises the steps of initializing a power distribution network system and setting simulation times according to topological relation and element information of the power distribution network system, wherein the element information comprises load point information;
s102: sampling the system state, generating a random number for each load point in the power distribution network system, respectively judging whether the random number of each load point is in a load point fault interval corresponding to the load point, and if so, judging that the load point has a fault; otherwise, the load point is in a normal state;
s103: analyzing the fault state, and calculating and recording the power failure time of each load point;
s104: repeating and sequentially executing the steps S102 to S103, and performing fault simulation until the preset simulation times are reached;
s105: and calculating the average power failure time of each load point, thereby calculating the reliability of the power distribution network system.
4. The method as claimed in claim 3, wherein in step S103, the analysis of the fault state is to classify the load points into three categories, including a load point at which the power supply is recovered after the fault element is repaired, a load point at which the power supply is recovered by switching the power supply, and a load point at which the power supply can be recovered after the section switch is operated.
5. The multi-target power distribution network reconstruction method considering power distribution network reliability as claimed in claim 4, wherein the power failure time T of the load point of power restoration after the fault element is repairedTTTR1The calculation expression of (a) is:
TTTTR1＝TTTR
in the formula, TTTRRepair time for failed components;
the power failure time T of the load point for recovering power supply through power supplyTTTR2The calculation expression of (a) is:
TTTTR2＝Ts
in the formula, TsFor the transfer of operating time;
the power failure time T of the load point capable of recovering power supply after the section switch is operatedTTTR3The calculation expression of (a) is:
TTTTR3＝Tcut
in the formula, TcutThe switch operation time is segmented.
6. The multi-target power distribution network reconstruction method considering power distribution network reliability as claimed in claim 1, wherein the calculation expression of the network loss objective function is as follows:
in the formula, PlossFor the active power loss of the system, kiIs the state of the load point i, ki1 or 0, ki1 is that the load point i is in a closed state, ki0 is the load point I in the off state, IiIs the current flowing at the load point i, riIs the resistance of the load point i and n is the total number of load points in the system.
7. The multi-target power distribution network reconstruction method considering the reliability of the power distribution network according to claim 1, wherein the power distribution network reconstruction model further comprises a line constraint, and the expression of the line constraint is as follows:
in the formula, pijAs active load of line ij, qijFor reactive loading of line ij, line ij being the line between load point i and load point j, pdjIs the active load of load point j, qdjIs the reactive load of the load point j, k is the number of load points, ViIs the voltage at the load point i, VjIs the voltage at the load point j, zijIs the impedance of line ij, iijIs the current of line ij, IiA current flowing at a load point Ij,maxMaximum current flowing at load point j, Sl' is the actual load of line ij, SlIs the maximum load of line ij.
8. The multi-target power distribution network reconstruction method considering the reliability of the power distribution network according to claim 1, wherein the power distribution network reconstruction model further comprises load point voltage offset constraints, and the load point voltage offset constraints have the expression:
Vj,min≤Vj≤Vj,max
in the formula, VjIs the voltage at the load point j, Vj,minLower voltage limit of load point j, Vj,maxThe upper voltage limit at load point j.
9. The method for reconstructing a multi-target distribution network considering distribution network reliability according to claim 1, wherein the step of determining the optimal reconstruction scheme further comprises:
a reconstruction scheme pre-screening step: acquiring the load rate of a power distribution network reconstruction scheme, if the load rate of part of lines in the power distribution network reconstruction scheme exceeds a preset heavy load operation threshold value, deleting the power distribution network reconstruction scheme, otherwise, reserving the power distribution network reconstruction scheme, and traversing each power distribution network reconstruction scheme; and executing the optimal reconstruction scheme determination step on the reserved power distribution network reconstruction scheme.
10. A multi-objective power distribution network reconfiguration device taking into account the reliability of a power distribution network, characterized by comprising a memory and a processor, wherein the memory stores a computer program, and the processor calls the computer program to execute the steps of the method according to any one of claims 1 to 9.
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