2. Department of Electrical, Computer and Biomedical Engineering, University of Rhode Island, Kingston, RI 02881, USA;
3. School of Electrical Engineering, Wuhan University, Wuhan 430072, China
SUSTAINABLE energy generation from renewable resources, such as wind power, are developing rapidly around the world in the past decades. Because of their highly randomness and volatility characteristics, largescale integration of these resources brings great challenges to the system secure and economic operation, as well as ancillary services and reserves scheduling [1]. Moreover, the relative poor prediction accuracy of wind generation causes a series of stability and economic issues, ranging from shortterm transient stability (e.g., frequency fluctuation [2]) to longterm generationdemand balancing problems (e.g., economic dispatch [3]). In this paper, we focus on the dayahead economic dispatch for wind generation integrated power system considering optimal reserve scheduling, and the design of effective solution algorithm.
In the power and energy community, the research for wind generation integrated power systems could be categorized into two folds. On one hand, the wind power holds the characteristic of high temporal variations, and its largescale integration will bring impact on the system transient stability and control [4]. Examples include impact of doublefed induction generator (DFIG) based wind turbine on power systems small signal stability [5], computational intelligence (CI) based wind farm active power and reactive power damping control [6][8]. On the other hand, the wind power acts as an external stochastic source in the power system economic operation, and should be carefully and dedicated addressed in the planning stage. To more strategically accommodate largescale intergraded wind power generation in economic scheduling, significant work has been finished in the market clearing modelling [9], [10] as well as stochastic optimization techniques [11][15]. The related investigations include stochastic power system operation by using chance constrained dayahead scheduling [16], [17], and risk based unit commitment (UC) for dayahead market clearing with wind power uncertainty [18], [19].
Moreover, with the decreased wind generation cost, the operators are required to fully accommodate the power from the wind without curtailment [20]. Therefore, the system operation and management will have higher requirement for the ancillary services and system reserves [21]. Along this direction, extensive studies have been carried out in the power and energy society (PES), such as procurement for loadfollowing reserves considering flexible demand and high wind penetration [22], securityconstrained scheduling based hourly reserve allocation versus demand response [23], and game theory based multiarea spinning reserve trading with wind power uncertainty [24]. Among all the proposed methods, the scenariobased stochastic programming has attracted the attention from the researchers in recent years [25]. However, we should notice that this method relies on the past experience, and its subjective and heuristic nature leaves many academics uncomfortable [26], [27].
Based on aforementioned discussion, in this paper a novel dayahead economic dispatch model for wind power integrated power system with optimal reserves scheduling is proposed and investigated. Considering the introduced stochastic factor from the wind, chance constrained model is proposed to address this issue, and the problem is formulated as a chance constrained stochastic nonlinear programming (CCSNLP). Based on the quantile and relaxation concepts, the CCSNLP in transformed into a deterministic nonlinear programming (NLP). Then a threestage solution framework based on particle swarm optimization (PSO), sequential quadratic programming (SQP) and Monte Carlo simulation (MCS) is proposed to solve this model effectively. The proposed model and algorithm is tested on IEEE 30bus system with wind power penetration, and the the results demonstrate the correctness of this dispatch model as well as the effectiveness of the threestage solution framework.
The rest of this paper is organized as follows. Section Ⅱ formulates the problem as chance constrained dayahead economic dispatch considering reserve scheduling. Section Ⅲ presents the methodology of the proposed threestage algorithm, including the PSO stage, the SQP solution and the verification stages. Section Ⅳ presents the simulation results of case study on IEEE 30bus system with wind power generation. Finally, conclusions are drawn in Section Ⅴ.
Ⅱ. PROBLEM FORMULATION A. Generator Output AnalysisThe power grid is a realtime system requiring the plants produce the right amount of electricity at the right time to consistently and reliably meet the load demand, such that the system frequency is maintained at the specified value. To fulfill this task, a certain amount of active power called control reserve is stored in the system [28], and the related control schemes could be categorized as primary, secondary and tertiary control.
