Unit Commitment Accommodating Large Scale Green Power

Abstract

As more clean energy sources contribute to the electrical grid, the stress on generation scheduling for peak-shaving increases. This is a concern in several provinces of China that have many nuclear power plants, such as Guangdong and Fujian. Studies on the unit commitment (UC) problem involving the characteristics of both wind and nuclear generation are urgently needed. This paper first describes a model of nuclear power and wind power for the UC problem, and then establishes an objective function for the total cost of nuclear and thermal power units, including the cost of fuel, start-stop and peak-shaving. The operating constraints of multiple generation unit types, the security constraints of the transmission line, and the influence of non-gauss wind power uncertainty on the spinning reserve capacity of the system are considered. Meanwhile, a model of an energy storage system (ESS) is introduced to smooth the wind power uncertainty. Due to the prediction error of wind power, the spinning reserve capacity of the system will be affected by the uncertainty. Over-provisioning of spinning reserve capacity is avoided by introducing chance constraints. This is followed by the design of a UC model applied to different power sources, such as nuclear power, thermal power, uncertain wind power, and ESS. Finally, the feasibility of the UC model in the scheduling of a multi-type generation unit is verified by the modified IEEE RTS 24-bus system accommodating large scale green generation units.

1. Introduction

Rapid growth in electrification together with more diversified clean power generation is causing an increasing disparity between peak and valley energy demands. In addition, there is a growing trend to integrate clean energy sources, like solar, wind and nuclear power, into the power grid. Most of China’s nuclear power plants are distributed in coastal areas, and their capacity is still increasing. The uncertainty of wind power and inflexible nuclear power generation means that the power grid frequency fluctuates easily and aggravates the burden of power system peak-shaving. Therefore, it is urgent to study how to schedule a multi-type generation unit that includes nuclear power and wind power for peak system operation.

Nuclear power plants are one of the key clean energy sources in China. In particular, the proportion of nuclear power in coastal provinces such as Fujian and Guangdong is still increasing. For safe operation, nuclear generation units usually operate at an invariant power to support the basic load and do not participate in load tracking or peak-shaving.

At present, nuclear power plants in the United States, Japan, and France are participating in peak-shaving. In [1], the feasibility and necessity of nuclear power plants participating in peak-shaving of power systems are described, including analysis of the modes and characteristics of peak operation. Fang et al. analyzed the feasibility of controlling the power of a pressurized water reactor nuclear power unit through the control rod in [2]. The core simulation and operating characteristics of AP1000 were studied in [3]. In [4], the ability and characteristics of peak-shaving with various nuclear generation units were analyzed, and existing problems in nuclear power generation and strategies for joint peaking operation of the other power sources were described. In [5], the advantage of nuclear power generation participation in load-following was studied, and it was shown that a nuclear power unit can directly participate in the system’s daily load peak-shaving based on the “12-3-6-3” mode, as verified by actual data from the grid.

In summary, there are few studies on joint optimization scheduling with nuclear power, non-Gaussian wind power generation, and energy storage. In [6], thermal and water joint optimization generation was proposed with the aim of minimizing the total power generation system cost, while also considering the constraints of pollutant emission and implementing suitable safety measures. In [7], the authors considered the peak regulation features of a security operation for nuclear power generation and used an objective function to quantify the total cost of thermal-nuclear-pumped storage; however, the integration of non-gaussian uncertain wind power units was not considered.

The use of ESS can mitigate some of the problems associated with the unpredictable nature of wind power generation [8]. ESS can function as a virtual power generation device by implementing peak-shaving and load following by absorbing and discharging energy to meet the demands of the grid. Energy storage technology, such as super conducting magnetic energy storage [9], thermal electric energy storage [10] and batteries [11] is currently developing rapidly. Large-scale ESSs have been considered for joint optimization scheduling [12,13].

The uncertainty associated with wind power causes difficulties in effective scheduling. A certain number of spinning reserves is needed in power generation plans to handle the problems of wind power and load uncertainty. Generally, reserve capacity is determined by a load demand ratio. However, given the non-gaussian distribution of prediction error of wind power, the reliability and economy needs of the power system cannot be guaranteed. There are two approaches to handle the unit commitment (UC) models associated with wind power prediction error. First, according to the uncertainty and probability distribution of wind power output, the system’s spin reserve capability and confidence interval can be set, respectively [14,15]. Multi-scene technology can be applied by analyzing the influence of wind power prediction error on generation scheduling results by simulating the discrete scenarios of various wind power output uncertainties [16,17,18]. However, this is a computationally expensive procedure.

