Proceedings of the 8th
World Congress on Intelligent Control and Automation
July 6-9 2010, Jinan, China
Optimal Operation of Hydropower Station Unit Based on New Algorithm *
Zhao Xuehua 1
, Huang Qiang 2,Wu Jianhua 1 1. Taiyuan University of Technology, Taiyuan, Shanxi Province, 030024, China; 2. Xi’an University of Technology, Xi’an, Shaanxi Province,710048, China Zhaoxh688@https://www.doczj.com/doc/0b6255620.html,
*
This work is partially supported by the National Natural Science Foundation of China Grant # 40901018, and the Youth Science and Technology Research Foundation of Shanxi Grant # 2007021025.
Abstract: When ensuring electric power system to supply power safely and reliably, economic and social benefit are very good which hydropower station economic operation brought. But selecting the optimal combination model of unit is a high-dimension, discrete, non-convex and non-linear optimization
problem, so it is very difficult to obtain optimal solution from theory. According to similarity between hydropower station economic operation problem and traveling salesman problem (TSP), by rationally treating constrain condition and object
function, hydropower station economic operation problem is converted into TSP, then it first establishes mathematics model to solve the optimum load distribution of units based on ant colony optimal algorithm at the hydropower station, it probes into a new approach and conducts the analysis and calculation in combination with a real example, the conclusion shows that this
method is feasible.
Key words: ant colony optimal algorithm; optimal operation;
hydropower station
I. I NTRODUCTION
With the implement of competitive power market and
increasing shortage of water resources, when ensuring electric
power system to supply power safely and reliably, economic operation of hydropower station can bring large economic benefit and save of precious water resources, so it is also of large social benefit. However, optimal operation of hydropower station unit is a high dimensional and non-linear optimization problem. If made economic benefit of hydropower station the best in an operation period, it is necessary for solving two problems of spatial optimization and time optimization. Spatial optimization is to most rationally select the number of the unit, and distribute the load to the unit, and time optimization is to further consider some
supplementary loss when the unit is started and stopped. This
problem may be solved by proportional allocation, Lagrangian Relaxation, dynamic programming and progressive optimality, et al. These methods have merits, but have obvious demerits.
For example, the demerits of proportional allocation are that the load is proportionally allocated to every unit, which neglects performance and feature of the unit, so this method is not used. Ones of Lagrangian Relaxation are that iterative times are many, and accuracy is low. Ones of dynamic programming are that computing speed is slow, and when the
number of unit is more, the system is larger, the curse of dimensionality can occur, it is difficult to meet the real-time operation demand. The convergence rate progressive optimality is restricted by the number of unit. In the recent
years, some new optimization methods are used to solve this
problem, such as Artificial Neural Network (ANN), Chaotic
Optimization Algorithm and Evolution Algorithm [1]. But at present it is very difficult to obtain optimal solution from theory. Ant Colony Optimal algorithm (ACO) was firstly
presented by an Italian Professor named M.Dorigo in the
1990s [2], based on self-organizing synergistic behaviors of the ant colony. It has been always applied to effectively solve some more difficult optimization problem in many fields, such as electric network planning, telecommunication route, channel wiring and military command etc [3],[4]. This paper will introduce this new algorithm to solve the problem of optimal operation of hydropower station unit.
II. THE PRINCIPLE OF ACO ACO is a kind of simulated evolutionary algorithm based on positive feedback principle of information. Its formulation was combined with the solving Traveling Salesman
Problem(TSP), so ACO will be introduce from TSP. Suppose there are n cities. Let
),(j i d be the distance
between city i and j . The TSP is to find the shortest closed path that contains every city exactly once. TSP can be
expressed by the following objective function: ∑?=++=11)1,()1,(min n i n d i i d D (1) Suppose m is the total amounts of ants. During the travel of ants, the probability of the ant k moving from city i to j is defined as the following:
??
???∈=∑∈otherwise k tabu j t t t P k tabu l ij il
ij ij
k ij 0)()()()()(β
αβ
αητητ (2) Where )(k tabu is the city set that ant k hasn’t visited yet at the moment, ij η
is the expecting level of ant moving from city i to j , usually it can be concretely determined by some kind of heuristic algorithm, )(t ij τ
is the pheromone concentration between city i and j
at a certain moment t . The initial value )0(ij τ of )(t ij τ
is a positive constant c given artificially. α, β
are the specific density of pheromone concentration and expecting factor respectively,
they reflect the relative significant level of pheromone and expecting. At t moment, each ant may choose a next city, and the
arriving moment that each ant moves the next city turns into 1+t . Thus after n moving, each ant will complete one tour, that is to say, it finishes one path search, now the time will be
n t +. Of all the paths that the whole ants travel, the shortest can be saved. At the same time, the pheromone on the paths among cities will be updated as:
),()()(n t t t n t ij ij ij +Δ+=+τρττ (3) Where ρ
is the residue coefficient, ρ?1 is the volatility degree of information. Let ),(n t
t k
ij +Δτ be pheromone per unit length that the
ant k releases on the path ij from the time t to n t +. So it can be shown as follows:
∑=+Δ=+Δm
k k
ij ij n t t n t t 1),(),(ττ (4)
If ant k travels the path ij , k
ij τΔ can be expressed by
Equation (5), and if not, k
ij τΔ is 0, so the equation can be
shown as
??
