Abstract
As the steady-state frequency of an actual power system decreases from its nominal value, the composite load of the system generally responds positively to lower power consumption, and vice versa. It is believed that this load frequency damping (LFD) effect will be artificially enhanced, i.e., sensitivities of loads with respect to operational frequency will increase, in future power systems. Thus, for wind-integrated power systems, this paper proposes a frequency-dependent chance constrained unit commitment (FDCCUC) model that employs the operational frequency as a dispatching variable so that the LFD effect-based load power can act as a supplemental reserve. Because the frequency deviation is safely restricted, this low-cost reserve can be sufficiently exerted to upgrade the wind power accommodation capability of a power system that is normally confined by an inadequate reserve to cope with uncertain wind power forecasting error. Moreover, when the FDCCUC model is applied to a bulk AC/DC hybrid power system consisting of several independently operated regional AC grids interconnected by DC tie-lines, a hierarchically implemented searching algorithm is proposed to protect private scheduling information of the regional AC grids. Simulations on a 2-area 6-bus system and a 3-area 354-bus system verify the effectiveness of the FDCCUC model and hierarchical searching algorithm.
Environmental and economic concerns have caused rapid growth in wind power generation (WPG) in recent decades all over the world. However, issues of wind curtailment (WC) and even load shedding (LS) could be serious if there are not adequate reserves prepared in advance to cover the wind power forecasting error [
Although various energy storage systems together with fast-acting gas turbines are high-quality reserve resources for balancing the wind power forecasting error, they generally suffer from immature technique, high expenditure, and insufficient net capacity [
It is pointed out in [
Generally speaking, wind farms are located in remote areas and integrated into local power grids. Owing to the needs of long transmission distances and large transmission capacities, a common engineering practice is to connect these wind-integrated power grids with power grids feeding load centers by high-voltage DC (HVDC) transmission lines [
The aim of this paper is to explore novel reserves that come from the LFD effect for day-ahead generation scheduling of a bulk AC/DC hybrid power system with WPG. Specifically, by allowing feasible deviations of operational frequencies for regional AC grids, the frequency-dependent extra power consumed or supplied by loads can act as partial reserves in an improved stochastic unit commitment problem to cope with the uncertain wind power forecasting error. Moreover, to meet risk preference, the proposed frequency-dependent chance constrained unit commitment (FDCCUC) model explicitly takes into account the wind curtailment expectation (WCE), load shedding expectation (LSE), and chance constrained power balance, which are computed based on the probability distribution of wind power forecasting error. Furthermore, the optimization of the FDCCUC model is approximately converted to a mixed-integer linear programming (MILP) problem and, accordingly, a hierarchical searching algorithm is proposed to find an optimal solution without requiring private scheduling information from the regional AC grids. The main contributions of this paper are summarized as follows.
1) This paper advocates the use of frequency-dependent load power as an auxiliary reserve for the day-ahead scheduling of a WPG-integrated power system. An FDCCUC model is elaborately constructed to adequately exert low-cost reserves based on LFD effects and to guarantee feasible deviations of system operational frequency simultaneously. Thus, from the viewpoint of risk operation, supplemental LFD effect based reserves can considerably increase the accommodation capability of a power system with respect to uncertain WPG.
2) A hierarchical searching algorithm is customized for the FDCCUC model of a bulk AC/DC hybrid power system that consists of several regional AC grids interconnected by DC tie-lines. The algorithm splits the searching process of the original large-scale MILP problem into mutually iterative computations of a master problem and several sub-problems with much smaller size. Each regional AC grid corresponds to one sub-problem that can be privately computed based on the information from the grid. Limited information needs to be exchanged between the system-level master problem and the sub-problems, allowing private scheduling information of the independently operated regional AC grids to be hidden and protected during the searching process.
The rest of paper is organized as follows. Section II formulates the FDCCUC model for bulk AC/DC hybrid power systems with WPG. Section III presents the details of the hierarchical searching algorithm for FDCCUC model to protect private scheduling information of regional AC grids. Numerical simulations are conducted in Section IV. Finally, Section V concludes the paper.
This paper deals with day-ahead scheduling of AC/DC hybrid power systems comprised of multiple large-scale regional AC grids interconnected by DC tie-lines. As shown in
(1) |

