A Two-Stage Robust Reactive Power Optimization ... - IEEE Xplore

IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 7, NO. 1, JANUARY 2016

301

A Two-Stage Robust Reactive Power Optimization Considering Uncertain Wind Power Integration in Active Distribution Networks Tao Ding, Member, IEEE, Shiyu Liu, Student Member, IEEE, Wei Yuan, Member, IEEE, Zhaohong Bie, Senior Member, IEEE, and Bo Zeng, Member, IEEE

Abstract—Traditional reactive power optimization aims to minimize the total transmission losses by control reactive power compensators and transformer tap ratios, while guaranteeing the physical and operating constraints, such as voltage magnitudes and branch currents to be within their reasonable range. However, large amounts of renewable resources coming into power systems bring about great challenges to traditional planning and operation due to the stochastic nature. In most of the practical cases from China, the wind farms are centrally integrated into active distribution networks. By the use of conic relaxation based branch flow formulation, the reactive optimization problem in active distribution networks can be formulated as a mixed integer convex programming model that can be tractably dealt with. Furthermore, to address the uncertainties of wind power output, a two-stage robust optimization model is proposed to coordinate the discrete and continuous reactive power compensators and find a robust optimal solution that can hedge against any possible realization within the uncertain wind power output. Moreover, the second order cone programming-based column-and-constraint generation algorithm is employed to solve the proposed two-stage robust reactive power optimization model. Numerical results on 33-, 69- and 123-bus systems and comparison with the deterministic approach demonstrate the effectiveness of the proposed method. Index Terms—Active distribution network, column-andconstraint generation algorithm, reactive power optimization, two-stage robust optimization, wind power.

m rij , xij bs,j π(j) δ(j) Cjmax /Cjmin ΩD Sj Ujmax /Ujmin Ilmax min Qmax c,j /Qc,j

Variables Hij , Gij Uj Pj , Q j tij Cj yj Qc,j

N OMENCLATURE Parameters B E T Ω

Set of buses Set of branches Set of branches with transformers Set of buses for reactive power compensators

Manuscript received May 21, 2015; revised September 09, 2015; accepted October 10, 2015. Date of publication November 20, 2015; date of current version December 11, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 51577147 and in part by the Fundamental Research Funds for the Central Universities and the Independence research project of State Key Laboratory of Electrical Insulation and Power Equipment in Xi’an Jiaotong University (EIPE14106). Paper no. TSTE-00443-2015. T. Ding, S. Liu, and Z. Bie are with the State Key Laboratory of Electrical Insulation and Power Equipment, Department of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, China (e-mail: [email protected]). W. Yuan and B. Zeng are with the Department of Industrial and Management Systems Engineering, University of South Florida, Tampa, FL 33613 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSTE.2015.2494587

Cardinality of Ω Resistance/reactance of branch (i, j) Shunt susceptance from j to ground Set of all parents of bus j Set of all children of bus j Upper/lower bound of shunt capacitors/reactors capacity at bus j Set of buses for shunt capacitors/reactors Step size of shunt capacitors/reactors at bus j Upper/lower bound of voltage magnitude at bus j Current capacity limit of branch l(i, j) Upper/lower bound of reactive power compensation for continuous reactive power compensators at bus j Active/reactive power flow from bus i to j voltage magnitude of bus j Active/reactive power of bus j Tap ratio of the transformer branch (i, j) Value of shunt capacitors/reactors at bus j Optimal step of shunt capacitors/reactors at bus j Value of reactive power compensation for continuous reactive power compensators at bus j I. I NTRODUCTION

T

HREE hierarchical structures of automatic voltage control (AVC) has been widely used in traditional power grid, which can be divided into tertiary voltage control (TVC) to obtain the global optimal power flow, secondary voltage control (SVC) to provide the area of network a coordinated control strategy for voltage support, and primary voltage control (PVC) to guarantee the local voltage in the assigned security region. Moreover, the operational time frame has been cooperated with one another [1], [2]. Therein, reactive power optimization, which is employed in the TVC, aims to minimize the total transmission losses by controlling reactive power compensators and transformer tap ratios under a given load level, while guaranteeing the physical and operating constraints, such as voltage magnitudes and branch currents, to be within their reasonable range. It is conducted by the automatic voltage control (AVC) from the practical