The conventional UC problem associated with thermal power is a lack of features that can be integrated with nuclear power and non-gaussian distributed random wind power combination participate in joint peak-shaving. The spinning reserve capacity constraints considering uncertainty and transmission line thermal security are derived with chance constraint conditions. The wind power forecast error will affect the system’s reserve capacity uncertainty. Introducing a spinning reserve chance constraint considering non-Gaussian wind power generation decreases redundant over-provisioning. The ESS for a power station has been designed in the form of a scheduling optimization model. Ultimately, a joint optimization model involving multi-type generation units is built and optimal day-ahead generation unit scheduling strategies are given and compared under different situations.

2. Modeling of Nuclear and Wind Power

2.1. UC Modeling of Nuclear Power Generation

Load following control of a nuclear power plant is obtained by regulating the nuclear reactor power. Generally, through controlling the concentration of the boron solution and the rod displacement height, load following control of the reactor is achieved.

At present, most nuclear power plants in China are second generation pressurized water reactors (PWRs). They have the capacity to participate in daily load tracking. Considering the restrictions imposed by nuclear power ramp rate constraints and peak-shaving depth constraints, the PWR nuclear power units can participate in daily peak-shaving in the “12-3-6-3” power output mode, i.e., they operate at rated power for 12 h, then drop to a light-load power level after 3 h, continue to run at lower power sustaining for 6 h, and then rise to rated power in 3 h before entering the next cycle. The peak-shaving mode of “12-3-6-3” is shown in Figure 1.

In addition, with the development of nuclear power technology, AP1000, EPR and other three generations of nuclear power units show more effective regulation performance, and participate in daily load tracking; peak-shaving operation is also more flexible. They can achieve “15-1-7-1” peak-shaving operation mode within 90% of the operating lifetime of the unit, as shown in Figure 1.

The nuclear power output model with peak-shaving characteristics should be considered in the UC problem. The mathematical model of the output power is expressed as [7]:

$$P_{N,it}=g_{it}(P_{N,imin}+\Delta P_{N,i})+h_{it}(P_{N,imin}+2\Delta P_{N,i})+e_{it}{P}_{N,imax}+f_{it}{P}_{N,imin}$$

(1)

where,

$$\begin{array}{c}{e}_{it}+f_{it}+g_{it}+h_{it}=1\hfill \\ e_{ik}\ge e_{it}-e_{it-1}k=t,t+1,\dots ,t+T_i^e-1\hfill \\ f_{ik}\ge f_{it}-f_{it-1}k=t,t+1,\dots ,t+T_i^f-1\hfill \\ h_{it+1}\ge f_{it}+g_{it-1}-1\hfill \\ e_{it+1}\ge h_{it}+g_{it-1}-1\hfill \\ g_{it+1}\ge h_{it}+e_{it-1}-1\hfill \\ f_{it+1}\ge g_{it}+h_{it-1}-1\hfill \end{array}$$

where, $e_{it}$, $f_{it}$, $g_{it}$, $h_{it}$ are all $\left\{0,1\right\}$ variables, and denote the operation state index with respect to four nuclear power operating levels: $T_i^e$ and $T_i^f$ are the minimum running time of the rated power and the low power modes, respectively; $P_{N,imax}$, $P_{N,imin}$ are the maximum and minimum injected power; and $\Delta P_{N,i}$ represents the power variation of a nuclear generation unit i within 1 h.

Under rapid variation of generation, traditional thermal power generation, creep fatigue may come up in the main steam line. Meanwhile, service life of water pump, deaerator, and high pressure heater will be reduced. In general, deep peak-shaving of nuclear power plant is arranged at the 65% of fuel capacity. After, the depth of peak-shaving will be lowered. Under rapid variation of nuclear generation, clad shell may bear limiting stress. Service life of nuclear generation will be reduced.

2.2. Modeling of Wind Power Generation Uncertainty

The random and fluctuating characteristics of wind power make it difficult to predict, and therefore plan, flexible scheduling. The analysis of wind power uncertainty is of great significance to UC models of wind farms. The forecast error of wind power output is expressed as:

$$P_w=P_f+\Delta P_e$$

(2)

where, $P_w$ represents the actual power, $P_f$ denotes the forecasted wind power, and $\Delta P_e$ represents the forecast error.

