A stochastic optimal power flow (SOPF) model considering uncertainties of load and wind power is developed based on chance constrained programming (CCP). The difficulties in solving the model are the nonlinearity and probabilistic constraints. In this paper, a limit relaxation approach and an iterative learning control (ILC) method are implemented to solve the SOPF model indirectly. The limit relaxation approach narrows the solution space by introducing regulatory factors, according to the relationship between the constraint equations and the optimization variables. The regulatory factors are designed by ILC method to ensure the optimality of final solution under a predefined confidence level. The optimization algorithm for SOPF is completed based on the combination of limit relaxation and ILC and tested on the IEEE 14bus system.
1. Introduction
Deterministic approaches to optimal power flow (OPF) problems have been well developed and widely used in power systems
[1

2]
. However, as renewable energy, such as wind power, has developed at a fast speed in recent years, it is of great significance to study the effects of uncertainty on the optimal power flow of the system containing renewable power supplies and other uncertain factors existing in power systems. These uncertainties mainly come from changes in network configuration, outages of system elements and forecast inaccuracies
[3]
. At present, the studies on OPF considering the stochastic factors can be classified as probabilistic optimal power flow (POPF) and stochastic optimal power flow (SOPF)
[4]
. POPF can obtain the probability distribution functions (PDFs) of state variables from the probability distributions of loads and other uncertain factors by MonteCarlo simulation
[5]
, cumulant method
[6]
, point estimate method
[7]
, etc. The uncertain factors generally do not affect the optimal results of POPF. SOPF differs from POPF in that it considers the uncertainty as part of the model’s constraints and objective
[4]
. Thus, uncertain factors affect the calculation process and the final results of SOPF. Present studies on SOPF also include two kinds: one formulates the model as deterministic OPF (or separates the model into subproblems as deterministic OPF) and mainly considers the uncertain changes in network configuration and component failures
[8]
, the other usually models the SOPF based on chance constrained programming (CCP)
[9]
.
The advantage of CCP is the comprehensive consideration between the profitability and reliability under uncertainty using chance constraint as inequality constraint satisfied with a predefined probability value to ensure a certain level of reliability. Recently, CCPbased methods have been applied to power system under uncertainty for transmission network expansion planning
[10]
, unit commitment
[11]
, evaluation of available transfer capability
[12]
, SOPF
[13]
, etc. For the CCPbased model with constraints including power flow equations and nonlinear chance constraints, different solving strategies have been developed. Hui Zhang et al.
[13]
solved the formulated CCPbased OPF under load power uncertainties by a backmapping approach and a linear approximation of the nonlinear model equations. Wang, Shuang et al.
[14]
transformed the chance constrains using sample average approximation (SAA) and solved the optimal model by genetic algorithm (GA). Qianfan Wang et al.
[15]
also used a combined SAA algorithm to solve the CCPbased problem. The disadvantage of SAAbased method is that it needs lots of samples to ensure the solution converges to the optimal one. Zhipeng Liu et al.
[16]
employed a Monte Carlo simulationembedded geneticalgorithmbased approach to solve the developed CCP model, which is rather computationally expensive for larger systems. Schellenberg, A. et al
[6]
and Z. Hu et al
[17]
combined the statistical step with the pure Newton step in a linear fashion to produce a step that was applied to optimization variables.
In this paper, the SOPF model is formulated based on CCP with stochastic inputs including load fluctuation and the forecast error of wind farm output. The solution of the formulated SOPF model is based on the classic interior point method of deterministic OPF with high convergence rate and less calculation. Since the SOPF is a highdimensional nonlinear model with a large number of uncertain inputs as well as multiple probabilistically constrained outputs, it is difficult to deal with the nonlinear chance constraints of CCP directly in SOPF model. Iterative learning control (ILC) is used to regulate the limits of chance constraints in this paper. This approach is very attractive in control area, because of its performance improving capabilities along repetitive trajectories, often met in practice
[18]
. A limit relaxation approach combined with ILC is proposed to solve the formulated CCPbased SOPF model. The proposed method can deal with multiple chance constraints simultaneously in the optimization process.
The remainder of this paper is organized as follows. In Section 2, the SOPF model based on CCP is formulated and the stochastic inputs are described. Section 3 proposes an optimization algorithm for the SOPF model. Results of case study are given in Section 4. The conclusion is presented in Section 5.
2. Model of SOPF
 2.1 OPF model based on CCP
Ignoring the wind power operation costs, the objective of the power system economic dispatch is to minimize the expected power generation cost of the system’s conventional generator sets. The deterministic OPF can be established as following
[1