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302

energy management system (EMS) in China, according to the performance of reactive power compensators [3]. Traditionally, the reactive power optimization could be formulated as a mixed-integer/nonlinear programming model and there were lots of techniques proposed to solve this model, which could be generally divided into two kinds of approaches [4]: one as the intelligent searches included simulated annealing [5], two-layer simulated annealing [6], evolutionary algorithms [7], [8], fuzzy clustering [9]–[11], genetic algorithms [12], tabu search [13], particle swarm optimization methods [14], [15], seeker optimization algorithm [16], etc.. The other as the conventional methods included gradient-based optimization algorithms [17], quadratic programming [18], successive linear programming [19], successive quadratic programming [20], Newton’s method [21], interior-point methods [22] and mixedinteger programming [23] as well as some decomposition methods [24]. It should be noted that continuous optimization algorithms were implemented by treating discrete variables as continuous ones and then rounded off to their nearest discrete values. This may only lead to inequality constraints violations when rounding off discrete variables with large step sizes such as for shunt capacitors/reactors. Nowadays, mixed integer programming approaches have been developed rapidly in research and study. Generally, mixed integer programming problems can be solved by branch-andbound, branch-and-cut or cutting plane methods. The first step of these methods is to relax the integer variables into continuous ones, and solve the relaxed model. Then, a search tree is dynamically generated by branching or adding cuts process. Finally, the bounding process shrinks the upper and lower bound of primary model according to the optimal solution of relaxed model at all leaf nodes until the optimal solution is achieved [25], [26]. Note that the global optimality of the relaxed model must be easily achieved. Usually, we hope the relaxed model is at least a convex problem, since there are great advantages to recognizing a problem as a convex optimization problem. One advantage is that the problem can be solved very reliably and efficiently. Another is related to theoretical or conceptual advantages of formulating a problem as a convex optimization problem, such that the global optimal solution can be guaranteed by Karush-Kuhn-Tucker (KKT) conditions [27], [28]. Recognizing the relaxed model as a convex problem, we call the original model as mixed integer convex programing. Nevertheless, the traditional reactive power optimization model is not a mixed integer convex programming due to the nonconvex power flow equations. That means, even after relaxing the integer variables, the model still takes on a nonconvex nature, whose global optimal solution is still difficult to achieve. Fortunately, the conic relaxation technique has been recently proposed and widely studied, which relaxed the nonconvex power flow equations by the use of second-order cones. Moreover, it was presented in [29], [30] that the conic relaxation usually had no gap or small gap to the original exact power flow equations in most distribution networks. Therefore, the reactive power optimization problem in distribution network can be relaxed to a mixed integer convex programming that can be tractably solved, comparing to the original nonconvex program. Meanwhile, large amounts of renewable resources coming into power systems brings about great challenges to traditional


Fig. 1. Centralized integration of wind farms in a Northern China.

planning and operation due to their stochastic nature. In most practical cases of China, the typically centralized integrated wind farms are widely used, as shown in Fig. 1. In each wind farm, lots of wind units are distributed and connected to the same 35 kV-level bus of the (35 kV/220 kV) voltage substation, called PCC (Point of Common Coupling) bus and the reactive power compensators are also installed to the PCC bus. Besides, the topology of 220 kV network is mostly radial structure in Northern China, termed as active distribution networks, where there are limited thermal generators and high penetration of wind power generation, and the reactive power compensators mainly including reactive-power outputs of wind units, SVG, SVC, shunt capacitors/reactors. At present, the shunt capacitors/reactors are still most commonly used in active distribution network due to their cheap costs, but they have a discrete nature. Even though wind units and SVG/SVC can be continuously adjusted and have better dynamic control performance, it has been investigated in the real-world Northern China that the capacity of SVC/SVG installed is limited at present due to its expensive costs. In practice, continuous and discrete reactive power compensators are both installed in the active distribution network. In addition, the transformer tap ratios also can be adjusted in the reactive power optimization to reduce the transmission losses. In fact, there are lots of realizations of wind power output due to the uncertain nature. In the practical reactive power dispatch, the system operator or the automatic voltage control (AVC) system can adjust the online or quickly available continuous reactive power compensators according to the real-time system operating status, whereas the discrete reactive power compensators are not required to be switched as frequently as the continuous ones. Due to the limitation of manufacturing techniques in China, there are physical limitations in the service life of circuit breakers and transformer tap changers. It is infeasible to operate these devices frequently [31]. Therefore, a two-stage robust optimization is employed in this work for reactive power optimization to coordinate the discrete and continuous reactive power compensators, while hedging against any possible realization within the uncertain

DING et al.: TWO-STAGE ROBUST REACTIVE POWER OPTIMIZATION

303

Fig. 2. A simple radial network with active power flow.

wind power output. Therein, the first stage decisions (i.e., discrete reactive power compensators) are served as the “hereand-now” decisions, i.e., they cannot be adjusted after the uncertainty is revealed, while the second stage decisions (i.e., continuous reactive power compensators) are represented as the “wait-and-see” decisions that can be adjusted after the first stage decisions are determined and the wind power uncertainty is revealed, which essentially provides the decision maker a recourse opportunity [32], [33]. The rest of the paper is organized as follows: Section II presents a general mixed integer second order cone based reactive power optimization model by the use of branch flow formulation for active distribution networks. In Section III, a two-stage robust reactive power optimization model is set up with the consideration of uncertain wind power integration into active distribution networks. Furthermore, the second order cone programming based column-and-constraint generation algorithm is presented to solve the proposed two-stage robust reactive power optimization model in Section IV. In Section V, numerical results on 33-, 69- and 123-bus systems and comparison with the deterministic approach demonstrate the effectiveness of the proposed method. Finally, conclusions are drawn in Section VI.