In theory, the forecast errors of wind power belong to Gaussian distribution. Due to the different forecast methods, time scales and geographical environments of wind farms, the probability distribution of the forecast error shows different distribution, non-universal characteristics.

The Gaussian mixture model (GMM) is a linear combination of single Gaussian probability density functions. It can accurately describe the probability density distribution of various shapes by adjusting either the linear combination weights or the parameter estimates. The Gaussian mixture distribution makes the forecast error modeling of wind power more accurate. The forecast error of wind power is regarded as a random variable subject to Gaussian mixture distribution, and then the probabilistic constraint model is transformed into the deterministic constraint model and solved using the chance constraint programming theory. A detailed solution is provided in the Appendix A.

The variance can be solved by the clustering algorithm [19]; the equivalent mean and covariance are expressed as:

$${\omega }_m=\sum _{j\in \mathrm{I}}{\omega }_j$$

(3)

$${\mu }_m=\frac{1}{{\omega }_m}\sum _{j\in \mathrm{I}}{\omega }_j{\mu }_j$$

(4)

$${\sigma }_m=\frac{1}{{\omega }_m}\sum _{j\in \mathrm{I}}{\omega }_j[{\sigma }_j+({\mu }_j-{\mu }_m){({\mu }_j-{\mu }_m)}^T]$$

(5)

where, ${\omega }_j$ and ${\omega }_m$ are the weight of the jth mixture component and the total weight, respectively, ${\mu }_j$ and ${\mu }_m$ are the mean of the jth mixture component and the total mean, respectively, ${\sigma }_j$ and ${\sigma }_m$ are the standard deviation of the jth mixture component and the total standard deviation, respectively, and T is determined by the ${\chi }^2$-test with 99% confidence.

Prediction accuracy is improving with the development of new technology. It is assumed that the forecast errors of load and wind power are subject to Gaussian distribution and Gaussian mixture distribution, respectively. Here, ${\sigma }_L$, ${\sigma }_W$ denote the standard deviation of the forecast error distribution of load and wind power, respectively [16,20]. ${\sigma }_W$ can be solved by Formulas (3)–(5). Assuming that the load and wind power forecast errors are uncorrelated random variables, the standard deviation of the total forecast error can be expressed as [21]:

$$\sigma =\sqrt{{({\sigma }_L^t)}^2+{({\sigma }_W^t)}^2}$$

(6)

Considering the uncertain factors, additional spinning reserve capacity is needed to ensure the security of generation scheduling. The spinning reserve capacity is mainly from thermal generation units, and the spinning reserve capacity for total power system at each time scale is expressed as:

$${\sum }_{i=1}^{m_0}{u}_{i,t}{P}_{G,imax}-{\sum }_{i=1}^{m_0}{P}_{G,it}\ge Re{s}_t$$

(7)

where $Re{s}_t$ denotes the spinning reserve capacity at time t.

${\tilde{P}}_{i,t}^W$ and ${\tilde{P}}_{L,t}$ are introduced, which are equal to the expected values $P_{i,t}^W$ and $P_{L,t}$ plus the error values $\Delta P_{i,t}^W$ and $\Delta P_{L,t}$, respectively. Formula (7) is expressed as:

$${\sum }_{i=1}^{m_0}{u}_{i,t}{P}_{G,imax}-{\sum }_{i=1}^{m_0}{P}_{G,it}\ge Re{s}_t+{\sum }_{i=1}^{m_2}\Delta P_{i,t}^W+\Delta P_{L,t}$$

(8)

Here, $z_t\stackrel{\Delta }{=}{\sum }_{i=1}^{m_2}\Delta P_{i,t}^W+\Delta P_{L,t}$ is a random variable, and $F_{z_t}^{-1}(·)$ denotes the inverse of the cumulative distribution function. With application of the chance constraints, (8) is written by:

$$P\left(z_t\le {\sum }_{i=1}^{m_0}\left(u_{i,t}{P}_{G,imax}-P_{G,it}\right)-Re{s}_t\right)\ge \alpha$$

(9)

Then, the conversion of probability constraints and deterministic constraints is achieved by $F_{z_t}^{-1}(·)$.

$${\sum }_{i=1}^{m_0}\left(u_{i,t}{P}_{G,imax}-P_{G,it}\right)-Re{s}_t\ge F_{z_t}^{-1}(\alpha )$$

(10)

where $\alpha$ denotes the confidence level.