Fig. 1 Bulk AC/DC hybrid power system with WPG.
where is the aggregated power adjustment of loads due to the frequency deviation ; and is the aggregated sensitivity coefficient of load power with respect to . Therefore, taking advantage of the additional power reserve associated with the LFD effect, this paper proposes that all regional AC grids can have different and time-varying operational frequencies as long as they remain within permitted safe ranges; random wind power forecasting errors can thus be partially covered by the LFD-based load reserve that, along with conventional reserves, is shared among the regional AC grids through DC tie-lines. Subsequently, a proposed FDCCUC model is formulated for AC/DC hybrid power systems with WPG.
Remark: it should be noted that the proposed auxiliary reserve from LFD effect could be conflict with existing secondary frequency control, i.e., eliminating frequency deviation. Whereas, the research in this paper is consistent with existing unit commitment research that provides thermal units with a preliminary reference operation strategy based on the forecasting load demand and renewable output in day-ahead time frame. Then the actual generation output is regulated by economical dispatch through secondary frequency control based on actual operational conditions with short-term load and renewable power prediction. Even so, this paper is an exploratory research focusing on potential operational strategy for the future power system with extreme high renewable penetration level. This paper attempts to facilitate the maximum renewable power without the unbridled expansion of traditional reserve services. Besides, introducing the proposed auxiliary reserve from LFD effect in unit commitment schedule could decrease the requirement for expensive fast-response reserve resources [
For the AC/DC hybrid power system studied in this paper, the objective of the FDCCUC model is presented as:
(2) |
(3) |
(4) |
(5) |
(6) |
where
The risk cost induced by overestimating and underestimating the WPG is specifically considered in this paper, as shown in (5). As mentioned previously, LFD effect based power reserves can simply be harnessed by shifting operational frequencies of regional AC grids from their nominal values. However, the frequency deviations must be minimized. So, and in (6) are utilized with two considerations: to express the frequency deviation (the detailed expression will be introduced later) and to enable the use of linear
In this paper, DC tie-lines of the AC/DC hybrid power system are treated as generating sources with respect to all regional AC grids, indicating that power as well as reserves of the DC tie-lines are actually injected to the AC grids if signs of these variables are positive, and vice versa. Thus, the following relationships should be naturally hold:
(7) |
(8) |
(9) |
(10) |
(11) |
where is the scheduled power received by the
Constraints (7)-(9) guarantee the power and upward/downward reserve at both ends of one single DC tie-line keep balance at any dispatching interval. Constraints (10) and (11) mean that, in practice, the reserve power delivered from a regional AC grid by the DC tie-lines should be less than the total reserve of this grid.
To safely operate the DC tie-lines engaged in transmission of scheduled power as well as reserves among regional AC grids, the following technical constraints should be included [
(12) |
(13) |
(14) |
(15) |
(16) |
(17) |
where is the permitted maximum power change of the DC tie-line. Constraints (12)-(17) commonly ensure that the actual power of the DC tie-line cannot be overloaded and very abruptly changed, taking into account the reserves.
2) Constraints on Power Balance and Chance Constraints Against Wind Power Uncertainties Within Regional AC Grids
The power balance must be held within all regional AC grids by the following constraints:
(18) |
(19) |
(20) |
(21) |
where is a deterministic value that represents the aggregated wind power forecasting of the
It can be observed from (7) and (18) that the power capacities contributed by the LFD effects are shared among all regional AC grids through the DC tie-lines. Furthermore, via the connections with thermal units in (16), the LFD-based power capacities can spare the capacities of some thermal units, which are then used as reserves. Therefore, the operational reliability of each regional AC grid against the uncertainties of wind power (forecasting error) is guaranteed by the following chance constraints:
(22) |
(23) |
where is a random variable that represents the aggregated wind power forecasting error of the
Chance constraints (22) and (23) can be straightforwardly interpreted as a given confidence to cover the uncertain overestimation () or underestimation () of wind power by the regional and cross-regional reserves. In other words, the probabilities of LS and WC events due to wind power forecasting error can be adequately restricted.
As usual, common technical constraints are imposed on thermal units as:
(24) |
(25) |
(26) |
(27) |
(28) |
(29) |
(30) |
(31) |
(32) |
where and are the binary variables (0/1) indicating the startup () and shutdown () operations of the
Constraints (24)-(32) are not further explained here, and the related descriptions can refer to [
Specifically, constraints (7)-(32) may need to be enumerated as:
(33) |
Clearly, detailed computations of , , and chance constraints (22) and (23) will complete the formation of the scheduling problem (2)-(33). These computations are essentially based on the probability distribution of wind power forecasting error. A general approach to derive the probability density function (PDF) of wind power forecasting error is to fit a large amount of historical error data to a parameterized distribution function, for example, the parameterized Gaussian function [
(34) |
where , , and are the tunable shaping parameters, and the details can be found in [
(35) |
(36) |

