valley2 valley2 v2 > : Ppeak2 þ ξp2 t∈T peak2
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where Pvalley1, Ppeak1, Pvalley2 and Ppeak2 are constant variables, ξv1, ξp1, ξv2 and ξp2 are random fuzzy variables, respectively. Taking t ∈ Tpeak1 for example, the process to analyze the power load error distribution is described as follows. Step 1: Suppose that ei follows the same distribution, where ei is the load error point when I
A
t ∈ Tpeak1 . Ppeak1 ¼ 1I ∑ ei , where I is the total number of e i . Then a new i¼1
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sequence{ei'} (i = 1, 2, ⋯ , I) is generated, where ei' = ei − Ppeak1. Step 2: Draw the frequency distribution histogram of ei', and compare the histogram with known distributions, like a uniform, normal or binomial distribution. The frequency distribution histogram in Fig. 3 suggests that ei' approximately obeys a normal distribution. Then the maximum likelihood estimation can be applied to determine the parameters of a normal distribution, that are mean μ1 and variance σ1.
Fig. 2 M-K result of the average daily load error in July
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Step 3: Determine the fuzzy membership grade of μ1 and σ1 by experiments. Taking μ1 as an example, Table 4 in Appendix is the experiment table to fill in. Then let *
M
μi1 ¼ μ1 ‐Δ21 þ ΔM1 i ; μ12 ¼ μ1 , where μi1 is the increased points on both sides of
n¼1
ð5Þ
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N s μi1 ¼ ∑ f n; μi1
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μ1, M + 1 is the total number of points, and ΔM1 is the distance between two points. Finally, calculate the average value x1 of L (L < I) points, which are randomly selected from {ei'} . Repeat N times and the experimental numbers are denoted as (1, x1) , (2, x2) , … , (N , xN) in turn. 1; if xn−δ ≤ μi1 ≤ xn þ δ ð4Þ f n ; μi1 ¼ 0; other
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i s μi1 ð6Þ A μ1 ¼ N Step 4: Calculate f n; μi1 , s μi1 and A μi1 , and A μi1 is the fuzzy membership distribution. The experiments results of μ1 and σ1 are displayed in Fig. 4. The distribution can be M =2 approximately described as a triangular fuzzy variable, μ1 ¼ μ01 ; μ1 ; μM 1 , M =2 σ1 ¼ σ01 ; σ1 ; σM 1 .
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By experiments and statistical analyses, these variables have approximately uniform and normal distribution, respectively. A fuzzy variable does express fuzziness of parameters and thus these random fuzzy variables can be expressed as: 8 ξv1 ∼U ða; bÞ > > < ξ ∼N μ ; σ2 p1 1 1 ð7Þ ξv2 ∼N μ2 ; σ22 > > : ξp2 ∼N μ3 ; σ23
Fig. 3 Distribution of ei' when t ∈ Tpeak1
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Fig. 4 Fuzzy membership distribution of μ1 and σ1
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3.2 Random Fuzzy Optimization Model
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where U(a, b) denotes a uniform distribution, and N μi ; σ2i ; i ¼ 1; 2; 3 denotes a normal distribution with meanμiand varianceσ2i . a, b, μi and σi are described as triangular fuzzy variables.
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3.2.1 Random Fuzzy Variable
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There are three kinds of measure in fuzzy sets: possibility measurePos(A), necessity measure Nec(A), credibility measure Cr{A}. Let f(ξ) ≤ 0 be a random-fuzzy event, then Ch{f(ξ) ≤ 0}(α) is defined as α ‐ chance of f(ξ) ≤ 0. Chf f ðξÞ ≤ 0gðαÞ ¼ supfβjCrfθ∈ΘjPrf f ðξðθÞÞ ≤ 0g≥ β g ≥ αg
ð8Þ
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where the function Pr{A} is indeed a probability measure on A, and Cr{A}is a credibility measure which plays the role of a probability measure in fuzzy set theory. 1 ðPosðAÞ þ NecðAÞÞ 2 1 ¼ PosðAÞ þ 1−Pos AC 2
CrfAg ¼
ð9Þ
where AC is complementary set of A. Let ξ be a random fuzzy variable, andα , β ∈ (0 , 1]. Then ξsup ðα; β Þ ¼ supfrjChfξ ≥ rgðαÞ≥ β g
ð10Þ
is called the(α, β)-optimistic value to ξ, and ξinf ðα; β Þ ¼ inf frjChfξ ≤ rgðαÞ ≥ β g is called the(α, β)-pessimistic value to ξ. Usually, we care about the value when α > 0.5andβ > 0.5, thenξsup(α, β) < ξinf(α, β).