However, it should be noted that the traditional branch flow model doesn’t take into account the transformer tap ratios. In order to study the impact of transformer tap ratios on the branch flow model, equations (1) can be rewritten as ⎧ 2 Hij +G2ij ⎪ P H , ∀j ∈ B = H − − r ⎪ 2 j jk ij ij Ui ⎪ ⎪ k∈δ(j) i∈π(j) ⎪ ⎪ 2 ⎪ Hij +G2ij 2 ⎪ ⎪ G Q = G − − x U ⎪ j jk ij ij i ⎪ ⎪ k∈δ(j) i∈π(j) ⎪ ⎪ ⎨ 2 +bs,j Uj ∀j ∈ B

H 2 +G2 2 2 ⎪ ⎪ + x2ij ijU 2 ij , Uj = Ui2 − 2 (rij Hij + xij Gij ) + rij ⎪ ⎪ i ⎪ ⎪ ∀(i, j) ∈ E\T ⎪ ⎪ ⎪ U2

H 2 +G2 2 ⎪ ⎪ ⎪ t2j = Ui2 − 2 (rij Hij + xij Gij ) + rij + x2ij ijU 2 ij , ⎪ ⎪ ij i ⎩ ∀(i, j) ∈ T (2) where (i, j) ∈ E\T denotes (i, j) ∈ E, but (i, j) ∈ / T. Furthermore, the reactive power optimization problem with considering the reactive power compensation to minimize transmission losses can be exactly written as

min

Qc,1 ,...,Qc,m ,C1 ,...,Cm (i,j)∈E

s.t Pj =

Hjk −

k∈δ(j)

Qj =

i∈π(j)

Gjk −

k∈δ(j)

k∈π(j)

(1) where (i, j) is a branch whose “from” bus is i and “to” bus is j.

(3)

Gij − xij

i∈π(j)

2 Hij

+ Ui2

G2ij

, ∀j ∈ B (4)

+ bs,j Uj2 ,

∀j ∈ /Ω 1 2 Qj + Uj Cj + Qc,j 2

2 Hij + G2ij Gij − xij = Gjk − Ui2 k∈δ(j)

It has been well recognized that a distribution network is often a radial network that can be formed by a set of recursive equations, which was proposed in [29], [30], called branch flow formulation. However, with wind power integration, the power flow becomes bidirectional, which is called active distribution network. Generally, for a radial network with n + 1 buses and n branches, we can set up a directional topology, where the positive direction of branch power flow is from “from” bus to “to” bus, shown in Fig. 2. Furthermore, the branch flow formulation can be formulated as ⎧ Uk (Gjk cos θjk + Bjk sin θjk ) ⎪ ⎨Pj = Uj ∀j ∈ B k∈π(j) , ∀(j, k) ∈ E ⎪ Uk (Gjk sin θjk − Bjk cos θjk ) ⎩Qj = Uj

2 Hij +G2ij Ui2

2 Hij + G2ij Hij − rij Ui2

(5)

i∈π(j)

+bs,j Uj2 , ∀j ∈ Ω

II. M IXED I NTEGER S ECOND O RDER C ONE BASED R EACTIVE P OWER O PTIMIZATION F ORMULATION

rij

(6) 2 2

Hij + Gij 2 + x2ij , Uj2 = Ui2 − 2 (rij Hij + xij Gij ) + rij Ui2 ∀(i, j) ∈ E/T (7) 2 2 2

Hij + Gij 2 Uj = Ui2 − 2 (rij Hij + xij Gij ) + rij + x2ij , 2 tij Ui2 ∀(i, j) ∈ T (8)

Ujmin ≤ Uj ≤ Ujmax ,

∀j ∈ B

(9)

2 + G2 Hij ij max ≤ Iij , Ui2

∀(i, j) ∈ E

(10)

max Qmin c,j ≤ Qc,j ≤ Qc,j ,

∀j ∈ Ω\ΩD

(11)

Cj = yj s j ,

∀j ∈ Ω ∩ ΩD

Cjmin ≤ Cj ≤ Cjmax ,

∀j ∈ Ω ∩ ΩD

(12) (13)

304


It can be easily seen that the model (3)–(13) is a mixed integer nonlinear nonconvex programming (MINNP) problem. That means, even after relaxing the integer variables, the model still takes on a nonlinear nature, whose global optimal solution is still difficult to achieve. It is fortunate that the conic relaxation technique has been proposed and widely studied to relax the nonconvex power flow equations (2) by the use of secondorder cones. Thus, the model (3)–(13) can be transformed into a mixed integer convex programming that can be tractably solved by some commercial solvers, comparing to the original mixed integer nonconvex programming. At first, we give a transformation, such that 2 Hij + G2ij = lij Ui2 ∀(i, j) ∈ E (14) Uj2 = uj ∀j ∈ B The constraints (4)–(8) by the use of (14) yields ⎧ Pj = Hjk − (Hij − rij lij ), ∀j ∈ B ⎪ ⎪ ⎪ ⎪ k∈δ(j) i∈π(j) ⎪ ⎪ ⎪ ⎪ Qj = (Gij − xij lij ) − Gjk + bs,j uj , ∀j ∈ /Ω ⎪ ⎪ ⎪ i∈π(j) k∈δ(j) ⎪ ⎪ ⎪ ⎨Qj + 12 uj Cj + Qc,j = Gjk − (Gij − xij lij ) k∈δ(j)

where xj,0 , xj,1 , . . . , xj,vj are 0–1 dummy variables for bus j where the reactive power compensators are installed; and each integer vj can be determined by

Cjmax − Cjmin log2 + 1 − 1 ≤ vj sj

Cjmax − Cjmin ≤ log2 +1 sj

(20)

(iii) As for the bilinear terms uj /t2ij , transformer tap ratio tij is a discrete variable and we assume the possible value of tij to be tij ∈ tij,1 , . . . ., tij,nij , where nij is the number of tap ratios for transformer branch ij. Furthermore, uj /t2ij can be expressed as (21) by introducing 0-1 binary variables rij,1 , rij,2 , . . . , rij,nij . uj /t2ij = uj

1

t2ij,1

rij,1 +

1 t2ij,2

rij,2 + · · · +

1 t2ij,nij

rij,nij (21)

i∈π(j)