The automatic generation control (AGC) in the power system consists of the primary and the secondary control, and the related reserve is called AGC reserve or frequencyresponse reserve. The primary control is provided by participated spinning generators, which response to disturbancescaused frequency deviations from the nominal value according to their speed droop characteristics. Meanwhile, the objective of the secondary control is to help the primary control to clear the frequency error, and bring the frequency back to its nominal value as soon as possible. Moreover, in the interconnected power system, the secondary control is also responsible for maintaining the power on the tieline to the predefined values. The third frequency control scheme is called tertiary control, and is usually activated manually such that the used primary and secondary control reserves are released after a large disturbance (e.g.,
In this paper, we are focusing on optimal reserve scheduling and we only consider the reserves available in the generators. Fig. 1 shows the AGCparticipated generator output power and the reserves in the normal and
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Fig. 1 Generator output with reserve scheduling at time period t. 
$ \begin{equation} \begin{cases} P_{Gi, t} + U_{Gi, t} \le \min ( P_{Gi, \max }, \, A_{Gi, \max } ) \\ P_{Gi, t}  D_{Gi, t} \ge \max ( P_{Gi, \min }, \, A_{Gi, \min } ) \end{cases} \end{equation} $  (1) 
where the meaning of the symbols are shown in the figure. For the
$ \begin{equation} P_{Gi, t} + U_{Gi, t} + R_{Gi, t} \le \min (P_{Gi, \max }, \, A_{Gi, \max }). \end{equation} $  (2) 
We should also notice that, the deployment of system reserves subject to the generator ramping up and ramping down capability, and could be modelled as:
$ \begin{equation} \begin{cases} U_{Gi, t} \le T_{1i} r_{ui} \\ D_{Gi, t} \le T_{1i} r_{di} \\ R_{Gi, t} \le T_{2i} r_{di} \end{cases} \end{equation} $  (3) 
where
Before we formulate the optimal AGC based reserve scheduling, we will first introduce the stochastic wind generation model and the load demand model. We assume the wind speed is composed as:
$ \begin{equation} v_t = v_{f, t} + e_{W, t} \end{equation} $  (4) 
where
$ \begin{equation} P_{W, t} = \begin{cases} {P_{Wr}, }&{v_r < v_t \le v_{\rm out} } \\ {\displaystyle\frac{{v_t^3  v_{in}^3 }}{{v_r^3  v_{in}^3 }} \cdot P_{Wr}, }&{v_{\rm in} < v_t \le v_r } \\ {0, }&\mbox{otherwise} \end{cases} \end{equation} $  (5) 
where
Similar as the stochastic wind power model, we formulated the total load demand as:
$ \begin{equation} P_{L, t} = P_{Lf, t} + e_{L, t} \end{equation} $  (6) 
where
$ \begin{equation} P_{Ls, t} = d_s \cdot P_{L, t}, \quad 1 \le s \le N_s \end{equation} $  (7) 
where
The load forecasting and wind power forecasting error will be the sources of the system frequency instability, and should be compensated by the AGC system, which is shown in Fig. 2. As we have mentioned above, there are multiple generators in the system participating in the AGC task. The power balancing task are sharing among all the generators according to the following equation:
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Fig. 2 Overall architecture of the AGC based system power balancing. 
