Economic Dispatch with Heavy Loading andMaximum Loading Identification Using Convex

Relaxation of AC OPFAbdullah Alassaf

Electrical Engineering DepartmentUniversity of South Florida, USA

University of Hail, KSAEmail: Alassaf@mail.usf.edu

Lingling FanElectrical Engineering DepartmentUniversity of South Florida, USA

Email: Linglingfan@usf.edu

Abstract— The objective of this paper is to explore twoengineering applications of convex relaxation of AC OPF: (i) toconduct economic dispatch at heavy loading conditions; and(ii) to identify the system’s loading limit. The conventionalAC OPF formulation solved by interior point method hasdifficulty to converge when the system is reaching its loadinglimit due to the singularity of the Jacobian matrix. Existingmethods to identify loading limits include continuous powerflow (CPF) method. For CPF, the direction of power transferhas to be identified first. Then, a PV nose is obtained andthe maximum loading level is at the nose point. Using convexrelaxation of alternating current optimal power flow (AC OPF),we will not encounter singularity issues. In this paper, wewill formulate a semi-definite programming (SDP) relaxationof AC OPF-based optimization problems to conduct economicdispatch and loading limit identification. Advantages of SDPOPF over MATPOWER are demonstrated. Further, loadinglimits identified by SDP OPF are compared with those foundusing the CPF method of MATPOWER. Our research indicatesthat two methods will lead to the same solutions for the testedsystems.

I. INTRODUCTION

This paper exploits an engineering application of convexrelation of AC OPF. The particular relaxation used in thispaper is the SDP relaxation [1], [2]. Conventional AC OPFis a nonconvex program and is usually solved by the interior-point method. MATPOWER [3] is one such software packagefor AC OPF using interior-point method. The conventionalpower flow equations are listed as the equality constraints inAC OPF. When evoking interior-point method-based solvers,those power flow equations have to be solved.

For power flow equations, Newton-Raphson method has aconvergence issue when the system loading reaches its limit.Similarly, when the system is reaching its limit, AC OPF alsohas a convergence issue and the interior-point method-basedsolver does not give a solution.

An advantage of SDP relaxation is that the power flowequations are no longer in terms of voltage magnitudeand angles. Instead, power flow equations are linear to theelements of a positive semi-definite (PSD) matrix. This newexpression makes the convergence issue disappear.

Thus, we can employ SDP relaxation of AC OPF to iden-tify loading limit or to find economic dispatch of generatorswhen the system is close to its loading limit. In addition,when the system is over its loading limit, SDP relaxationwill indicate that there is no feasible solution.

Using SDP relaxation of AC OPF and second order coneprogramming relaxation of AC OPF to identify loading limithas been addressed in [4], [5].

In this paper, we investigate the two optimization problemformulation based on SDP OPF to investigate both economicdispatch under heavy loading and loading limit identification.Comparison with MATPOWER will be given to demonstratethe advantages of SDP OPF.

This paper is organized as follows. In Section II, we giveAC OPF formulation in the conventional and SDP relaxationforms. In Section III, we show the advantage of semi-definiteprogramming using CVX over nonlinear programming usingMATPOWER in solving optimal power flow problems. InSection IV, the maximum loading level identification iscarried by SDP OPF and MATPOWER’s runcpf function.Conclusion of this paper in found in Section V.

II. AC OPF

The objective functions and the constraints of powersystem depend on the applications that need to be optimized.The essential active power objectives and constraints of ageneral OPF problem are given in Table I [6].

TABLE IESSENTIAL ACTIVE POWER OPF OBJECTIVES AND CONSTRAINTS [6]

Objectives Constraints• Economic dispatch • Generator output in MW(minimum cost losses, • Operating limits of MW and MVArMW generation, or • MW and MVAr interchangestransmission losses)• Environmental dispatch• Maximum power transfer

A. Mathematical AC OPF Formulation

The typical AC OPF formulation is to minimize thegeneration cost while considering the system’s physicaloperation limits. For a system with n buses, the parametersand decision variables are listed as follows.

• Vk = Vdk + jVqk: The voltage phasor in rectangularcoordinates.

