CN104063808A - Trans-provincial power transmission cascade hydropower station group peak-shaving dispatching two-phase search method - Google Patents

Abstract

The invention belongs to the field of hydropower dispatching operation, and discloses a trans-provincial power transmission cascade hydropower station group peak-shaving dispatching two-phase search method. The characteristics of load complementation between receiving-end power grids can be fully utilized and different peak power demands of multiple receiving-end power grids can be taken into consideration to realize reasonable allocation of quality resources between the multiple power grids. The technical scheme is as follows: the trans-provincial power transmission cascade hydropower station group peak-shaving goal is realized in two phases according to the characteristics of the trans-provincial power transmission peak-shaving issue; in the optimization calculation process, load curves with great difference of a plurality of provincial power grids are reconstructed into an overall load by adoption of a load reconstruction strategy, and the hydropower station power output process is determined on the basis by adoption of a variable-scale optimization method; and then the allocation process of output power of the station between the receiving-end power grids is optimized by adoption of an exterior point search method, an initial solution is generated through successive load shedding, and the peak power sizes of the receiving-end power grids are corrected in an iterative mode by taking load level as heuristic information to meet the peak-shaving demands of the multiple power grids. Peak-shaving demands of multiple provincial power grids can be taken into consideration, and a peak-shaving result better than actual dispatching can be obtained. The method is feasible.

Description

One transprovincially power transmission Hydropower Stations peak regulation is dispatched two stage Search methods

Technical field

The present invention relates to water power management and running field, particularly one transprovincially power transmission Hydropower Stations peak regulation dispatch two stage Search methods.

Technical background

Through the basin step rolling development of more than 20 years, China has formed especially big basin and master stream Hydro Power Systems with Cascaded Reservoirs general layout, the Wujiang River, Hongsuihe River two large Hydropower Base master stream steps substantially all build up, Jinsha jiang River middle and lower reaches, Middle-lower Reaches of The Lancang River, Yalongjiang River, the Dadu River four especially big basin of large Hydropower Base Hydropower Stations have also entered comprehensive completion and concentrated operation period, and Chinese water power has marched toward on a large scale transregional dissolving scheduling and operation phase transprovincially.Simultaneously, follow putting into operation that south electric network " two cross direct current " and national grid " give direct current " again, water power transprovincially transregional transmission of electricity scale has been realized rapid growth, has become the important electric power resource of the developed area such as Yangtze River Delta, Delta of the Pearl River receiving end electrical network.Different from on-site elimination mode, present stage single basin step need to be simultaneously for multiple provincial power networks or regional power grid transmit electric power, meet the particularly peak power demand of electricity consumption in these receiving end areas.How to utilize the complementary characteristic of loading between each receiving end electrical network, give full play to the adjusting function of these high-quality hydroelectric resources, alleviate better the ubiquitous peak regulation pressure of provincial power network such as the regional power grid such as East China, south and Shanghai, Jiangsu, Zhejiang, Guangdong and become theory and practice problem urgently to be resolved hurrily in China's electrical network and Model for Cascade Hydroelectric Stations, its Study on Problems is the key point of China's flood TV university scope resource distribution.

This respect research at present there is not yet pertinent literature report both at home and abroad.Transprovincially power transmission peaking problem relates to multiple electrical networks, need to process the more complicated interconnected constraint of electric power and optimization aim function, face complex conditions and the difficult problem such as distributing electric power between the great multiple network loads of difference and net, Optimization Solution difficulty is larger, needs effective method for solving.Achievement of the present invention taking master stream, CHINA SOUTHERN POWER Hongsuihe River Hydropower Stations transprovincially power transmission problem as background, taken into account many peaking demand of power grids that Hydropower Stations faces in power transmission transprovincially, there is important popularizing value.

Summary of the invention

The technical problem to be solved in the present invention be to provide one transprovincially power transmission Hydropower Stations peak regulation dispatch two stage Search methods, can make full use of the load complementary characteristic between receiving end electrical network, take into account the differentiation peak value electricity needs of multiple receiving end electrical networks, realize the Reasonable Regulation And Control configuration of high-quality resource between multiple electrical networks.