$ \begin{equation} P_{Gi, t}' = P_{Gi, t} + k_{i, t} \left( {P_{L, t}  P_{W, t}  \sum\limits_{j = 1}^{N_G } {P_{Gj, t} } } \right) \end{equation} $  (8) 
where
$ \begin{equation} k_{1, t} + k_{2, t} + \cdots + k_{N_{\rm G}, t} = 1, \quad 0 \le k_{i, t} \le 1. \end{equation} $  (9) 
When the system is in
$ \begin{array}{l} {P_{Gi,{t^\prime }}} = {P_{Gi,t}} + \frac{{{r_{i,t}}}}{{1  {r_{l,t}}}}{P_{Gl,t}}\\ \;\;\; + \frac{{{k_{i,t}}}}{{1  {k_{l,t}}}}\left( {{P_{L,t}}  {P_{W,t}}  \sum\limits_{\begin{array}{*{20}{c}} {j = 1}\\ {j \ne l} \end{array}}^{{N_G}} {{P_{Gj,t}}} } \right) \end{array} $  (10) 
where the
$ \begin{equation} r_{1, t} + r_{2, t} +\cdots + r_{N_{\rm G}, t} = 1, \quad 0 \le r_{i, t} \le 1. \end{equation} $  (11) 
In current ancillary service market, we assume the transmission system operator (TSO) purchases the AGC reserve and contingency reserve according to the distribution vector
$ \begin{equation} \begin{cases} {U_{Gi, t} = k_{i, t} U_t } \\ {D_{Gi, t} = k_{i, t} D_t } \\ {R_{Gi, t} = r_{i, t} R_t } \end{cases} \end{equation} $  (12) 
where
In this paper, we consider dayahead economic dispatch with optimal reserve scheduling, therefore the optimization horizon is set as
$ \begin{equation} \begin{split} \min&\sum_{t = 1}^{N_{\rm T} } {\sum_{i = 1}^{N_{\rm G} } {[a_i (P_{{\rm G}i, t} )^2 + b_i P_{{\rm G}i, t} + c_i]} } \\ & + \sum_{t = 1}^{N_{\rm T} } {\sum_{i = 1}^{N_{\rm G} } {[\alpha _i U_{Gi, t} + \beta _i D_{Gi, t} + \gamma _i R_{Gi, t}]} } \end{split} \end{equation} $  (13) 
where
The operational constraints considered in this paper are as follows:
1) System Power Balance:
$ \begin{equation} \sum\limits_{i = 1}^{N_G } {P_{Gi, t} + P_{Wf, t} = P_{Lf, t} }. \end{equation} $  (14) 
2) Regular Generation Limits:
$ \begin{equation} P_{Gi, \min } \le P_{Gi, t} \le P_{Gi, \max }. \end{equation} $  (15) 
3) Regular Generation Ramping Limits:
$ \begin{equation}  r_{di} T_{di} \le P_{Gi, t}  P_{Gi, t  1} \le r_{ui} T_{ui} \end{equation} $  (16) 
where
4) System Reserves Constraints: the system reserves (i.e., AGC and contingency reserves) limits have been introduction in Section \ref{sec:2.1} and \ref{sec:2.2} as shown in (1)(3) and (8)(12). Moreover, the reserves limits should also consider the
$ \begin{equation} \begin{cases} {\Pr \left\{ {\sum\limits_{\scriptstyle i = 1 \atop \scriptstyle i \ne j}^{N_G } {\left( {U_{Gi, t} + P_{Gi, t} } \right) + } P_{W, t} \ge P_{L, t} } \right\} \ge \eta } \\ {\Pr \left\{ {\sum\limits_{\scriptstyle i = 1 \atop \scriptstyle i \ne j}^{N_G } {\left( {D_{Gi, t}  P_{Gi, t} } \right)  } P_{W, t} \ge  P_{L, t} } \right\} \ge \eta } \\ {\sum\limits_{\scriptstyle i = 1 \atop \scriptstyle i \ne j}^{N_G } {R_{i, t} \ge P_{Gj, t} } } \end{cases} \end{equation} $  (17) 
where the
5) Network Security Constraints:
$ \begin{equation} \begin{cases} {\Pr \left\{ {\pmb{A}_j \left( {P_{G, t}^{'}  P_{L, t} + P_{W, t} } \right) \le \overline P _{Linej} } \right\} \ge \eta } \\ {\Pr \left\{ {\pmb{A}_j \left( {P_{G, t}^{'}  P_{L, t} + P_{W, t} } \right) \ge  \overline P _{Linej} } \right\} \ge \eta } \end{cases} \end{equation} $  (18) 
where
In summary, (1)(3) and (8)(18) is the proposed CCSNLP economic dispatch model considering the generation and reserve cost. The stochastic source comes from the load and wind forecasting error and the model is given in (4)(7). The variables need to be optimized are the planned generator output
We usually transform the probabilistic chance constraints to deterministic constraints based on the following principles:
1) If the chance constraint has the following decoupled form:
$ \begin{equation} \Pr \left\{ {f\left( x \right) \ge h\left( \xi \right)} \right\} \ge \eta \end{equation} $  (19) 
where
$ \begin{equation} f\left( x \right) \ge Q_\eta \left( {h\left( \xi \right)} \right). \end{equation} $  (20) 
Obviously, the chance constraints in (17) could be transformed based on this situation.
