In the new round of electricity market reforms in China,power exchange (PX) and system operators (SOs) are being relatively separated.Medium- and long-term (MLT)electricity trading is organized by PXs while the physical power system is operated by the SOs [1].However,access to physical power grid information for PXs is limited.It is difficult for PXs to consider the detailed requirements of secure operation of power grids when organizing electricity trading in the MLT market [2].Currently,these MLT electricity trading contracts are physically binding,indicating that by the end of the contract date,the generators are due to finish the contracted electricity generation.Therefore,the trading results in the MLT market,in turn,reduce the dispatching space for SOs.

Presently,the electricity trade in China is still dominated by MLT electricity trading.The functional separation between SOs and PXs makes it necessary to coordinate the relation between dispatching and trading so that the PXs can consider the security constraints of power grids in advance when organizing MLT market transactions [3].In this case,the balance between the secure and stable operation of power grids and the economic benefits of the market can be realized [4,5].This is the starting point of the security precheck of bilateral transactions.

The security pre-check of bilateral transactions focuses on the quantity of electricity.However,a typical solution of security constrained economic dispatch (SCED) [6,7]is the dispatch feasible region (DFR) [8]that focuses on the power output of generation units instead of the electricity quantity [9].This paper shows a transformation of the solution from power output to electricity quantity,i.e.,from the DFR to the generation feasible region (GFR).

Solid works on the distribution system security region(DSSR) [10]have been demonstrated by researchers.The clear mechanism of pre-checking for MLT power market [11]has been extensively studied.Most studies focus on the steady-state system security region in the power space [12],whereas research in the electricity quantity space has received little attention.

Converting power constraints into simple and clear electricity quantity constraints can facilitate security preverification of bilateral transactions,and feasible pre-check models [13]were presented.

However,uncertainties related to numerical boundary conditions during system operation,such as load forecasting,renewable energy forecasting [14-16],line contingency,and maintenance scheduling [17],will directly influence the calculation accuracy of the security constraints of transaction electricity and consequently affect the conclusion of security checking of MLT transactions.Thus,investigations into security region fluctuations caused by these uncertainties are necessary to enhance grid reliability.In our study,we put the most emphasis on the uncertainties from net nodal load power,i.e.,nodal load forecasting minus renewable power generation.We herein focus on numerical boundary fluctuations and demonstrate the geometrical characteristics of the electricity security boundary (ESB)under the fluctuations from the load prediction errors.One way to characterize this boundary distortion is presented,and the effectiveness of our proposed method is validated and evaluated herein.

The remainder of this paper is organized as follows.In Section 2,the economic dispatch model,GFR model,and visualization method are introduced.In Section 3,uncertainties affecting the GFR are modeled.In Section 4,sensitivity analysis results based on a three-generator system are presented.The concluding statements are presented in Section 5.

SCED is a simplified optimal power flow problem,which is widely used in the power industry [18-21].It focuses on generating the most economic generation schedule while considering key system operation constraints such as power balance constraint,transmission security constraints,and generation limitations,e.g.,ramp rates and minimum and maximum output levels.

The objective function of SCED is aimed at minimizing the operating cost:

where H is the number of time intervals during the dispatch,G is the maximal number of generators,i is the number of generation units,pi(t) denotes the active power per unit i during time interval t,Ci[pi(t)]and is the operating cost of unit i during time interval t.SCED takes only on-line units into consideration,i.e.,the startup and shutdown(commitment) of the generating unit is neglected.

The constraints are listed below:

1.System load balancing constraint:

where pk(t) denotes the power injection from unit k at time interval t,dj(t) is the load demand of load j at time interval t,and NG and ND are the number of generators and loads,respectively.Renewable generation,for example,photovoltaic power generation and wind power generation,can be modeled as a negative load.

2.Output power limit constraints:

whereand pUi represent the lower and upper limits of unit i,respectively.

Unit ramping rate constraints are as follows:

where Δpi denotes the allowed maximum load ramping rate.

3.Active power flow constraints:

The abovementioned inequalities are network security constraints based on DC power flow assumptions that refer to capacity limits of active power flows in transmission lines.Tl,j is the relative element in the power transfer distribution factor (PTDF) [22],and is the corresponding generator nodes in the submatrix of PTDF [23].Fl is the active power transmission limit of line l [13].

In this paper,we provide intuitive graphics of uncertainty influences on the ESB solved through SCED.Based on our previous work [13],this is feasible.

The visualization principle in [13]is utilized herein,which is elaborated as follows.

Based on the constraints listed above,we are able to describe a power system with power flow limits and can thus obtain a feasible solution containing exact power output by certain units at corresponding time intervals,namely DFR,as shown below,where pi,j represents the power output from unit i at interval j.G is the maximal number of generators,and H is the maximal number of time intervals.

