energies Article
Optimization Scheduling Method for Power Systems Considering Optimal Wind Power Intervals Mengyue Hu
ID
and Zhijian Hu *
School of Electrical Engineering, Wuhan University, Wuhan 430072, China;
[email protected] * Correspondence:
[email protected]; Tel.: +86 136-3862-9108 Received: 7 May 2018; Accepted: 17 June 2018; Published: 1 July 2018
Abstract: Wind power intervals with different confidence levels have an impact on both the economic cost and risk of dispatch plans for power systems with wind power integration. The higher the confidence level, the greater the bandwidth of corresponding intervals. Thus, more reserves are needed, resulting in higher economic cost but less risk. In order to balance the economic cost and risk, a unit commitment model based on the optimal wind power confidence level is proposed. There are definite integral terms in the objective function of the model, and both the integrand function and integral upper/lower bound contain decision variables, which makes it difficult to solve this problem. The objective function is linearized and solved by discretizing the wind power probability density function and using auxiliary variables. On the basis, a rolling dispatching model considering the dynamic regulation costs among multiple rolling plans is established. In addition to balancing economic cost and risk, it can help to avoid repeated regulations among different rolling plans. Simulations are carried on a 10-units system and a 118-bus system to verify the effectiveness of the proposed models. Keywords: unit commitment; rolling dispatch; wind power interval; optimal confidence level
1. Introduction Due to its inherent intermittency and volatility, the integration of large-scale wind power brings great uncertainty to power systems, and it challenges the formulation of scheduling plans for power systems. Traditional scheduling models use wind power point prediction information to arrange the units’ output and cannot fully take the uncertainty of wind power into account. More reserves are needed to keep power balance in power system, which certainly impose higher operation costs. In order to deal with the wind power uncertainty, a lot of researches have been conducted. Stochastic optimization (SO) [1], chance-constrained optimization (CCO) [2], robust optimization (RO) [3] and interval optimization (IO) [4] are the most commonly used methods. The main idea of SO is to generate a large set of scenarios by sampling the probability distribution of wind power, and then eliminate the part of these scenarios with low probability. Thus, the remaining typical scenarios along with their probabilities are used to model the wind power uncertainty. The disadvantage of SO is that it depends on the accuracy of probability distribution of wind power uncertainty and the selected scenarios. In the CCO method, the solution can violate the system security constraints to some extent, but the probability that the solution should satisfy the constraints is not less than a certain predefined confidence level. The optimality of a CCO solution also relies on the probability distribution of uncertain variables which is problematic in real-world applications [5]. RO does not require any assumptions about the probability distribution of uncertain variables and allows fluctuations of wind power within a given range. However, its solution tends to be conservative for that its objective is to minimize the cost of the worst-case scenario. IO generates the upper and lower
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boundaries of generation dispatch. It requires less computation, but the results are still conservative and not precise enough [6]. The above methods have their own advantages and disadvantages, but these methods are all based on given wind power confidence levels. For example, the SO method in [7] generates the scenarios within upper and lower boundaries of the 95% confidence intervals. RO and IO usually only focus on the circumstances that may occur within the interval boundary, regardless of the confidence level, or only consider the interval with a given confidence level [5,8,9]. In summary, these methods usually set the interval boundary with a fixed confidence level such as 90% [10], or set the interval boundary to ±20% [11] of predictive values, or 20% [12] of the installed capacity and so on. Generally speaking, the wind power forecasting interval with a high confidence level tends to have a greater bandwidth. That is, the possible fluctuation range of wind power is larger, and thus more reserves are needed, but the risks of wind curtailment and load shedding are smaller. In a word, the higher confidence level leads to greater economic cost but smaller risk. How to balance the economic cost and risk of system is a problem that worth studying. A RO dispatch model with adjustable uncertainty sets is established in [13] to reduce the conservativeness, and the dispatch costs along with the wind power confidence levels varying from 60% to 95% are calculated. It can be seen that the economic cost and risk of the solutions vary with the confidence levels. A wind power adjustable interval based model for scheduling the energy and reserve is proposed in [6], and it shows better performance than traditional models based on prediction intervals. However, it is pointed out in [14] that this study regards wind power unschedulable and it uses a Gaussian distribution to simulate wind power uncertainty, which may lead to a more expensive schedule. Since there exists a complementary relationship between the economic cost and risk of the power system dispatch plan as the wind power confidence level changes, there should be an optimal confidence level that balance the two. Therefore, from the perspective of wind power confidence level, a unit commitment (UC) model based on optimal wind power intervals is proposed in this paper. The results can provide reference for the selection of wind power optimal confidence intervals in the UC problem and the selection of confidence level in wind power interval forecasting. On the basis of day-ahead UC model, an intra-day rolling dispatch model is established to revise the deviations of the pre-day plan for that the forecast accuracy of wind power increases as the time scale decreases. A look-ahead economic dispatch approach is proposed in [15] with the objective of minimizing the total production cost. For each sub-period, based on the latest forecast information for a prespecified time horizon, the look-ahead dispatch model formulates a dispatch plan in a rolling manner [16], and only the results of the first one or several periods are actually executed. However, the conventional look-ahead dispatch models always aim at the optimal operation cost and lack the overall optimality within the entire scheduling periods. Therefore, this paper introduces the dynamic adjustment costs, which takes into account the man power and material resources required for the regulation of the dispatch plans, and it can help to reduce the regulation quantity among different rolling dispatch plans. In summary, the major contributions of this paper are as follows: (1) (2)
(3)
A UC model considering the optimal wind power confidence intervals is established to balance the economic costs and risk of the dispatch plan for the power system with wind power integration. The UC model is a mixed integer nonlinear programming (MINLP) problem with integral terms in its objective, and both the integrand function and integral upper/lower bound contain decision variables. The objective function is linearized and solved by discretizing the wind power probability density function and using auxiliary variables. Based on the UC model, in addition to optimal wind power confidence intervals, the dynamic adjustment cost is introduced into intra-day rolling dispatch model to reduce the amount of adjustments among different rolling dispatch plans.
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2. Overall Framework and Representation of Wind Power Uncertainty 2. Overall Framework and Representation of Wind Power Uncertainty 2.1. Overall Framework 2.1. Overall Framework A dispatch strategy with two time-scale modules is developed in this paper, including the A dispatch strategy with two time-scale modules is developed in this paper,is including the dayday-ahead UC and the intra-day rolling dispatch. The framework of the strategy shown in Figure 1 ahead UC and the intra-day rolling dispatch. The framework of the strategy is shown in Figure 1 and and depicted below. depicted below. (1) Day-ahead UC. is aismixed integer nonlinear programming problem problem with the goal minimizing (1) Day-ahead UC.It It a mixed integer nonlinear programming withof the goal of the economic cost and risk cost. The UC for the next operating day is performed in current minimizing the economic cost and risk cost. The UC for the next operating day is performed in operation day, andday, it provides the UC plan of 24 h for day.next day. current operation and it provides the UC plan of the 24 hnext for the (2) Intra-day rolling information, it isit (2) Intra-day rolling dispatch. dispatch. Based Basedon onthe thelatest latestload loadand andwind windpower powerforecasting forecasting information, activated every 1 h to modify the power generation output of the next 4 h in a rolling manner. is activated every 1 h to modify the power generation output of the next 4 h in a rolling manner. In each each rolling rolling plan, plan, only only the In the dispatch dispatch plan plan of of the the first first hour hour is is actually actually executed. executed. Current operating day
Day-ahead unit commitment
1 hour Rolling dispatch 1 Rolling dispatch 2 ……