• Ykj = Gkj + jBkj : The admittance between Bus k andBus j.

• Pk and Qk are the active and reactive injected powerat Bus k, respectively.

• PGk and QGk are the active and reactive generatedpower at Bus k, respectively.

• PDk and QDk are the active and reactive demand powerat Bus k, respectively.

• Subscripts max and min notate maximum and minimumlimits, respectively.

• αk , βk, and γk are the constant, linear and quadraticfuel cost coefficients of unit k.

The conventional AC OPF formulation for minimizing thegeneration cost of a system is shown in (1) and (2).

min Fcost(PGk) =∑kεG

(αk + βkPGk + γkP2Gk) (1)

subject to:

Pk = Vdk

(∑nj=1(VdjGk,j − VqjBk,j)

)+ Vqk

(∑nj=1(VqjGk,j − VdjBk,j)

)(2a)

Qk = Vqk

(∑nj=1(VdjGk,j − VqjBk,j)

)− Vdk

(∑nj=1(VqjGk,j + VdjBk,j)

)(2b)

Pk − PGk + PDk = 0 (2c)Qk −QGk +QDk = 0 (2d)Vk,min ≤ Vk ≤ Vk,max (2e)PGk,min ≤ PGk ≤ PGk,max (2f)QGk,min ≤ QGk ≤ QGk,max (2g)

B. Mathematical AC OPF Formulation in SDP Form

In this subsection, AC OPF formulation is set to be in theSDP form [2]. e1, e2, e3, . . . , and en are the standard basisvectors in Rn. M1, M2, M3, . . . , and Mn are the standardbasis matrices. k is the bus index. < and = are the real and

1

1

)(

1 1)

Fig. 1. The Convexification Process.

imaginary components, respectively.

Yk = ekeTk Y (3a)

Yk =1

2

[<(Yk + Y Tk ) =(Y Tk − Yk)=(Yk − Y Tk ) <(Yk + Y Tk )

](3b)

Yk = −1

2

[=(Yk + Y Tk ) <(Yk − Y Tk )<(Y Tk − Yk) =(Yk + Y Tk )

](3c)

Mk =

[eke

Tk 0

0 ekeTk

](3d)

X = [<(V )T =(V )T ]T (3e)

W = XXT (3f)

Rank(W ) = 1 (3g)W � 0 (3h)

Pk = Trace{YkW} (3i)

Qk = Trace{YkW} (3j)

Pmink − PDk

≤ Trace{YkW} ≤ Pmaxk − PDk

(3k)

Qmink −QDk

≤ Trace{YkW} ≤ Qmaxk −QDk

(3l)

(V mink )2 ≤ Trace{MkW} ≤ (V max

k )2 (3m)

The bus reference angle is set to be 0o by forcing Vqk tobe 0. This is obtained by the following equations [7].

Nk =

[0 00 eke

Tk

](4a)

Trace{N1W} = 0 (4b)

The convexification of AC OPF form is realized by drop-ping the rank constraint, Rank(W ) = 1. The convexificationprocess is shown in Fig. 1.

III. ECONOMIC DISPATCH UNDER HEAVING LOADING

In this section, we go through study cases and solve themusing MATPOWER for conventional AC OPF and CVX

Fig. 2. Three-Bus System Case Study [9]

[8] with the SDPT3 solver for SDP OPF. MATPOWERsolves the original nonlinear programming based AC OPFconstraints in (2). On the other hand, CVX toolbox withthe SDPT3 solver solves the SDP relaxation of AC OPFconstraints in (3). We start with a 3-bus system, then weconduct 6- and 14- bus systems. We have tested larger scalesystems up to 118-bus system, however, they are not shownin this paper due to space restrictions.

A. Three-Bus System

This system is adopted from [9]. The system is shownin Fig. 2. The optimal solution of the system using MAT-POWER, which applies Interior Point Method in nonlinearprogramming, is shown in Table II. CVX optimal solution,which is based on semidefinite programming, is found inTable III. We can see that the two solutions match fromTable II and Table III.