ð11Þ
Yuan L. et al.
A triangular fuzzy distributionξcan be represented asξ = (r1, r2, r3), then 8 if r2 ≤ 0 > < r1 ; 1 ; if r1 ≤ 0 ≤ r2 posðξ ≤ 0Þ ¼ > : r1 −r2 0 ; other
ð12Þ
3.2.2 Random Fuzzy Optimization Model
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The conventional deterministic optimization model of STHS for cascade stations is to dispatch power load to stations and units with an objective of minimizing the cascaded total energy consumption while satisfying power load demands and some prevailing constraints during the given time. The objective function is formulated as follows: E s ¼ min ∑ ∑ K i ⋅H it ⋅Qit ⋅Δt t¼1 i¼1
ð13Þ
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where Es denotes the minimum of energy consumption; T and I denote the number of periods and hydropower stations, respectively; Ki denotes the comprehensive efficiency coefficient of i station; Qit and Hit denote the outflow and effective waterhead of i station in period t, respectively;Δt denotes the interval of time period. Then power balance must be obeyed as formula (14) to satisfy the power grid, and other constraints including water balance, maximum and minimum limits should be considered. Other limits like hydro unit start-up costs, shut-down costs, the spinning reserve requirements and restricted operating zones can be flexibly taken into account for the model. ∀t Pt ¼ P1;t þ ⋯ þ Pi;t þ ⋯ þ PI;t ð14Þ Pi;t ¼ Pi;1;t þ ⋯ þ Pi; j;t þ ⋯ þ Pi; J i ;t ∀i ∀t
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where Pt denotes the real-time load demand of hydropower system in period t; Pi , t and Pi , j , t denote power output of i station and power output of j unit of i station in period t, respectively; Ji denotes the number of units of i station. Since the daily power load error is uncertain, deterministic models of short-term hydropower scheduling aren’t suitable to solve such a problem, and the conventional deterministic optimization model should be improved to some extent. On the basis of the random fuzzy load error, it is appropriate to introduce random fuzzy programming into hydropower scheduling, which contributes to coping with uncertainty. The view that random fuzzy load error affects solutions of unit commitment has been widely accepted (Bertsimas et al. 2013; Ruiz et al. 2009), for the maximum power load and power demand will determine generation schedules in the coming day. As for the power generation side, hydropower stations must have sufficient adjustable spare capacity to track the load error at any time (Gan and Litvinov 2003). The random fuzzy optimization also can transform an uncertain problem into a certain one to reduce the calculation time. Therefore, we attempt to develop a random fuzzy optimization model (RFO) for cascade hydropower stations paying more attention to unit spare capacity. By using the model, an eligible power generation schedule is expected to obtain and report to the power grid, and an optimal unit schedule is achieved simultaneously for guiding
Random fuzzy optimization model for short-term...