1 hour Rolling dispatch 21 15 minutes
Figure 1. 1. Diagram Diagram representation representation of of the the overview overview framework. framework. Figure
2.2. Representation Representation of of Wind Wind Power Power Uncertainty Uncertainty 2.2. At present, systems provide predictive results in the of point At present, most mostwind windpower powerforecasting forecasting systems provide predictive results inform the form of forecasting. To estimate the confidence intervals of wind power, it is necessary to obtain the point forecasting. To estimate the confidence intervals of wind power, it is necessary to obtain the conditional distribution distribution of of the the wind wind power power output output with with respect respect to to the the point point forecasting value, that conditional forecasting value, that is, is, the conditional probability distribution of the possible output under the condition of “known point the conditional probability distribution of the possible output under the condition of “known point forecasting value”. value”. forecasting In order order to to obtain obtain the the probability probability distribution, distribution, the the idea idea of of “forecast “forecast bin” bin” in in [17] [17] is is used used in in this this In paper. At first, the historical wind power data along with its predictive value are formed as a row paper. At first, the historical wind power data along with its predictive value are formed as a row vector in in the there areare NN sets of data, an an N ×N2 vector the form form of of [predictive [predictivevalue, value,measured measuredvalue]. value].Assuming Assuming there sets of data, matrix is eventually formed. TheThe matrix is is normalized, and ranked in in × 2 matrix is eventually formed. matrix normalized, andallallthe therow rowvectors vectors are are ranked ascending order in accordance with the predictive values. Then the wind power forecast domain (i.e., ascending order in accordance with the predictive values. Then the wind power forecast domain [0, 1][0, p.u.) is divided into into M evenly distributed binsbins which are are called forecast bins (FB). The value of (i.e., 1] p.u.) is divided M evenly distributed which called forecast bins (FB). The value M is determined by the size of N, and it is set as 50 in this paper. The width of each FB is 0.02 p.u., of M is determined by the size of N, and it is set as 50 in this paper. The width of each FB is 0.02 p.u., that is, is, the p.u., and that in in thethe FBFB m (m = 1,=2,1,…, that the range range of of the theforecast forecastvalues valuesininthe thefirst firstFB FBisis[0,[0,0.02] 0.02] p.u., and that m (m 2, M) is [(m − 1)/50, m/50] p.u.. Thus, the values of the predictive values in each FB are similar, while the . . . , M) is [(m − 1)/50, m/50] p.u.. Thus, the values of the predictive values in each FB are similar, actualthe values are generally quite different. Then the probability distribution of the of possible wind while actual values are generally quite different. Then the probability distribution the possible power output in each FB can be obtained by counting the actual measured values. When the wind power output in each FB can be obtained by counting the actual measured values. When the predictivevalue valueisisknown, known,ititisiscompared comparedwith withthe therange range each FB. If falls it falls within a certain predictive ofof each FB. If it within a certain FB,FB, it’sit’s in in accordance with the probability distribution of the FB. accordance with the probability distribution of the FB. In the the distribution distribution statistics statistics of of the the data, data, there there are are two two kinds kinds of ofmodels: models: parameter parameter distribution distribution In models and non-parameter distribution models. The parameter distribution models include normal models and non-parameter distribution models. The parameter distribution models include normal distribution, beta distribution, etc. However, in reality, it is difficult to find a reasonable parameter distribution, beta distribution, etc. However, in reality, it is difficult to find a reasonable parameter distribution for for that that different different prediction prediction methods methods are are applied applied on on different different wind wind farms, farms, and and the the distribution distribution of prediction errors are different as well. Therefore, it is not reasonable to simply assume distribution of prediction errors are different as well. Therefore, it is not reasonable to simply assume that the measured values in the FB obey a certain probability distribution. A non-parametric empirical distribution model is used to simulate the distribution of the measured values. The main
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that the measured values in the FB obey a certain probability distribution. A non-parametric empirical distribution model is used to simulate the distribution of the measured values. The main advantage of this method is that it is not necessary to assume any specific distribution of random variables. Sufficient historical data are needed to approximate the probability distribution. If the measured values in a FB are ranked in ascending order as x1 , x2 , · · · , xn , then the empirical probability distribution function (PDF) of the wind power in the FB is as follows[17]: Fn ( p) =
1 n θ ( p − xi ) n i∑ =1 (
θ ( p − xi ) =
(1)
1, i f p ≥ xi 0, i f p < xi
(2)
When the number of samples n in Equation (1) is large enough, the empirical distribution function is close to the theoretical distribution. Based on the empirical probability distribution, the intervals with different confidence levels can be calculated. For example, the quantiles corresponding to 0.025 and 0.975 in the PDF are the lower and upper interval boundaries with 95% confidence level. Thus, the intervals with confidence levels from 1% to 99% can be obtained. 3. The Proposed Model 3.1. UC Model Based on Optimal Conficence Level 3.1.1. Objective The objective function of the UC model which does not consider the wind power confidence interval is to minimize fuel cost, on-off cost and reserve cost, etc. In addition to the plan for wind power point predictive value, the UC scheme based on optimal wind power confidence level needs to consider any circumstances that may occur in the interval, as well as the risk that the actual value falls outside the interval. Taking into account the impact of wind power intervals with different confidence levels on the UC scheme, an objective function is proposed and shown in Equation (3): NT NG
F = min
NT NW
∑ ∑ (Ci,t + Ci,tr + Ci,tSU ) + ∑ ∑ CW j,t g
t =1 i =1
(3)
t =1 j =1
where i is the number of thermal power units. j is the number of wind farms. t is the period number. NT is the total number of the periods. NG is the total number of thermal power units. NW is the g number of wind farms. Ci,t is the fuel cost of thermal power units i in period t and its expression is r is the reserve cost of unit i in period t, and the expression is shown in shown in Equation (4). Ci,t SU Equation (5). Ci,t is the start-up cost of unit i in period t, and the expression is shown in Equation (6). CW j,t is the cost incurred by wind farm j in period t, and the expression is shown in Equation (9): g
2 Ci,t = ( ai Pi,t + bi Pi,t + ci ) Ii,t up
(4)
r dr dn Ci,t = cur i ri,t + ci ri,t
(5)
SU Ci,t = csu i Ii,t (1 − Ii,t )
(6)
where ai , bi and ci are the cost coefficients of the thermal power unit i. Pi,t is the dispatching power of dr the thermal power unit i in period t. cur i and ci are the upward and downward reserve cost coefficient up dn are the upward and downward reserves provided by unit i in period t, csu is the for unit i. ri,t and ri,t i starting cost constant for unit i, Ii,t is a binary variable which indicates the unit’s on/off status.
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g
Ci,t in Equation (4) is a nonlinear term, and it is linearized by piecewise linearization. The expression is as follows: Nm
∑
g
Ci,t = f imin Ii,t +
Pi,t,m Ki,t,m
(7)
m =1
where number of segments. f imin is the fuel cost corresponding to the minimum output Energies Nm 2018, is 11,the x FOR PEER REVIEW 5 of of 19 the unit i. Ki,t,m is the slope of the unit i at the segment m in period t. Pi,t,m is the output of unit i at SU SU segment m inmperiod t. Binary variables ui,t and i,t arevintroduced to linearize Ci,t inCEquation (6). ui,t the segment in period t. Binary variables ui,tvand i,t are introduced to linearize i,t in Equation is the start-up status variable, for which 1 means that the unit is in the boost process. For the shut(6). ui,t is the start-up status variable, for which 1 means that the unit is in the boost process. For the down statusstatus variable vi,t, 1 represents that the unit in the process. Thus, Equation (6) can shut-down variable vi,t , 1 represents that theisunit is shut-down in the shut-down process. Thus, Equation be transformed into theinto linear shown shown in Equation (8): (6) can be transformed theexpression linear expression in Equation (8): CiSU cisu ui ,t su ,t =SU Ci,t = ci ui,t = IIi,ti ,t − − IIi,ti ,t−−11 uui,ti ,t−−vvi,ti ,t = u +ui,tv +≤vi,t 1 ≤1
i ,t
(8) (8)
i ,t
and Figure 22 shows ofof wind power, in in which Figure shows the theprobability probabilitydensity densitydistribution distributioncurve curve wind power, whichwtw, t , wwt t and wtt represent representthe the scheduled scheduled wind wind power, upper t, w upper and and lower lowerbound boundofofconfidence confidenceinterval intervalatattime time t, respectively. respectively. Probability density
Load shedding Upward Downward reserve reserve 0
wt
wt
Wind curtailment Power/p.u.