⎪ ⎪ + bs,j uj , ∀j ∈ /Ω ⎪ 2 ⎪ ⎪ ⎪ = u − 2 (r H + xij Gij ) + rij + x2ij lij , ∀(i, j) ∈ E\T u ⎪ j i ij ij ⎪ ⎪ u 2 j ⎪ ⎪ = ui − 2 (rij Hij + xij Gij ) + rij + x2ij lij , ∀(i, j) ∈ T ⎪ t2 ⎪ ij ⎪ 2 ⎩ Hij + G2ij = lij ui , ∀(i, j) ∈ E

(15) It can be observed that the first two and fourth equations in (15) are all affine equalities, but the third and fifth equations contain bilinear terms uj Cj and uj /t2ij . Besides, the last equations are nonconvex quadratic equalities. We can simplify them as follows: (i) For the last equations, conic relaxation is performed by relaxing the quadratic equalities into inequalities. Thus, it yields: 2 + G2ij ≤ lij ui , Hij

∀(i, j) ∈ E

n ij k=1

rij,k = 1.

Furthermore, the bilinear terms in (21) can be exactly linearized by the use of big M approach, such that ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

uj /t2ij =

n ij k=1

hj,k /t2ij,k

−M (1 − rij,1 ) + uj ≤ hj,k ≤ uj + M (1 − rij,1 ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −M rij,1 ≤ hj,k ≤ M rij,1

rij,1 , . . . , rij,nij ∈ {0, 1}

n ij k=1

rij,k = 1 (22)

(16)

Note that, (16) is equivalent to (17), and therefore (16) can be reformulated as a standard second order cone formulation (18). 2

with rij,1 , rij,2 , . . . , rij,nij ∈ {0, 1} and

2

(2Hij )2 + (2Gij )2 + (lij − ui ) ≤ (lij + ui ) , ∀(i, j) ∈ E (17) 2Hij 2Gij ≤ lij + ui , ∀(i, j) ∈ E (18) lij − ui 2 (ii) In addition, the integer variables in (12) can be equivalently expressed as a combination of 0–1 binary variables, such that ⎧

Cj = Cjmin + sj 20 xj,0 + 21 xj,1 + · · · + 2vj xj,vj ⎪ ⎪ ⎪ ⎨ C max −C min 0 ≤ 20 xj,0 + 21 xj,1 + · · · + 2vj xj,vj ≤ j sj j , ⎪ ∀j ∈ Ω ∩ ΩD ⎪ ⎪ ⎩ xj,0 , xj,1 , . . . , xj,vj ∈ {0, 1} (19)

where M is a large number and hj,1 , . . . , hj,nij are dummy variables. (iv) The bilinear terms uj Cj can be formulated as uj Cj = Cjmin + sj (20 xj,0 uj + 21 xj,1 uj + · · · + 2vj xj,vj uj )

(23)

Similarly, each bilinear terms xj,k uj in (22) can be transformed into linear expressions by the use of big M approach. By introducing dummy variables σj,1 , . . . , σj,vj , it attains ⎧

⎨ uj Cj = Cjmin + sj 20 σj,0 + 21 σj,1 + · · · + 2vj σj,vj u − M (1 − xj,k ) ≤ σj,k ≤ uj + M (1 − xj,k ) ⎩ j −M xj,k ≤ σj,k ≤ M xj,k (24) After the conic relaxation (i) and the transformation (ii)–(iv), the original reactive power optimization model (3)–


305

(13) is relaxed into a 0–1 mixed integer second order cone programming as follows:

min

Qc,1 ,...,Qc,m ,x,r (i,j)∈E

s.t Pj =

(rij lij )

(Hij − rij lij ) −

i∈π(j)

Qj =

(25)

Hjk ,

∀j ∈ B (26)

k∈δ(j)

(Gij − xij lij ) −

i∈π(j)

Gjk + bs,j uj ,

∀j ∈ /Ω

k∈δ(j)

(27) 0

1 min C + sj 2 σj,0 + 21 σj,1 + · · · + 2vj σj,vj Qj + 2 j Gjk − (Gij − xij lij ) + bs,j uj , + Qc,j = k∈δ(j)

III. T WO -S TAGE ROBUST R EACTIVE P OWER O PTIMIZATION C ONSIDERING U NCERTAIN W IND P OWER I NTEGRATION Traditionally, reactive power optimization is only conducted under one deterministic snapshot. To address the uncertainties of wind power output, a two-stage robust optimization model is set up to coordinate the discrete and continuous reactive power compensators, where discrete reactive power compensators are served as the “here-and-now” decisions, i.e., they cannot be adjusted after the uncertainty is revealed, while the continuous reactive power compensators are represented as the “wait-andsee” decisions that can be adjusted to actual wind power output. The robust optimal solution can hedge against any possible realization within the uncertain wind power output. Furthermore, the two-stage robust reactive power optimization model can be formulated as

i∈π(j)

∀j ∈ Ω (28)

2 uj = ui − 2 (rij Hij + xij Gij ) + rij + x2ij lij , ∀(i, j) ∈ E\T (29) nij

2 hj,k /t2ij,k = ui − 2 (rij Hij + xij Gij ) + rij + x2ij lij ,

s.t Pj =

∀(i, j) ∈ T

(30) Qj = (31)