2]
:
s.t
The objective is a quadratic cost function. It can also be the transmission losses, economic benefits, the environmental benefits, etc. Power flow Eqs. (23) can be rewritten as
where,
X
= [
P_{G}
,
Q_{G}
,
θ
,
V
].
The conventional unit output constraint (4), generator ramp rate constraint (5), spinning reserve constraints (6), reactive power constraint of reactive source (7), nodal voltage constraint (8), and branch flow constraint (9) can be represented in the manner of vector as:
Inequality constraints (11) can be further expressed as:
where,
H
(
X
) = [
H
^{'}
(
X
) −
h
^{max}
,
h
^{min}
−
H
^{'}
(
X
)]
^{T}
.
The deterministic OPF (DOPF) is established with (1) as the objective and (10) and (12) as the constraints. However, the solution obtained with the deterministic OPF which does not consider stochastic factors may lead to inferior or even infeasible decision under stochastic environment. When considering the uncertainties of load and wind power,
P_{W}
,
P_{D}
,
Q_{W}
and
Q_{D}
are stochastic variables. Therefore,
H
(
X
) is also stochastic according to (2) and (3). Chance constrained method is considered as a useful tool to evaluate the probability or reliability of holding boundary conditions
[10

13]
. In CCPbased OPF model, the inequality constraint is replaced by the chance constraint which is defined in the form of probability as:
The chance constraint requires the inequality should be satisfied at a predetermined level of probability
β^{n}
( 0 ≤
β^{n}
≤ 1 ).
β^{n}
= 1 means that the inequality should always keep true in the supposed stochastic conditions. The chance constraint does not work if its confidence level is zero. The elements of
β
can be set as the same or defined separately according to different reliability requirements of inequality constraints. As a result, the OPF under uncertainty can be reformulated into a chance constrained OPF problem with (1) as the objective function subjected to constraints (2)(3) and (13).
 2.2 Stochastic Inputs
The stochastic inputs discussed in this paper include the uncertainty of wind farm outputs and loads. The load forecast is more precise than wind power forecast of a wind farm. Therefore, we focus on analyzing the uncertainty of wind forecast here. The analysis can also be applied to describe loads.
Based on a specific forecast method, the actual wind power output can be divided into two parts: the forecasted wind power (deterministic part) and the forecast error (uncertain part), as shown in Eq. (14). From (14), the PDF of the actual wind power can be obtained by the probability distribution of the forecast error.
The forecast methods of
includes: the physical forecast based on weather information, the forecast method based on historical data, the combined forecast method, etc.
[19
,
20]
. Due to the random fluctuation of wind speed as well as the precision and adaptability of the forecast methods, precise forecast of wind power is impossible. Gaussian distribution is usually adopted to describe the distribution of forecast error in analysis. However, it has already been found that the distribution of actual forecast error has the characteristic of skewing either to the left or to the right. Bludszuweit, H. et al
[21]
and Fabbri, A. et al
[22]
have proposed that Gaussian distribution is not valid to describe the wind power forecast error and the Beta distribution gives more reasonable and accurate results. In this paper, cumulant is used to describe the probabilistic characteristics of stochastic inputs no matter which distribution the stochastic inputs should fit. According to the history data of wind power forecast, the cumulants of the measured forecast error can be estimated by maximum likelihood estimation (MLE)
[23]
.
Usually, correlations of these stochastic inputs may exist
[14
,
24]
. Correlations can be included in the model under study, but it is not the main concern in this paper. The correlated stochastic inputs can be represented as (15) applying a Gram–Schmidt orthogonalization.
The elements of Δ
W
is
.
ρ
_{1 }
ρ
_{2}
…
ρ_{n}
is a group of uncorrelated variables.
3. A Combined Solution
 3.1 Limit relaxation
The difficulty in solving the CCPbased SOPF is the processing of chance constraints as shown in (13). Due to complicated nonlinearity of the chance constraints, the scope of
X
is impossible to be expressed explicitly in mathematical formula. Therefore, it is hard to adjust the optimization variables directly. This paper proposes a limit relaxation method to regulate the optimization variables indirectly, taking advantage of the correlation between optimization variables and constraints. The limit relaxation introduces regulatory factor
a
(
a