?=Δotherwise ij path the travels k ant L Q k k
ij
0/τ (5) In which Q is a constant, k L is the whole distance that the ant k travels in this travel.
III. THE MODEL OF ECONOMIC OPERATION OF
HYDROPOWER STATION
The criterion of economic operation of hydropower station is that the economic benefits of hydropower station operation are maximal, on the premise of meeting various restricted condition [5]. For example, if hydropower station can be regulated, given daily load, the total water consumption of hydropower station should be minimum. Its object function is expressed by equation (6):
???
??????????+Δ?=∑∑==T t M k b kt kt t t W t P Q W 11
),1()(min (6)
Subject to:
Power balance constraint
∑==
M
k kt
t P
P 1
(7)
Unit power constraint
k kt k
P P P max min ≤≤ (8)
Rotating reservation constraint
t t M k t kt k R P P P ≥??
?????∑=1max λ (9) Where )(kt kt P Q is the average discharge at the time t when the power of the unit k is kt P . t Δ is the length of time-interval. ),1(t t W b ? shows water loss of the unit starting and stopping from the time 1?t to t . k k P
P max min ， is minimum
and maximum power of the unit k respectively. kt λ is 1 when the unit k operates at the time t , otherwise is 0. t R is
rotating reservation rate (assumed the maximum power of water turbine and electric generator is equivalent), T is the number of time-interval. M is the amount of the unit.
IV. ECONOMIC OPERATION OF HYDROPOWER
STATION BASED ON ACO
Economic operation of hydropower station is in fact a multistage optimal decision problem. The unit can be combined by various means at the any time, and TSP solved by ACO is a multistage optimal decision problem too. So it is a key that unit combination of economic operation of hydropower station should be converted to TSP when it is solved by ACO.
A. Object Function Conversion
In order to realize the conversion from optimal combination of the unit to city mode of TSP, the concept of state and decision of dynamic programming is introduced [6], and defined as follows:
1) Any unit combination which met constraint condition is termed as a kind of state in each time interval.
2) The state from 1?t time interval to t is termed as a decision.
So the route may be defined as decision set constituted by any state of unit combination that meet constraint condition in each time interval, when is from 1 to T .
According to the above concept, the effective combination state of the unit corresponds to the city of TSP, and the decision between the states corresponds to the route between the cities, so object function equation (6) can be expressed by equation (10).
∑
=?=T
t t t zy c c W W 11),(min (10) Where ∈t c all the set of status available in t time interval, ),(1t t zy c c W ? shows the transfer water from the 1?t c status in
t -1 time interval to the t c in t time interval. The following need be noticed: 1) it can be seen that ),(1t t zy c c W ? should
include two parts of water by comparison between eq. (6) and
eq. (10), the first part of water may not be obtained in fact
before economic distribution, so the second part of water is
decided in this paper, then is accumulated the first part. 2)
)(k tabu expresses the city set that ant k hasn’t visited yet at the moment, however, optimal combination of the unit need include all the set of status available met constraint condition. These problems will be further formulated when treating constraint condition. B. Constraint Condition Treatment
Economic operation of hydropower station is a constrained optimal problem, but TSP is the unconstrained one. So it is a key how to convert a constrained optimal problem to the unconstrained one. (1) Treatment of power balance constraint Power balance constraint is in fact equality one, the unit power is distributed by means of Lagrangian Relaxation when economic operating. Because it is necessary that iteration times are excessive and calculation is much, the approach in reference [7] is introduced in this paper.
(2) Treatment of unit power and rotating reservation
constraint The two constraints are in fact inequality constraints. All the states of unit combination are recorded in )(k tabu , the
ones that can not meet constraint is excluded in )(k tabu . Rotating reservation constraint may be directly treated by )(k tabu . Then for unit power constraint, the combinations that do not meet bound demand may be excluded in )(k tabu , in order to reduce the number of searching route. On the other side when a unit power exceeds the bound, the power is equal to the bound value. Then the other unit will not be readjusted
until met conditions.