We can first obtain an electricity quantity vector Q throughout the entire time horizon for each generator unit by summing up the columns in P,which can be formulated as follows:

This leads to another constraint of electricity quantity:

where qB represents the contract electricity quantity of each unit,which have been determined at the beginning of the transaction in the MLT electricity trading,and qS denotes the planned electricity quantity.A similar transformation can be applied to other constraints above into the electricity quantity space.

According to the contract electricity quantity portion β over the whole demand,we can obtain an optimization target for generators:

We call the solution boundaries of this optimization as qkGmin and qkGmax.Therefore,the ESB bounds for the generators are

Similarly,the optimization for the power lines is

The solution boundaries of this optimization are identified as qlLmin and qlLmax.Therefore,the ESB bounds for the generators are

Therefore,we can obtain a couple of constraints on the GFR according to their ESB.The optimization solving can be performed on a system with any number of generators.

Tan has proved [13,24]that for a system with NG units of generators,a convex polyhedron with NG degrees of freedom is derived from the inequality constraints,and a hyperplane with NG-1 degrees of freedom can be derived from the equality constraints.To intuitively show the uncertainty impacts,we demonstrate it using a three-unit system.

The region where the hyperplane intersects the convex polyhedron is the GFR of the NG-unit system (shown in Fig.1).

The three planes M1,M2,and M3 denote the electricity quantity constraints of the three generators.The blue pentagon represents the GFR considering only the output power limit constraints.

This region is our focus,and it is derivable with the multi-parametric toolbox (MPT) as for SCED.

Fig.1 Projection to produce GFR for a three-unit system

Long-term load forecasting is a crucial boundary condition for power system operation,especially for the emerging trading market [25].Precise load forecast statistics are required before the dispatch,which is listed in the system load balancing constraint.As stated in [26],load overestimation may lead to the excessive operation of generation units,while underestimation would probably lead to supply-demand imbalance and insufficient reserves,which may impact the security of power system operations.

Extensive progress has been made on load forecasting in both the short term and medium term with SVM [27],neural networks [28,29],and other techniques [30].

Errors in forecasting are unavoidable [8].Thus,it is reasonable that the GFR derived may not be accurate.The first step is to determine how the GFR may vary if these errors are taken into consideration.

A key precondition that we should assume is the manner in which we model this uncertainty [31].According to Tong’s validation [26],the t-distribution is the best to depict the distribution of short-term load forecasting errors.Other approaches suggest that normal distribution can also be a good constructor [32,33].

After the MLT trading is completed,transmission lines and transformer outages are still uncertain in power system operations.Considering the N-1 contingency condition,critical line outage may considerably influence the security level of the overall system.

In this study,we aimed at evaluating how this outage of separate lines may affect the GFR and to what extent the decrease in line capacity would change the original GFR.

We implemented the mathematical model on MATLAB with MATPOWER,CPLEX,YALMIP,and MPT.The following cases are based on a typical 3-generator-and-9-line system extracted from the MATPOWER IEEE case9 in Fig.2 and applied to a daily load curve for 24 h.

In the load forecasting error cases,we assume that the errors follow a normal distribution [32,33].

In this case,the hourly load forecasting curve is generated at the beginning of each bilateral transaction,but the actual load curve inevitably deviates from this forecast curve.To study how this fluctuation affects the GFR,we add small positive and negative random numbers that follow the normal distribution (with a maximum 10% of the peak value deviation from the forecasting value) to the forecast curve to generate a new curve as for the actual load.The Monte Carlo method was adopted to generate 500 actual load curves.All GFR boundaries are plotted in Fig.3.

Fig.2 A typical 3-generator-and-9-line system

Fig.3 500 Monthly GFRs of three-generator system under load forecasting errors

It is proved in 2.2 that the convex polyhedron is of NG-1 degrees of freedom (in this case of three generators,it is a convex polygon).As shown in Fig.3,the GFRs exhibit a stepped discontinuity in its boundaries when the load curve errors are taken into consideration.

Each polygon is made of 24 constraint boundaries,including several redundant ones,and the smallest region by these boundaries is the GFR.

Fig.4 shows the 24 boundaries’ 500-sample distribution in the histogram whose heights show frequencies from the 500 samples,and Fig.4 shows the intercepts of the boundaries’ expressions.In other words,the heights of the bars show frequencies of the Y-axis intercepts of the boundaries’ descriptive expressions.It is obvious that the boundaries have disparate distribution patterns,and continuous distributions and gaps occur.

Figs.5 and 6 show two typical boundaries’ distribution patterns.