Fig. 2 Illustration of WC and LS according to PDF of a wind farm.
where and are the monotonic functions of the total upward reserve and total downward reserve , respectively.
Chance constraints (22) and (23) can be calculated based on (34) as:
(37) |
(38) |
where is the inverse function of the cumulative distribution function (CDF) of . Interestingly, is also analytical [
The optimization (2)-(36) is a mixed-integer nonlinear programming (MINLP) problem due to the nonlinearities introduced by (3), (35), and (36). Solving a large-scale and general MINLP problem directly is not easy nor convenient, with MILP problems being more favorable. According to (35), (36), and
(39) |
(40) |
(41) |
(42) |
(43) |
(44) |
(45) |
(46) |

Fig. 3 Piecewise linearization of LSE and WCE . (a) . (b) .
where and are the initial values of and when the upward and downward reserves are zero, respectively; NLS and NWC are the numbers of linear line segments employed for approximating and , respectively; and denote the vertices of linear line segments for and , respectively; and are the increments with respect to these vertices; and and are the slopes of the linear line segments. Analogously, (3) can also be piecewise linearized as:
(47) |
(48) |
(49) |
(50) |
(51) |
where is the fuel cost of the thermal unit located at the regional AC grid as it outputs the minimum power; denotes the vertex of linear line segments for fuel cost of the
The combination of (2), (4)-(33), and (37)-(51) comprises the proposed FDCCUC model for multiple regional AC grids interconnected by DC tie-lines. Although the formed optimization is an MILP problem, direct computation in a centralized manner conflicts with the fact that global information cannot be fully accessed for the centralized dispatching because regional AC grids are independently managed by different system operators. Therefore, the next section proposes a hierarchical searching algorithm to deal with this issue.
III. Hierarchical Searching Algorithm for FDCCUC Model to Protect Private Scheduling Information of Regional AC Grids
The proposed FDCCUC problem can be arranged in the following specific MILP form:
(52) |
s.t.
(53) |
(54) |
(55) |
(56) |
(57) |
where is the vector of decision variables whose information should be hidden internally within the
Due to the binary variables, the conventional branch-and-bound (BB) method is employed to solve the MILP problem (52)-(57) [
(58) |

Fig. 4 Illustration of proposed hierarchical searching algorithm. (a) Branching process. (b) Hierarchical distributed iteration process.
By fixing the binary variables along the path according to (58) and relaxing all other binary variables to be continuous within the range [0, 1], a linear programing (LP) problem associated with the node can be constructed as:
(59) |
where is the vector of continuous variables after fixing and relaxing binary variables for the node; and is the minimized objective value of the node, which also indicates the lower bound of the objective values of child nodes originating from the node; and , , , and are the coefficient matrices with proper dimensions of the node.
In this paper, Dantzig-Wolfe (DW) decomposition is employed to compute the large-scale LP problem (59) [
(60) |
(61) |
(62) |
where is the
Step 1: a restricted master problem (RMP) is first formed by using partially known extreme points, as follows:
(63) |
s.t.
(64) |
(65) |
(66) |
where is the minimized objective value of the RMP; and is the subset of . The decision variables of the RMP are the weights . In particular, the RMP is solved with the conventional simplex method to derive the optimal basis multipliers Γ0 and Γ1, which are two row vectors with proper dimensions.
Step 2: a sub-problem is then constructed based on Γ0 for each area m as:
(67) |
s.t.
(68) |
(69) |
(70) |
where θm is the minimized objective value of the sub-problem. Vectors and contain the decision variables of this optimization, and the optimal solution based on the simplex method can be represented by . This solution is actually an extreme point of the polyhedron described by (60) and (61).
Step 3: after solving the above RMP and sub-problems, the following test numbers are computed:
(71) |
where is the
Thus, if (), the optimal solution of the original LP problem (59) is found and ; otherwise, the newly obtained extreme points () are added to the RMP () and Steps 1 to 3 are repeated.
Once is derived, it will be compared with the global lower bound
This paper proposes that the searching algorithm employed for the FDCCUC problem is implemented in an efficient hierarchical manner, as shown in
1) Build and maintain the BB searching tree, and master the searching progress.
2) According to the path set of the node, pass on the triplet (m, ii, state) to dispatching center m to determine how the binary variables belonging to these regional grids are fixed and slacked.
3) Receive θm and from the regional dispatching centers, compute the RMP, and pass on the derived Γ0 and Γ1 to all regional dispatching centers.
4) Identify convergence of the searching process for the LP problem (59) associated with the node.
The manipulations performed by the regional dispatching centers include:
1) Inform the system-level dispatching center how many binary variables they have and determine fixed and slacked binary variables according to the received triplet (m, ii, state).
2) Compute the sub-problems and pass on the derived θm and to the system-level dispatching center.
This section validates the proposed FDCCUC model, which is hierarchically solved through numerical simulations. First, the effectiveness of the FDCCUC model in harnessing the reserve capacities of loads based on the LFD effects is demonstrated and compared with the existing models in terms of wind power accommodation capability. Then, the computational efficiency of the hierarchical searching algorithm is tested.
All simulations are performed within the environment of MATLAB 2020a running on a PC with an Intel Cor
Two systems, i.e., the 2-area 6-bus system and the 3-area 354-bus system, are used as the testbeds.