−M rij,1 ≤ hj,k ≤ M rij,1 , ∀(i, j) ∈ T rij,1 , . . . , rij,nij ∈ {0, 1} ,

nij

(32)

rij,k = 1, ∀(i, j) ∈ T

(33)

min

(Gij − xij lij ) −

i∈π(j)

Ujmin

2

2 ≤ uj ≤ Ujmax , ∀j ∈ B

max 2 lij ≤ Iij , ∀(i, j) ∈ E

k∈δ(j)

max Qmin c,j ≤ Qc,j ≤ Qc,j , ∀j ∈ Ω\ΩD uj − M (1 − xj,k ) ≤ σj,k ≤ uj + M (1 − xj,k ) , −M xj,k ≤ σj,k ≤ M xj,k

∀j ∈ Ω ∩ ΩD k = 1, . . . , vj

(38)

0 ≤ 20 xj,0 + 21 xj,1 + · · · + 2vj xj,vj ≤ ∀j ∈ Ω ∩ ΩD xj,0 , xj,1 , . . . , xj,vj ∈ {0, 1} ,

(37)

i∈π(j)

2 rij

+

x2ij

(44) lij , (45)

2 hj,k /t2ij,k = ui − 2 (rij Hij + xij Gij ) + rij + x2ij lij ,

k=1

∀(i, j) ∈ T −M (1 − rij,1 ) + uj ≤ hj,k ≤ uj + M (1 − rij,1 ) , ∀(i, j) ∈ T

(46) (47)

−M rij,1 ≤ hj,k ≤ M rij,1 , ∀(i, j) ∈ T

(48)

2Hij 2Gij ≤ lij + ui , ∀(i, j) ∈ E lij − ui

(49)

2

Cjmax − Cjmin , sj (39)

∀j ∈ Ω ∩ ΩD

Gjk + bs,j uj , ∀j ∈ /Ω

∀(i, j) ∈ E\T nij

(36)

(42)

(43)

1 C min + sj 20 σj,0 + 21 σj,1 + · · · + 2vj σj,vj Qj + 2 j + Qc,j = Gjk − (Gij − xij lij ) + bs,j uj , ∀j ∈ Ω

(34)

(35)

Hjk , ∀j ∈ B

k∈δ(j)

uj = ui − 2 (rij Hij + xij Gij ) +

2

(41)

k∈δ(j)

k=1

2Hij 2Gij ≤ lij + ui , ∀(i, j) ∈ E lij − ui

(rij lij )

(Hij − rij lij ) −

i∈π(j)

k=1

∀(i, j) ∈ T − M (1 − rij,1 ) + uj ≤ hj,k ≤ uj + M (1 − rij,1 ),

max

min

x∈X,r∈R Pj ∈Δ Qc,1 ,...,Qc,m (i,j)∈E

(40)

Ujmin

2

2 ≤ uj ≤ Ujmax , ∀j ∈ B

max 2 lij ≤ Iij , ∀(i, j) ∈ E

(50)

(51)

306


max Qmin c,j ≤ Qc,j ≤ Qc,j , ∀j ∈ Ω\ΩD

(52)

∀j ∈ Ω ∩ ΩD uj − M (1 − xj,k ) ≤ σj,k ≤ uj , +M (1 − xj,k ) − M xj,k ≤ σj,k ≤ M xj,k k = 1, . . . , vj (53) with Δ=

Y=

max

min aT y

y|Cy ≤ f , Qi y + qi 2 ≤ Dy = g − Gz, Ey = u

(57)

umin ≤u≤umax

n

(Qwi + λi qi ) = a

i=1

π1 ≤ 0 min

(67) (68)

≤u≤u

max

(69)

where π1 , π2 and π3 are dual variables for constraints (61), (62) and (63); (λi , wi ) is the conic dual variables for i-th second order cone constraints (64). It can be easily seen from (65) that the bilinear terms of the objective function uT π3 are not convex, but fortunately, these bilinear terms are separable, i.e., the feasible regions of u and π3 are disjoint, which gives the fact that the optimality of model (65)–(69) must be achieved at one extreme point of the uncertainty set, i.e., the lower or upper bound of u. As a result, bilinear terms can be reformulated as uT π3 =

s

us π3,s =

s

max min min us π3,s u s δs + π3,s − π3,s (70)

where δs is a binary dummy variable. If δs = 1, us = umin s and if δs = 0, us = umax s . Furthermore, the bilinear term us δs can be linearized by the use of the big-M approach, such that ⎧ T

max min min us π3,s rs us π3,s = + π3,s − π3,s ⎪ ⎨ u π3 = s s , ∀s −M δs ≤ rs ≤ M δs ⎪ ⎩ −M (1 − δs ) + us ≤ rs ≤ us + M (1 − δs ) (71) where rs is a continuous dummy variable and M is a large number. As a result, the subproblem (72)-(78) can be reformulated as a mixed integer second order cone programming.