wt
1
Figure density function function and and confidence confidence interval. interval. Figure 2. 2. Wind Wind power power probability probability density
If the actual output of wind power is located within the boundary of the interval w , w , the If the actual output of wind power is located within the boundary of the interval wt , wt , tthe twind wind power prediction error be compensated the upward and downward reserves. If the power prediction error can becan compensated by the by upward and downward reserves. If the actual actual output exceeds the boundary, the wind power uncertainty cannot be covered by system output exceeds the boundary, the wind power uncertainty cannot be covered by system reserves, and reserves, and wind curtailment or will loadbeshedding will bethe used ensure the safesystem. operation the wind curtailment or load shedding used to ensure safeto operation of the The of wider system. The wider the confidence interval is, the more reserves are needed for the system, but the the confidence interval is, the more reserves are needed for the system, but the smaller the risks of smaller the risks of wind curtailment load shedding. Therefore, the costand of actual reserves wind curtailment and load shedding. and Therefore, the cost of actual reserves risk cost causedand by risk cost caused by wind power uncertainty are introduced into the objective function in this paper. wind power uncertainty are introduced into the objective function in this paper. It can can be be seen seen from from the the figure figure that that the the three three variables variables divide divide the the probability probability density density curve curve into into It four parts, thearea areaof ofthe theshaded shadedpart partrepresents represents the the expected expected load load shedding shedding four parts, the the first first part part is is [0, [0, wwtt],],the w w probability due and the the wind wind power power probability due to to wind wind power power under under estimation. estimation.The Thesecond secondpart partisis[ [wt t, , wtt ],], and errors in this part which is caused by under estimation estimation can can be be covered covered by by upward upward reserves reserves of of system. system. The third part is [[w wtt ,, ww andthe theuncertainties uncertaintiesininthis thispart partcaused causedby by the the wind wind power overestimation t ], t ],and can be covered by downward reserves. The fourth part is [w , 1], the area of shaded part represents can be covered by downward reserves. The fourth part t is [ wt , 1], thethe area of the shaded part the expected wind curtailment probability caused by wind power overestimation. Therefore, the cost represents the expected wind curtailment probability caused by wind power overestimation. caused by wind power uncertainty can be expressed as follows: Therefore, the cost caused by wind power uncertainty can be expressed as follows: ov,ur un,dr un,wc C Wt =ov,ls K ov,ls E ov,ls + K ov,urEEov,ur + K un,drEEun,dr ++ K un,wc E un,wc CW Eov,ls Kun,wc j ,j,t t j ,tE j,t j,t = j ,K j,t j ,t + Kov,ur j,tj ,t + Kun,dr
ov,ls E j ,t = E ov,ur = j ,t E un,dr j ,t =
(9) (9)
wt
0 (wt − x) f ( x)dx w w 0 (wt − wt ) f ( x)dx + w (wt − x) f ( x)dx t
t
t
1
wt
t
t
w (wt − wt ) f ( x)dx + w 1
( x − wt ) f ( x)dx
(10)
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Rw Eov,ls = 0 t (wt − x ) f ( x )dx j,t R R Eov,ur = wt (wt − wt ) f ( x )dx + wt (wt − x ) f ( x )dx j,t 0 R1 R wwtt Eun,dr = ( w − w ) f ( x ) dx + t t j,t wt wt ( x − wt ) f ( x ) dx R1 un,wc Ej,t = wt ( x − wt ) f ( x )dx
(10)
ov,ur where f (x) is the probability density distribution function. Eov,ls , Eun,dr and Eun,wc are the j,t , E j,t j,t j,t expected load shedding caused by wind power overestimation (OVLS), the expected overestimation wind power that can be covered by upward reserve (OVUR), the expected underestimation wind power that can be covered by downward reserve (UNDR), the expected wind curtailment caused by wind power underestimation (UNWC), respectively. Kov,ls , Kov,ur , Kun,dr and Kun,wc are their corresponding cost coefficients. It is worth noting that the OVUR include both [wt , wt ] part and [0, wt ] part. Since the uncertainty in [0, wt ] is also covered by upward reserve, and the upward reserve is used up so that partial load has to be cut to maintain the power balance of power system. Similarly, the UNDR also include the [wt , 1] part.
3.1.2. Constraints The constraints are as follows: (1)
Supply-demand balance constraints: NG
NW
i =1
j =1
∑ Pi,t + ∑ w j,t = PtL ∀t
(11)
where Pi,t is the active output of unit i in period t. wj,t is the schedule value of wind farm j in period t. PtL is the load forecast value in period t. (2)
Thermal power plants output constraints: Pimin Ii,t ≤ Pi,t ≤ Pimax Ii,t
(12)
where Pimax and Pimin are the upper and lower limits of the output of unit i. (3)
Unit segment output constraints: Nm
Pi,t =
∑
Pi,t,m ∀i, t
(13)
m =1 max 0 ≤ Pi,t,m ≤ Pi,m ∀i, t, m
(14)
max is the maximum output of unit i on segment m. where Pi,m
(4)
Unit ramping up and down constraints: Pi,t − Pi,t−1 ≤ URi (1 − ui,t ) + Pimin ui,t ∀i, t
(15)
Pi,t−1 − Pi,t ≤ DRi (1 − vi,t ) + Pimin vi,t ∀i, t
(16)
where URi , DRi are the maximum up and down ramp rate of unit i. In addition to Equations (15) and (16), it is necessary to consider any circumstances where the wind power output changes within the confidence intervals. For example, when wind power jumps from the confidence interval boundary of the current time period to the boundary of next time period, it is also necessary to ensure that the conventional units have sufficient adjustment capability to ensure
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the safe operating of the system. Therefore, Equations (17)–(20) are also considered in this model, representing the two extreme cases shown in Figure 3: up
dn ( Pi,t − ri,t ) − ( Pi,t−1 + ri,t−1 ) ≤ URi (1 − uit ) + Pimin uit ∀i, t up
dn ) ≤ DRi (1 − vi,t ) + Pimin vi,t ∀i, t ( Pi,t−1 + ri,t−1 ) − ( Pi,t − ri,t up
dn min ( Pi,t + ri,t ) − ( Pi,t−1 − ri,t ui,t ∀i, t −1 ) ≤ URi (1 − ui,t ) + Pi up
dn min ( Pi,t−1 − ri,t vi,t ∀i, t −1 ) − ( Pi,t + ri,t ) ≤ DRi (1 − vi,t ) + Pi
(5)
Unit minimum ON and OFF time limits:
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n
o
Ii,t OFF − Ii,ttime Ii,τ ∀τ ∈ [t + 1, min NT, t + upmin −1 ] −1 ≤ limits: (5) Unit minimum ON and i o I t − Ii ,t −1 ≤ Ii,τ ∀τ ∈ [t + 1, min{NTn, t + upimin − 1}] Ii,t−1i,− Ii,t ≤ 1 − Ii,τ ∀τ ∈ [t + 1, min NT, t + dwimin − 1 ]
(21)
(21) (22)
min Idw − Ii,t are ≤ 1 − the Ii,τ ∀ τ ∈ [t + 1, min{ NT , t + dw −and 1}] shut-down time(22) i ,t −1min i where upmin and minimum start-up time for i i min min unit i, respectively. where up and dw are the minimum start-up time and shut-down time for unit i, i
i
(6) respectively. Unit upward and downward reserves constraints:
(6) Unit upward and downward reserves constraints: up ri,t ≤ min{URi Ii,t , Pimax Ii,t − Pi,t } ∀i, t max riup ,t ≤ min{UR n i Ii,t , Pi Ii,t − Pi,t } ∀i,ot
(23) (23) (24) (24)
dn ri,t ≤ min DRi Ii,t , Pi,t −minPimin Ii,t ∀i, t ridn ≤ min{DRi Ii ,t , Pi ,t − Pi Ii ,t } ∀i, t ,t
The power uncertainty uncertainty within within the the lower lower and and upper upper boundaries boundaries of of the the confidence confidence intervals intervals The wind wind power is is considered considered to to be be covered covered by by system system reserves, reserves, so so that that Equations Equations (25) (25) and and (26) (26) are are also alsoincluded: included: NW NW
NG NG
j=j =11
i =i = 11
NW NW
NG NG
∑ ((wwj,tj ,t −− wwjj,t,t )) ==∑riup,rt i,t, ,∀∀tt
∑
up
(25) (25)
∑ridn,rt i,tdn, ,∀∀tt
wj,t −w ((w wjj,t j ,t − ,t )) ==
j=j =11
(26) (26)
i =i = 11
Wind power Installed capacity wt −1
Upper boundary
wt
wt −1
wt
Lower boundary
0
t-1
t
Time
Figure confidence intervals. intervals. Figure 3. 3. Schematic Schematic diagram diagram of of ramp ramp constraints constraints within within confidence
3.2. Rolling 3.2. Rolling Dispatch Dispatch Model Model As the thewind windpower powerprediction prediction accuracy increases decreasing time scale, the intra-day As accuracy increases withwith decreasing time scale, the intra-day rolling rolling schedule based on the latest wind power forecasting value can help to reduce the deviation of schedule based on the latest wind power forecasting value can help to reduce the deviation of the the day-ahead plan. As shown in Figure 4, Δt = 15 min, T = 4 h, ΔT = 1 h. The rolling plan is formulated every 15 min before the scheduled time, and a plan for 4 h is formulated. The plan is executed every 1 h, and the plan for the first 1 h out of 4 h will be actually executed every time. Generally, the rolling dispatch model only optimizes the cost of each rolling plan, and lacks the consideration of coordination among multiple rolling plans, which affects the global optimality in
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day-ahead plan. As shown in Figure 4, ∆t = 15 min, T = 4 h, ∆T = 1 h. The rolling plan is formulated every 15 min before the scheduled time, and a plan for 4 h is formulated. The plan is executed every 1 h, and the plan for the first 1 h out of 4 h will be actually executed every time. Generally, the rolling dispatch model only optimizes the cost of each rolling plan, and lacks the consideration of coordination among multiple rolling plans, which affects the global optimality in the overall scheduling periods. Therefore, the coordination among various rolling plans is considered in this paper. Taking tD –tE period as an example, when formulating the first rolling plan at time t A − ∆t, the wind power prediction accuracy for this period is not high enough so that the confidence level is not strictly required. When formulating the fourth rolling plan at time t D − ∆t, the confidence level needs to be strictly met. In addition to the fuel cost and risk cost, the regulation cost is also introduced in the objective function of the model. The purpose is to reduce the overall amount of regulations and avoid repeated regulations among the rolling plans. Energies 2018, 11, x FOR PEER REVIEW 8 of 19 ΔT
Δt
T IV
A B C D
III
IV
II
III
IV
I
II
III
IV
I
II
III
IV
I
II
III
I
II
E
F G
tA
tB
tC
tD
tE
tF
tG
tH
tI
t
Figure Figure 4. 4. Overview Overview of of the the intra-day intra-day rolling rolling dispatch. dispatch.