According to the standard deviation of the total forecast error (6), (10) is further approximated as:

$${\sum }_{i=1}^{m_0}\left(u_{i,t}{P}_{G,imax}-P_{G,it}\right)\ge Re{s}_t+\eta (\alpha ){\left({({\sigma }_D^t)}^2+{({\sigma }_W^t)}^2\right)}^{1/2}$$

(11)

where $\eta (\alpha )$ denotes cumulative distribution function at confidence level $\alpha$, which can be acquired by looking up the Gaussian distribution table.

3. UC Program Considering Multiple Generation Types

The UC program involving various generation types considers the constraints of each generation unit’s operation characteristics together with the security constraints of the whole system. The optimization goal is to minimize the operating cost of the grid by determining the operational status of each generation unit. This paper mainly considers thermal power, wind power, nuclear power, and energy storage.

3.1. Objective of UC

The objective of UC is to minimize the operating cost of the whole system, which mainly comprises thermal and nuclear power expenses.

$$\begin{array}{c}minOF={\displaystyle \sum _{t=1}^T}{\displaystyle \sum _{i=1}^{m_0}}(F{C}_{G,it}+ST{C}_{G,it}+SD{C}_{G,it})\hfill \\ +{\displaystyle \sum _{t=1}^T}{\displaystyle \sum _{i=1}^{m_0}}{g}_{G,i}·C_G+{\displaystyle \sum _{t=1}^T}{\displaystyle \sum _{i=1}^{m_1}}F{C}_{N,it}\hfill \\ +{\displaystyle \sum _{t=1}^T}{\displaystyle \sum _{i=1}^{m_1}}(P_{N,imax}-P_{N,it})·C_N\hfill \end{array}$$

(12)

where, T represents the time horizon in one day (i.e., T = 24 h), t denotes the time period (hours), and $m_0$ and $m_1$ denote the number of thermal power units and nuclear power units, respectively. $F{C}_{N,it}$ is the fuel cost of the nuclear power units, $F{C}_{G,it}=a_i{P}_{G,it}^2+b_i{P}_{G,it}+c_i$ is the quadratic function, which represents the fuel cost of thermal power units; $ST{C}_{G,it}=s{t}_i{y}_{it}$ and $SD{C}_{G,it}=s{d}_i{z}_{it}$ represent the up/down cost of thermal generation units, respectively, ${\displaystyle \sum _{t=1}^T}{\displaystyle \sum _{i=1}^{m_0}}{g}_{G,i}·C_G$ and ${\displaystyle \sum _{t=1}^T}{\displaystyle \sum _{i=1}^{m_1}}(P_{N,imax}-P_{N,it})·C_N$ denote the peak-shaving costs of thermal power units and nuclear power units, respectively; and $a_i$, $b_i$, $c_i$, $C_G$, $C_N$, $s{t}_i$, $s{d}_i$ denote the known cost coefficients of generators.

There, the quadratic function of the fuel cost should be linearized. Firstly, the interval $[P_i^{min},P_i^{max}]$ is divided into n equal-sized subintervals. Secondly, for a particular subinterval k, we define a variable $p_{i,t}^k$ that varies between zero and the subinterval length $\Delta P_i^k$. Furthermore, the relevant parameters of subinterval k should satisfy the following constraints.

$$\begin{array}{c}0\le p_{i,t}^k\le \Delta P_i^k{u}_{i,t}\forall k=1:n\hfill \\ \Delta p_i^k=\frac{P_i^{max}-P_i^{min}}{n}\hfill \\ P_{i,ini}^k=\left(k-1\right)\Delta P_i^k+P_i^{min}\hfill \\ P_{i,fin}^k=\Delta P_i^k+P_{i,ini}^k\hfill \\ P_{i,t}=P_i^{min}{u}_{i,t}+{\displaystyle \sum _k}{p}_{i,t}^k\hfill \\ C_{i,ini}^k=a_i{(P_{i,\mathrm{int}}^k)}^2+b_i{P}_{i,\mathrm{int}}^k+c_i\hfill \\ C_{i,fin}^k=a_i{(P_{i,fin}^k)}^2+b_i{P}_{i,fin}^k+c_i\hfill \\ s_i^k=\frac{C_{i,fin}^k-C_{i,ini}^k}{\Delta P_i^k}\hfill \end{array}$$

(13)

where, $s_i^k$ is the slope of the line segment between the start point and the end point in the subinterval k.