(59)

n

T λi di +wiT ci f T π1 +(g−Gz ∗ ) π2 − i=1 (SP) π ,πmax

max min min 1 2 ,π3 ,u, us π3,s rs + π3,s − π3,s λ1 ,...,λn ,w1, ...,wn +

min aT y

s

(72)

(60)

y

s.t. Cy ≤ f

(61)

Dy = g − Gz ∗ Ey = u Qi y + qi 2 ≤ cTi y + di ,

(65)

(58) + di , i = 1, . . . , n

For a given first-stage decision variables z ∗ , we can obtain the following subproblem: max

s.t. C T π1 + D T π2 + E T π3 +

u

s.t. Az ≥ b, z ∈ {0, 1} cTi y

wi 2 ≤ λi , i = 1, . . . , n

The column-and-constraint generation (CCG) algorithm is essentially a decomposition algorithm designed for two-stage robust optimization, which is implemented as a master problem (MP) and sub-problem (SP) scheme similar to Benders’ decomposition method. But the convergence performance of this method outperforms Bender’s algorithm [32], [33]. The MP dynamically generates constraints with recourse decision variables for an identified scenario in the uncertainty set in each iteration. The SP seeks the worst case scenario for a given first stage decisions. For simplicity, the two-stage robust reactive power optimization model (41)–(56) can be expressed in compact matrix format as umin ≤u≤umax y∈Y

λi di + wiT ci

i=1

(56)

IV. C OLUMN - AND -C ONSTRAINT G ENERATION A LGORITHM

z

n

f T π1 + (g − Gz ∗ ) π2 + uT π3

(66)

where Δ is the uncertainty set that denotes the uncertain active power injection which combines wind power and load demand together; Pj and Pj are the lower and upper bound of injected active power with the consideration of uncertain wind power; X and R are the feasible region of discrete control variables.

min

−

(54) (55)

T

max

π1 ,π2 ,π3 ,u, λ1 ,...,λn ,w1, ...,wn

Pj | Pj ≤ Pj ≤ Pj , j ∈ B ⎧ ⎫ 0 1 ⎪ ⎪ j,1 + · · · ⎨0 ≤ 2 xj,0 + 2 xmax ⎬ min Cc,j −Cc,j tj , ∀j ∈ Ω ∩ Ω X= +2 xj,tj ≤ D sj ⎪ ⎪ ⎩ ⎭ xj,0 , xj,1 , . . . , xj,tj ∈ {0, 1} ⎫ ⎧ nij ⎬ ⎨ r =1 , ∀(i, j) ∈ T R = k=1 ij,k ⎭ ⎩ rij,1 , . . . , rij,nij ∈ {0, 1}

In order to solve the above “max-min” bi-level program, we can take dual for the inner “min” linear program model according to strong duality, which yields a single level “max” model, such that

(62) (63) i = 1, . . . , n

(64)

s.t. C T π1 + D T π2 + E T π3 +

n

(Qwi + λi qi ) = a

i=1

(73) wi 2 ≤ λi , i = 1, . . . , n −M δs ≤ rs ≤ M δs , ∀s

(74) (75)


−M (1 − δs ) + us ≤ rs ≤ us + M (1 − δs ) , ∀s π1 ≤ 0 u

min

≤u≤u

307

(76) (77)

max

(78)

For the robust reactive power optimization problem, the MP includes a subset of discrete reactive power compensators that can be deployed to achieve an optimal objective value. Furthermore, the whole procedure of the column-and-constraint generation method for the robust reactive power optimization can be described as the following five steps with a given convergence error ε: Step 1: Set LB = −∞, U B = +∞, k = 0; Step 2: Solve the following master problem by a mixed integer second order cone solver: (MP) min η s.t.Az ≥ b, z ∈ {0, 1}

(80)

l

η ≥ a y , ∀l ≤ k (81) l T l Qi y + qi ≤ ci y + di , i = 1, . . . , n, ∀l ≤ k (82) 2 l

Cy ≤ f , ∀l ≤ k l

Dy = g − Gz, ∀l ≤ k l

∗

Ey = u , ∀l ≤ k

(83) (84) (85)

Derive an optimal solution (z ∗ , η ∗ , y l∗ ) with l = 1, 2, .., k and update the lower bound as LB = η ∗ ; Step 3: Fix y ∗ and solve the sub-problem (72)-(78). If the sub-problem is feasible, derive an optimal solution (u∗ , π1∗ , π2∗ , π3∗ , λ∗1 , . . . , λ∗n , w1∗ , . . . , wn∗ ) as well as the optimal objective value ϑ(z * ); otherwise set ϑ(z ∗ ) = +∞. Furthermore, update the upper bound as U B = min{U B, ϑ(y * )}; Step 4: If (U B − LB) ≤ ε, return y * and stop. Otherwise, fix u* and add the cuts as (a) If the sub-problem in step 3 is feasible, create variables y l+1 and add the following constraints to MP η ≥ aT y l Qi y l + qi ≤ cTi y l + di , i = 1, . . . , n 2 Cy l ≤ f

Dy = g − Gz l

Ey = u

(86) (87) (88)

l

∗

(89) (90)

(b) If the sub-problem in step 3 is infeasible, create variables y l+1 and add the following constraints to MP Qi y l + qi ≤ cTi y l + di , i = 1, . . . , n (91) 2 Cy l ≤ f

(92)

l

Dy = g − Gz l

It was proven in [32] that given a finite binary set, the column-and-constraint generation method could converge in finite iterations to obtain a global optimal solution. In practice, the algorithm usually derived an optimal solution after a small number of iterations.

(79)

z,η,y l

T

Fig. 3. A 33-bus radial network topology.

Ey = u

∗

Step 5: Update k = k + 1 and go to Step 2.