Technical scheme of the present invention is: the present invention disclosed one transprovincially power transmission Hydropower Stations peak regulation dispatch two stage Search methods, comprise load reconstruct and output distribution two large divisions, complete power transmission peak regulation scheduling process transprovincially according to following step (1)-(8):

(1) initial calculation condition, comprises hydroelectric station operation condition and constraint, and each network load process and be subject to electric weight demand;

(2) make power station numbering m=1;

(3) what build m power station faces workload demand { C ₁, C ₂..., C _t, wherein C ₁, C ₂..., C _tbe respectively the period 1,2 ..., the load value of T;

(4) with { C ₁, C ₂..., C _tbe workload demand, under given objective function and constraint condition, adopt DFP-method to determine day 96 processes of exerting oneself in m power station;

(5), under given controlled condition, adopt outer point searching method to determine the output distribution process of m power station at G electrical network, i.e. { p _{m, 1,1}, p _{m, 1,2}..., p _{m, 1, T}, { p _{m, 2,1}, p _{m, 2,2}..., p _{m, 2, T}..., { p _{m, G, 1}, p _{m, G, 2}..., p _{m, G, T}, wherein, { p _{m, 1,1}, p _{m, 1,2}..., p _{m, 1, T}, { p _{m, 2,1}, p _{m, 2,2}..., p _{m, 2, T}..., { p _{m, G, 1}, p _{m, G, 2}..., p _{m, G, T}represent that respectively m power station is the 1st, 2 ..., the 1st, 2 of G electrical network ..., the load value of T period;

(6) recalculate each electrical network residue load;

(7) make m=m+1, if m is less than or equal to M, skip to step (3); Otherwise go to step (8);

(8) calculate the desired value of current solution, judge whether target is improved, the if so, initial solution using current solution as next round iteration, and skip to step (2); Otherwise, judge whether to reach to maximum iteration time, if so, algorithm search stops, output optimal result, otherwise, skip to step (2).

The present invention contrasts prior art following beneficial effect: one of the present invention transprovincially power transmission Hydropower Stations peak regulation is dispatched two stage Search methods, realizes in two stages power transmission Hydropower Stations peak regulation target transprovincially; Optimizing in computation process, first adopt load reconstruction strategy that the great load curve of multiple provincial power network differences is reconstructed into a total load, adopt on this basis DFP-method to determine the power station process of exerting oneself; Then adopt outer point searching method to optimize the assigning process of output of power station between receiving end electrical network, in assigning process, adopt successively cutting load to generate initial solution, and the height of loading is as heuristic information, the peak power size of the each receiving end electrical network of iteration correction, to meet multiple peak load regulation network requirements.Contrast prior art, the present invention can take into full account the complementary characteristic of loading between each receiving end electrical network, takes into account the electricity needs of many receiving ends electrical network in power transmission peak regulation transprovincially, realizes the Reasonable Regulation And Control configuration of high-quality resource between many electrical networks.

Brief description of the drawings

Fig. 1 is that the inventive method totally solves block diagram.

Fig. 2 is two provincial power network schematic diagram of single power station power transmission.

Fig. 3 is distributing electric power problem initial solution schematic diagram between the net that obtains of outer point searching method.

Fig. 4 (a) be in the Hydropower Stations optimum results of master stream, Hongsuihe River Guangxi Power Grid be subject to electric process.

Fig. 4 (b) be in the Hydropower Stations optimum results of master stream, Hongsuihe River Guangdong Power Grid be subject to electric process.

Embodiment

Below in conjunction with drawings and Examples, the invention will be further described.

The extensive transregional conveying transprovincially of water power is in order to regulate and control high-quality resource between the multiple electrical networks of configuration to alleviate day by day serious peak regulation pressure, this problem has obviously different from the water power scheduling problem of single electrical network in the past, need to process the more complicated interconnected constraint of electric power and optimization aim function, face complex conditions and the difficult problem such as distributing electric power between the great multiple network loads of difference and net, Optimization Solution difficulty is larger.For how making full use of the complementary characteristic of loading between each receiving end electrical network, take into account the electricity needs of many receiving ends electrical network in power transmission peak regulation transprovincially, rarely have bibliographical information both at home and abroad.The present invention disclose one transprovincially power transmission Hydropower Stations peak regulation dispatch two stage Search methods, will be transprovincially power transmission peaking problem be decomposed into two crucial subproblems (the one, considers multiple provincial power network workload demands, optimize the process of exerting oneself in definite power station; The 2nd, exert oneself process and each electrical network of stationary electric power plant is subject to electric weight, determine the distributing electric power process of power station between receiving end electrical network), and for these two subproblems, adopt stage by stage DFP-method and outer point searching method based on load reconstruct to solve respectively, to coordinate, Hydropower Stations is exerted oneself and each electrical network is subject to electric process, meets multiple peak load regulation network requirements.