TABLE IITHREE-BUS SYSTEM MATPOWER SOLUTION

Bus |V |(p.u.) δ (degrees) PG(MW) QG(MVAr)Bus 1 1.1 0 128.46 42.52Bus 2 1.1 9.001 188.22 49.13Bus 3 1.1 -11.641 0 70.29

Total Generation Cost = 5694.54 $ /hour

TABLE IIITHREE-BUS SYSTEM CVX SOLUTION

Bus |V |(p.u.) δ (degrees) PG(MW) QG(MVAr)Bus 1 1.1 0.0031 128.4570 42.5181Bus 2 1.1 9.0006 188.2194 49.1240Bus 3 1.1 -11.6410 0 70.2930

Total Generation Cost = 5694.5366 $ /hour

Next, the system gets tested with very high power de-mands. The power demands are shown in Table IV. MAT-POWR could not solve the problem, it does not convergeafter 20 iterations. On the other hand, CVX solves theproblem, the solution is given in Table V. As the load at

Fig. 3. Six-Bus System Case Study [10]

Bus 3 increases to 329.76, CVX can not solve the problembecause the solution is in the infeasible area, which is beyondthe AC OPF problem relaxation. In this case, infeasibilitymeans the system is exceeding the operation limits. Thus,SDP OPF gives the ability to depict the system operation atits physical limits.

TABLE IVTHREE-BUS SYSTEM WITH POWER DEMANDS

Bus PD(MW) QD(MVAr)Bus 1 300 150Bus 2 315 150Bus 3 329.73 150

TABLE VTHREE-BUS SYSTEM CVX SOLUTION

Bus |V |(p.u.) δ (degrees) PG(MW) QG(MVAr)Bus 1 1.1 0.0011 499.7572 302.9615Bus 2 1.1 4.3729 491.5208 300.5785Bus 3 1.1 -83.8065 0 503.7369

Total Generation Cost = 51,097.3 $ /hour

B. Six-Bus SystemThe 6-Bus system is taken from [10], it is shown in Fig.

3. The optimal solution of the system using CVX matchesMATPOWER, and it is shown in Table VI.

Again, the power demands dramatically increase to provethat CVX can solve the problem when it operates at its

TABLE VISIX-BUS SYSTEM CVX AND MATPOWER SOLUTION

Bus |V |(p.u.) δ (degrees) PG(MW) QG(MVAr)Bus 1 1.050 0 50 35.33Bus 2 1.050 -0.45 89.63 55.88Bus 3 1.070 -0.545 77.07 84.88Bus 4 0.987 -2.051 0 0Bus 5 0.985 -2.745 0 0Bus 6 1.005 -2.555 0 0

Total Generation Cost = 3126.36 $ /hour

physical limits, and MATPOWER cannot. The demand loadsare found in Table VII, and the solution is shown in TableVIII.

TABLE VIISIX-BUS SYSTEM WITH INCREASING POWER DEMANDS ON BUS 4, 5

AND 6

Bus PD(MW) QD(MVAr)Bus 4 150 84Bus 5 100.5 90Bus 6 101 80

TABLE VIIISIX-BUS SYSTEM CVX SOLUTION

Bus |V |(p.u.) δ (degrees) PG(MW) QG(MVAr)Bus 1 1.0500 0.0825 107.0510 51.7093Bus 2 1.0500 -1.8654 116.4908 99.9985Bus 3 1.0700 -0.1343 144.9349 99.9979Bus 4 0.9500 -5.6386 0 0Bus 5 0.9526 -4.5743 0 0Bus 6 0.9883 -3.8457 0 0

Total Generation Cost = 5013.52 $ /hour

C. IEEE 14-Bus System

Now, we solve IEEE 14-Bus system [11], shown in Fig. 4.We do a little adjustment on the system, and that is we ignorethe compensation on bus 9. The reason is that we show theeffect of reactive power on maximum loading level on thebus in the following section. We use CVX and MATPOWERto solve the system. Both methods give the same optimalsolution that is given in Table IX.

If the active power load at bus 14 only increases to 79.657MW, MATPOWER cannot solve the problem because it doesnot converge. CVX, on the other hand, gives the solution inTable X. Beyond this value, CVX shows that the solution isinfeasible.