3.2.1. 3.2.1. Objective Objective Taking D–tH periods as an example, the objective function of this rolling plan includes the Taking ttD –tH periods as an example, the objective function of this rolling plan includes the regulation costs DI, CIII CIII to to DII, DII, CIV CIV to to DIII, DIII, and and the the day-ahead day-ahead plan plan of of ttG–t H period to DIV, regulation costs of of CII CII to to DI, G –tH period to DIV, in addition to fuel costs costs and and risk risk costs costs of of DI, DI, DII, DII, DIII, DIII, and and DIV DIV periods. periods. Similarly, the objective objective in addition to the the fuel Similarly, the function of the next rolling plan includes the fuel costs and risk costs of the EI, EII, EIII, function of the next rolling plan includes the fuel costs and risk costs of the EI, EII, EIII, and and EIV EIV periods, the regulation costs of DII-EI, DIII-EII, DIV-EIII, and the day-ahead plan of t H–tI period to periods, the regulation costs of DII-EI, DIII-EII, DIV-EIII, and the day-ahead plan of tH –tI period to EIV. EIV. It is worth that this the is only cost of the function, objective rather function, thanscheduling the actual It is worth notingnoting that this is only cost the of the objective thanrather the actual scheduling cost within the overall scheduling periods. At the time t D, only the wind power and load cost within the overall scheduling periods. At the time tD , only the wind power and load forecasting forecasting in DI, andare DIV periods known, therisk fuel costofand risk costs of these values in DI,values DII, DIII, andDII, DIVDIII, periods known, theare fuel cost and costs these four periods are four periods are included in the objective function. However, the actual total cost within tD–tH included in the objective function. However, the actual total cost within tD –tH includes the fuel costs includes the fuel costs of DI, EI, and GI periods, and the corresponding regulation and risk costs of DI, EI,and FI, risk and costs GI periods, andFI,the corresponding regulation costs for each period, costs forthe each such as the tD–tE period with four regulation costs (day-ahead plan of tD–tE such as tD –tperiod, E period with four regulation costs (day-ahead plan of tD –tE period to AIV, AIV to BIII, period to AIV, BIII, BIIIthere to CII, to DI). Therefore, there are 16 regulations in the tD–tH BIII to CII, CII toAIV DI).to Therefore, areCII 16 regulations in the tD –tH periods. periods. The intra-day rolling plan does not change the day-ahead unit commitment, and it is essentially a The intra-day plan does not change the day-ahead unit commitment, and it is essentially dynamic economicrolling dispatch problem. The objective function includes the fuel cost, reserve cost, risk acost, dynamic economic dispatch problem. The objective function includes the fuel cost, reserve cost, risk and regulation cost, and it is expressed as follows: cost, and regulation cost, and it is expressed as follows: NT NG
NT NG
F= min F= min∑
NT NW
NT NW
∑ (C(Ci,tig,t++CCi,trir,t ++ CCiMi,tM,t ))++ ∑ ∑C WjC,t W j,t
t =1 i =1
t =1 i =1
g
t =1 j =1
t =1 j =1
(27) (27)
M
where Ci,t is the regulation cost of the unit i in period t, and its expression is shown in Equation (28). W The expression of Cj,t in Equation (27) is different from Equation (9) for that a penalty factor μ is introduced to coordinate the risks among different rolling plans, and it is shown in Equation (29): CiM ,t
=
clm
ΔPk ,i ,t
clm Pk ,i ,t − Pk −1,i ,t +ΔT , if 1 ≤ t ≤ T − Δt = m ' cl Pk ,i ,t − Pk ,i ,t , if T − Δt < t ≤ T
(28)
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M is the regulation cost of the unit i in period t, and its expression is shown in Equation (28). where Ci,t The expression of CW j,t in Equation (27) is different from Equation (9) for that a penalty factor µ is introduced to coordinate the risks among different rolling plans, and it is shown in Equation (29):
( M Ci,t
=
cm l ∆Pk,i,t
=
cm l Pk,i,t − Pk−1,i,t+ ∆T , i f 1 ≤ t ≤ T − ∆t 0 cm l Pk,i,t − Pk,i,t , i f T − ∆t < t ≤ T
ov,ls CW + Kun,wc Eun,wc ) + Kov,ur Eov,ur + Kun,dr Eun,dr j,t = µl Kov,ls E j,t j,t j,t j,t
(28)
(29)
where cm l is the cost coefficient of output regulation, l = 1, 2, 3, 4 which represent the I, II, III, IV periods in a rolling plan. ∆Pk,i,t is the output regulation amount of the unit I at time t in the k-th rolling plan. Pk,i,t is the output of the unit i at time t in the k-th rolling plan. Pk−1,i,t+∆T is the output of the last 0 rolling plan at time t + ∆T. Pk,i,t is the day-ahead output corresponding to the time t in the k-th rolling plan. This formula reflects that the regulation amount for each rolling plan is the output of this plan minus the output of the previous plan, and if there is no plan for the previous time, the minuend is the output of day-ahead plan. 3.2.2. Constraints Considering that the scheduled power of the rolling plan is related to the daily plan and the previous rolling plan, the regulation quantity of the rolling plan should not be too large, and the deviation of the correction value needs to be controlled within a certain range: ( ∆Pk,i,t ≤ ∆Pmax_up = min( Pmax − Pk,i,t , ∆Pu ) i i k,i,t ∆Pk,i,t ≤ ∆Pmax_dn = min( Pk,i,t − Pmin , ∆Pd ) i
k,i,t
(30)
i
max_up
max_dn where ∆Pk,i,t , ∆Pk,i,t are the maximum values of the upward regulation and the downward regulation in the k-th rolling plan of unit i at the time t. ∆Piu and ∆Pid are the amounts of maximum upward and downward regulation allowed for the unit i. The penalty factor µl needs to satisfy the following constraint:
µ l > µ l +1
(31)
Other constraints can refer to the content of the UC model which is shown in Equations (9)–(24). up dn , The decision variables of the model are the output Pk,i,t , the upward/downward reserve rk,j,t /rk,j,t and the wind power dispatch value wk,t . 3.3. Model Transformation and Solution The decision variables of the proposed UC model are the binary variables Ii,t , ui,t and vi,t which indicate the unit’s on/off, start-up and shut-down status, the scheduled power Pi,t , up dn the upward/downward reserve ri,t /ri,t , the scheduled wind power w j,t , and the upper/lower boundary of confidence interval w j,t /w j,t . The UC model is a mixed integer nonlinear programming problem (MINLP). There are definite integral terms in the objective function of the model, and both the integrand function and integral upper/lower bound contain decision variables, which makes it difficult to solve this problem. Therefore, it is necessary to linearize the integral terms of the objective function. Take the first two terms in Equation (9) as an example, set E1 j,t = Kov,ls · Eov,ls + Kov,ur Eov,ur . j,t j,t Divide the probability density function into S segments, then the following equivalent expression can be obtained: Kov,ls (w j,t − x j,t,s ) + Kov,ur (w j,t − w j,t ), i f 0 ≤ x j,t,s ≤ w j,t φj,t ( x j,t,s ) = (32) Kov,ur (w j,t − x j,t,s ), i f w j,t < x j,t,s ≤ w j,t 0, i f w j,t < x j,t,s ≤ 1
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E1 j,t =
Z 1 0
S
φj,t f ( x )dx =
∑ φj,t,s f j,t,s dx j,t,s
(33)
s =1