The present invention is made up of load reconstruct and output distribution dual stage process.First stage adopts load reconstruction strategy that the great load curve of multiple provincial power network differences is reconstructed into a total load, adopts on this basis DFP-method to determine the power station process of exerting oneself; Subordinate phase proposes outer point searching method and optimizes the assigning process of output of power station between receiving end electrical network, adopt successively cutting load to generate initial solution, and will load height as heuristic information, the peak power size of the each receiving end electrical network of iteration correction, to meet multiple peak load regulation network requirements.

The concrete operation method in each stage is according to following process (a)-(g) be achieved:

(a) DFP-method based on load reconstruct

DFP-method based on load reconstruct mainly solves first subproblem, i.e. the optimization problem of exerting oneself of Hydropower Stations under multiple network load demands, and solution procedure is mainly divided into following two steps.

The first step: reconstruct power station faces workload demand

Because multiple provincial power network load curves aspect such as between the order of magnitude, peak and low ebb number, peak are current exists very big-difference conventionally, directly determine that the peaking operation plan in power station is very difficult.So, adopt load reconstruction strategy that many load curves are reconstructed into a total load herein, to reduce output of power station search difficulty; For making total load retain as far as possible each network load Variation Features, be especially subject to load peak and the low ebb information of the larger electrical network of electric weight proportion, in literary composition, using power transmission amount ratio as weight, adopt the method for normalizing reconstruct of loading, see following formula.

$C_t=\underset{g=1}{\overset{G}{\mathrm{\Σ}}}{R}_{m,g}\×C_{g,t}\×\frac{C_{g,t}-C_{g,\min }}{C_{g,\max }-C_{g,\min }},$ Wherein, 1≤t≤T

In formula: C _tfor the load of the residue in the t period of electrical network, MW; C _g,tfor the g electrical network load value of t period; C _{g, max}for the load maximal value of g electrical network; C _{g, min}for the load minimum value of g electrical network; G is electrical network numbering, 1≤g≤G, and G is electrical network sum; T is period sequence number, 1≤t≤T, and T is period sum.

Second step: determine output of power station process

The total load obtaining taking the first step is basis, this subproblem can be converted in the single electrical network that is easy to solve to the GROUP OF HYDROPOWER STATIONS optimization problem of exerting oneself, then adopt DFP-method to solve, referring to document: the GROUP OF HYDROPOWER STATIONS short-term change dimensional optimization dispatching method that complicated period coupled mode constraint is lower, Shen Jianjian etc., Proceedings of the CSEE .2014,34 (1): 87-95.

(b) the outer point searching method of distributing electric power between net

Second subproblem is the fixing output of power station process obtaining on last stage, utilizes the load difference between provincial power network, optimizes its assigning process at each receiving end electrical network, meets the peak regulation demand of these electrical networks.Rarely have at present the research report about this respect both at home and abroad, carry out problem solving so the inventive method according to problem feature, proposes a kind of outer point searching method.The method is taking peak load regulation network as target, and using the height of loading as heuristic information, from an initial infeasible solutions, the allocated size of iteration correction day part output of power station between receiving end electrical network, until converge to the feasible solution of problem.Because method search procedure is progressively to converge to feasible solution from an infeasible solutions, therefore be referred to as outer point searching method, its solution procedure comprises following two major parts:

(1) determine initial solution.Under stationary electric power plant is exerted oneself the condition of process, adopt following formula can obtain each electrical network and be subject to electric weight, and then to be subject to electric weight as controlling target, face load taking power station as basis, adopt successively cutting load method to determine that the ideal of each electrical network is subject to the electricity process of exerting oneself.