Finally, linear expression of the quadratic function of the fuel cost is as follows.

$$F{C}_{\mathrm{i},t}=(a_i{(P_i^{min})}^2+b_i{P}_i^{min}+c_i)u_{i,t}+\sum _k{s}_i^k{p}_{i,t}^k$$

(14)

3.2. Constraints for Generation Units

The constraints of the conventional UC program include network safety, power balance and minimum up/down time, etc. If the nuclear power, wind power and the energy storage participate in day-ahead-scheduling, the impact of nuclear power plant peak-shaving operation characteristics and wind power uncertainty on spinning reserve should be considered. These constraints are expressed as follows.

3.2.1. Load and Generation Power Balance Constraints

The power balance constraints are expressed as:

$${\sum }_{i=1}^{m_0}{P}_{G,it}+{\sum }_{i=1}^{m_1}{P}_{N,it}+{\sum }_{i=1}^{m_2}{P}_{i,t}^W=P_{L,t}$$

(15)

where, $P_{G,it}$ and $P_{N,it}$ represent the output power of the thermal generation unit and the nuclear power at time t, respectively, $P_{i,t}^W$ represents the predicted power of the wind generation unit i at time t, and $P_{L,t}$ represents the system’s total load demand at time t.

3.2.2. Minimum and Maximum Injected Power Constraints

The minimum and maximum injected power constraints are expressed as:

$$u_{i,t}{P}_{G,imin}\le P_{G,it}\le u_{i,t}{P}_{G,imax}$$

(16)

where, $P_{G,imin}$ and $P_{G,imax}$ represent the minimum power generation and maximum power generation of the thermal power unit i, respectively; $u_{i,t}$ represents the operating status of unit i at time t; and $u_{i,t}=0/1$ represents that unit i is in the off/on state at time t.

3.2.3. Reserve Capacity Constraints

According to Section 2.2, the spinning reserve capacity considering both the load and wind power prediction uncertainty is expressed as:

$${\sum }_{i=1}^{m_0}\left(P_{G,im\mathrm{ax}}-P_{G,it}\right)+{\sum }_{i=1}^{m_1}\left(P_{N,im\mathrm{ax}}-P_{N,it}\right)\ge Re{s}_t+\eta (\alpha ){\left({({\sigma }_D^t)}^2+{({\sigma }_W^t)}^2\right)}^{1/2}$$

(17)

3.2.4. Ramp Rate Constraints

In order to cope with changes in load and wind power output, thermal power units need to adjust their output in time. The ramp rate constraints are expressed as:

$$\begin{array}{c}{P}_{it}-P_{it-1}\le R{U}_i\hfill \\ P_{it-1}-P_{it}\le R{D}_i\hfill \end{array}$$

(18)

where, $R{D}_i$ and $R{U}_i$ denote the down/up active power limit of unit i under normal operating status, respectively.

3.2.5. Minimum Thermal Generation up/down Time Constraints

By discretizing the nonlinear startup cost, we can get a piecewise linear function. The formula of the minimum up time is expressed as [22]:

$$\begin{array}{c}\begin{array}{c}{\sum }_{t=1}^{{\zeta }_i}1-u_{i,t}=0\\ {\sum }_{t=k}^{k+U{T}_i-1}{u}_{i,t}\ge U{T}_i{y}_{i,\mathrm{t}},\forall k={\zeta }_i+1\dots T-U{T}_i+1\\ {\sum }_{t=k}^T{u}_{i,t}-y_{i,t}\ge 0,\forall k=T-U{T}_i+2\dots T\end{array}\hfill \\ {\zeta }_i=min\{T,(U{T}_i-U_i^0)u_{i,t=0}\}\hfill \end{array}$$

(19)

The formula of the minimum down time is expressed as:

$$\begin{array}{c}\begin{array}{c}{\sum }_{t=1}^{{\xi }_i}{u}_{i,t}=0\\ {\sum }_{t=k}^{k+D{T}_i-1}1-u_{it}\ge U{T}_i{z}_{i,k},\forall k={\xi }_i+1\dots T-D{T}_i+1\\ {\sum }_{t=k}^{T}1-u_{i,t}-z_{i,t}\ge 0,\forall k=T-D{T}_i+2\dots T\end{array}\hfill \\ {\xi }_i=min\{T,(D{T}_i-S_i^0)(1-u_{i,t=0})\}\hfill \end{array}$$