where s = 1, 2, · · · , S. f j,t,s is the probability of the s-th segment of discretized probability density function f(x). dx j,t,s is the horizontal axis corresponding to the s segment of the probability density function. φj,t . is a segmented function, and its function image is shown in Figure 5. The slope of the first2018, segment in PEER Figure 5 is greater than the slope of the second one because the penalty cost Energies 11, x FOR REVIEW 10 offor 19 the load shedding is greater than the cost for the upward reserve. Since the objective function is the minimization problem of E1 j,t , the auxiliary variable η j,t,s can be introduced to convert Equation (33) S E1 j ,t = η j ,t , s f j ,t , s dx j ,t , s to Equation (34): (34)
S s =1
where η j ,t , s
E1 j,t = ∑ η j,t,s f j,t,s dx j,t,s (34) needs to satisfy the condition shown in Equation (35). Thus, the first two terms in s =1
Equation can be Equations (34)in and (35). Similarly, thethe lastfirst twotwo terms in Equation (9) where η j,t,s(9) needs to expressed satisfy the by condition shown Equation (35). Thus, terms in Equation cancan be expressed by Equations (36)(34) andand (37)(35). in which is the auxiliary (9) variable: γ j ,t , s the (9) be expressed by Equations Similarly, lastcorresponding two terms in Equation can be expressed by Equations (36) and (37) in which γ j,t,s is the corresponding auxiliary variable: η j ,t , s ≥ K ov,ls ( w j ,t − x j ,t , s ) + Kov,ur ( w j ,t − w j ,t ), ∀j , t , s ηj,t,s K K ov,ur (w( w x j ,t ,)s ),+∀Kjov,ur η j ,t≥ , t , s (w j,t − w j,t ), ∀ j, t, s j,tj ,− (35) , s ≥ ov,ls t − xj,t,s (35) η j,t,s ≥ Kov,ur (w j,t − x j,t,s ), ∀ j, t, s η j ,t , s ≥ 0, ∀j , t , s η j,t,s ≥ 0, ∀ j, t, s S
E 2 j ,t = S γ j ,t , s f j ,t , s dx j ,t , s E2 j,t = s∑ =1 γ j,t,s f j,t,s dx j,t,s
(36) (36)
γ j ,t , s ≥ K un,dr ( w j ,t − w j ,t ) + K un,wc ( x j ,t , s − w j ,t ), ∀j , t , s γj,t,s ≥ Kun,dr (w j,t − w j,t ) + Kun,wc ( x j,t,s − w j,t ), ∀ j, t, s ( x , s − w j(,tx), ∀j− , t ,w s ), ∀ j, t, s γ j ,t , s ≥ Kγun,dr j,t,s ≥j ,tK j,t un,dr j,t,s γ j ,t , s ≥ 0, ∀j , t , s γ j,t,s ≥ 0, ∀ j, t, s
(37) (37)
s =1
φ j，t Slope
K ov ,ls Slope
0
K ov ,ur
w j ,t
w j，t
x j，t，s
Figure5.5. Function Function image imageof ofφϕj,t.. Figure i,t
In addition, the decision variables w j ,t / w j ,t which correspond to the boundaries of 99 different In addition, the decision variables w j,t /w j,t which correspond to the boundaries of 99 different confidence levels are discrete, and a binary variable ICP is introduced to handle them. Set ICP as a 1 × confidence levels are discrete, and a binary variable ICP is introduced to handle them. Set ICP as 99 vector, if the 90th element is 1 and the other elements are 0, the confidence level is 90%. Thus, a 1 × 99 vector, if the 90th element is 1 and the other elements are 0, the confidence level is 90%. through Equations (38) and wind powerpower confidence level can associated with the wind Thus, through Equations (38)(39), andthe (39), the wind confidence levelbecan be associated with the power interval boundaries . That is, the actual decision variable is I CP : w / w j ,t w j ,t/w j,t . That is, the actual decision variable is I CP : wind power interval boundaries j,t up ( w j ,t = I CP .W up w = I .W j,t CP .W dn dn ww j ,t== IICP CP .W j,t
(38) (38)
99
ICP (k ) = 1
(39)
k =1
where Wup and Wdn are T × 99 matrices, representing the upper and lower boundaries of wind power intervals with 99 different confidence levels. The calculation method can refer to Section 2.3. All the nonlinear terms are linearized, and the model established in this paper is converted into
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99
∑ ICP (k) = 1
(39)
k =1
where W up and W dn are T × 99 matrices, representing the upper and lower boundaries of wind power intervals with 99 different confidence levels. The calculation method can refer to Section 2.2. All the nonlinear terms are linearized, and the model established in this paper is converted into a mixed integer linear programming problem (MILP) that can be solved using commercial optimization software CPLEX. 4. Simulation and Discussion 4.1. Data and Parameters Setting The 10-units system and 118-bus system are tested to verify the effectiveness and superiority of the proposed models. The wind power and load data are shown in Appendix A. The upper and lower limits of the generator output, the minimum start-up and shut-down time, the fuel cost coefficients, the reserve cost coefficients and so on can refer to [6,18]. The wind power data in 2015 and 2016 from [19] is used to simulate the empirical PDF. The capacity of the wind farm is set as 350 MW, and the cost coefficients Kov,ur , Kov,ls , Kun,dr and Kun,wc are 120, 200, 60 and 120 USD/MW, respectively. The simulations are conducted based on Matlab 2013a, and CPLEX 12.6.1. 4.2. Results of UC Model Simulation of four cases are conducted in this section. Cases 1 and 3 are tested on the 10-units system. Cases 2 and 4 are tested on the 118-bus system. In cases 1 and 2, the confidence levels of the 24 h are set to be the same. In cases 3 and 4, the 24-h wind power confidence levels are different. 4.2.1. Case 1 and Case 2 Table 1 shows the output of conventional units. It can be seen from Table 1 that the output of the units is in accordance with the distribution of the load. Units 1 and 2 are always on because they have the largest capacity and are used to ensure the supply of power to basic load. Table 1. Conventional units output (Case 1). Hour
G1
G2
G3
G4
G5
G6
G7
G8
G9
G10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
285.7 316.3 385.5 455 446.6 455 455 455 455 455 455 455 455 455 455 391.5 381.3 408.8 403.8 455 455 408.2 338.2 358.4
150 202.7 262.7 322.7 382.4 442.4 455 455 455 455 455 455 455 455 424.6 364.6 304.6 335 395 455 455 395 335 275
0 0 0 0 0 0 0 20 90 130 130 130 90 20 0 0 0 0 0 0 0 0 0 0
0 0 0 0 20 90 130 130 130 130 130 130 90 20 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 25 85 145 162 162 130.8 140.4 80.4 25 25 42 102 162 115.7 65.8 25 0
57.1 49.5 53.1 51 49.6 39.9 60.8 79.2 49.2 49.2 43.8 64.4 48.3 53 53.1 53.3 49.9 55.8 54.4 80 57.1 49.5 53.1 52.3
0 0 0 0 0 0 0 0 0 0 25 75.1 25 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 10 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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The calculated optimal confidence of case 1 is 73%, and the corresponding optimal wind power confidence intervals is shown in Figure 6. The costs are shown in Table 2. In order to verify the effectiveness of the proposed UC model based on the optimal confidence intervals, the UC model with a specified confidence level is compared with it. The total costs with confidence levels from 1% to 99% is shown in Figure 7. Energies 2018, 11, x FOR PEER REVIEW 12 of 19
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300 300
Wind power / WM Wind power / WM
250 250 200 200
upper boundary of intervals upper boundary of intervals predictive values predictive values
150 150 100 100
lower boundary of intervals lower boundary of intervals
50 50 0 00 0
5 5
10 15 10 Time / h 15 Time / h
20 20
25 25
Figure 6. Optimal wind power confidence intervals (Case 1). 1). Figure 6. 6. Optimal Optimal wind wind power power confidence confidence intervals intervals (Case (Case Figure 1).