operation in practice. The objective of the model is to minimize the cascaded total energy consumption as the same as conventional deterministic optimization model. While the power load constraint formula (14) is improved as the formula (15) and formula (16), which are more related to maximum and minimum output limits of stations. Equality constraint: I
J
∑ ∑ U i; j;t Pi; j;t ¼ Pplan t
t ¼ 1; 2; …; T
ð15Þ
i¼1 j¼1
8 I J > plan max > > < ∑ ∑ U i; j;t Pi; j;t ≥ Pt þ et i¼1 j¼1 > I J
ð16Þ
t ¼ 1; 2; …; T
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i¼1 j¼1
t ¼ 1; 2; …; T
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plan min > > : ∑ ∑ U i; j;t Pi; j;t ≤ Pt þ et
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where Ui , j , t is a binary variable indicating whether j unit of i station is on or off in time period t; Pi , j , t is power output of j unit of i station in time period t; Pplan is planned power load t assigned by the dispatching department of grid in time period t. Inequality constraint:
A
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max where Pmin i; j;t and Pi; j;t are the minimum and maximum power output limits of j unit of i station in period t, respectively; et is power load error in time period t. According to the formula (10) and (11), if α > 0.5 and β > 0.5, formula (16) can be rewritten as: 8 I J > max plan > > t ¼ 1; 2; …; T < ∑ ∑ U i; j;t Pi; j;t ‐Pt ≥ f 1;t i¼1 j¼1 ð17Þ I J > plan > > ∑ ∑ U i; j;t Pmin ≤ f 2;t t ¼ 1; 2; …; T : i; j;t ‐Pt
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i¼1 j¼1
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where f1 , t and f2 , t are pessimistic and optimistic values to et at time t, respectively. Other conventional constraints are as follows: 1) Water balance equation V i;t ¼ V i;t−1 þ I i;t −Qi;t ⋅Δt
∀i ∀t
ð18Þ
where Vi , t and Vi , t − 1 represent the water storage volume of i station in period t and t − 1, respectively; Ii , t and Qi , t represent the inflow and outflow of i station in period t, respectively.
2) Hydraulic connection I i;t ¼ Qf i−1;t‐τ þ S i−1;t‐τ þ Ri;t
∀i ∀t
ð19Þ
where Qfi − 1 , t − τ and Si − 1 , t − τ represent discharge and discarded water through the turbine of i − 1 station in period t − τ, respectively;τ represents water delay of i − 1 to istation;Ri , t represents inter-zone inflow between i − 1 and i station.
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3) Maximum and minimum constraints max Z min i;t ≤ Z i;t ≤ Z i;t
∀i
∀t
ð20Þ
max Qmin i;t ≤ Qi;t ≤ Qi;t
∀i
∀t
ð21Þ
max Pmin i;t ≤ Pi;t ≤ Pi;t
∀i
∀t
ð22Þ
max where Z min i;t and Z i;t represent the minimum and maximum water level limits of istation in max period t, respectively; Qmin i;t and Qi;t represent the minimum and maximum water discharge
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4) Variation amplitude constraints 8 < Z i;t −Z i;t−1 ≤ ΔZ i : Q −Q ≤ ΔQ i;t i;t−1 i
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max limits ofi station in period t, respectively; Pmin i;t and Pi;t represent the minimum and maximum power output limits of istation in period t, respectively.
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where ΔZi and ΔQirepresent the water level and discharge constraints of adjacent time intervals t and t − 1, respectively. 5) Units minimum on/off-time
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(
down T off i; j;t ≥ T i; j on T i; j;t ≥ T up i; j
ð24Þ
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down where T up are the minimum required time periods of j unit of i station for on/off i; j and T i; j off status, respectively. T on =T i; j;t i; j;t is the continuous time duration of j unit on/off status of i station before t.
6) Marginal values constraints ∀i
ð25Þ
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Z i;0 ¼ Z i;start
where Zi , start is initial water level of i station and it is given.
3.3 Solution Searching Strategy
Short-term optimization of hydropower stations aims at finding out optimal daily schedules for stations and units with a fifteen minutes’ or one hour’s time interval. The above optimization problem is divided into iterative hierarchical sub-problems, and a framework that integrates actual requirements from power grid, stations and units is developed to find a comprehensive solution for the overarching problem, showing in Fig. 5. The detailed solution searching strategy includes four steps. Step 1: Analyze load error distribution in the last few days as description in 3.1 Section and simulate groups of load error data. Based on random fuzzy programming, power constraints for hydropower stations are reinforced as formula (15) and (17).
Random fuzzy optimization model for short-term... Begin Data preparation Obtain the planned load of the cascade from Grid Reinforce stations constraints
Analyze load error distribution
Generate 500 groups of error dada for further use
Load dispatching Set t=1 Discretize load within the limits at time t
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Power Grid constraints
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Dispatch the planned load among cascade stations
Stations constraints Unit Commitment
Optimize unit commitment and loading
Units constraints
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Generate daily generation schedules t++
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Check chance constraints
No
Increase running units
Meet upper bound ? Yes
No
Yes
No
Meet lower bound ?