E _m,g＝E _m×R _m,g

In formula: E _m,gfor m power station is to the day power transmission amount of g electrical network; E _mrepresent the total electric weight of power station m in schedule periods; R _m,gfor sending the electric weight ratio of g electrical network in m power station,

Without loss of generality, Fig. 2 has provided the single power station examples of problems of two provincial power networks of power transmission simultaneously that adopts said method to obtain, due in search procedure to reach given target electric weight as the condition of convergence, so meeting electrical network, the initial solution in Fig. 3 retrained by electric weight demand for control, but violate period power balance constraint, the electricity sum of exerting oneself that is subject to of corresponding periods of all electrical networks is not equal to this period output of power station, as can be seen from Figure 3.

(2) initial solution obtaining taking the first step is basis, exert oneself and preferentially increase the rule that the high remaining lotus period exerts oneself according to preferentially reducing the low remaining lotus period, redistribute the day part electrical network that does not meet period power balance constraint and exerted oneself by electricity, until all periods all reach power balance.

In service at short term scheduling; the peak regulation task of system is born in power station mostly; object is to utilize the advantages such as Hydropower Unit start and stop are rapid, Ramp Rate is fast; electrical network peak value and load process are followed the tracks of to adjusting; to reduce the remaining load peak-valley difference of electrical network, and level and smooth as far as possible remaining load process, and then can reduce the frequent startup-shutdowns of power supply such as the poor thermoelectricity of adjusting function; save generating and consume, realize power grid security, energy-conservation, economical operation.So this method adopts the minimum Hydropower Stations peak regulation optimization aim that represents of the remaining load mean square deviation of electrical network, sees following formula:

$\min F=\underset{t=1}{\overset{T}{\mathrm{\Σ}}}{((C_t-\underset{m=1}{\overset{M}{\mathrm{\Σ}}}{p}_{m,t})-\underset{t=1}{\overset{T}{\mathrm{\Σ}}}(C_t-\underset{m=1}{\overset{M}{\mathrm{\Σ}}}{p}_{m,t})/T)}^2$

In formula: F is the remaining load mean square deviation of electrical network; P _m,tfor m power station is at the exerting oneself of t period, MW; C _tfor the load of the residue in the t period of electrical network, MW; 1≤m≤M, M is power station sum; 1≤t≤T, T is period sum.

For power transmission Hydropower Stations transprovincially, single power station need to meet the peak regulation requirement of multiple provincial power networks simultaneously, this class GROUP OF HYDROPOWER STATIONS scheduling problem is typical multi-objective optimization question, needs the remaining load mean square deviation of the multiple electrical networks of simultaneous minimization, and its objective function can be expressed as:

$\min F_g=\underset{t=1}{\overset{T}{\mathrm{\Σ}}}{((C_{g,t}-\underset{m=1}{\overset{M}{\mathrm{\Σ}}}{p}_{m,g,t})-\underset{i=1}{\overset{T}{\mathrm{\Σ}}}(C_{g,t}-\underset{m=1}{\overset{M}{\mathrm{\Σ}}}{p}_{m,g,t})/T)}^2$

g＝1，2，...，G

In formula: F _gfor the remaining load mean square deviation of g electrical network; P _{m, g, t}for delivering to exerting oneself of g electrical network, MW in the t period in m power station; C _g,tfor the g electrical network load value of t period; 1≤g≤G, G is electrical network sum.

For reducing problem solving difficulty, adopt conventional target weight method that multi-objective optimization question is converted into single goal problem, see following formula:

$\min F=\underset{g=1}{\overset{G}{\mathrm{\Σ}}}{w}_g\×\frac{F_g}{C_{g,\max }^2}$

In formula: w _gfor the target weight of g electrical network, consider the equal importance of each peaking demand of power grid, in literary composition, get w _g=1/G; C _{g, max}for the load maximal value of g electrical network, the object of introducing this variable is that the residue load in objective function is normalized, to avoid each network load level difference to cause optimization solution deviation.

Solve the constraint condition expression formula that power transmission peaking problem need to be satisfied transprovincially as follows:

(1) water balance constraint

V _m,t+1＝V _m,t+3600×(Q _m,t-q _m,t-Qd _m,t)Δ _t

In formula: V _{m, t+1}for power station m is at the storage capacity of period t+1; V _m,tfor power station m is at the storage capacity of period t; Q _m,tfor power station m is at the reservoir inflow of period t, k is total number in power station, m upstream, power station; for power station, upstream k after considering when stagnant flows into the flow of power station m at period t; Qn _m,tfor power station m is at the interval flow of period t; Qd _m,tfor power station m is in the discharge of abandoning of period t; q _m,tfor power station m is at the generating flow of period t; Δ _tfor t period hourage.