Total cost / USD Total cost / USD
5
x 10 5.95 x 10 5 5.95 5.9 5.9 5.85 5.85 5.8 5.8 5.75 5.75 5.7 5.7 5.65 5.65 5.6 5.6 5.55 5.55 0 0
Optimal Optimal confidence level confidence level 67 69 71 73 75 77 79 67 69 71 73 75 77 79
20 20
40 60 40 60 / % Confidence level Confidence level / %
80 80
100 100
Figure 7. Total cost under different confidence levels for 10-units system (Case 1). Figure levels for for 10-units 10-units system system (Case (Case 1). 1). Figure 7. 7. Total Total cost cost under under different different confidence confidence levels Table 2. The costs for Case 1 and Case 2. Table 2. The costs for Case 1 and Case 2. Case Total Cost (USD) Fuel Cost (USD) Reserve Capacity Cost (USD) Risk Cost (USD) Table 2. The costs for Case 1Capacity and CaseCost 2. (USD) Risk Cost (USD) Case Total Cost (USD) Fuel Cost (USD) Reserve 1 555,574 491,300 19,158 40,516 12 555,574 491,300 19,158 40,516 1,721,658 1,664,158 12,275 44,421 Case Total Cost (USD) Fuel Cost (USD) Reserve Capacity Risk Cost (USD) 2 1,721,658 1,664,158 12,275 Cost (USD) 44,421 1 555,574 491,300 19,158 40,516 It can be seen from Figure 7 that the overall cost of the UC model shows a trend of decreasing 1,664,158 12,275 It 2can be seen1,721,658 from Figure 7 that the overall cost of the UC model shows a trend 44,421 of decreasing
first and then increasing with the gradual increase of the confidence level. There are few local first and then increasing with the gradual increase of the confidence level. There are few local minimum points in the curve, and it is found that the global minimum point is the confidence level minimum points in the Figure curve, 7and is found minimum point is theofconfidence level It can be seen thatitthe the overallthat costthe of global the UC shows a model trend decreasing of 73%, which is from consistent with simulation results of model the proposed and proves first the of 73%, which is consistent with the simulation results of the proposed model and proves the and then increasing with the gradual increase of the confidence level. There are few local minimum effectiveness of the proposed model. In addition, the confidence level of the traditional UC model effectiveness of the proposed model.that In addition, the confidence level traditional UCof model points in the and it is found the global minimum is of thethe confidence level 73%, based on the curve, wind power intervals is usually set to be more point than 90%, which will lead an overbased on the wind power intervals is usually set to be more than 90%, which will lead an overwhich is consistent with simulation results of the proposed and proves the effectiveness conservative scheme, andthe thereby increase the overall cost of themodel system. conservative scheme, andInthereby increase the overalllevel costof of the the traditional system. of theToproposed model. addition, the confidence UC model based on the further verify the effectiveness of the proposed model, the test on 118-bus system is carried To further verifyisthe effectiveness of the than proposed model,will the lead test on 118-bus system is scheme, carried wind power intervals usually set to be more 90%, which an over-conservative out. The optimal confidence level of case 2 is 60% and the costs are shown in Table 2. The total costs out. The optimal confidence level of case 2 system. is 60% and the costs are shown in Table 2. The total costs and thereby increase the overall the under different confidence levelscost areofshown in Figure 8, in which the global minimum point is the under different confidence levels are shown in Figure 8, in which the global minimum point is the confidence level of 60% and it is consistent with the simulation results of the proposed model. Thus, confidence level of 60% and it is consistent with the simulation results of the proposed model. Thus, the same conclusions can be drawn. the same conclusions can be drawn.
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To further verify the effectiveness of the proposed model, the test on 118-bus system is carried out. The optimal confidence level of case 2 is 60% and the costs are shown in Table 2. The total costs under different confidence levels are shown in Figure 8, in which the global minimum point is the confidence level of 60% and it is consistent with the simulation results of the proposed model. Thus, the same Energies 2018, 11, x FOR PEER REVIEW 13 of 19 conclusions can be drawn. 1.745
x 10
6
Total cost / USD
1.74 optimal confidence level
1.735 1.73
50
60
70
1.725 1.72
0
20
40 60 Confidence level / %
80
100
Figure levels for for 118-bus 118-bus system system (Case (Case 2). 2). Figure 8. 8. Total Total cost cost under under different different confidence confidence levels
4.2.2. 4.2.2. Case Case 33 and and Case Case 44 The 3. The The output output results results of of Case Case 33 are are shown shown in in Table Table 3. The optimal optimal confidence confidence intervals intervals and and optimal optimal confidence levels for Case 3 are shown in Figures 9 and 10. confidence levels for Case 3 are shown in Figures 9 and 10. The 4. Compared 2, it can be The costs costs of of Case Case 33 and and Case Case 44 are are shown shown in in Table Table 4. Compared with with Table Table 2, it can be seen seen that that the total cost of Case 3 is lower than that of Case 1 because Case 3 increases the complexity of the total cost of Case 3 is lower than that of Case 1 because Case 3 increases the complexity of the the problem on the problem on the basis basis of of Case Case 1, 1, that that is, is, setting setting the the confidence confidence levels levels for for 24 24hhdifferent differentfrom fromeach eachother, other, and space is is expanded. expanded. and the the solution solution space Table 3. Conventional units output output (Case (Case 3). 3). Table 3. Conventional units Hour G1 G2 G1 G2 1 285.6 150 1 2 285.6 317.1 150 199.4 2 3 317.1387 199.4 259.4 3 387 259.4 4 454.9 319.4 4 454.9 319.4 448.7 379.4 379.4 5 5 448.7 439.4 6 6 455 455 439.4 7 7 455 455 455455 8 8 455 455 455455 9 9 455 455 455455 10 455 455 10 455 455 11 455 455 11 455 12 455 455455 12 455 13 455 455455 14 13 455 455 455455 15 14 455 455 427.9 455 16 15 386.3455 367.9 427.9 17 374.4 307.9 16 386.3 367.9 18 422.6 335 374.4 395 307.9 19 17 416.6 18 422.6 20 455 455335 19 416.6 21 455 455395 22 20 406.5455 395455 23 21 336.5455 335455 24 22 356.4 406.5 275395 23 336.5 335 24 356.4 275
Hour
G3 0 00 00 0 0 0 00 00 00 2020 9090 130 130 130 130 130 130 90 2090 020 00 0 0 0 00 00 00 00 00 0 0 0 0 G3
G4 G4 0 00 00 0 0 0 20 20 90 90 130 130 130 130 130 130 130 130 130 130 130 130 90 90 20 0 20 00 0 0 0 00 00 00 00 00 00 0 0
G5 G5 0 00 00 0 0 0 00 00 00 2525 8585 145 145 162 162 162 162 125.4 125.4 136.7 76.7 136.7 25 76.7 25 25 26.8 25 86.8 26.8 146.8 86.8 115.3 64.9 146.8 25 115.3 64.90 25 0
G6 G7 G6 20 0 20 20 0 20 20 0 20 20 0 20 20 20 0 20 20 0 36.836.8 0 59.859.8 0 27.327.3 0 27.3 27.3 0 72.3 72.334.7 25 34.720 77 20 20 25 20 20 0 20 20 0 20 20 0 20 20 20 0 2061.7 0 20 20 0 61.720 0 20 20 0 20 20 0 20 0 20 0
G8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 77 0 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 10 0 0 0 0
G7
G9 G10 G9 G10 0 0 00 00 0 00 00 0 0 0 0 0 0 0 0 0 00 00 0 00 00 0 00 00 0 00 00 0 00 00 0 0 0 0 0 0 0 0 0 00 00 0 00 00 0 00 00 0 00 00 0 00 00 0 0 0 0 0 0 0 0 0 00 00 0 100 00 0 100 00 0 00 00 0 00 00 0 0 0 0 0 0 0 0 0 0
G8
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Table 4. The costs for Case 3 and Case 4. TableCost 4. The costs forReserve Case 3 and Case 4.Cost (USD) Risk Cost (USD) Fuel (USD) Capacity Fuel Cost (USD) Reserve Capacity Cost (USD) Risk Cost (USD) 475,488 21,365 Fuel 475,488 Cost (USD) Reserve Capacity Cost40,101 (USD) 21,365 Cost (USD) Risk 40,101 1,663,199 12,461 44,112 475,488 21,365 40,101 1,663,199 12,461 44,112 1,663,199 12,461 44,112
Case Total Cost (USD) Case Total Cost (USD) 3Case Total541,554 Cost (USD) 3 541,554 4 3 1,720,576 541,554 4 1,720,576 4 1,720,576
Confidence level Confidence level / %/ %
74 74 72 72 70 70 68 68 66 66 64 640 0
5 5
10 15 10 Time / h 15 Time / h
20 20
25 25
Figure 9. The optimal wind power confidence levels (Case 3). Figure levels(Case (Case3). 3). Figure9.9.The Theoptimal optimal wind wind power power confidence confidence levels 300 300
upper boundary of intervals upper boundary of intervals
Wind power / WM Wind power / WM
250 250 200 200
predictive values predictive values
150 150 100 100
lower boundary of intervals lower boundary of intervals
50 50 0 00 0
5 5
10 15 10 Time / h 15 Time / h
20 20
25 25
Figure 10. The optimal wind power confidence intervals (Case 3). Figure10. 10.The Theoptimal optimal wind wind power power confidence confidence intervals Figure intervals(Case (Case3). 3).
