VSC-HVDC OPF

**VSC**

HVDC system requires HVDC converter which converts electric power from high voltage AC to high voltage DC, or vice versa. Insulated-gated bipolar transistor(IGBT) type semiconductor device, giving usage as self-commutated converter, is used for HVDC converter usually referred to as voltage-source converter(VSC). VSC is differentiated from line commutated converters by ability to improve harmonic performance so that the converter no longer depends on synchronous machines.

***VSC-HVDC***[1], [2]

HVDC operates at high voltage so that current amount becomes lower when same power is being transmitted. Therefore loss, which is proportional to amount of current, occurs less in HVDC system. There are various uses of HVDC line in continuously changing worldwide transmission system, power transfer between systems in long distance or differently rated frequency, demand of cheaper electricity, stronger influence of environmental organizations or local residents. Encountering need for more efficiently managed power system, HVDC system, especially that consist of VSC is highly required. Voltage source converter – high voltage direct current(VSC-HVDC) technology cannot only control active power flow but also provide an independent voltage regulation to the connected AC system due to the accommodation of surplus or deficit in reactive power done by VSC. This brings advantage from reduction of AC filters and shunt capacitors in the transmission system.

the basic form, two terminal VSC-HVDC system consists two VSC converters is shown above.

where i = k, m, power flow equations in the above figure are given as follows,

**VSC-HVDC system for Newton-Raphson OPF** [2], [6]

Appearance of HVDC means the conventional optimal power flow solution that considering only AC power flow will not properly reach optimal output when applied to system consist of HVDC transmission line. The new paradigm requires OPF model considering VSC converters between AC and DC high voltage transmission line. VSC-HVDC OPF suggested below is concentrated on forming Lagrangian function including equality and inequality constraints. In this case, the loss at the VSC is neglected, only loss at DC transmission line is considered.

the general Lagrangian function is,

where (i=k, m; j=k,m; i≠j), λpk, λpm, λqk, λqm are Lagrange multipliers at nodes k and m, the power mismatch equation at nodes i=(k,m) is,

where λti, λv, λdc are Lagrange multipliers, Mci is amplitude modulation index of insulated gate bipolar transistor controller(PWM), the equations below represent equality constraints in Lagrangian function,

where λcpij, λcqij are Lagrange multipliers, and Pijspec, Qijspec are specified active and reactive power reference, the equations below represent controlled active and reactive power from node to in Lagrangian function,

the equation below represents quadratic penalty term imposed on voltage magnitude to be controlled at one of the VSC-HVDC AC terminals,

the inequalities constraints given below are checked in order to control variables within limits,

The Lagrangian minimization is achieved by three steps, the initialization of all system variables, main and inner iteration loop by Newton process, and solving a linear equation including Hessian matrix W, gradient vector ∇L, correction vector △z of the form,

Solving the optimization problem lets exploiting the controller capabilities most economically with supply and demand, keeping within bounds all the constraints.Though the solution is limited to a two terminal configuration.

***Multiterminal VSC-HVDC OPF***[3], [4]

Comparing to two terminal VSC-HVDC system, multiterminal VSC-HVDC system can cover across larger area due to additional controllability in transmission system. Reduction of power losses is the best benefit that could be earned from increasing number of HVDC transmission lines. The benefit can be evaluated from the trade-off between the reduced power loss from AC transmission lines by embedding DC links and extra losses coming from VSC stations, which means that increasing number of terminals does not mean reduction of total loss. However, the fact that there isn’t only one HVDC line for each system all the time in reality brings the need for multiterminal VSC-HVDC OPF.

In multiterminal VSC-HVDC OPF, HVAC and HVDC mixed modelling is done in three parts, HVAC grid, HVDC grid, the converters and the DC grid each postulated as steady-state. Figure below shows the converter model.

A. HVAC grid

where k≠m, active and reactive power flow from node k to m are,

active and reactive power injections in node k are,

B. Converters

the equation below guarantees the power balance between the AC and DC side considering loss, where are constant coefficients,

C. HVDC grid

where power flow can be solved by Ohm’s law and Joule’s law,

Using the model as the basis for soliving the nonlinear optimization problem, state variables are set in state vector x. AC node voltage and angle, converter AC bus voltage and angle, active and reactive power at each bus, reactive power at SVC, and DC node voltage are included in x.

where objective function is set as meaning of cost minimization as below,

power balance at each node is,

power balance at converter is,

active and reactive power limit, voltage limit at each node are,

**VSC-HVDC OPF application on transmission loss evaluation** [5]

To obtain more accurate optimal output for the system in VSC-HVDC system where AC and DC line both exist, loss in every part should be modeled, not neglected. Therefore loss considered optimal power solution can reach optimal cost minimizing solution. The method below suggests minimization of total power loss through the objected function considering the modelling of loss in each part.

where Ali, Bli, Cli are approximated VSC loss coefficient of VSC i,

considering bus voltage, voltage angle, line conductance,

Even further, for better optimal output to be obtained, constraints are specified into AC and DC as bus voltage limit to AC bus voltage limit and DC bus voltage limit, and additionally considers AC, DC transmission line capacity limit.

**Summary**

Future power transmission networks are indicated to be consist of more DC transmission line than ever before, the key seems to be how to find optimal power flow solution in VSC-HVDC system. VSC-HVDC OPF with two-terminal had been researched at early stage, but as power systems were expected to be more complicated containing more than one HVDC lines, multiterminal VSC-HVDC OPF had been under research recently. Still, there are lots of number of cases of systems and better optimizing method is required, whether improvement will be done in modelling, evaluation part, or other, however, it's obvious that more effort should be made.

**Reference**

[1] Z. Xiao-Ping, “Multiterminal voltage-sourced converter-based HVDC models for power flow analysis,” IEEE Trans. Power Del., vol. 19, no.4, pp. 1877–1884, Oct. 2004.

[2] A. Pizano-Martinez, C. R. Fuerte-Esquivel, H. Ambriz-Perez, and E. Acha, “Modeling of VSC-based HVDC systems for a Newton-Raphson OPF algorithm,” IEEE Trans. Power Syst., vol. 22, no. 4, pp. 1794–1803, Nov. 2007.

[3] M. Baradar, M. R. Hesamzadeh, and M. Ghandhari, “Modelling of multi-terminal HVDC systems in optimal power flow formulation, presented at the Elect. Power Energy Conf., London, ON, Canada, 2012.

[4] R. Wiget and G. Andersson, “Optimal power flow for combined AC and multi-terminal HVDC grids based on VSC converters,” presented at the IEEE Power Energy Soc. Gen. Meeting, San Diego, CA, 2012.

[5] W. Feng, A. Tuan, L.B. Tjernberg, A. Mannikoff, and A. Bergman, “A new approach for benefit evaluation of multiterminal VSC-HVDC using a proposed mixed AC/DC optimal power flow,” IEEE Trans. Power Del., vol. 29, no. 1, pp. 432–443, Feb. 2014.

[6] D. I. Sun, B. Ashley, B. Brewer,A. Hughes, and W. F. Tinney, “Optimal power flow by Newton approach,” IEEE Trans. Power App. Syst., vol.PAS-103, no. 10, pp. 2864–2880, Oct. 1984.