(2) power station day is controlled electric weight demand

E _m＝E' _m

In formula: E _m, E' _mrepresent respectively the total electric weight of power station m in schedule periods and given demand electric weight.

(3) electrical network is subject to electric weight demand for control

$\underset{t=1}{\overset{T}{\mathrm{\Σ}}}{p}_{m,g,t}\×{\mathrm{\Δ}}_t=E_m\×R_{m,g}$

In formula: R _m,gfor sending the electric weight ratio of g electrical network in m power station,

(4) period power balance constraint

$\underset{g=1}{\overset{G}{\mathrm{\Σ}}}{p}_{m,g,t}=p_{m,t}$

(5) generating flow constraint

${\underset{\‾}{q}}_{m,t}\≤q_{m,t}\≤{\stackrel{\‾}{q}}_{m,t}$

In formula: q _m,t, q _m,trepresent respectively the generating flow at period t and the bound thereof of power station m.

(6) outbound traffic constraints

${\underset{\‾}{S}}_{m,t}\≤S_{m,t}\≤{\stackrel{\‾}{S}}_{m,t}$

In formula: S _m,t, s _m,trepresent respectively outbound flow and the bound thereof of power station m at period t.

(7) output of power station constraint

${\underset{\‾}{p}}_{m,t}\≤p_{m,t}\≤{\stackrel{\‾}{p}}_{m,t}$

In formula: p _m,trepresent respectively the average output bound of power station m at period t.

(8) reservoir level constraint

${\underset{\‾}{Z}}_{m,t}\≤Z_{m,t}\≤{\stackrel{\‾}{Z}}_{m,t}$

In formula: Z _m,t, z _m,trepresent respectively upper pond level and the bound thereof of power station m at period t.

(9) output of power station climbing restriction

$\left\{\begin{array}{ccc}(1+{\mathrm{\μ}}_m){\stackrel{\‾}{p}}_{m,t-1}-p_{m,t}\≥0& \mathrm{if}& p_{m,t}>p_{m,t-1}\\ (1-{\mathrm{\μ}}_m){\stackrel{\‾}{p}}_{m,t-1}-p_{m,t}\≤0& \mathrm{else}& \end{array}\right.$

In formula: for power station m is in the average output upper limit of period t-1; μ _mfor the climbing capacity of power station m.

(10) power station startup-shutdown sustained periods of time requirement

$\left\{\begin{array}{ccc}{p}_{m,t}>0& \mathrm{if}& p_{m,t-t{g}_m}=0\mathrm{and}{p}_{m,t-1}>0\\ p_{m,t}=0& \mathrm{if}& p_{m,t-t{s}_m}>0\mathrm{and}{p}_{m,t-1}=0\\ p_{m,t}\≥0& \mathrm{else}& \end{array}\right.$

In formula: for power station m is at period t-tg _maverage output; for power station m is at period t-ts _maverage output; t _gm, t _smrepresent respectively the minimum start and shutdown sustained periods of time of power station m.

(11) vibrating area constraint

$(p_{m,t}-{\stackrel{\‾}{\mathrm{ps}}}_{m,t,k})(p_{m,t}-{\underset{\‾}{\mathrm{ps}}}_{m,t,k})>0$

In formula: ps _{m, t, k}represent respectively the bound of power station m at k the vibrating area of period t.

(12) minimum start units limits

(p _m,t-pmin _m)p _m,t≥0

In formula: pmin _mfor the minimum service capacity of power station m.

According to above-mentioned thought, once complete Optimized Operation process, is achieved according to following step (1)-(8):

(1) initial calculation condition, comprises hydroelectric station operation condition and constraint, and each network load process and be subject to electric weight demand;

(2) make power station numbering m=1;

(3) what build m power station faces workload demand { C ₁, C ₂..., C _t, wherein C ₁, C ₂..., C _tbe respectively the period 1,2 ..., the load value of T;

(4) with { C ₁, C ₂..., C _tbe workload demand, under given objective function and constraint condition, adopt DFP-method to determine day 96 processes of exerting oneself in m power station;