IV. IDENTIFYING MAXIMUM LOAD LEVEL

As the bus demand load increases, the voltage and the fre-quency decrease for both the bus and the system [12]. Controlsystems can be used to restore them, however, there arecertain limitations of the system to handle power demands.In this section, we investigate the maximum loading level onbus and determine the bus voltage while the maximum loadoccurs. In our examination, we release the power generation

Fig. 4. IEEE 14-Bus System Case Study [11]

TABLE IXIEEE 14-BUS SYSTEM CVX AND MATPOWER SOLUTION

Bus |V |(p.u.) δ (degrees) PG(MW) QG(MVAr)Bus 1 1.0600 0 194.2590 0Bus 2 1.0418 -4.0410 36.7089 32.0250Bus 3 1.0159 -9.9429 28.7408 26.6046Bus 4 1.0099 -8.5832 0 0Bus 5 1.0125 -7.3788 0 0Bus 6 1.0501 -10.2621 0 14.8189Bus 7 1.0332 -11.1178 0 0Bus 8 1.0600 -10.3331 8.7387 16.1913Bus 9 1.0179 -12.9168 0 0Bus 10 1.0160 -13.1979 0 0Bus 11 1.0293 -13.1433 0 0Bus 12 1.0336 -13.6974 0 0Bus 13 1.0276 -13.7176 0 0Bus 14 1.0035 -14.3278 0 0

Total Generation Cost = 8088.22 $ /hour

TABLE XIEEE 14-BUS SYSTEM CVX SOLUTION

Bus |V |(p.u.) δ (degrees) PG(MW) QG(MVAr)Bus 1 1.0600 -0.0081 37.8290 0Bus 2 1.0600 -0.2932 105.4574 17.2126Bus 3 1.0600 -2.7409 83.3948 35.5711Bus 4 1.0302 -3.3345 0 0Bus 5 1.0287 -3.1624 0 0Bus 6 1.0600 -9.1917 13.0089 24.0000Bus 7 1.0332 -2.3175 0 0Bus 8 1.0600 6.3088 93.2573 23.1423Bus 9 1.0175 -7.3617 0 0Bus 10 1.0176 -7.9776 0 0Bus 11 1.0352 -8.6963 0 0Bus 12 1.0360 -10.6632 0 0Bus 13 1.0184 -11.1546 0 0Bus 14 0.9400 -16.0732 0 0

and the bus voltage limits of the tested bus. By these steps,we find the maximum demand power on the tested bus, aswell as the voltage at that level. Identifying the maximumloading level is equivalent to determining the voltage stabilitymargins. That is, increasing the load above the maximumlimit causes voltage collapse and system instability. We alsoshow that the ability of loading increases as the tested busreactive power raises. We go through the same study casesin the previous section. The solution is obtained by usingCVX.

A. Maximum Loading Level Formulation

The following are the objective function, which is max-imizing load on the tested bus, and the physical systemconstraints.

max PDi (5)

Subject to:(2)

Note: the voltage constraint on the tested bus is relaxed.

B. Three-Bus System

Starting with ignoring the synchronous condenser on bus3, we find the bus voltage and the maximum power loadingusing (2) and (5), the result is found in Table XI. As bus 3is compensated by a synchronous condenser with a value of50 MVAr, the bus voltage and the maximum power loadingare enhanced. The result with compensation is also shownin Table XI.

TABLE XITHREE-BUS SYSTEM MAXIMUM LOADING LEVEL

Bus |V |(p.u.) δo PD(MW) CompensationBus 3 0.7153 -63.0728 113.3347 QG3 = 0 MVArBus 3 0.8465 -69.9686 141.7333 QG3 = 50 MVAr

The solutions from SDP OPF are compared with contin-uation method using MATPOWER CPF. The PV curves areshown in Fig. 5. As bus 3 is compensated with 50 MVAr,the nose, critical, point gets upper-right shifted. The reasonis that the loading level and the voltage become higher. Thetwo methods lead to the same results.