A
t=T
Yes
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Calculate objective function value
Accuracy requirement ?
Cut down running units
No
Yes
Obtain the optimal schedules End
Fig. 5 Solution framework of the random fuzzy optimization model
Step 2: Discretize and dispatch the given planned power load of the cascade to every single stations at every time period considering power grid constraints, including transmission capability, line maintenance and spinning reserve. It’s a repeatedly iterative step until the optimization objective is satisfied within accuracy requirement. Step 3: Optimize unit commitment of a single station, taking the minimized water consumption as the objective by taking an advantage of characteristic curves of the turbines. Meanwhile, unit priority and maintenance can be set and controlled
Yuan L. et al.
voluntarily. In particular, output limits of stations that are related to load error work well at this step. Step 4: Water discharge, output and water level at time t will be obtained by the random fuzzy optimization model. When T time intervals are cycled, the total hydroenergy consumption of cascade stations is calculated and saved. Repeat step 2 ~ 4 until a termination condition is satisfied.
4 Results and Discussion
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The model is applied in Qing Cascade for short-term scheduling and implemented by Java on a personal computer. The simulation results and discussion are shown as follows.
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4.1 Impact of Uncertain Load Error
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Whether a daily generation schedule is good enough depends on indicators we care most. When a daily generation schedule is figured out, all the indicators such as power generation, generation benefit, water/hydroenergy consumption, times of unit switches and running time in restricted operating zones should be come out to help dispatchers make decisions. However, prior to that, we’d better make it clear which indicators we care about so as to reduce the computational work on indifferent indicators. If an indicator has no or little change among all schedules, it will be of no value for selecting a satisfactory schedule. In other words, we may care indicators that are greatly affected. Therefore, a simulation experiment is carried out to find out which indicator the uncertain power load error affects most on daily generation schedules. Firstly, operators have suggested three indicators that they care, that are hydroenergy consumption of the cascade, daily power generation and total times of unit switches. Secondly, we take a typical day, May 18, 2015 as an example, and get the planned power load of the cascade. Then, the load error is analyzed and based on the distribution characteristics of load error, 500 groups of real-time actual power load data on May 18, 2015 are simulated by Monto Carlo Method. Finally, 500 groups of data are applied to the conventional deterministic optimization model and the three indicators are figured out. The experiment statistics are shown in Table 2. The historical data on May 18, 2015 are also listed for comparison. It can be seen that the minimum, maximum and average value of hydroenergy consumption are very close to the historical data, and so is the power generation, which means that the load error has little impact on hydroenergy consumption and daily power generation of the cascade. While regarding the times of units’ starts and stops, simulation and historical value differ Table 2 The experiment statistics of the conventional deterministic optimization model Indicators
History Simulation
Min Max Avg Avg
Hydroenergy consumption (106kWh)
Power generation (106kWh)
Unit switches times
25.91 24.55 26.79 26.20
25.44 24.20 26.53 26.03
17 27 60 37.7
Random fuzzy optimization model for short-term...
greatly. Therefore, the load error affects times of unit switches most for the cascade of Qing River, and we care the indicator most.
4.2 Results of Random Fuzzy Optimization Model
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According to the results from the impact of uncertain load error, more attention should be paid to unit switches. Also the operators hold the same view that load error does bring starts and stops costs in practice. It’s well known that frequent unit switches reduce service life and increase maintenance costs (Nilsson and Sjelvgren 1997). Therefore, take the actual power load on May 18, 2015 as the input of conventional deterministic and RFO optimization model, the results of the unit status are shown in Fig. 6, in which dark and light blue represent unit status of on and off, respectively, and the number represents time periods. The comparative results above show that the major difference lies in unit status of Shuibuya. For Shuibuya, times of unit switches significantly reduces from 11 to 5. Further analysis establishes that some units keep running longer to avoid frequent unit switches so as to cope with the power load error, thus indicating that the proposed RFO model works well. The unit schedules can be used for guiding operation in practice even if actual real-time load is uncertain. Meanwhile, the day-ahead power generation schedule for cascade stations follows the planned power load very well, which is fundamental for power system operations. Then the power generation schedules can be reported to the dispatching department of grid.