C. The Steps of Calculation
According to the above the model conversion and the treatment of constraint condition, the steps solving economic operation of hydropower station is as following: (1) The maximum and minimum load is selected among them. All the unit are combined to various states, then calculating the sum of maximum and minimum power of the unit at every state, the upper and lower limit value of active power at this state is obtained, compared with the load, if the lower limit value is greater than maximum load or the upper
limit value is less than minimum load at a state, this state
should be get rid of. Else the upper and lower limit value of
active power and this state should be recorded.
(2) Given initial state 0c .
(3) ACO is initialized. (4) Access next time interval, judge if it is final time
interval, if it is, turn into step (6), if not, access next step.
(5) Treatment of inequality constraints and calculation
result of step (1) form )(k tabu at present time interval. Water
loss of the unit starting and stopping is taken as route length,
next state is selected by transfer regulation of ant colony, and
then ants transfer the state. As computing the power and water
loss of every unit at this state, and then computed water loss is
accumulated in route length of ant, turn into step (4). (6) The shortest route is recorded in this cycle, and the information of route is refreshed by refresh regulation of global information. (7) if it does not meet the condition of unit stopping, let t is equal to 0, and then turn into step (4) and start next cycle, if not, it is stopped.
Ⅴ. APPLICATION There are three units at a hydropower station, rated power of 1＃and 2＃ unit is both 50MW, one of 3＃ unit is 20MW.
Performance parameter of each unit is given in TABLE I at a head. Load of three time-intervals is 73MW, 87MW and 59MW respectively, and one time-interval is an hour. TABLE I
UNIT PERFORMANCE PARAMETER unit
Consumption performance/(m 3/s) Bound of
power/(MW) water loss of the unit
starting and stopping
/(m 3) 1# 0.1279+0.928P +0.0022P 2 54.2，5.3 13327
2# 0.2964+1.122P +0.0007P 2
54.2，5.3 13542
3# 0.693+1.0229P +0.0078P 2 22.1，4.1 8754
Stochastically suppose the parameter α=2, β=4, ρ=0.7 and Q =30, the number of ants m =10, the maximum iteration times is 100, and detention times is 10[8]. It is shown in TABLE Ⅱ that the power and water consumption of each
unit is computed by ACO, the total water consumption is
843760.2m 3.
It is shown in TABLE Ⅲthat the power and water
consumption of each unit is computed by Genetic Algorithm,
the total water consumption is 843870.8m 3. It can be seen that accuracy of the result is high by ACO, the mode of unit combination can be rationally selected in economic operation
of hydropower station.
TABLE Ⅱ
UNIT POWER AND WATER CONSUMPTION IN EACH TIME-
INTERVAL BY ACO
unit
1st time-interval 2nd time-interval 3rd time-interval Active
power
/(MW)
water
consumpt
ion
/(m3)
Active
power
/(MW)
water
consumpti
on
/(m3)
Active
power
/(MW)
water
consumpti
on
/(m3)
1# 49.188 183952 52.352 197066.4 46.063 170996.2 2# 16.021 66426.33 25.964 107643 6.0701 25707.95 3# 7.7904 30641.16 8.6827 34340.1 6.8972 26986.95 The
total
/(m3
)
843760.2
TABLE Ⅲ
UNIT POWER AND WATER CONSUMPTION IN EACH TIME-
INTERVAL BY GENETIC ALGORITHM
Unit
1st time-interval 2nd time-interval 3rd time-interval Active
power
/(MW)
water
consumpt
ion
/(m3)
Active
power
/(MW)
water
consumpt
ion
/(m3)
Active
power
/(MW)
water
consump
tion
/(m3)
2# 15.441 64040.7 27.45813 113875.9 8.5481 35778.85 3# 7.9172 31164.3 7.99463 31483.93 7.2859 28570.04 The
total
/(m3
)
843870.8
Ⅵ.C ONCLUSION
(1) ACO is a kind of simulated evolutionary algorithm based on positive feedback principle of information, and searches the solution space by synergistic behaviors of
artificial ants among which the information is continuously exchanged, which can find better solution very easily, and is
of nature parallelism. Its application prospect is very widespread. (2) Economic operation of hydropower station and TSP
are both a multistage optimal decision problem. According to similarity between them, by rationally treating constrain condition and object function, hydropower station economic operation problem is converted into TSP, then it establishes mathematics model to solve the problem about economic operation based on ant colony optimal algorithm in the hydropower station. The conclusion of real example shows that this method is feasible. A new algorithm on optimal operation of hydropower station unit based on ACO is found.
A CKNOWLEDGEMENT
This research is supported by the National Natural Science Foundation of China (Grant No. 40901018), and the Youth Science and Technology Research Foundation of Shanxi (Grant No. 2007021025).
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