(5), under given controlled condition, adopt outer point searching method to determine the output distribution process of m power station at G electrical network, i.e. { p _{m, 1,1}, p _{m, 1,2}..., p _{m, 1, T}, { p _{m, 2,1}, p _{m, 2,2}..., p _{m, 2, T}..., { p _{m, G, 1}, p _{m, G, 2}..., p _{m, G, T}, wherein, { p _{m, 1,1}, p _{m, 1,2}..., p _{m, 1, T}, { p _{m, 2,1}, p _{m, 2,2}..., p _{m, 2, T}..., { p _{m, G, 1}, p _{m, G, 2}..., p _{m, G, T}represent that respectively m power station is the 1st, 2 ..., the 1st, 2 of G electrical network ..., the load value of T period;

(6) recalculate each electrical network residue load;

(7) make m=m+1, if m is less than or equal to M, skip to step (3); Otherwise go to step (8);

(8) calculate the desired value of current solution, judge whether target is improved, the if so, initial solution using current solution as next round iteration, and skip to step (2); Otherwise, judge whether to reach to maximum iteration time, if so, algorithm search stops, output optimal result, otherwise, skip to step (2).

Now, using basin, Hongsuihe River, Southwestern China area Tianshengqiao-I (being called for short sky one), natural bridge secondary (being called for short sky two), Ping Ban and all the people present power station, imperial beach as research object, adopt the inventive method to make its operational plan a few days ago.It one, days two, San Zuo power station, imperial beach is the controlled power station of whole basin step, directly adjusts pipe by CHINA SOUTHERN POWER, to Guangdong, Guangxi two provinces provide electric power, bear the peak regulation task of two provincial power networks.Power station basic data is as shown in table 1, and result of calculation as shown in Figure 4.From map analysis, in the situation that meeting given power transmission amount, the inventive method, by distributing electric power between coordinating net, makes Guangdong and Guangxi Power Grid be subject to electricity all to concentrate on peak period separately, the differentiation peak load demand that has effectively met two electrical networks, has realized good peak regulation effect.Because Guangdong Power Grid payload is 6.3 times of left and right of Guangxi Power Grid, so identical be subject to electric weight in the situation that, Guangxi Power Grid peak regulation range is obviously greater than Guangdong Power Grid, the more former load of the former remaining load peak-valley difference reduces 44%, and the latter reduces 6%.Meanwhile, with the actual running results contrast, the peak regulation range of two electrical networks on average improves approximately 5%, and wherein Guangxi Power Grid peak regulation range is up to 12%, and electrical network residue load is more level and smooth, and peak regulation successful improves.

Table 1

Claims (1)

1. transprovincially power transmission Hydropower Stations peak regulation is dispatched two stage Search methods, and its feature comprises the steps:

(1) initial calculation condition, comprises hydroelectric station operation condition and constraint, and each network load process and be subject to electric weight demand;

(2) make power station numbering m=1;

(3) what build m power station faces workload demand { C ₁, C ₂..., C _t, wherein C ₁, C ₂..., C _tbe respectively the period 1,2 ..., the load value of T;

(4) with { C ₁, C ₂..., C _tbe workload demand, under given objective function and constraint condition, adopt DFP-method to determine day 96 processes of exerting oneself in m power station;

(5), under given controlled condition, adopt outer point searching method to determine the output distribution process of m power station at G electrical network, i.e. { p _{m, 1,1}, p _{m, 1,2}..., p _{m, 1, T}, { p _{m, 2,1}, p _{m, 2,2}..., p _{m, 2, T}..., { p _{m, G, 1}, p _{m, G, 2}..., p _{m, G, T}, wherein, { p _{m, 1,1}, p _{m, 1,2}..., p _{m, 1, T}, { p _{m, 2,1}, p _{m, 2,2}..., p _{m, 2, T}..., { p _{m, G, 1}, p _{m, G, 2}..., p _{m, G, T}represent that respectively m power station is the 1st, 2 ..., the 1st, 2 of G electrical network ..., the load value of T period;

(6) recalculate each electrical network residue load;

(7) make m=m+1, if m is less than or equal to M, skip to step (3); Otherwise go to step (8);

(8) calculate the desired value of current solution, judge whether target is improved, the if so, initial solution using current solution as next round iteration, and skip to step (2); Otherwise, judge whether to reach to maximum iteration time, if so, algorithm search stops, output optimal result, otherwise, skip to step (2).