C. Six-Bus System

In this system, the investigated buses are 4, 5, and 6.We have examined each bus separately. Then, every bus iscompensated with a synchronous condenser with a value of20 MVAr. The results of maximizing the load level on eachbus with and without compensation are shown in Table XII.

The comparison between optimization and continuationmethods on the 6-bus system is applied on bus 4 withoutcompensation. Bus 4 PV curve is shown Fig. 6. The twomethods lead to the same results of 487 MW as Bus 4’sloading limit.

Active Demand Power (MW)

Fig. 5. Bus 3 PV Curve

TABLE XIISIX-BUS SYSTEM MAXIMUM LOADING LEVEL

Bus |V |(p.u.) δ (degrees) PD(MW) CompensationBus 4 0.6735 -56.9789 486.7798 QG4 = 0 MVArBus 4 0.6842 -58.1005 497.6326 QG4 = 20 MVArBus 5 0.6646 -63.4751 427.5265 QG5 = 0 MVArBus 5 0.6770 -64.8659 438.0877 QG5 = 20 MVArBus 6 0.7879 -56.7647 450.0843 QG6 = 0 MVArBus 6 0.8370 -87.5602 453.3528 QG6 = 20 MVAr

D. IEEE 14-Bus System

The PQ buses in IEEE 14-Bus System are 4, 5, 7, 9, 10, 11,12, 13, and 14. By repeating the same procedure, each bus istested alone, the solution is found in Table XIII. Afterward, asynchronous condenser of values of 19, 30, 50, 100, 150 and200 MVAr are step-by-step added to bus 9 only to prove thatthe maximum loading level increases as the reactive powerraises. The solution of bus 9 maximum loading level withcompensation is found in Table XIII.

By using bus 9 of the 14-bus system to compare optimiza-tion and continuation methods. The PV curve of bus 9, withQG9 = 0 MVAr, is shown Fig. 7. Both methods show thatfor Bus 9, 352 MW is the loading limit.

V. CONCLUSION

This paper gives a review of applying SDP AC OPF to findeconomic dispatch at heavy loading condition and identifyloading limits. Relaxation of AC OPF problem in SDP formgives an exact or an approximate global solution withoutiterations. We show that the AC OPF problem in SDP formcan be solved using CVX. We test the CVX on variouspower system types. Then, we compare the solution withMATPOWER, which applies the interior point method innonlinear programming. We find the solutions match eachother. Moreover, CVX has the ability to give a solution of a

Active Demand Power (MW)

Fig. 6. PV Curve of Bus 4

TABLE XIII14-BUS SYSTEM MAXIMUM LOADING LEVEL

Bus |V |(p.u.) δ (degrees) PD(MW) CompensationBus 4 0.6766 -81.0280 715.7236 QG4 = 0 MVArBus 5 0.6754 -71.9308 682.4810 QG5 = 0 MVArBus 7 0.7530 -88.7925 353.3889 QG7 = 0 MVArBus 10 0.6440 -79.6868 253.2913 QG10 = 0 MVArBus 11 0.6361 88.9029 251.2200 QG11 = 0 MVArBus 12 0.6014 86.4068 206.7138 QG12 = 0 MVArBus 13 0.7246 61.9912 314.0533 QG13 = 0 MVArBus 14 0.6164 -67.9183 160.4554 QG14 = 0 MVArBus 9 0.6733 -86.7417 351.7682 QG9 = 0 MVArBus 9 0.6885 -89.3901 363.6388 QG9 = 19 MVArBus 9 0.6974 89.1735 370.1267 QG9 = 30 MVArBus 9 0.7139 86.7336 381.2615 QG9 = 50 MVArBus 9 0.7560 81.5239 405.8269 QG9 = 100 MVArBus 9 0.7990 77.4211 426.5157 QG9 = 150 MVArBus 9 0.8422 74.2203 444.1128 QG9 = 200 MVAr

convex relaxation of AC OPF problem when the system isoperating at the physical constraint limits.

Further, we introduced identifying the maximum powerloading level. We show that the results from SDP OPF wellmatch the results obtained from MATPOWER’s CPF method.

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