Fig. 5 Topology of 2-area 6-bus system.

Fig. 6 Prediction values of wind power and load in 2-area 6-bus system for a typical summer day.
The 3-area 354-bus system is composed of three modified IEEE 118-bus systems; the two with wind power integrated are regarded as sending-end systems (SESs), and the remaining one is regarded as a receiving-end system (RES). The aggregated capacities of wind power are 2500 MW at SES#1 and 3000 MW at SES#2. There are two DC tie-lines interconnecting these three regional AC grids: HVDC#1 connects SES#1 and the RES, while HVDC#2 connects SES#2 and the RES.
In addition, according to data records from EirGird, Ireland, the LS cost and wind curtailment penalty (WCP) are set to be 3500 $/MWh and 80 $/MWh, respectively. The upward and downward reserve prices of thermal units used in this paper are 30 $/MWh and 15 $/MWh, respectively [
To verify the effectiveness of the proposed FDCCUC model, economical indices including the thermal unit OC, RC, load shedding penalty (LSP), and WCP are introduced for the 2-area 6-bus system according to (2)-(5). The FDCCUC model will be compared with two other scheduling models commonly found in references in terms of these indices [
1) Model 1: DC tie-line power is adjustable within a permitted range, but no reserve is shared between regional AC grids.
2) Model 2: both power and reserve can be transmitted by DC tie-lines among multiple regional AC grids.
3) Model 3: the proposed FDCCUC model, which considers not only sharing of power and reserve among multiple AC grids, but also extra reserve capacities due to the LFD effects.
The comparison of economical indices for three models in the 2-area 6-bus system are listed in
Model | TC ($) | RC ($) | LSP ($) | WCP ($) | Total ($) |
---|---|---|---|---|---|
1 | 438000 | 38400 | 3160 | 1050 | 480000 |
2 | 429000 | 38400 | 3160 | 1050 | 472000 |
3 | 425000 | 18300 | 2850 | 690 | 447000 |
Sensitivity analysis of the penalty coefficient , which is used to confine the frequency deviation, is conducted. As shown in

Fig. 7 Sensitivity analysis results of penalizing coefficient .
Another role of penalty coefficient is to avoid a large operational frequency deviation which is introduced as a decision variable in this paper to serve as auxiliary reserve source. The maximum permitted frequency deviation for Areas 1 and 2 are set to be 0.5 Hz and 0.2 Hz, respectively [

Fig. 8 Simulation results of operational frequency with different penalty coefficients.
As mentioned in the sensitivity analysis, system safety compromises the economy. However, by setting the penalty coefficient to be , the investigating system operates at a minor deviation from nominal values, which maintains a safety margin to frequency limitation (the maximum deviation within one day is less 0.3 Hz for Area 1 and frequency excursion events become less frequent). Note that this frequency limitation is stipulated by Chinese grid operational code for normal condition and smaller than the threshold value of emergency measurement like under frequency load shedding (UFLD). Though the hard constraints (19)-(21) could restrict operational frequency deviation, the frequency deviation penalty could reduce the duration of frequency excursion. The constant value of may be not suitable when the load side participates in auxiliary service market. Therefore, a more theoretical method of determination of should be taken into consideration in the future work.