Fig.7 shows that some boundary conditions are less affected by load forecasting errors such as the upper left boundary (the boundary indicated by the orange arrow),while those on the right are greatly affected,and some constraints leap (the boundary by the yellow arrow) and others even disappear (the boundary by the blue arrow).This is quite an interesting phenomenon because we only imposed fluctuations within ±10% of the load curve;however,it causes a significant change in the GFR,thereby showing the importance of load forecasting accuracy to security pre-check in MLT electricity transactions.Highprecision load forecasting in typical urban areas is possible.However,we should also recognize the limitations of the pre-check.Slight load fluctuation in a small-scale power system is likely to cause serious deformation of feasible region results.With the expansion of the power grid structure and the increase in line redundancy,the impact of load fluctuation on the feasible region will be smaller than the small-scale test results.

To explore the causes of the boundary jump in the GFR,we remove the output power limit constraints and retain all other constraints (as shown in Fig.8).The original leaps disappear under the same load forecasting errors,and the distribtion of each boundary tends to be normally distributed (Fig.9),thereby demonstrating that the previous leaps are caused by the hyperplane of the equality constraints,beginning to cross the edges of the power output limit polyhedron.

Fig.4 24 boundaries’ 500-sample distribution

Fig.5 Typical boundary distribution

Fig.6 Typical boundary distribution

Fig.7 500 Monthly GFRs of 3-generator system under load forecasting errors (with arrow indicators)

Fig.8 500 Monthly GFRs of the three-generator system under load forecasting errors (power output limits removed)

Fig.9 24 boundaries’ 500-sample distribution (power output limits removed)

All our cases are based on the first three generator units in the MATPOWER IEEE case 9 (its topological structure is shown in Fig.2).First,the nine lines are disconnected once each time,and the corresponding GFRs are evaluated.

Once one of the lines is disconnected,the closed-loop system will become a multi-terminal power supply,and the reliability will decline.The following figure shows that the feasible areas after the first six lines are completely disconnected.

It can be inferred that the leaping is quite serious,with nearly no regularity.When the connection between node 2 and node 8 is offline,the constraint conflicts;thus,the GFR does not exist.From the above-mentioned cases,it is too radical to disconnect one entire line directly.

Fig.10 GFR fluctuation under respective line outages

Therefore,we consider a more ordinary case:instead of a serious line outage,we assume that one of the corridors contains multiple lines,some of which have a smaller fault,resulting in a decrease in transmission capacity.

First,we reduce the transmission capacity of branch 2 connecting node 4 and node 5 by multiplying it by a factor between 0 and 1.The results are shown in Fig.11.When the factor decreases from 0.8,the GFR shrinks along the direction of the arrow,and there is no solution after the factor reaches 0.08.

When the transmission capacity of the branch connecting nodes 7 and 8 is reduced,a similar trend can be observed in the GFR,as shown in Fig.12.SCED becomes infeasible when the factor drops to 0.03.

Fig.11 GFRs under transmission capacity reduction between nodes 4 and 5

Fig.12 GFRs under transmission capacity reduction between nodes 7 and 8

Fig.13 GFRs under transmission capacity reduction between nodes 2 and 8

In summary,the loss of line transmission capacity leads to a reduction in the GFR,but there is considerable flexibility,indicating that abundant capacity loss is allowed until SCED becomes infeasible.An infeasible situation will not occur until the transmission capacity is significantly reduced.However,in this situation,the area of the GFR has been significantly reduced,posing a serious threat to the secure operation of the system.

Then,the transmission capacity of the branch connecting nodes 2 and 8 reduces in proportion similarly.From the figure below,the feasible area dramatically shrinks even more.This acceleration in shrinking is in line with the intuition:as the power output of a certain generator unit is limited,the output of other units should increase rapidly,and the GFR area of the system meeting the constraints will be greatly reduced (as shown by the remarkable jump in the lower right corner of the figure below).This indicates that in the actual system,ensuring the stability of the unit’s power output is crucial.

In this study,a transformation of DFR into GFR is implemented to depict an electricity quantity space in the long-term transaction.Furthermore,we constructed and evaluated a visualization model of SCED for a three-unit system.We proposed and emulated two key uncertainty sources that may distort the original GFR before the transaction,which is a novel empirical analysis of sensitivity.Typical GFR deformations under load forecasting errors or transmission contingencies are graphically presented in our approach for a three-generatorunit system with a detailed explanation.We show that a relatively minor error in load forecasting may lead to great fluctuations in GFR boundaries,and the fluctuations are mainly induced by the output power limit constraints of the generation units.Capacity reduction of line transmission would also lead to GFR shrinking,whose distortion patterns depend on the position and criticality of the lines.With the burgeoning of renewable energy penetration,greater challenges in system security have emerged.Hopefully,we hope our results can provide a perspective for the impact of uncertainties on power system operations.

Future works include elaborating uncertainty modeling such as incorporating the spatial and temporal correlation of uncertainties.

Acknowledgments

This work was supported in part by the National Key R&D Program of China under Grant No.2020YFB0905900,and in part by the State Grid Corporation of China project“Research on inter-provincial price coupling mechanism of national unified electricity spot market”.

Declaration of Competing Interest

We declare that we have no conflict of interest.

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