(93) (94)

V. N UMERICAL A NALYSIS A. Configuration of Three Test systems Three test systems, including 33-bus, 69-bus and 123busdistribution network available from [34], were analyzed in this section by applications of the proposed methodology with uncertainties on wind power generation. The computational tasks were performed on a 2.0 GHz personal computer with 4 GB RAM, and the proposed method was programmed in MATLAB where the mixed integer conic programming were solved using CPLEX 12.5. A 33-bus radial system is presented in Fig. 3, where bus 1 represents the substation bus whose voltage magnitude is assumed to be constant and there are 4 wind farms connected at buses 13, 21, 24 and 31 respectively. The step of tap ratio of the transformer (TR) in the substation is 0.01 and the range is [0.94, 1.06]. There are two switchable capacitors/reactors (SCRs) connected to bus 3 and bus 6 whose capacity are both [−0.06, +0.06] MVar. Besides, each wind farm installs SVG/SVC that can be continuously adjusted, with each having the capacity of 0.01 MVar. The uncertainty interval of each wind power output is given by [1 − α, 1 + α] × P f , where P f is the forecasted wind power output and α reflects the forecasted wind power confidence interval. Moreover, we consider three scenarios for comparison: (i) Low penetration of wind power: forecasted wind power are 0.1 MW, 0.3 MW, 0.3 MW and 0.3 MW respectively and the step sizes of the two SCRs are 0.03 MVar and 0.02 MVar; (ii) High penetration of wind power with large step size of SCRs: forecasted wind power are 1 MW, 3 MW, 3 MW and 3 MW respectively and the step sizes of the two SCRs are 0.03 MVar and 0.02 MVar; (iii) High penetration of wind power with small step size of SCRs: forecasted wind power are 1 MW, 3 MW, 3 MW and 3 MW respectively and the step size of both SCRs is 0.01 MVar. For the 69-bus system, as shown in Fig. 4, there are 5 wind farms with each having 3 MW forecasted wind power output and the step of tap ratio of the transformer (TR) in the substation is 0.01 and the range is [0.94, 1.06]. There are three switchable

308


TABLE I C OMPARISON OF D ISCRETE R EACTIVE P OWER C OMPENSATORS B ETWEEN D ETERMINISTIC AND ROBUST M ODEL U NDER THE F IRST S CENARIO


TABLE II C OMPARISON OF T RANSMISSION L OSSES B ETWEEN D ETERMINISTIC AND ROBUST M ODEL U NDER THE F IRST S CENARIO


capacitors/reactors (SCRs) connected to bus 3, 8 and 11, where the capacity SCRs are both [−0.1, +0.1] MVar and the step size is 0.02 MVar. Besides, each wind installs SVG/SVCs that are continuously adjustable, each having the capacity of 0.01 MVar. For the 123-bus system, as shown in Fig. 5, there are 10 wind farms with each having 5 MW forecasted wind power output and there are five switchable capacitors/reactors (SCRs) connected to bus 12, 35, 54, 76, and 108.The information of TR, SCRs and SVG/SVC for each wind farm are the same as those in the 69-bus systems.

B. Comparison of Solution Property Between Two-Stage Robust Optimization and Deterministic Approach Traditionally, the reactive power optimization was conducted using a deterministic approach by (25)–(40). To evaluate the performance of the proposed two-stage robust reactive power optimization (RRPO) approach and the traditional deterministic approach, we randomly generate 10000 scenarios via Monte Carlo simulation (MCS) to find the worst-case scenario (WC) by fixing the discrete variables computed from determin, where istic approach. Furthermore, let η = O(RRPO)−O(WC) O(RRPO) O(RRPO) and O(WC) are the transmission losses from robust model and the worst case of the deterministic model. Firstly, the comparisons of robust and the traditional deterministic approach on 33-bus system under three scenarios are shown in Tables I–VI. The results show that the transmission losses for higher penetration of wind power (second and third scenarios) are larger and increase significantly with the increase of α and the power flow reverses from the active distribution network to grid code, whereas the optimal transmission losses from robust and worst case of deterministic approach

TABLE III C OMPARISON OF D ISCRETE R EACTIVE P OWER C OMPENSATORS B ETWEEN D ETERMINISTIC AND ROBUST M ODEL U NDER THE S ECOND S CENARIO

TABLE IV C OMPARISON OF T RANSMISSION L OSSES B ETWEEN D ETERMINISTIC AND ROBUST M ODEL U NDER THE S ECOND S CENARIO

are the same for low wind power penetration scenario (first scenario). Most importantly, the deterministic approach may be confronted with much larger transmission losses under the worst-case scenario with large α. Secondly, the solution from the deterministic model is not robust, since the delivery of power is not possible without violating the physical and operational constraints under some


TABLE V C OMPARISON OF D ISCRETE R EACTIVE P OWER C OMPENSATORS B ETWEEN D ETERMINISTIC AND ROBUST M ODEL U NDER THE T HIRD S CENARIO

309

TABLE VII C OMPARISON OF T RANSMISSION L OSSES B ETWEEN D ETERMINISTIC AND ROBUST M ODEL ON 69- AND 123-B US S YSTEMS

TABLE VIII M AXIMUM G AP OF C ONIC R ELAXATION ON THE T HREE T EST S YSTEMS TABLE VI C OMPARISON OF T RANSMISSION L OSSES B ETWEEN D ETERMINISTIC AND ROBUST M ODEL U NDER THE T HIRD S CENARIO