(20)

where, $D{T}_i$ and $U{T}_i$ denote the minimum down/up time of thermal generation unit i, respectively; $U_i^0$ and $S_i^0$ denote the initial down/up time, respectively; and $y_{i,t}$ and $z_{i,t}$ denote the start-up/shut-down status, respectively. The constraint that $y_{i,t}$ and $z_{i,t}$ satisfy is expressed as:

$$\begin{array}{c}{y}_{i,t}-z_{i,t}=u_{i,t}-u_{i,t-1}\\ y_{i,t}+z_{i,t}\le 1\\ y_{i,t},z_{i,t},u_{i,t}\in \{0,1\}\end{array}$$

(21)

3.2.6. Nuclear Power Peak Regulation Depth Constraints

The nuclear power peak-shaving depth constraints are expressed as:

$${\eta }_i=\frac{P_{N,imax}-P_{N,i}}{P_{N,imax}}\le {\eta }_{max}$$

(22)

where, ${\eta }_i$ represents peak-shaving depth, $P_{N,i}$ denotes power generation of nuclear plant i, $P_{N,imax}$ is maximum power generation of nuclear plant, i.e., the capacity of nuclear plant. ${\eta }_{max}$ is maximum peak-shaving depth. For example, the capacity of nuclear plant is 100 MW, the maximum peak-shaving of 0.8, $P_{N,i}$ can not be lower than 20 MW.

The nuclear generation unit has various installed capacities, reactor types and thresholds, ${\eta }_{max}$, of maximum peak-shaving depth.

The constraint to be satisfied during depth peak-shaving of the thermal power unit is expressed as:

$$P_{G,imin}^{\prime }\le P_{G,it}\le P_{G,imax}$$

(23)

$$g_{G,it}=\left\{\begin{array}{cc}{P^{\prime }}_{G,imin}-P_{G,it}\hfill & P_{G,it}<{P^{\prime }}_{G,imin}\hfill \\ 0\hfill & P_{G,it}\ge {P^{\prime }}_{G,imin}\hfill \end{array}\right.$$

(24)

The basic and depth peak-shaving states are two states in which thermal power participates in peak-shaving. The injected power threshold is denoted by $P_{G,imin}^{\prime }$, which is used to identify the two states, i.e., peak-shaving cost compensation is only given when a certain peak-shaving depth is reached. Where, $P_{G,imin}^{\prime }$ is the minimum limit of the injected power of the thermal generation that taking part in the basic peak-shaving; and $P_{G,it}$ is the injected active power with respect to the thermal generation unit i at time t.

3.2.7. Load Following Constraints of Nuclear Generation Units

According to Section 2.1, the nuclear generation unit considering the peak-shaving characteristics is expressed as:

$$P_{N,it}=e_{it}{P}_{N,imax}+f_{it}{P}_{N,imin}+g_{it}(P_{N,imin}+\Delta P_{N,i})+h_{it}(P_{N,imin}+2\Delta P_{N,i})$$

(25)

3.2.8. ESS Constraints

The ESS cannot exceed the maximum and minimum limits capacity:

$$SO{C}_{i,t}=SO{C}_{i,t-1}+(P_{i,t}^c{\eta }_c-P_{i,t}^d/\phantom{P_{i,t}^d{\eta }_d}{\eta }_d)\Delta t$$

(26)

$$SO{C}_{min}\le SO{C}_t\le SO{C}_{max}$$

(27)

where, $SO{C}_{max}$ and $SO{C}_{min}$ denote the maximum and minimum limits of the residual capacity of the ESS, respectively; $SO{C}_t$ is the storage capacity at time t. $P_{i,t}^c$ and $P_{i,t}^d$ denote the charge/discharge power at time t, respectively, ${\eta }_c$ denotes the charging efficiency; and ${\eta }_d$ represents discharging efficiency; $\Delta t$ denotes the interval time scale, is equal to 1 h.

Considering the current limitations of converters, the injected power of the ESS at each moment cannot violate a certain power limit.

$$\begin{array}{c}{P}_{i,min}^c\le P_{i,t}^c\le P_{i,max}^c\hfill \\ P_{i,min}^d\le P_{i,t}^d\le P_{i,max}^d\hfill \end{array}$$

(28)

where $P_{i,max}^c$ and $P_{i,max}^d$ denote the maximum active power limits during charging and discharging respectively. Here, the lower limits we set are $P_{i,min}^c=P_{i,min}^d=0$, $P_{i,max}^c=P_{i,max}^d=SO{C}_{max}$.

Besides, the depth and timing of charge and discharge affect the life of the battery; relevant constraint models need to be introduced for studying their influence on battery life.