In addition, the fuel cost of the thermal power units is 475,488 USD which is less than 491,300 In addition, the fuel cost of the the overall thermal powerofunits is 475,488 USDunits which is less than USD Case 1 and the thermal power is less than that491,300 of the In of addition, theindicates fuel costthat of the thermaloutput power units is 475,488 USD which is less than 491,300 USD of Case 1 and indicates that the overall output of the thermal power units is less than thatpower of the Case 1. Because the scheduled wind power in Case 3 is increased, that is, the capacity of wind USD of 1. Case 1 andthe indicates thatwind the overall output ofisthe thermalthat power units is lessofthan that of the Case Because scheduled power in Case 3 increased, is, the capacity wind power consumption inthe Case 3 is improved. In conclusion, Case 3 further reduces overall costs and increases Case 1. Because scheduled wind power in Case 3 is increased, that is, the capacity of wind power consumption Case power 3 is improved. In conclusion, Case 3 further costs and increases the capacityin ofinCase wind consumption. Comparing 2 andreduces Case 4,overall the same conclusions can consumption 3power is improved. In conclusion, CaseCase 3 further reduces overall costs and increases the capacity of wind consumption. Comparing Case 2 and Case 4, the same conclusions can be obtained. thebecapacity of wind power consumption. Comparing Case 2 and Case 4, the same conclusions can obtained. be obtained. 4.3. Rolling Dispatch Model Results 4.3. Rolling Dispatch Model Results 4.3. Rolling Dispatch Results In this section,Model the rolling plan simulation experiment is carried out on the basis of the day-ahead In this section, the rolling plan simulation is carried out on the basis the day-ahead unit commitment plan. The scheduling time isexperiment 4 h. The system parameters and costofcoefficients are Incommitment this section, the rolling plan simulation is carried out on the the day-ahead unit plan. The scheduling time isexperiment 4 h. The system parameters andbasis cost of coefficients are μl and cost thecommitment same as thoseplan. in Section 4.2. The values of risk penalty factorparameters coefficient of output unit The scheduling time is 4 h. The system and cost coefficients are the same as those in Section 4.2. The values of risk penalty factor μl and cost coefficient of output to Appendix A. clmm can theregulation same as those in refer Section 4.2. The values of risk penalty factor µl and cost coefficient of output regulationm cl can refer to Appendix A. regulation cl cases can refer to Appendix Three are studied in this A. section. Both Case 5 and Case 6 consider fuel cost, reserve cost, Three cases are studied thissection. section. Both Case 55 and Case consider fuel cost, are studied ininthis Both Case and Case66Case consider fuel cost, reserve cost, andThree risk cases cost, excluding regulation cost. The difference between 5 and Case 6reserve is thatcost, the and risk cost, excluding regulation cost. The difference between Case 5 and Case 6 isconfidence that the and risk cost, levels excluding regulation cost.of The difference between and Case is thatinthe confidence in all four periods Case 5 are limited to Case more5 than 90% 6while Case 6 this confidenceislevels inonly all four periods of Case 5 are limited regulation to more than 90%basis while Case this condition limited the 7 includes of in Case 6.is 6limited levels in all four periods of in Case 5first are period. limited Case to more than 90% while in cost Caseon 6 this condition condition is limited only in the first period. Case 7 includes regulation cost on basis of Case 6. only in the first period. Case 7 includes regulation cost on basis of Case 6.
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4.3.1. Comparison of Case 5 and Case 6 4.3.1. Comparison of Case 5 and Case 66 The rolling plan results performed in the 14th to 17th h for Case 5 and Case 6 are shown in The rolling plan results performed performed in in the the 14th 14th to to 17th 17th hh for for Case Case 55 and and Case Case 66 are areshown showninin Appendix A, corresponding to the outputs of the DI, EI, FI and GI periods in Figure 4. Taking tD –tH Appendix A, corresponding to the outputs outputs of of the the DI, DI, EI, EI, FI FI and and GI GI periods periodsin inFigure Figure4.4.Taking TakingtDtD–t–tHH periods as an example, when formulating the rolling plan at time t D − ∆t, the optimal confidence periods as an example, when formulating formulating the the rolling rolling plan plan at at time time tt −−ΔΔtt,, the the optimal optimalconfidence confidence levels for Case 5 and Case 6 are shown in Figure 11. It can be seen thatDDthe confidence levels in 4 h for Case 6 are shown in Figure 11. It can be seen that the confidence levels in levels for Case 5 and in Figure 11. It can be seen that the confidence levels in4order 4hhfor forto Case 5 are greater than 90%, and the confidence levels for Case 6 decrease with the time. In Case 5 are greater than 90%, and the confidence confidence levels levels for for Case Case 66 decrease decreasewith withthe thetime. time.In Inorder ordertoto observe the difference between the results of Case 5 and Case 6, the costs of the actual executed plans observe the difference between the results results of of Case Case 55 and and Case Case6, 6,the thecosts costsof ofthe theactual actualexecuted executedplans plans for tD –tH periods are shown in Table 5. It can be seen that the total cost of Case 6 is smaller than that of for tD–tH periods are shown in Table 5. 5. It It can can be be seen seen that that the the total total cost costof ofCase Case66isissmaller smallerthan thanthat that Case 5. The reason for this difference is that Case 5 aims at optimizing the fuel cost and risk cost of of Case 5. The reason for this difference difference is is that that Case Case 55 aims aims at atoptimizing optimizingthe thefuel fuelcost costand andrisk riskcost costofof each rolling plan, but lacks consideration of the overall optimality within within thedispatching dispatching periods. each rolling plan, but lacks consideration consideration of of the the overall overall optimality optimality withinthe the dispatchingperiods. periods.
Confidencelevel level//% % Confidence
90 90 80 80 70 70 60 60 50 50
22
case case 55 case case 66 44
66
88 10 10 Time Time // 15min 15min
12 12
14 14
16 16
Figure 11. The optimal wind Figure11. 11.The Theoptimal optimal wind wind power power confidence confidence levels. Figure power confidencelevels. levels. Table Comparison of simulation Case 6.6. Table5.5. 5.Comparison Comparisonof ofsimulation simulation results results for forCase Case555and and Table results for Case and Case Case 6. Total Cost Fuel Cost Upward Reserve Cost Downward Reserve Total Cost Fuel Cost Upward Reserve Cost Downward ReserveCost Cost Case Case Case Total Cost(USD) (USD) Fuel Cost (USD) Upward Reserve Cost (USD) Downward(USD) Reserve Cost (USD) (USD) (USD) (USD) (USD) (USD) (USD) 5 5 423,525 364,359 10,749.4 18,800.6 423,525 364,359 10,749.4 18,800.6 5 423,525 364,359 10,749.4 18,800.6 6 6 421,114.3 361,927.3 12,713.8 16,836.2 421,114.3 361,927.3 12,713.8 16,836.2 6 421,114.3 361,927.3 12,713.8 16,836.2
Risk RiskCost Cost (USD) 29,616 29,616 29,616 29,637 29,637 29,637
Risk Cost (USD) (USD)
4.3.2. Comparisonofof Case and Case 4.3.2. Comparison 4.3.2. Comparison ofCase Case666and andCase Case777 The outputregulations regulationsfor for different rolling plans in which the The output are shownin inFigures Figures12 12and and13, 13, which the The output regulations fordifferent differentrolling rolling plans plans are are shown shown in Figures 12 and 13, inin which the quantity of each regulation in Case 7 is greatly reduced compared with that in Case 6. quantity of each regulation in Case 7 is greatly reduced compared with that in Case 6. quantity of each regulation in Case 7 is greatly reduced compared with that in Case 6.
Regulationquantity quantity/ MW / MW Regulation
700 700
The first time The first time The second time The second time The third time The third time The fourth time The fourth time
600 600 500 500 400 400 300 300 200 200 100 100 0 00 0
2 2
4 4
6 6
8
10
12
14
16
12 14 16 Time / 815min10 Time / 15min Figure 12. The output regulations for each period (Case 6). Figure12. 12.The Theoutput outputregulations regulations for for each each period Figure period (Case (Case6). 6).