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4.3 Simulation Operation of Uncertain Load Error
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Even if the RFO model provides us a better daily schedule than the conventional deterministic model mainly because of its fewer times of unit switches, we may still wonder if the unit schedule works well when the actual real-time power load is quite different from the planned values. To further verify the validity of RFO model, the generated 500 groups of load error data in 4.1 Section are applied to simulate the actual uncertain load. Assuming that stations run
(a) Unit schedules from conventional deterministicoptimization model.
(b) Unit schedules from random fuzzy optimization model. Fig. 6 Unit schedules from different models
Yuan L. et al. Table 3 Times of unit switches by 500 times simulation Stations Times
Model 1 Model 2
SBY
GHY
GBZ
27.17 19.23
14 14
3 3
Model1 and Model2 refer to the conventional deterministic and random fuzzy optimization model, respectively
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as the unit schedules from two models, respectively. Thereafter, we analyze the indicators of cascade stations and the comparison results are shown in Table 3 and Fig. 7. Times of unit switches shown in Table 3 makes clear that Model 2 can avoid frequent unit switches, and that Shuibuya plays an important role in bearing load basis because of its capacity. It’s speculated that if Shuibuya reaches its maximum power output, Geheyan will take over the job. In addition, Fig. 7 indicates that Model 2 doesn’t suffer a disadvantage in hydroenergy consumption in statistical terms. Therefore, it proves that Model 2 works much better than Model 1 in reducing unit switches. In fact, this paper aims at short-term scheduling for only 96 time periods a day. While more time periods in actual operation and the impact of reducing unit switches will be more dramatic.
5 Conclusions
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In this paper, we have studied one of the most intriguing questions related to short-term hydropower scheduling considering uncertainty of power load, including randomness and fuzziness. We define this kind of uncertainty as the error between planned and actual real-time power load for cascade stations. The error has a negative effect on short-term operation of stations, especially on unit switches, which is a challenging and pressing problem for shortterm hydropower scheduling. Based on random fuzzy programming, the load error is described as a random fuzzy variable and its distribution is analyzed. Meanwhile, unit spare capacity is considered to deal with load error and a random fuzzy optimization model for short-term hydropower scheduling is established. Finally, a practical solution searching strategy is proposed to solve the optimization model.
Fig. 7 Hydroenergy consumption of the cascade by 500 times simulation
Random fuzzy optimization model for short-term...
As main regulating stations, the Qing cascade plays a crucial role in peak regulation and frequency modulation, which is taken as an example. The results reveal that load error affects times of unit switches most and the random fuzzy optimization model help reduce times of unit switches significantly, compared with the conventional deterministic model. Meanwhile, the validity of unit schedules of random fuzzy optimization model is verified by using simulated groups of load error data. Therefore, the proposed random fuzzy optimization model is a simple practical solution to avoid frequent unit switches in short-term hydropower scheduling. In addition, it’s expected to be a promising alternative for other dispatching centers and hydropower stations when dealing with uncertain load error.
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Acknowledgements This work is supported by the National Key R&D Program of China (2016YFC0402205), the National Natural Science Foundation of China (No. 51579107), and the Key Program of the Major Research Plan of the National Natural Science Foundation of China (No. 91547208), and special thanks are given to the anonymous reviewers and editors for their constructive comments.
Appendix
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Table 4 is the experiment table to fill in to help determine a fuzzy membership grade of μ1.
Num
μ01
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μi1
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…
…
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f 1 ; μ01 f 2; μ01 … f n; μ01 … f N ; μ01 0 s μ1 A μ01
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(1, x1) (2, x2) … (n , xn) … (N , xN) sum frequency
μ11
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Table 4 The experiment table to determine the fuzzy membership grade of μ1
… … …
… f n; μi1 … s μi1 A μi1
…
‐1 μM 1
…
… … …
μM 1 f 1 ; μM 1 f 2; μM 1 … f n; μM 1 … f N ; μM M 1 s μ1 A μM 1
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