> 0 ) as the margin to the limits of the inequality constraints to narrow the solution space. Usually, the optimal solution achieves (or closes to) one (or more than one) of the limits of the inequality constraints in deterministic OPF. The regulatory factor defined in advance can prevent the optimal solution from reaching the limits of the inequality constraints. And it is regulated iteratively by ILC, described in section 3.3, until all of the chance constraints are satisfied. Therefore, the problem of directly calculating the optimization variables is transformed into a problem on how to narrow the solution space to ensure the optimal solution of the formulated SOPF.
Regulatory factor is added to the inequality constraints as shown in (16).
Eq. (16) can also be represented as
When
a
is added to the deterministic OPF, the OPF model applying barrier function can be written as follows
[2]
:
The interior point method can be adopted to solve thisdeterministic model. A given regulatory factor
a
is added in Eq. (20) to relax the limits, which is different from the interior point method in the traditional optimal power flow.
The Lagrange function of the optimization model (18)(21) is as following:
The Newton iteration step of the Karush–Kuhn–Tucker (KKT) equation of (22) is
where,
, [
μ
] is used to denote a diagonal matrix with vector
μ
on the diagonal. Eq. (23) shows the relationship of
H
(
X
) , Δ
X
and
a
. The regulatory factor can indirectly change theoptimization variables through (23). The output of the optimization model (18)(21) includes:
X
,
Z
,
γ
,
λ
and
μ
.
 3.2 Calculation ofhβ
The CornishFisher series is developed based on cumulant theory to approximate the probability density function. The principles of cumulant and CornishFisher series can be referred in
[25]
and
[26]
.
The linearized representation of
H_{m}
is as:
The uncertainty of the inequality constraints can be represented as:
The Jacobian matrix
J_{m}
at each iteration can be derived from the elements of
G_{X}
when the deterministic optimization process ends. According to the cumulant theory, cumulants of Δ
H_{m}
at each order can be calculated based on (25). Then the probability density functions of Δ
H_{m}
can be obtained with CornishFisher series. A crossboundary vector ,
h_{β,m}
can be obtained with elements calculated as
where,
is the point of
at which the probability of
equals to
β^{n}
.
indicates that
n
th chance constraint is not satisfied at
m
th iteration. For nonchance constraints (or the chance constraints which are not considered in the model), the corresponding elements of
h_{β}
are considered as zeros.
 3.3 Iterative learning control method
Iterative Learning Control (ILC) method can track the reference exactly in a finite time interval without the knowledge of the system models. (It applied to nonlinear systems.) It finds a suitable control action using the information from previous iterative results. A typical learning law of ILC is of the form:
where,
L
(·) is the learning function in ILC method. It is usually a function of the tracking errors in the learning process. Many control algorithms, ranging from simple scalar gains to sophisticated optimization computations, have been designed and applied in different areas
[27

30]
. Qing Wu, et al
[30]
used a PIDtype iterative learning algorithm to detect, isolate, and estimate faults effectively in the fault diagnosis of a satellite. The PIDType ILC method is also adopted in this paper to regulate the chance constraints. The discrete expression of
a
at (m+1) th iteration can be written as
According to (28), regulatory factor at (m+1) th iteration is updated using the information of (m1) th iteration and m th iteration. The PID parameters are also updated adaptively by ILC to ensure good convergence behavior. The elements of regulatory factor corresponding to nonchance constraints and the satisfying chance constraints at 1st iteration are zeros. The reference of the model in the study keeps zero. It is a class of timeinvariant nonlinear system with fixed target trajectory. Usually, the optimal solution can be obtained in several iterations by PIDbased ILC.
The optimization framework for SOPF with ILC is shown in
Fig.1
.
The optimization framework for SOPF
The memory block stores
h
_{β,1}
,…,
h
_{β,m}
and
a
_{1}
,…,
a_{m}
for later calculation. The probabilistic calculation is based on cumulant method and CornishFisher series. The model (18)(21) is solved by the classic interior point method. ILC method calculates the regulatory factors for the model (18)(21). Usually, the chance constraints can be satisfied in finite iterative steps.
 3.4 Optimization process
The optimization process is as follows:

1) Initialize: seta0= 0 ;

2) Solve the deterministic optimization model (18)(21) with forecast inputs based on (22) and (23);

3) Calculate the cumulants of ΔHmfrom (25) with outputs of step 2) and uncertain inputs;

4) Generate PDFs ofHmfrom (24) based on cumulants and CornishFisher series;

5) If the iterations reach the preset maximum number, go to step7); else, go to next step;

6) Calculatehβ,mfrom (26). If max(hβ,m) ≤ le− 4 , go to the end; else, updateaaccording to (28) and go to step 2). The number of iterations (m) increases with one in value.

7) Choose one of the selections: (i) go to the end, output the results of the maximum iteration and show “the optimal solution that satisfies the given probability constraints does not exist”; (ii) decreaseβand go to step 2) to find a suboptimal solution. The confidence levels (β) are determined by the reliability requirements of studied system or by compromised consideration between objective and reliability.
4. Case Study
The proposed optimization method is tested on a modified IEEE 14bus system with wind farms. POPF is also considered and compared with the SOPF in this paper. The POPF is solved by the steps 1) to 4) of the proposed optimization process. The solution of POPF equals to that of deterministic OPF.
The 14bus test system is developed based on the standard IEEE 14bus system and has two voltage levels: 110kV and 220kV. The connection and parameters of lines, voltage upper and lower bounds, and parameters of conventional generators are the same as those given in the standard IEEE 14bus system
[31]
. Bus 1 is considered as the slack bus. The rated power is 100 MVA. The mean values of load powers equal to their nominal values and the parameters of the wind farms are from the actual wind farm data. In Table A.1 (in Appendix), the forecast values of loads are given in p.u.. The Std. (%) value is the ratio between the Std. of load forecast error and
. The load of node 9 is discrete with distribution as shown in Table A.2. The forced outage rate of generator 1 is 0.09 and that of generator 2 is 0.08. Two wind farms are connected to bus 9 and bus 12 with forecasted power 0.45 and 0.30, respectively. The wind farm is modeled from a statistical point of view to analyze its impact on the dispatch of electrical network. The cumulants of forecast error of wind powers are shown in Table A.3. No. in Table A.3 is the number of the bus the wind farm connected to.
The chance constraints on (8) and (9) are considered. Therefore there are sixtyeight chance constraints. In order to explain the proposed optimization algorithm, the upper bound of apparent power in the branch between node 2 and node 3 is supposed to be 50MVA. When all elements of are set to 95%, there are six constraints which do not meet Eq. (13) using the proposed POPF. That is to say, the solution of deterministic OPF may be infeasible in stochastic environment. The six constraints can be represented as follows:
After iteration 10, the optimal solution can be achieved by the proposed optimization algorithm for SOPF. Elements of
a
corresponding the six constraints are 0.0681, 0.00462, 0.00705, 0.00683, 0.00808, and 0.0712, respectively (other elements of are zeros).
p^{n}
of the six constraints on each iteration regulated by the ILC method are shown in
Fig. 2
. It can be seen form
Fig. 2
that C
_{1}
and C
_{6}
are the key ones which affect the ending condition.
p^{n} of C_{1}C_{6} regulated by ILC
Optimal results obtained with POPF and SOPF are compared in
Tables 1

3
. According to the proposed optimization processes for POPF and SOPF, the results of POPF are the solution of model (18)(21) when
a
= 0 and that of SOPF are the solution of model (18)(21) with the final solution of
a
optimized by ILC.
Table 1
shows the regulation of active and reactive power of the generators by the SOPF compared with the POPF. The voltage magnitudes are shown in
Table 2
and power flow of branch 3 in
Table 3
. It can be concluded from
Tables 1