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The first time The second time The third time The fourth time
Regulation quantity / MW
70 60 50 40 30 20 10 0
0
2
4
6
8
10
12
14
16
Time / 15min Figure 13. The output regulations for each period (Case 7). Figure 13. The output regulations for each period (Case 7).
The first regulation quantity in Figure 13 is greater than the other regulations. This is because first regulation quantity in Figure 13 is greater the forecast other regulations. This than is because theThe deviation between the day-ahead and intra-day windthan power results is larger that thebetween deviation day-ahead and intra-day wind power results larger that the between intra-daythe neighboring rolling forecast results, and theforecast regulations areismade tothan correct between the intra-day neighboring rolling forecast results, and the regulations are made to correct the the deviations. In addition, the regulation quantity is gradually reduced in Case 7, which indicates deviations. In addition, the regulation quantity is gradually reduced in Case 7, which indicates that that the intra-day rolling plans can gradually correct the deviations of day-ahead plan. If all the theregulations intra-day rolling plans can correct the be deviations day-ahead plan. If all the regulations are executed in advance, it will difficult of and risky. From this perspective, the ΔTgradually areresult executed ∆T7in it Case will be of Case is advance, superior to 6. difficult and risky. From this perspective, the result of Case 7 is The superior tocosts Caseand 6. regulation quantity of Case 6 and Case 7 are shown in Table 5. It can be seen that theThe regulation quantity of Casequantity 7 is significantly and the fuel cost isin smaller costs and regulation of Case smaller, 6 and Case 7 are shown Table than 6. ItCase can 6. beThe seen reason for this difference Case lacks consideration thethe regulation which leads to 6. that the regulation quantityisofthat Case 7 is 6significantly smaller, of and fuel cost costs, is smaller than Case repeated regulations among the rolling plans. The lower fuel cost of Case 7 indicates that the total The reason for this difference is that Case 6 lacks consideration of the regulation costs, which leads of thermal power units the is less, and plans. thus theThe wind power greater. That is, the capacity to output repeated regulations among rolling lower fueloutput cost ofisCase 7 indicates that the total of consuming wind power hasisbeen output of thermal power units less, improved. and thus the wind power output is greater. That is, the capacity of consuming wind power has been improved. Table 5. Comparison of simulation results for Case 6 and Case 7.
Case 6
Case 7 6 7
6. Comparison of simulation results for and Case Regulation 1 Table Regulation 2 Regulation 3 Regulation 4 Case Fuel6Cost Risk7.Cost (MW) (MW) (MW) (MW) (USD) (USD) 6727.2 1574.5 1987.5 2363.3 29637 Regulation Regulation Regulation Regulation Fuel361927.3 Cost Risk Cost 699.4 392.3 166.8 109.4 359407.1 29819 1 (MW)
2 (MW)
3 (MW)
4 (MW)
(USD)
(USD)
6727.2 699.4
1574.5 392.3
1987.5 166.8
2363.3 109.4
361927.3 359407.1
29637 29819
5. Conclusions
Reserve Cost (USD) 29550 Cost Reserve 28739.6
(USD)
29550 28739.6
A unit commitment model considering the optimal wind power confidence intervals is proposed to balance the economic cost and risk of power system with wind power integration. On the basis, in 5. Conclusions order to achieve the global optimality of the overall cost, regulation cost is taken into account to A unit an commitment model dispatch considering the based optimal power confidence intervals is proposed establish intra-day rolling model onwind optimal confidence intervals. Through the tosimulation balance the economic cost and risk of power system with wind power integration. On the basis, of the models, the following conclusions can be drawn: in order to achieve the global optimality of the overall cost, regulation cost is taken into account to (1) The simulation results show that as the wind power confidence level changes, there is a tradeoff establish an intra-day rolling dispatch model based on optimal confidence intervals. Through the between the economic cost and risk cost. The global optimal point which balance the economic simulation of the models, the following conclusions can be drawn: cost and risk cost of the system can be effectively obtained by the proposed model. (2) The simulationresults resultsshow of intra-day rolling model shows that this model to (1) The simulation that as the winddispatch power confidence level changes, therecan is ahelp tradeoff reduce the regulations, thereby reducing thethe overall cost between theregulation economicquantity cost andand riskavoid cost. repeated The global optimal point which balance economic within the scheduling periods. cost and risk cost of the system can be effectively obtained by the proposed model.
(2)
The simulation results of intra-day rolling dispatch model shows that this model can help to reduce the regulation quantity and avoid repeated regulations, thereby reducing the overall cost Funding: received no external funding. withinThis theresearch scheduling periods. Author Contributions: Data curation, M.H.; Methodology, Z.H.; Writing—review & editing, M.H.
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Author Contributions: Data curation, M.H.; Methodology, Z.H.; Writing—review & editing, M.H. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.
Appendix A Table A1. The day-ahead predicted wind power data (p.u.). Time Period
1
2
3
4
5
6
7
8
9
10
11
12
Wind power Time period Wind power
0.57 13 0.32
0.49 14 0.43
0.42 15 0.52
0.36 16 0.59
0.28 17 0.68
0.2 18 0.74
0.15 19 0.68
0.11 20 0.64
0.1 21 0.57
0.12 22 0.49
0.16 23 0.41
0.23 24 0.33
Table A2. The intra-day rolling predicted wind power data (p.u.).
Number 1 2 3 4 5 6 7
Time Period 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0.12 0.22 0.25 0.38 0.52 0.62 0.74
0.14 0.23 0.27 0.41 0.56 0.62 0.77
0.16 0.25 0.3 0.43 0.56 0.64 0.77
0.18 0.25 0.33 0.47 0.57 0.68 0.78
0.23 0.25 0.39 0.53 0.64 0.76 0.8
0.24 0.27 0.42 0.57 0.64 0.79 0.77
0.26 0.31 0.44 0.58 0.65 0.8 0.72
0.26 0.34 0.48 0.58 0.7 0.8 0.69
0.26 0.4 0.55 0.66 0.78 0.83 0.65
0.28 0.43 0.59 0.66 0.82 0.8 0.64
0.32 0.46 0.6 0.68 0.83 0.75 0.67
0.35 0.5 0.61 0.73 0.83 0.72 0.67
0.41 0.57 0.68 0.77 0.85 0.67 0.65
0.44 0.61 0.68 0.81 0.82 0.66 0.61
0.47 0.62 0.7 0.81 0.77 0.69 0.54
0.51 0.62 0.75 0.82 0.74 0.68 0.52
Table A3. The day-ahead predicted load data (MW). Time Period
1
2
3
4
5
6
7
8
9
10
11
12
Load Time period Load
700 13 1400
750 14 1300
850 15 1200
950 16 1050
1000 17 1000
1100 18 1100
1150 19 1200
1200 20 1400
1300 21 1300
1400 22 1100
1500 23 900
1550 24 800
Table A4. Parameters of rolling dispatch model. µ1
µ2
µ3
µ4
cm 1
cm 2
cm 3
cm 4
4
3
2
1
5
1
0.5
0.1
Table A5. Executed rolling plans for Case 1. Period
G1
G2
G3
G4
G5
G6
G7
G8
G9
G10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
455 455 455 455 358.4 314.2 261.2 222.2 168.3 150 150 150 150 150 150 154.5
247.1 210.2 270.2 330.2 455 455 455 455 455 451.7 454.7 455 455 455 455 455
130 130 60 20 0 0 0 0 0 0 0 0 0 0 0 0
130 130 107.5 70 0 0 0 0 0 0 0 0 0 0 0 0
162 162 162 144.3 162 162 162 162 162 162 133.6 73.6 93.4 74.4 102 162
50.1 48.8 48.1 43.2 41.8 38.1 44.8 44.8 43.1 42.7 43.1 74.2 47.7 80 78.4 42.9
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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Table A6. Executed rolling plans for Case 2. Period
G1
G2
G3
G4
G5
G6
G7
G8
G9
G10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
455 455 455 428.1 358.4 314.2 261.2 222.2 168.3 150 150 150 455 455 455 455
455 455 455 395 455 455 455 455 455 451.7 427.1 455 200.7 199 225 254
130 107.4 78.6 70 0 0 0 0 0 0 0 0 0 0 0 0
20 20 20 70 0 0 0 0 0 0 0 0 0 0 0 0
43.7 25 25 25 162 162 162 162 162 162 133.6 73.6 25 25.4 25.4 25.4
70.5 73.6 69.2 74.6 41.8 38.1 44.8 44.8 43.1 42.7 70.7 74.2 65.5 80 80 80
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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11. 12. 13. 14.
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