TABLE IX C OMPARISON OF C OMPUTATIONAL P ERFORMANCE B ETWEEN D ETERMINISTIC AND ROBUST A PPROACH ON T HREE T EST S YSTEMS


scenarios within the uncertainty set. In contrast, the robust model can find the optimal solution that is robustly feasible. Furthermore, comparison between Tables III and V shows that small size of SCRs can reduce transmission losses as well as the gap between robust model and worst case of deterministic model (i.e., η). Moreover, Fig. 6 depicts that for low penetration of wind power, SCRs needs to switch capacitors to improve voltage profile, whereas for high penetration of wind power, SCRs needs to switch reactors to mitigate voltage spike. In addition, the comparison of transmission losses between deterministic and robust model for 69- and 123-bus systems is shown in Table VII, where the similar results can be observed as that for 33-bus system: the deterministic approach obtains

larger transmission losses under the worst-case scenario than that from the robust approach. Moreover, the solution from the deterministic model is not robust, since the delivery of power is not possible without violating the physical and operational constraints under some scenarios within the uncertainty set. In contrast, the robust model can find the optimal solution that is robustly feasible. Finally, Table VIII depicts the maximum gap of conic relaxation on the three test systems, where it is obvious that the gap on any test system is smaller than 10−6 , which implies that the conic relaxation is actually exact to the original nonconvex model in most cases of the distribution network. C. Comparison of Computational Performance Between TwoStage Robust Optimization and Deterministic Approach The comparison of computational performance between robust and deterministic approach on the three radial systems is shown in Table IX, which summarizes the number of iterations and the total time to solve the two models. Although the number of iterations for two-stage robust optimization model is small (i.e., less than 10), which suggests that

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the column-and-constraint algorithm can rapidly identify the worst-case scenarios, the total time for robust reactive optimization is longer than the deterministic method. It is found that the main computational cost is spent on solving the sub-problem (72)–(78), which aims to find the global optimality of the inner “max-min” model during the step 3. It should be noted that the robust optimization approach needs longer time than the traditional deterministic approach, but can achieve better and more robustly feasible solution. Finally, take the 69-bus system with α = 0.4 for illustration. The CCG algorithm via step 1–5 only needs 5 iterations to converge (see Fig. 7), where the upper bound is increasing and the lower bound is decreasing, and the algorithm stops until the gap between the upper and lower bound is smaller than the given value. VI. C ONCLUSION This work proposes a two-stage robust reactive power optimization model to address high penetration of uncertain wind power integrated into active distribution networks, which aims to coordinate the discrete and continuous reactive power compensators and find a robust optimal solution that can hedge against any possible realization within the uncertain wind power output. Then, conic relaxation based branch flow formulation is employed to set up a mixed integer convex programming model and the second order cone programming based column-and-constraint generation algorithm is utilized to solve the proposed two-stage robust reactive power optimization model. The comparison with the deterministic approach on several test systems shows the effectiveness of the proposed method. Specifically, the robust optimization approach can achieve better and more robustly feasible solution than the traditional deterministic approach, although it needs longer time. R EFERENCES [1] S. Corsi et al., “Coordination between the reactive power scheduling function and the hierarchical voltage control of the EHV ENEL system,” IEEE Trans. Power Syst., vol. 10, no. 2, pp. 686–694, May 1995.

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Tao Ding (S’13–M’15) received the degree in mathematics from Southeast University, Nanjing, China, the B.S.E.E. and M.S.E.E. degrees from Southeast University, Nanjing, China, and the Ph.D. degree from Tsinghua University, Beijing, China, in 2007, 2009, 2012 and 2015, respectively. From 2013 to 2014, he was a Visiting Scholar at the Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, TN, USA. He is currently an Associate Professor with the State Key Laboratory of Electrical Insulation and Power Equipment and the School of Electrical Engineering, Xi’an Jiaotong University, Xi’an, China. His research interests include electricity markets, power system economics and optimization methods, and power system planning and reliability evaluation. He was the recipient of the Outstanding Graduate Award of Beijing City

311

Shiyu Liu (S’15) received the B.S.E.E. degree from Hohai University, Nanjing, China, in 2014. She is currently pursuing the Master’s degree at Xi’an Jiaotong University, Xi’an, China. Her research interests include reactive power optimization and renewable energy integreation.

Wei Yuan (S’13–M’15) received the B.S. degree in electrical and computer engineering from Xi’an Jiaotong University, Xi’an, China, the Ph.D. degree in industrial engineering from the University of South Florida, Tampa, FL, USA, in 2010 and 2015, respectively. He was a Research Aide Intern with the Energy System Division, Argonne National Laboratory, Lemont, IL, USA, in 2013. His research interests include robust optimization and algorithm development, optimization methods with applications in various engineering systems including the energy systems.

Zhaohong Bie (M’98–SM’12) received the B.S. and M.S. degrees in electric power from Shandong University, Jinan, China, and the Ph.D. degree from Xi’an Jiaotong University, Xi’an, China, in 1992, 1994, and 1998, respectively. Currently, she is a Professor with the State Key Laboratory of Electrical Insulation and Power Equipment and the School of Electrical Engineering, Xi’an Jiaotong University. Her research interests include power system planning and reliability evaluation, as well as the integration of the renewable energy.

Bo Zeng (M’11) received the Ph.D. degree in industrial engineering from Purdue University, West Lafayette, IN, USA. He currently is an Assistant Professor with the Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, PA, USA. His research interests include polyhedral study and computational algorithms for stochastic and robust mixed integer programs, coupled with applications in energy, logistics, and healthcare systems. He is a member of IIE and INFORMS.