3
: (i) The voltage magnitudes obtained with SOPF have a certain margin to the limit. (ii) The system’s active loss is lowered using SOPF. (iii) Adjusted by the proposed method for SOPF, the apparent power of branch 3 are smaller than that of POPF. The final probabilities of C
_{1}
C
_{6}
solved by POPF and SOPF are shown in
Table 4
. It can be seen from
Table 4
that the probabilities of SOPF are equal to or greater than the set confidence level. That is to say, the chance constraints are satisfied by the regulation of ILC in proposed algorithm for SOPF.
Table 4
shows the probability of satisfying C
_{2}
is more than that of C
_{1}
. The reason is that the constraints C
_{1}
and C
_{2}
are correlated and they cannot reach the limits simultaneously due to the line loss on branch 3. With consideration of accuracy comparison, Monte Carlo Simulation (MCS) with 10000 samples is used as reference.
Fig. 3
shows the PDFs of C
_{1}
solved by POPF and SOPF using the proposed method and MCS, respectively. The PDFs of C
_{2}
C
_{6}
change in the same way as C
_{1}
.
Output power of generators
Output power of generators
Bus voltage
Power flow of branch 3
Probabilities of the two methods
Probabilities of the two methods
Probability density curves of C_{1}
Average Root Mean Square (ARMS) error is computed as
where,
PM_{i}
is the
i
th point’s value on the cumulative distribution curve calculated using the proposed method.
MC_{i}
is the
i
th point’s value on the cumulative distribution curve calculated using the Monte Carlo method;
N
represents the number of points and
N
= 30 here.
The relevant ARMS results of C
_{1}
C
_{6}
are shown in
Table 5.
It can be seen from
Table 5
that the proposed methods for POPF and SOPF can approximate Cumulative Distribution Function (CDF) of voltage magnitudes (C
_{3}
C
_{6}
) more accurately than that of line flows (C
_{1}
C
_{2}
).
ARMS of C_{1}C_{6}
Ignoring the stochastic factors, including load fluctuation, the failure rate of conventional generator and the forecast error of wind farms, the minimum cost of the deterministic calculation (or POPF) is $4063.250. When considering the stochastic factors mentioned above, the minimum cost of SOPF with
β
= 0.95 solved by the proposed method is $4189.763 which is higher than that of POPF.
Fig. 4
shows the objective function values of SOPF when all the predefined probability levels increase from 0.8 to 0.98. The cost increases as
β
becomes greater. Although the generation cost of the SOPF is greater than that of the POPF, the scheduling scheme of SOPF can better withstand the error and fluctuation of the forecast loads and wind power.
Objective values of SOPF with different confidence levels.
Although the sensitivity term
is difficult to be expressed in mathematical formula, it can be evaluated based on the experimental results. Take C
_{1}
and C
_{6}
as examples to analyze the responses of the objective value to different confidence levels of a single constraint. Set the confidence level of C
_{1}
or C
_{6}
as a specific value and other elements of
β
as 0.95, the results are shown in
Fig. 5
. The objective value increases as the confidence level of C
_{1}
or C
_{6}
varying from 0.80 to 0.95. The main increment of objective value occurs in [0.84 0.87] when the confidence level of C
_{6}
changes and in [0.90 0.95] when that of C
_{1}
changes. That is to say, sensitivities of objective value to constraints’ confidence levels have different sensitive areas and changing rules. The results of
Fig. 4
and
Fig. 5
are useful in determining suitable confidence levels by tradeoff between objective and reliability.
Objective values when a single confidence level varies.
All of the calculation is performed by a computer equipped with Intel(R) Core(TM) i52410M CPU@ 2.3GHz. The calculation time of POPF is 0.932 seconds and that of SOPF is 9.128 seconds. The IEEE118 case is also tested as the 14bus, the calculation time of POPF and SOPF are 2.125 seconds and 23.159 seconds, respectively.
5. Conclusion
Due to the complexity in solving the CCPbased SOPF model, an indirect method is proposed based on limit regulation and ILC. Taking into account the relationship between the constraint equations and the optimization variables, the direct calculation of optimal solution is replaced by narrowing the solution space with minimum regulation through adding regulatory factors to the inequality constraints. The regulatory factors are controlled by ILC method to ensure the optimality of the solution. The final results verify the effectiveness of the proposed algorithm and the related solution validation process shows that the proposed model can help increase the reliability of power system. The results are also conducive to further consider the economic and securityconstraint compromised dispatch and the risk management with the actual operational data.
Nomenclature
 Parameters:

C2,i,C1,i,C0,iCoefficients of production cost function

CGThe group of adjustable generators

Minimum active output of generatori

Maximum active output of generatori

URiRampup rate limit of generatori

DRiRampdown rate limit of generatori

USRtUpper limit on system spinning reserve

DSRtLower limit on system spinning reserve

Minimum reactive output of generatori

Maximum reactive output of generatori

Lower limit on voltage magnitude of busi

Upper limit on voltage magnitude of busi

Lower limit on apparent power of transmission line between busiand busj

Upper limit on apparent power of transmission line between bus i and bus j

kp,ki,kdPID parameters of ILC method
 Variables:

PGiActive outputs ofCG

PWiActive outputs ofith wind farm

QGiReactive outputs ofCG

QWiReactive outputs of ith wind farm

Sij,tApparent power of transmission line between bus i and bus j

Pw(t) Actual wind power output

Forecast wind power

NbNumber of boundarycrossing constraints at 1st iteration

Percentile ofHncorresponding toβn

Δ Stochastic part of variable

tTime period

mCounter of iterations

nIndexing number of the chance constraints

ummth input in the iterative process of ILC method
 Vectors:

XOptimization variable

θ= [θit] Angle vector of bus voltages

V=[Vit] Magnitude vector of bus voltages

aRegulatory factor

aminSubvector ofacorresponding to lower limits

amaxSubvector ofacorresponding to upper limits

hminLower limit vector of inequality constraints

hmaxUpper limit vector of inequality constraints

βPredefined confidence level

γPerturbation parameter

ZPositive slack variables when the inequality constraints are converted into equality

λLagrange multipliers of equality constraints

μLagrange multipliers of inequality constraints

eA vector of all ones

PdForecast active inputs

QdForecast reactive inputs

XmSolution of the deterministic OPF at m th iteration

ΔWStochastic part of inputs

ρA group of uncorrelated variables

rTracking error vector during the iterative process in ILC method
 Matrix:

JmJacobian matrix atmth iteration

TCorrelation matrix
Acknowledgements
The authors would like to acknowledge the financial supports from “National 863 Hitech Project (No. 2012AA050204)” funded by Chinese government.
BIO
Jinxia Gong She received the M.S. degree from Shanghai Jiaotong University, Shanghai, China in 2009. Now she pursues the Ph.D. degree in the Dept. of Electrical Engineering, Shanghai Jiao Tong University. She mainly focuses her research on renewable energy and power system analysis.
Da Xie He received the M.S. degree from Ha’erbin Institute of Technology in 1996 and Ph.D. degree from Shanghai Jiao Tong University in 1999. He is an associate professor with the Dept. of Electrical Engineering, Shanghai Jiao Tong University. He mainly focuses his research on FACTS and power system simulation.
Chuanwen Jiang He received the M.S. and Ph.D. degrees from Huazhong University of Science and Technology, Wuhan, China, in 1996 and 2000, respectively, and completed his postdoctoral research in the Dept. of Electrical Engineering, Shanghai Jiao Tong University, Shanghai, China, in 2002. He is a Professor with the Dept. of Electrical Engineering, Shanghai Jiao Tong University. He is currently researching reservoir dispatch, load forecast in power systems, and the electric power market.
Yanchi Zhang He received the B.S. from Beijing University, Beijing, China in 1989 and M.S. degrees from Dalian Institute of Technology, Dalian, China in 1997. His employment experience included Xinjiang University and Goldwind Science and Technology Company. Now he is a professor of Shanghai DianJi University. His special fields of interest include wind power and its control.
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