2) If the chance constraint has the following coupled form:
$ \begin{equation} \Pr \left\{ {f\left( x \right) \ge h\left( \xi, x \right)} \right\} \ge \eta \end{equation} $  (21) 
where
$ \begin{equation} \left {A_j \left( {P_{G, t}'  P_{L, t} + P_{W, t} } \right)} \right \le \overline P _{Linej} \end{equation} $  (22) 
and the solution based on this relaxation need to be verified by using Monte Carlo simulation (MCS). Obviously, the network security chance constraints in (18) fit the form in (21). Because the transmission line power flow limit
To calculate the quantile of the random variable
Step 1: Set
Step 2: Carry out the
Step 3: Then calculate
Step 4: Calculate the mean value
Step 5: If the deviation
Step 6: Otherwise,
In the literature, there are also many other methods could be used to find the quantile, such as the kernel density estimation based method [32] and the convolution based method [33]. However, we should notice that, the function
Based on the aforementioned technique, the original CCSNLP is transformed to a NLP. Then a threestage solution framework is proposed and the flow chart is shown in Fig. 3. The three stages are PSO optimization stage to search the line power flow limits
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Fig. 3 Flow chart of the proposed threestage solution framework. 
Before we present the detailed design of the PSO stage, we will first introduce this optimization method briefly. PSO belongs to the natureinspired CI firstly brought forward by Kennedy and Eberhart [34]. It represents the swarm's behavior by mimicking the moving organisms in a fish school or a bird flock for food or resource searching. The main idea is to judge the adaptability of each particle in each generation based on the fitness function. The global and local best particles are selected as the examples to guide the direction and velocity updating of the rest particles. The searching capability of this method is highly related to the swarm size, generation number and the fitness function design [35], [36]. The procedure can be generalized in the following four steps.
Step 1: Initialization.
The position dimension of each particle in the swarm corresponding to the number of the transmission line limits in the whole time span
$ \begin{equation} \begin{cases} {v_{i, \max } = 0.1(x_{i, \max }  x_{i, \min } )} \\ {v_{i, \min } =  v_{i, \max }, \quad i = 1, 2, \ldots, N_{\rm Line}\times T} \end{cases} \end{equation} $  (23) 
where
Step 2: Updating the local and global best.
After the PSO initialization, the particles are send to NLP solution stage and the MCS verification stage, and then the following fitness function is feedback:
$ \begin{align} \nonumber Fitness =& \sum\limits_{t = 1}^{N_{\rm T} } {\sum_{i = 1}^{N_{\rm G} } {[a_i (P_{{\rm G}i, t} )^2 + b_i P_{{\rm G}i, t} + c_i]} } \\ \nonumber &+ \sum_{t = 1}^{N_{\rm T} } {\sum_{i = 1}^{N_{\rm G} } {[\alpha _i U_{Gi, t} + \beta _i D_{Gi, t} + \gamma _i R_{Gi, t}]} } \\ & + m\sum_{t = 1}^{N_{\rm T} } {\sum_{j = 1}^{N_{\rm Line}' } {(\eta  \eta _j)} } \end{align} $  (24) 
where
Step 3: Position updating.
In the third step, the velocity and position of each particle is updated as:
$ \begin{equation} \left\{ \begin{split} v_{i, j} (t + 1) =\, &w(t)v_{i, j} (t)+ c_1 r_1 (x^L  x_{i, j} (t)) \\ & + c_2 r_2 (x^G  x_{i, j} (t)) \\ x_{i, j} (t + 1) =\,&x_{i, j} (t) + v_{i, j} (t + 1) \end{split}\right. \end{equation} $  (25) 
where
Step 4: Determining whether to finish procedure.