Fig. 9 Scheduled power and reserve transmitted through DC tie-line for Models 1 and 3.

Fig. 10 Simulation results of different models with increasing penetration levels of wind power. (a) Total upward reserve. (b) Total OC.
Provided the same amounts of wind power curtailment result, the maximum wind power is increased from 120 MW in Model 1 to 160 MW in Model 2, and finally to 182 MW in Model 3. In other words, the proposed FDCCUC model can integrate approximately 51% more wind power than the other two models. Although the required reserves increase in each model as the penetration level of wind power increases, the total OC still drops because more free wind energy is utilized by Model 3.
In this subsection, Models 2 and 3 will be further employed to demonstrate the quality of the optimal solution derived by the proposed hierarchical searching algorithm. First, the optimization problems of both scheduling models are solved in a centralized manner using the Gurobi package. The solutions obtained are then used as benchmarks for comparisons with the solutions derived by the hierarchical searching algorithm. Based on the 3-area 354-bus system, comparisons of the economical indices are detailed in
Model | Algorithm | TC () | RC () | LSP () | WCP () | Total () |
---|---|---|---|---|---|---|
2 | Centralized | 2217900 | 489630 | 39552 | 17204 | 2764300 |
Hierarchical | 2217800 | 489610 | 39550 | 17203 | 2764200 | |
3 | Centralized | 2184700 | 223300 | 35672 | 11305 | 2458300 |
Hierarchical | 2184600 | 223290 | 35671 | 11305 | 2458200 |
In the 3-area 354-bus system, three regional AC grids are deployed with three regional dispatching centers that carry out computations of sub-problems and communicate with a system-level dispatching center that computes the RMP. Specifically, one computational iteration is defined as solving the RMP and then each of the three individual sub-problems (bearing in mind that several such iterations may be required to accomplish computations of each node during the BB process). Moreover, because the sub-problems can be solved by the regional dispatching centers in parallel, the time consumption of an iteration is defined as , where is the solution time of the RMP and is the solution time of the

Fig. 11 CDF of computation time of each iteration.
The total time cost of the hierarchical searching algorithm in terms of the sum of the time consumed by all iterations is about 97 min. Notably, the current version of the hierarchical searching algorithm is somewhat inefficient compared with the commercial package Gurobi, which can directly solve the formulated MILP problem in around 5 min. This huge gap in computational efficiency is primarily due to the BB method used by the hierarchical searching algorithm, which is still far from optimal; future studies could considerably improve the searching efficiency of the proposed hierarchical searching algorithm. Even so, the current version of the hierarchical searching algorithm is obviously suitable for day-ahead scheduling and, most importantly, it keeps regional information hidden during the searching process.
Mining operational flexibilities of loads is a hot trend for power systems to confront the problems caused by the increasing integration of WPG. Thus, by introducing an auxiliary reserve released from LFD effect, this paper proposes an FDCCUC model together with a customized hierarchical searching algorithm to promote the ability of bulk AC/DC hybrid power systems to accommodate uncertain wind power. Optimization model derivations and numerical simulations lead to the following conclusions: ① the proposed reserve is practically feasible and low-cost, and can be released by changing the system operational frequency within a permitted safe range; ② the additional reserve based on the LFD effect is effective when reducing the WCE and LSE, i.e., it allows systems to endure more uncertain wind power forecasting error and permits the installation of larger wind power capacities; ③ the necessity of scheduling reserves transmitted by DC tie-lines in the FDCCUC model is demonstrated, otherwise the cross-regional reserve share may be unexpectedly restricted by technical constraints of the DC tie-lines; and ④ in contrast with centralized algorithm, the proposed hierarchical searching algorithm derives almost the same high-quality solutions but requires quite limited information exchange between distributed dispatching centers. Although this exploratory research facilitates renewable penetration levels without expansion of traditional reserve resources, future work should focus on theoretical methods to restrict frequency deviation.
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