3.2.9. Line Transmission Power Constraints

The line transmission power constraints are expressed as:

$$\sum _{\mathrm{g}\in {\Omega }_G^i}{P}_{g,t}+\sum _{h\in {\Omega }_H^i}{P}_{h,t}+P_{i,t}^w-L_{i,t}-P_{i,t}^c+P_{i,t}^d=\sum _{j\in {\Omega }_{\ell }^i}{P}_{ij,t}$$

(29)

$$P_{ij,t}=\frac{{\theta }_{i,t}-{\theta }_{j,t}}{x_{ij}}$$

(30)

$$-P_{ij,t}^{max}\le P_{ij,t}\le P_{ij,t}^{max}$$

(31)

where ${\theta }_{i,t}$ denotes bus i voltage angle at time t, and $x_{ij}$ denotes the branch reactance between bus i and bus j. $P_{ij,t}^{max}$ represents the maximum active power flow limits for branch $ij$. ${\Omega }_G^i$ represents all thermal generation units at bus i, ${\Omega }_H^i$ represents all nuclear power units at bus i, and ${\Omega }_{\ell }^i$ represents all buses linked to bus i.

Introducing the probability constraint, the line transmission capacity constraint is the joint opportunity constraint when the line power is regarded as a random variable.

$$F{C}_{\mathrm{i},t}=(a_i{(P_i^{min})}^2+b_i{P}_i^{min}+c_i)u_{i,t}+\sum _k{s}_i^k{p}_{i,t}^k$$

(32)

Equation (32) is transformed into a deterministic constraint as follows:

$$\begin{array}{c}{P}_{ij,t}\le P_{ij,t}^{max}-F_{\Delta P_{ij,t}}^{-1}({\beta }_1)\hfill \\ P_{ij,t}\ge P_{ij,t}^{min}+F_{\Delta P_{ij,t}}^{-1}({\beta }_2)\hfill \end{array}$$

(33)

where ${\beta }_1$ and ${\beta }_2$ are the confidence level.

4. Case Study

The effectiveness of the proposed UC model is verified by the modified IEEE RTS-24 bus system, which consists of 10 thermal generation units, two wind farms, two nuclear power stations and two energy storage power stations. The cost coefficients of the generators are listed in

Table 1. Cost coefficients of thermal power units.

Number	${\mathit{a}}_{\mathit{i}}(\mathbf{\$}/\mathbf{M}{\mathbf{W}}^2\mathbf{h})$	${\mathit{b}}_{\mathit{i}}(\mathbf{\$}/\mathbf{MWh})$	${\mathit{c}}_{\mathit{i}}(\mathbf{\$}/\mathbf{h})$	$\mathit{s}{\mathit{t}}_{\mathit{i}}(\mathbf{\$})$	$\mathit{s}{\mathit{d}}_{\mathit{i}}(\mathbf{\$})$
g1	0.00048	16.19	1000	52,000	13,000
g2	0.00048	16.19	1000	52,000	13,000
g3	0.0021	16.8	720	17,667	4367
g4	0.0021	16.8	720	17,667	4367
g5	0.002	16.6	700	17,667	4367
g6	0.002	16.6	700	16,667	4167
g7	0.001	16.19	800	50,000	12,000
g8	0.00068	16.19	850	50,000	12,000
g9	0.00068	16.19	850	50,000	12,000
g10	0.001	16.19	800	0	0

and

Table 2. Cost coefficients of nuclear power units.

Number	${\mathit{a}}_{\mathit{i}}(\mathbf{\$}/\mathbf{M}{\mathbf{W}}^2\mathbf{h})$	${\mathit{b}}_{\mathit{i}}(\mathbf{\$}/\mathbf{MWh})$	${\mathit{c}}_{\mathit{i}}(\mathbf{\$}/\mathbf{h})$	${\mathit{P}}_{\mathit{N},\mathbf{i}\mathit{Max}}(\mathbf{\$})$
h1	0	9.33	7320	300
h2	0	9.33	7320	100

. The daily load demand and wind power generation are shown in Figure 2. The quadratic function of the fuel cost and the constraints of minimum up/down time are linearized by reference to the literature [22]. The proposed UC mathematical model is solved with GAMS MINLP solver.

The total installed capacity of the wind farms connected to lines 8 and 21 is 200 and 50 MW, respectively. The installed capacity of the nuclear power plant connected to buses 1 and 18 is 100 and 300 MW, respectively.