The process of optimization will be finished once one of the following termination criteria is satisfied:
1) The maximum generation reached.
2) The fitness function converged to a predefined small value.
C. NLP Solution StageAs can be observed from the flow chart, the NLP solution stage consists of two parts. The fist part is the initialization to find proper initial value. This is carried out by randomly giving the coefficient vector
The second part is the solution part based on the initial value provided by the first part. This is carried out by using sequential quadratic programming (SQP), which uses a serials of quadratic programming to approximate the solution for KuhnTucker equations [40]. The converged solution will send to the third stage for verification.
D. MCS Verification StageAs we have mentioned above, the chance constraints have been relaxed to deterministic constraints, therefore the Monte Carlo simulation (MCS) could be carried out to verify the solution obtained in stage two. The procedure is summarized as follows:
Step 1: Set
Step 2: In time period
Step 3: If
Step 4: Otherwise,
In this threestage framework, the NLP solution and MCS verification stages are implemented as functions called by the PSO. We should notice that, in the literature, the NLP and verification stages are usually calculated iteratively. After the verification, the line limits will be adjusted manually (e.g., decrease the limits by a mandatory value), and then calculate the NLP again with the updated limits. The proposed PSO stage in this paper could provide a heuristic technique to facilitate the line limits adjustment, therefore speed up the algorithm convergence.
Ⅳ. NUMERICAL STUDY A. Simulation SetupIn this section, numerical simulation is carried out on the modified IEEE 30bus system with wind power integration. The system parameters are shown in Table Ⅰ and the system structure is shown in Fig. 4. As can be observed from this figure, a wind farm with rated capacity of 300 MW is integrated to the system from bus 15. The cutin, rated, and cutout wind speeds are set as
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Fig. 4 Oneline diagram of the IEEE 30bus system with wind generation. 
The forecasted load demand and wind power generation for the dayahead dispatch are shown in Fig. 5. As can be observed from this figure, the wind power is abundant around 8:00 am and decreased to a valley at 19:00 pm. However, the pattern of the load changing is just opposite to the wind, where the demand peak is between 12:00 pm to 20:00 pm.
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Fig. 5 The forecasted values of the wind power and load demand. 
Based on these forecasted values, the threestage algorithm is carried out on the benchmark system, and the convergence curves with 20 independent trials are shown in Fig. 6 and 7. Specifically, the convergence of the punishment part and generation and reserve cost part are shown in Fig. 6, and the mean fitness value is shown in Fig. 7. It is interesting to notice that as the punishment decreasing (i.e., more lines are satisfied the confidence interval in the verification stage), the generation and reserve cost increasing. This result is in consistent with our intuition that the higher the system security, the greater the operating cost.
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Fig. 6 The punishment part, generation and reserve cost part during the optimization. 
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Fig. 7 The mean fitness value on 20 independent trials. 
The results of generation dispatch, distribution coefficient
The total AGC reserve and contingency reserve purchased by the system operator is shown in Fig. 8. To accommodate the increased load demand with changing wind power generation, the AGC ramping up reserve is increasing all the way to the end. However, the AGC ramping down reserve is decreasing during 7:0012:00 am and 21:0024:00 pm. The reason for this result is that the wind power is abundant during these two time periods, and the wind generator could be operated at the rated power with relative higher confidence. Moreover, we should notice that the contingency reserve has the same trend with the load demand changing. This is because once a
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Fig. 8 The total AGC and contingency reserves. 
In this paper, dynamic economic dispatch model for wind power penetrated system with optimal reserve scheduling was proposed and investigated. The problem was formulated as a CCSNLP problem, and transformed into a deterministic NLP problem by using Bootstrap based sampling to find the quantile and the relaxation methodology. Then a threestage algorithm framework, with PSO optimization, SQP solution and MCS verification, was developed to tackle this problem. Simulation results on IEEE 30bus system with wind power penetration demonstrated the convergence characteristics and the effectiveness of the proposed threestage algorithm.
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