In the test system, the wind generation is linked to both bus 8 and bus 21. The Gaussian mixture parameters of wind power forecast error are $c_1^{(1)}=0.7$, $u_1^{(1)}=0$, ${\sigma }_1^{(1)}=8$, $c_2^{(1)}=0.3$, $u_2^{(1)}=4$, ${\sigma }_2^{(1)}=2$, $c_1^{(2)}=0.8$, $u_1^{(2)}=1$, ${\sigma }_1^{(2)}=4$, $c_2^{(2)}=0.2$, $u_2^{(2)}=5$, ${\sigma }_2^{(2)}=10$. On lines 12 and 13, the corresponding wind turbine output power transfer distribution factors ${\beta }^{(1)}$ and ${\beta }^{(2)}$ are 0.26 and 0.42, respectively. The detailed derivation of line transfer power probability is provided in the Appendix A.

4.1. Impact on Cost of Peak Regulation with Nuclear Power

In this part, three schemes are designed for comparative analysis.

(1)

Scheme one

Nuclear power units do not participate in peak-shaving scheduling and maintain stable operation at full power.

(2)

Scheme two

Nuclear power units participate in peak-shaving scheduling of the power system according to the output pattern of “12-3-6-3”. The peak regulation depth of two nuclear power units is set in advance at 30%.

(3)

Scheme three

The method of peak-shaving scheduling of the power system is the same as in scheme two. The optimization solution is based on the joint optimal peak-shaving scheduling model. The optimization cost results are shown in Figure 3.

As can be seen in Figure 3, scheme three has the lowest total operating cost. At this time, the optimal peak-shaving depth of the two nuclear power units is 0% and 29.8%, respectively. Shorter start-stop operation times are the main reason for the reduced operational costs. For example, compared to scheme one, the one-time stop operation cost of thermal power unit 3 (g3) is reduced in schemes two and three with peak-shaving of nuclear power units.

4.2. Optimal Scheduling of the Proposed Multi-Type Generation Unit

Considering the joint dispatching model presented, including nuclear generation, the optimized peak shaving depth of nuclear power is set to ${\eta }_i=30\%$, The injected power and up/down status of 10 thermal power units are shown in Figure 4, where we can see that the No. 5 and No. 6 thermal power units (g5 and g6, respectively) are always in a stop state, which avoids the high up/down cost.

The optimized peak regulation output curves of the No. 1 and No. 2 nuclear power plants (h1 and h2) are shown in Figure 5, respectively. The output of nuclear generation follows the system demand, which effectively decreases the peak-valley load distinction. Figure 6 shows the charging and discharging active power of ESS on both bus 8, 21, and the total power at each time.

4.3. Energy Storage Parameters Impact on System Cost

The ESS has both charging and discharging active power characteristics, which play an important role in peak-shaving.

For the UC model considering ESS, different energy storage parameters have different influences on the total cost.

Table 3. Total cost under different ESS parameters.

ESS Parameters (Energy Storage Upper Limit/MWh, Power Upper Limit/MW)	Total Cost/$
80, 16	1.3337 × 10${}^6$
100, 20	1.3336 × 10${}^6$
150, 30	1.3264 × 10${}^6$
200, 40	1.3239 × 10${}^6$

illustrates that the overall system cost decreases when the active power capacity charging and discharging power limit increase.

4.4. Impact of Capacity Confidence Coefficients on System Cost

Taking into account uncertainty when modelling reserve capacity gives more dexterity spare for power system scheduling.

To capture different degrees of net power fluctuation, different confidence levels are set. Figure 7 shows the hourly deterministic reserve capacity in comparison with the probabilistic reserve constraint at various confidence levels. Besides, as shown in Figure 8, the overall cost rises with confidence level increasing.

5. Conclusions

This paper presented a UC program that incorporated various generation types, such as thermal power, nuclear power, wind generation and ESS. The introduction of a nuclear power plant eases the peak-shaving pressure, decreases the up/down frequency of the conventional thermal power unit, and reduces the operational expenses. This ESS model makes scheduling more flexible, thus serving as an important adjunct function to generation system scheduling. This allows the grid to make full use of clean energy, such as solar and wind energy. In addition, a GMM has been applied to model wind power with non-gaussian uncertainty, and the spinning reserve setting of the system is more rational; this provides the dispatcher with more comprehensive choices. Regarding the influence of uncertainty on system scheduling, some alternatives have not yet been considered, and these will be studied in our future work.